Theory of groups and symmetries : finite groups, Lie groups, and Lie algebras 9789813236868, 9813236868

Table of contents :
Groups and transformations --
Lie groups --
Lie algebras --
Representations of groups and Lie algebras --
Compact Lie algebras --
Root systems and classification of simple Lie algebras --
Homogeneous spaces and their geometry --
Solutions to selected problems.

Citation preview

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

THEORY  OF  GROUPS  A ND  SYMMETRIES Finite Groups, Lie Groups, and Lie Algebras Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-3236-85-1

For any available supplementary material, please visit http://www.worldscientific.com/worldscibooks/10.1142/10898#t=suppl Desk Editor: Ng Kah Fee Typeset by Stallion Press Email: [email protected] Printed in Singapore

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Contents

Preface 1

xiii

Groups and Transformations 1.1

1.2

1.3

Groups: Basic Concepts and Definitions . . . . . . . 1.1.1 Group and subgroup. Examples . . . . . . . . 1.1.2 Normal subgroups, cosets, quotient (factor) groups . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Direct products of groups, conjugacy classes, center . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Example. Group of permutations (symmetric group) Sn . . . . . . . . . . . . . Matrix Groups. Linear, Unitary, Orthogonal and Symplectic Groups . . . . . . . . . . . . . . . . . 1.2.1 Vector spaces and algebras . . . . . . . . . . . 1.2.2 Matrices. Determinant and Pfaffian, direct product . . . . . . . . . . . . . . . . . . . . . 1.2.3 Linear operators and matrix groups. Groups of linear transformations GL and SL . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Matrix groups associated with bilinear and Hermitian forms . . . . . . . . . . . . . . 1.2.5 Matrix groups of types O, Sp and U . . . . . Homomorphisms . . . . . . . . . . . . . . . . . . . . 1.3.1 Mapping . . . . . . . . . . . . . . . . . . . . . 1.3.2 Group homomorphism. Kernel and image . . 1.3.3 Exact sequence . . . . . . . . . . . . . . . . . vii

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Groups of Transformations. Conformal Group . . . . . . 1.4.1 Groups of transformations. Linear inhomogeneous groups . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Semidirect product of groups . . . . . . . . . . . 1.4.3 Conformal group Conf(Rp,q ) . . . . . . . . . . . .

Lie Groups 2.1

2.2

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Manifolds and Lie Groups . . . . . . . . . . . . 2.1.1 Smooth manifolds . . . . . . . . . . . . . 2.1.2 Lie group manifolds. Examples . . . . . 2.1.3 Manifold of conformal group Conf(Rp,q ). Isomorphism of Conf(Rp,q ) and O(p + 1, q + 1) . . . . . . . . . . . . 2.1.4 Compact Lie groups . . . . . . . . . . . Tangent Spaces. Haar Measure . . . . . . . . . 2.2.1 Tangent spaces of smooth manifolds . . 2.2.2 Invariant metric on Lie group. Haar measure . . . . . . . . . . . . . . .

54 54 58 61 67

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82 86 89 89

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Lie Algebras

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3.1

103

3.2

3.3

Matrix Lie Algebras . . . . . . . . . . . . . . . . . . . . 3.1.1 Tangent spaces to manifolds of matrix Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Matrix Lie groups and matrix Lie algebras . . . . 3.1.3 Examples of matrix Lie algebras . . . . . . . . . . 3.1.4 Tangent spaces of matrix Lie group manifolds (continued) . . . . . . . . . . . . . . . . . . . . . General Construction . . . . . . . . . . . . . . . . . . . . 3.2.1 General definition of Lie algebra. Homomorphisms of Lie algebras and exponential mapping A(G) → G . . . . . . . . . . . . . . . . . . . . . 3.2.2 Structure constants. Simple and semisimple Lie algebras, direct sum of Lie algebras . . . . . . . . 3.2.3 Realifications and real forms of complex Lie algebras . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Killing metric in Lie algebra. Cartan’s criterion for semisimplicity . . . . . . . . . . . . . . . . . . . . Lie Algebras of Classical Series . . . . . . . . . . . . . .

103 105 107 116 117

117 125 129 136 138

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3.4

3.5 4

3.3.1 Examples of defining commutation relations of some Lie algebras . . . . . . . . . . . . . . . . 3.3.2 Real forms of Lie algebras s(n, C), so(n, C) and sp(2r, C) . . . . . . . . . . . . . . . . . . . . 3.3.3 Examples of isomorphisms and automorphisms of Lie algebras. “Accidental” isomorphisms . . . Lie Algebra of Conformal Group Conf(Rp,q ) . . . . . . . 3.4.1 Conformal algebra in more than 2 dimensions . . . . . . . . . . . . . . . . . . . . . 3.4.2 Digression. Isometries and conformal isometries of manifolds, Killing vectors and Liouville theorem . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Isomorphism conf(Rp,q ) = so(p + 1, q + 1), for p + q > 2 . . . . . . . . . . . . . . . . . . . . . Locally Isomorphic Lie Groups. Universal Covering . . .

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163 170 172

Representations of Groups and Lie Algebras

179

4.1

179

4.2

4.3

4.4

4.5

Linear (Matrix) Representations of Lie Groups . . . . . 4.1.1 Definition of representation of a group. Examples . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Regular and induced representations of finite groups. Faithful representations . . . . . . . . . . 4.1.3 Equivalent representations. Equivalence of defining representation and its conjugate for SU (2), characters . . . . . . . . . . . . . . . . Representations of Lie Algebras . . . . . . . . . . . . . . 4.2.1 Definition of Lie algebra representation . . . . . . 4.2.2 Examples of Lie algebra representations . . . . . Direct Product and Direct Sum of Representations . . . 4.3.1 Direct (tensor) product of representations. Tensors . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Direct sum of representations . . . . . . . . . . . Reducible and Irreducible Representations . . . . . . . . 4.4.1 Definition of reducible and irreducible representations . . . . . . . . . . . . . . . . . . . 4.4.2 Schur’s lemma . . . . . . . . . . . . . . . . . . . . Representations of Finite Groups and Compact Lie Groups. Group Algebra and Regular Representations . . . . . . . . . . . . . . . . . . . . . . .

179 184

188 190 190 193 196 197 201 203 203 209

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Elements of Character Theory for Finite Groups and Compact Lie Groups . . . . . . . . . . . . . . . . . . 4.6.1 Examples. Irreducible representations and characters of C3 and S3 . . . . . . . . . . . . 4.6.2 Characters of finite groups and compact Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Irreducible representations and characters of SO(2) = U (1) . . . . . . . . . . . . . . . . . . Universal Enveloping Algebra. Casimir Operators, Yangians . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Definition of universal enveloping algebra U(A) of Lie algebra A . . . . . . . . . . . . . . . . . . . 4.7.2 Representations of U(A). Center of U(A) and Casimir operators . . . . . . . . . . . . . . . 4.7.3 Finite-dimensional representations of Lie algebras. su(2) and s(2, C) . . . . . . . . . . . . . . . . . 4.7.4 Coproduct in universal enveloping algebra U(A). Yangians . . . . . . . . . . . . . . . . . . . . . .

Compact Lie Algebras 5.1 5.2 5.3

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220 224 232 233 233 236 248 253 271

Definition and Main Properties of Compact Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Structure of Compact Lie Algebras . . . . . . . . . . . . 275 Relation of Compact Lie Algebras to Compact Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

Root Systems and Classification of Simple Lie Algebras 6.1

220

Cartan Subalgebra. Rank of Lie Algebra and Cartan–Weyl Basis . . . . . . . . . . . . . . . . . . 6.1.1 Regular elements. Cartan subalgebra and rank of Lie algebra . . . . . . . . . . . . . . . . . . . 6.1.2 Cartan–Weyl basis . . . . . . . . . . . . . . . . Root Systems of Simple Lie Algebras . . . . . . . . . . 6.2.1 Properties of roots of simple Lie algebras . . . . 6.2.2 Weyl group and simple roots . . . . . . . . . . . 6.2.3 Dynkin diagrams. Root systems of classical Lie algebras s(n, C), so(n, C), sp(2n, C) . . . .

281 . 281 . . . . .

282 283 292 293 303

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Contents

6.2.4 Dynkin diagrams and classification of finite-dimensional simple Lie algebras . . . . . 323 6.2.5 Root systems of exceptional Lie algebras . . . . . 332 7

Homogeneous Spaces and their Geometry

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7.1 7.2

345

7.3 7.4 7.5

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Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . Examples of Homogeneous Spaces. Parameterizations of Groups SO(n) and U (n) . . . . . . . . . . . . . . . . Action of Group G in Coset Space G/H. Induced Representations . . . . . . . . . . . . . . . . . . Models of Lobachevskian Geometry and Geometry of Spaces AdS and dS . . . . . . . . . . . . . . . . . . . Metrics and Laplace Operators in Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Elements of differential geometry on smooth manifolds . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Invariant metrics in homogeneous spaces . . . . . 7.5.3 Regular representations and invariant vector fields on Lie groups . . . . . . . . . . . . . . . . . . . . 7.5.4 Laplace operators on Lie groups and homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . 7.5.5 Spherical functions on homogeneous spaces . . . .

Solutions to Selected Problems 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15

Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem

1.1.9, Section 1.1.1 . . 1.1.27, Section 1.1.4 . 1.2.22, Section 1.2.5 . 2.1.8, Section 2.1.2 . . 3.1.2, Section 3.1.2 . . 3.1.4, Section 3.1.3 . . 3.4.3, Section 3.4.2 . . 3.3.16, Section 3.3.1.2 3.3.17, Section 3.3.1.2 3.3.18, Section 3.3.1.2 4.7.8, Section 4.7.2 . . 4.7.9, Section 4.7.2 . . 4.7.14, Section 4.7.2 . 4.7.26, Section 4.7.4 . 4.7.30, Section 4.7.4 .

349 363 367 376 376 386 397 399 405 409

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8.16 8.17 8.18 8.19 8.20

Problem 6.2.14, Section 6.2.5 . . Problems 7.1.1, 7.1.3, Section 7.1 Problem 7.2.4, Section 7.2 . . . . Problem 7.2.6, Section 7.2 . . . . Problem 7.5.8, Section 7.5.2 . . .

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Selected Bibliography

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References

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Index

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Preface

The word “symmetry” descends from Greek “συμμτ ρια”, meaning commensurability or proportionality. Symmetries are sometimes viewed as “harmonies of proportions”; geometric symmetries include mirror symmetry, translation symmetry, symmetry of patterns and crystals, etc. In general, symmetry transformations are such that they keep intact properties of an object or set of objects (their shapes, etc.). On the set of all symmetry transformations under which given objects are invariant, one naturally defines composition (multiplication), i.e., an operation which associates to two consecutive symmetry transformations another symmetry transformation. In this way one arrives at the notion of a set of symmetry transformations endowed with a multiplication operation, i.e., at the definition of a group. ´ Foundation of the group theory is attributed to Evariste Galois (1811– 1832). When studying the solvability of algebraic equations, he introduced the notions of finite field and group, and invented what is now known as Galois theory. In particular, a group is formed by n! permutations of n roots x1 , x2 , . . . , xn of nth-order algebraic equation (x − x1 )(x − x2 ) . . . (x − xn ) = 0. Clearly, these permutations do not modify this equation and hence constitute a symmetry group of this equation. Presently, the theory of groups and symmetries is an important integral part of theoretical and mathematical physics. In physics of elementary particles, cosmology and related areas, as well as many branches of condensed matter physics, particularly important are Lie groups and

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Lie algebras, which describe continuous symmetries. It suffices to recall that the relativistic physics is based on Lorentz and Poincar´e groups, while the cornerstone of the current theory of elementary particles, the Standard Model, is the local (gauge) symmetry with the gauge group SU (3) × SU (2) × U (1). In view of this, the book concentrates mainly on Lie groups and algebras and related topics. It is foreseen that this book will be supplemented with an accompanying book dedicated to the detailed study of issues which are particularly important for physics and have to do with representations of the groups SU (2) and SL(2, C); SU (n) and SL(n, C); Poincar´e group, as well as their Lie algebras. This book is aimed at advanced undergraduate students and graduate students specializing in theoretical and/or mathematical physics, although discussions of a number of topics go somewhat beyond this level. There are numerous problems suggested, whose purpose is to enable one to gain working knowledge of the theory. Solutions to problems marked with asterisk are collected in the end of this book. Some of the sections begin with concrete examples of constructions emerging in the theory, and only after that the general definitions of these constructions are given. This way of presentation is in accordance with the advice of Felix Klein given in his “Vorlesungen u ¨ ber die Entwicklung der Mathematik im 19 Jahrhundert” [2]. In our book, we tried to prove, or at least give hints of proofs, of majority of statements made. We emphasize, however, that in places, the level of rigor is not to the standard of a mathematically oriented reader. Text written in small font may be skipped at first reading. Presenting systematic and comprehensive bibliography on the topics we discuss would be way out of the scope of this book. To orient the reader, we collected an incomplete list of monographs and reviews in the end of the book. In the course of presentation, we give references to papers and/or books where a concrete topic is studied in more detail. The reader should also benefit from a comprehensive index. This book grew from lectures read at Departments of Theoretical and Nuclear Physics of Dubna International University, Department of Quantum Theory and High Energy Physics, Department of Quantum Statistics and Field Theory and Department of Particle Physics and Cosmology of Physics Faculty of M.V. Lomonosov Moscow State University. We are indebted to S.E. Derkachev, S.A. Fedoruk, E.A. Ivanov, D.V. Kirpichnikov, S.O. Krivonos, S.A. Mironov, A.I. Molev, O.V. Ogievetsky, Ya.M. Shnir, A.V. Silantiev, A.O. Sutulin, V.O. Tarasov, S.V. Troitsky and N.A. Tyurin,

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as well as many other colleagues from the Theory Division of the Institute for Nuclear Research of the Russian Academy of Sciences and N.N. Bogoliubov Laboratory of Theoretical Physics of the Joint Institute for Nuclear Research, Dubna, for numerous helpful discussions and comments.

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Chapter 1

Groups and Transformations

1.1. 1.1.1.

Groups: Basic Concepts and Definitions Group and subgroup. Examples

Definition 1.1.1. Finite or infinite set G is a group provided that there is an operation (called group operation or group multiplication) defined in G which maps any pair of elements g1 and g2 from G to an element g3 ∈ G (one writes g1 · g2 = g3 ), and which has the following properties: (a) Associativity. For any three elements g1 , g2 , g3 , the following relation holds: (g1 · g2 ) · g3 = g1 · (g2 · g3 ). (b) Existence of unit element e ∈ G, such that for all g ∈ G one has g · e = e · g = g. (c) Existence of an inverse element. For any element g ∈ G there exists g −1 ∈ G such that g · g −1 = g −1 · g = e. Definition 1.1.2. If the number of elements in a group G is finite, then the group is finite. The number of elements in finite group G is called order of the group and is denoted by ord(G). Finite groups as well as groups with infinite but countable sets of elements are discrete. Otherwise a group is continuous. Definition 1.1.3. If all elements of a group G commute with each other, i.e., ∀g1 , g2 ∈ G one has g1 · g2 = g2 · g1 , then the group is Abelian. Below are some examples of groups. Example 1. Set of real numbers with removed zero, R\{0}, is a group with the group operation coinciding with conventional multiplication of 1

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numbers. The unit element of this group is 1, while the inverse element to x ∈ R\{0} is 1/x. Problem 1.1.1. Why is zero removed from R? Example 2. Set of complex numbers z of unit modulus, |z| = 1, is a group denoted by U (1). The group multiplication in U (1) is the multiplication of complex numbers (for any z1 , z2 ∈ U (1) one has z1 z2 ∈ U (1), since |z1 z2 | = 1 for |z1 | = |z2 | = 1). Unit element in U (1) is z = 1, while inverse element to z ∈ U (1) is z −1 (z −1 ∈ U (1) since |z −1 | = 1 for |z| = 1). Example 3. (Z, +), or simply Z, is group of integer numbers {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}, where the group operation is addition of numbers. The group of real numbers with addition as the multiplication operation is denoted by (R, +). Problem 1.1.2. What are the unit elements in groups (Z, +) and (R, +)? Example 4. Set of integers modulo an integer number n (i.e., any two numbers k and k + n are identified; these objects are denoted as k mod(n)) is a cyclic group Zn under addition modulo n, i.e., k mod(n) + m mod(n) = (k + m) mod(n). The order of Zn equals n. Example 5. Cyclic group Cn of symmetry transformations of regular polygon with n vertices. This group consists of rotations around the origin O of circumscribed circle by angles multiple of 2π/n (see Fig. 1.1.1). 2

1

3

n n−1

O

Fig. 1.1.1.

Regular polygon with n vertices and circumscribed circle.

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The vertices of a regular polygon in Fig. 1.1.1 are labeled by numbers {1, 2, . . . , n}. The counterclockwise rotations by angles multiple of 2π/n (elements of the group Cn ) are in one-to-one correspondence with cyclic permutations of vertices, or, in other words, cyclic permutations of n numbers {1, 2, . . . , n}. The rotation gk ∈ Cn by angle 2πk/n, {1 → k + 1, 2 → k + 2, . . . , n − 1 → k − 1, n → k},

(1.1.1)

is identical to the rotation gk+n by angle 2π(k + n)/n. The order of group Cn equals n. The group multiplication in Cn is defined as two consecutive rotations: gk · gm = gk+m .

(1.1.2)

Unit element is g0 , while the inverse element to gk is g−k = gn−k . Note that the rotation by angle 2πk/n in complex plane can be associated with complex number zk = ei2πk/n equal to one of the nth-order roots of 1. Thus, the group Cn can be viewed as the set of all complex numbers z = zk (k = 0, 1, . . . , n − 1) such that z n = 1. In this representation, the multiplication of elements gk ↔ zk is defined as the multiplication of complex numbers: zk · zm = zk+m . Definition 1.1.4. One-to-one correspondence (mapping) ρ : G → G between all elements of two groups G and G consistent with multiplications in these groups, i.e., ρ(g1 ) · ρ(g2 ) = ρ(g1 · g2 ),

∀g1 , g2 ∈ G,

is called group isomorphism. In this case we say that groups G and G are isomorphic and identify them as algebraic objects: G = G . As an example, two numbers {+1, −1} make cyclic group C2 which is isomorphic to Z2 , i.e., C2 = Z2 . Problem 1.1.3. Show that if ρ : G → G is isomorphism, then ρ(e) = e , where e and e are unit elements in G and G , and ρ(g −1 ) = [ρ(g)]−1 . Problem 1.1.4. Let ρ : G → G be isomorphism. Show that the inverse mapping ρˆ : G → G (such that ρˆρ is identical operation) is also consistent with multiplication operations, i.e., ρˆ(g1 ) · ρˆ(g2 ) = ρˆ(g1 · g2 ). Hence, ρˆ : G → G is also isomorphism. Problem 1.1.5. Show that the group Zn is isomorphic to Cn .

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Example 6. Dihedral group Dn is a group of all symmetries of a regular polygon with n vertices (Fig. 1.1.1; let us call this polygon n ), which includes not only all rotations from the group Cn , but also all mirror reflections against the symmetry axes passing through O and one of the vertices. To describe the group Dn , we extend the group Cn by a single extra transformation (element r), which is one of the mirror reflections transforming the polygon n into itself. We choose the latter transformation r as the mirror reflection against the vertical axis passing through the origin O. In the case of Fig. 1.1.1 this axis runs through the vertex 1, so the reflection r keeps the vertex 1 at its place, 1 → 1, while other vertices move in accordance with the rule {1 → 1, 2 → n, 3 → n − 1, . . . , n − 1 → 3, n → 2}.

(1.1.3)

One more mirror reflection against the vertical axis takes the vertices of n back to their places, so that we have r2 = e.

(1.1.4)

If, before the reflection r, the polygon n has experienced rotation g1−k , the vertex 1 has been replaced by the vertex k (the vertical axis runs now through the vertex k). Then the reflection r is described by the permutation k → k, (k−1) ↔ (k+1), (k−2) ↔ (k+2), etc. Let us adopt a convention that the multiplication gm ·r denotes two consecutive operations, first reflection r and then rotationa gm (“word” gm · r is read from right to left): (gm · r)n = gm (rn ). Problem 1.1.6. Derive the identity gm · r = r · g−m .

(1.1.5)

Prove that the mirror reflection rk of the polygon n against the axis (k, O), which runs through the origin O and vertex k in Fig. 1.1.1, is a composition of the reflection against the vertical axis and rotations, rk = gk−1 · r · g1−k ,

(1.1.6)

which first moves the vertex k to the position 1, then reflects the polygon against the vertical axis and in the end takes the vertex k back to its original place. a Another way of reading the word g m · r gives the transformation n · (gm · r), with the rotation gm applied first and the reflection r afterwards. This case does not yield anything new.

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Problem 1.1.7. Regular polygons with even number of vertices 2n have additional symmetry axes which pass through the middle points of the opposite sides. Show that reflections against these additional axes are reduced to transformations of the type r · gk . Hint: derive the formula r ∗ = r · g1 ,

(1.1.7)

for reflection r∗ against the axis that runs through the middle points of the sides (1, 2n) and (n, n + 1). Making use of the relations (1.1.5), (1.1.6) and (1.1.7), one can reduce all elements of the dihedral group Dn to the form {gk , r · gk } (k = 0, . . . , n − 1) and arrange them in a sequence {g0 = e, g1 , . . . , gn−1 , r, r · g1 , . . . , r · gn−1 }.

(1.1.8)

Thus, the order of the group Dn equals 2n. The rotations gm ∈ Dn obey the identity gm = g1m . Then, in view of (1.1.8), any element of the group Dn has the representation rk g1m (k = 0, 1; m = 0, 1, . . . , n − 1). In this sense the group Dn is generated by just two elements (g1 , r) which are called generators. They obey the relations (1.1.4), (1.1.5) and g1n = e. This characterization of a finite group in terms of its generators is called presentation. The presentation of Dn (a = g1 , b = r) is written as follows: Dn :

a, b | a · b = b · a−1 ,

b2 = e , an = e.

(1.1.9)

The group Dn is non-Abelian, which is obvious from (1.1.5). Definition 1.1.5. Elements {s1 , s2 , s3 , . . .} of a discrete group G are generators of this group if all elements of G can be expressed as products of s1 , s2 , s3 , . . . and their inverse elements. Problem 1.1.8. Write all elements and find the order of di-cyclic group Q2n which is defined by its presentation Q2n :

a, b | a · b = b · a−1 ,

b 2 = an ,

a2n = e.

Example 7. Group of all permutations of n objects, or symmetric group Sn (see also Section 1.1.4) can be viewed as a group of all one-to-one mappings of a set of integers {1, 2, . . . , n} to itself: A:

{1 → a1 , 2 → a2 , 3 → a3 , . . . , (n − 1) → an−1 , n → an },

(1.1.10)

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where {a1 , a2 , . . . an } is a new arrangement of numbers {1, 2, . . . , n}. The order of Sn equals n!. The permutations (1.1.10) are conveniently denoted as follows: 

1 a1

A=

2 a2

3 a3

... n− 1 . . . an−1

 n . an

(1.1.11)

A useful property here is that the columns in (1.1.11) can be arbitrarily interchanged, with the permutation A remaining intact. This, in particular, is convenient when defining the product of two permutations. The product A · B of two permutations A and B is their consecutive action, first B and then A:  A·B = 

1 a1

b1 ab1  1 = ab1 =

   ... n 1 2 ... n · . . . an b1 b2 . . . bn    b2 . . . bn 1 2 ... n · ab2 . . . abn b1 b2 . . . bn  2 ... n . ab2 . . . abn

2 a2

(1.1.12)

Unit element is the identical permutation {1 → 1, 2 → 2, . . . , n → n}, while the permutation inverse to (1.1.10) is A−1 : {a1 → 1, a2 → 2, a3 → 3, . . . , an → n},

(1.1.13)

or −1

A

 a1 = 1

a2 2

a3 3

 . . . an . ... n

(1.1.14)

The permutation (i, j) which permutes only two numbers i and j out of {1, 2, . . . , n}, while other numbers remain at their places, is called transposition. Clearly, any permutation can be obtained as a certain number of consecutive transpositions, so the transpositions are generators of the group Sn . Definition 1.1.6. Permutations which are products of even (odd) number of transpositions are even (odd).

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Permutations of the form (here k ≤ n) {a1 → a2 , a2 → a3 , a3 → a4 , . . . , ak−1 → ak , ak → a1 }, while remaining (n − k) objects stay at their places, are called cycles of length k and are denoted as follows:   a1 a2 a3 . . . ak ak+1 . . . an (a1 , a2 , . . . , ak ) ≡ a2 a3 a4 . . . a1 ak+1 . . . an = (a1 , a2 , . . . , ak ) · (ak+1 ) · · · (an ), (in what follows, cycles with one element are often omitted in notations for brevity). Recall that the columns in permutation (1.1.11) can be interchanged arbitrarily, so the elements involved in a cycle can be placed anywhere in the upper row. As an example, the following permutation is a cycle:   a1 a2 a3 a4 a5 a6 = (a1 , a3 , a2 , a4 ) a3 a4 a2 a1 a5 a6 (the cycle involves the first, third, second and fourth elements, while the rest stay intact). The transposition (i, j) is a cycle of length 2. Cycles with nonoverlapping sets of symbols do not affect each other, so they commute. As an example (1, 3, 4) · (2, 6, 7, 5) = (2, 6, 7, 5) · (1, 3, 4) ∈ S7 . Problem 1.1.9. b Show that any permutation belonging to group Sn can be decomposed into product of cycles with nonoverlapping sets of symbols. As an example, find such a decomposition for the permutation   1 2 3 4 5 6 7 8 9 10 ∈ S10 . (1.1.15) 3 6 4 1 2 8 10 5 9 7

Example 8. Group SO(2) of proper rotations (no reflections) in 2dimensional Euclidean space R2 around a given point O ∈ R2 , i.e., group of continuous symmetries of a circle S 1 in R2 centered at point O. The product of rotations is their consecutive action. The group SO(2) may be viewed as a continuum limit of cyclic groups, SO(2) = limn→∞ Cn . Its elements b Solutions

to problems marked with asterisk are collected in Chapter 8.

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gφ ∈ SO(2) (counterclockwise rotations by angles φ) can be written as matrices   cos φ, − sin φ , (1.1.16) Oφ = sin φ, cos φ which act on column with coordinates in R2 :     x x cos φ − y sin φ = . Oφ · y y cos φ + x sin φ These matrices obey Oφ OφT = I2 and hence are called orthogonal (hereafter T denotes transposition of a matrix and I2 is unit (2 × 2) matrix). They obey also special condition det(Oφ ) = 1, hence the letter S in the notation SO(2). Problem 1.1.10. Show that groups SO(2) and U (1) are isomorphic. Hint: write the transformation in C: w → w = z · w, ∀z ∈ U (1) in terms of linear transformation in R2 , by setting z = eiφ and w = x + iy. Example 9. Group O(2) of rotations and reflections in R2 (O from “orthogonal”), i.e., group of symmetries of a circle in R2 which includes both rotations and reflections against the axes running through the center. The group O(2) may be viewed as continuum limit of the dihedral groups, O(2) = limn→∞ Dn . Elements of O(2) describing proper rotations are given by matrices (1.1.16), while other elements of O(2) can be written as compositions (products) R · Oφ , where the matrix R makes reflection against the vertical symmetry axis of the circle (cf. (1.1.5)),   −1 0 (1.1.17) R= , R · Oφ = O−φ · R. 0 1 All elements O ∈ O(2) obey the orthogonality condition OOT = I2 , but det(R · Oφ ) = −1. Example 10. General linear group GL(n, C) is a set of all nondegenerate n × n matrices M = ||Mij || (det(M ) = 0) with complex elements, Mij ∈ C. The group operation in GL(n, C) is matrix multiplication. Problem 1.1.11. Prove associativity of multiplication in the group GL(n, C) (one of the properties required by the definition 1.1.1). Why does one impose nondegeneracy of matrices M ∈ GL(n, C)? Describe the group GL(1, C).

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The group GL(n, R) is defined in a similar way, as a set of all nondegenerate n×n matrices ||Mij || (det(M ) = 0) with real elements Mij ∈ R and matrix multiplication as the group operation. Matrices M obeying det(M ) = 1 are called special. The set of all special matrices SL(n, K), where K denotes either R or C, is a group with matrix multiplication as group operation. This group is called special linear. Problem 1.1.12. Show that SL(n, K) is a group. Example 11. Group O(n, R) ≡ O(n) (orthogonal) of rotations and reflections in n-dimensional real space Rn coincides with the group of symmetries of an (n − 1)-dimensional sphere S n−1 . A sphere S n−1 of radius ρ in Rn is a set of points x ∈ Rn with coordinates {x1 , x2 , . . . , xn } obeying x21 + x22 + · · · + x2n ≡

n 

xi xi = ρ2 .

(1.1.18)

i=1

In what follows we often omit the summation sign in similar formulas: summation over repeated indices is always assumed (unless the opposite is explicitly stated). Consider a linear transformation xi → xi = Oij xj ,

∀x ∈ S n−1 ,

(1.1.19)

such that xi xi = ρ2 . This transformation takes points of the sphere (1.1.18) to points x  belonging to the same sphere. The matrix O of this transformation, because of the condition xi xi = xi xi , has to obey the orthogonality relation OT · O = In ,

(1.1.20)

where In is unit n × n matrix. Thus, the transformation (1.1.19) with orthogonal matrix O is a symmetry of a sphere S n−1 , and the group O(n) of all these symmetries coincides with the set of all n × n real orthogonal matrices with matrix multiplication as the group operation. O(n) satisfies all group axioms. Indeed, ∀O1 , O2 ∈ O(n) one has (O1 · O2 )T · O1 · O2 = O2T · (O1T · O1 ) · O2 = In , In ∈ O(n) is the unit element, and if O ∈ O(n) then (O−1 )T ·O−1 = In , i.e., O−1 ∈ O(n). It follows from (1.1.20) that det(O) = ±1, i.e., the group O(n) consists of two nonoverlapping subsets O+ (n) and O− (n) with det(O) = +1 and det(O) = −1, respectively.

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Problem 1.1.13. Show that if OT · O = In then (O−1 )T · O−1 = In . Example 12. Group SO(n) (special orthogonal). The subset O+ (n) of orthogonal matrices with det(O) = +1 is denoted by SO(n) and is a group (with matrix multiplication as group operation), since ∀O, O ∈ SO(n) one has det(O · O ) = +1, det(O−1 ) = +1 and hence (O · O ) and O−1 belong to SO(n). SO(n) is the group of proper rotations (without reflections) in Rn . Problem 1.1.14. Show that the subset O− (n) ⊂ O(n) of orthogonal n × n matrices O obeying det(O) = −1 is not a group. Show that any element O ∈ O− (n) can be cast into the form O = R · O where O ∈ SO(n), and R is a fixed element of O− (n) (e.g., R = diag(−1, 1, . . . , 1)). Groups R\{0}, Z, Zn = Cn , U (1) = SO(2) and GL(1, C) are Abelian. The dihedral group Dn (n > 2), di-cyclic group Q2n (n > 1), symmetric groups Sn (n > 2), groups O(n) (n = 2, 3, . . . ), SO(n) (n = 3, 4, . . . ), GL(n, R), GL(n, C) (n = 2, 3, . . . ) are non-Abelian. All information on finite groups is contained in the multiplication table or Cayley’s table (introduced in 1854 by Arthur Cayley). As an example, this table for group D5 is e

g1 g2 g3 g4

r rg1 rg2 rg3 rg4

e g1 g2 g3 g4

e g1 g2 g3 g4

g1 g2 g3 g4 e

g2 g3 g4 e g1

g3 g4 e g1 g2

g4 e g1 g2 g3

r rg4 rg3 rg2 rg1

rg1 r rg4 rg3 rg2

rg2 rg1 r rg4 rg3

rg3 rg2 rg1 r rg4

rg4 rg3 rg2 rg1 r

r rg1 rg2 rg3 rg4

r rg1 rg2 rg3 rg4

rg1 rg2 rg3 rg4 r

rg2 rg3 rg4 r rg1

rg3 rg4 r rg1 rg2

rg4 r rg1 rg2 rg3

e g4 g3 g2 g1

g1 e g4 g3 g2

g2 g1 e g4 g3

g3 g2 g1 e g4

g4 g3 g2 g1 e

One way to establish an isomorphism of finite groups is to compare their Cayley’s tables. Problem 1.1.15. Establish the isomorphism D3 = S3 . Hint: write all elements of D3 as permutations of vertices of triangle.

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H

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e

G Fig. 1.1.2.

Subgroup H in group G.

Cayley’s table of D5 shows that five elements {e, g1 , g2 , g3 , g4 } make a subset C5 in the group D5 , which by itself has group properties. Indeed, according to Cayley’s table, the product of any two elements of C5 is again an element of C5 ; all other group properties are also satisfied. Clearly, the same property holds for a subset Cn in the group Dn for any n ≥ 2. Let us also recall that the actions of elements from Cn and Dn are special permutations of n vertices {1, 2, . . . , n}, i.e., these groups form subsets of the general group Sn of all permutations of n elements. Importantly, these subsets are closed under the group operation defined in Sn . In this way one arrives at an important notion of a subgroup — “group in a group” (see Fig. 1.1.2). Definition 1.1.7. Subset H of a group G is a subgroup, if H is a group with respect to the group operation (multiplication) defined in G, i.e., unit element e belongs to H, and for all h1 , h2 , h from H one has h1 · h2 ∈ H and h−1 ∈ H, where the multiplication is the group operation in G. Examples of subgroups 1. Any group G contains two trivial subgroups, the group G itself and subgroup with a single unit element e. 2. Cyclic group Cn = Zn is an Abelian subgroup of dihedral group Dn . 3. Groups Cn and Dn are subgroups of Sn (for n > 2). 4. Group Cn = Zn is a subgroup of U (1) = SO(2). 5. Groups Dn and SO(2) are subgroups of O(2). 6. Group SO(n) is a subgroup of O(n). 7. Group O(n) is a subgroup of GL(n, R). 8. Group GL(n, R) is a subgroup of GL(n, C). 9. Group O(k) is a subgroup of group O(n) (n > k). The subgroup O(k) may be embedded in O(n) in various ways. One of them is

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block-diagonal: ⎛ ⎜ ⎜ Og = ⎜ ⎝

g 0 ...

⎞ 0 .. ⎟ . ⎟ ⎟ ∈ O(n), 0 ⎠

g ∈ O(k).

(1.1.21)

0 In−k

10. Group SL(n, C) is a subgroup of GL(n, C), while SL(n, R) is a subgroup of GL(n, R). 1.1.2.

Normal subgroups, cosets, quotient (factor) groups

Definition 1.1.8. Let H be a subgroup of a group G. If g · h · g −1 ∈ H for any element g ∈ G and any element h ∈ H (this property is written as gHg −1 ⊂ H), then H is normal subgroup of G. Example. Subgroup Cn of group Dn is normal: ∀h ∈ Cn and ∀g ∈ Dn one has g · h · g −1 = h ∈ Cn .

(1.1.22)

Indeed, this fact is obvious for g ∈ Cn ⊂ Dn . If, on the other hand, g ∈ Dn involves reflection, g = r · gk , then g −1 = g−k · r and, making use of the identities (1.1.4) and (1.1.5), we obtain g · gm · g −1 = r · gk · gm · g−k · r = r · gm · r = gn−m ∈ Cn . Problem 1.1.16. Show that SO(n) is a normal subgroup of O(n). ´ Galois. His profound obserNormal subgroups were introduced by E. vation is that a group can be “divided” by its normal subgroup, and the result of the “division” is again a group. To describe this “division”, one introduces a notion of coset. Definition 1.1.9. Left coset of a subgroup H ⊂ G with respect to an element g ∈ G is a set of elements in G of the form g · h, where h runs through all elements of H. In other words, left coset is a subset of elements gH = {g · h | h ∈ H} in G. Similarly, a subset Hg = {h · g | h ∈ H} is called right coset of a subgroup H ⊂ G with respect to an element g ∈ G.

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Sets of all left and right cosets (coset spaces) of a subgroup H ⊂ G are denoted by G/H and H\G, respectively. Example. Consider cosets of a subgroup Cn ⊂ Dn with respect to elements g ∈ Dn . If we take an element g ∈ Dn belonging to the subgroup Cn , then the left coset g Cn coincides with the set e Cn = Cn . On the other hand, if we take an element g = r · gm which involves reflection, then the left coset coincides with the set r · Cn . It follows from (1.1.8) that the two cosets e Cn and r Cn do not overlap and together make the whole group Dn . Likewise, by making use of (1.1.5), one finds that Dn is decomposed into two nonoverlapping right cosets Cn e and Cn r. Moreover, Cn e = e Cn = Cn and, in view of (1.1.5), Cn r = r Cn , i.e., left and right cosets of a normal subgroup Cn ⊂ Dn coincide. Problem 1.1.17. Describe cosets of a subgroup SO(n) in O(n). Proposition 1.1.1. Left (right) cosets of a subgroup H ⊂ G either coincide or do not overlap. Left cosets of a normal subgroup H with respect to an element g ∈ G are the same as right cosets for any g. Proof. Let the overlap of two cosets g1 H and g2 H be nonvanishing, i.e., let there exist an element g ∈ G such that g ∈ g1 H and g ∈ g2 H. This means that there exist elements h1 , h2 ∈ H such that g1 · h1 = g = g2 · h2 .   This implies that g1 = g2 · h2 · h−1 1 = g2 · h , where h ∈ H, i.e., g1 ∈ g2 H, g2 ∈ g1 H, and hence the cosets g1 H and g2 H coincide. The proof for right cosets is the same. Let H be a normal subgroup. Then, by definition, ∀g ∈ G one has gHg −1 ⊂ H and g −1 Hg ⊂ H. Upon multiplying these relations by g on the right and left, respectively, we get gH ⊂ Hg and Hg ⊂ gH, i.e., gH = Hg.

Let H be a subgroup of a group G. Since any element g ∈ G belongs to some coset — namely, the coset gH (or Hg), the entire group G is a union of nonoverlapping left (right) cosets of a subgroup H ⊂ G. Definition 1.1.10. The number of cosets of a subgroup H in a group G is index of the subgroup H in G, denoted by indG (H).

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Problem 1.1.18 (Lagrange theorem). Show that the order (number of elements) and index of a subgroup H of a finite group G are divisors of the order of the group G: ord(G) = ord(H) · indG (H). Consider a set G/H of cosets (coset space) of normal subgroup H ⊂ G and define the product of two cosets g1 ≡ g1 H and g2 ≡ g2 H as the subset of G with elements f1 · f2 , where f1 ∈ g1 , f2 ∈ g2 : {f1 · f2 |f1 ∈ g1 , f2 ∈ g2 }.

(1.1.23)

This product is again a coset (g1 · g2 )H ≡ g 1 ·g2 , since by definition of normal subgroup and in view of Proposition 1.1.1 we have (g1 H) · (g2 H) = g1 (H g2 )H = g1 (g2 H)H = (g1 · g2 ) H.

(1.1.24)

Thus, the coset space G/H, where H is a normal subgroup of G, is endowed with the group operation induced from the group G. Associativity of the product (1.1.24) follows from associativity of the group operation in G. Unity in G/H is the coset eH, which coincides with the subgroup H, while the inverse coset to gH is g −1 H. Hence, we have proven the following proposition. Proposition 1.1.2. Cosets of a normal subgroup H of group G make a group G/H, which is called quotient (factor) group. Example. We have seen that the group Dn is decomposed into two cosets e˜ = e · Cn and r˜ = r · Cn of the normal subgroup Cn . Cayley’s table for elements e˜, r˜ ∈ Dn /Cn coincides with that of the group C2 : e˜ · e˜ = e˜,

e˜ · r˜ = r˜ · e˜ = r˜,

r˜ · r˜ = e˜.

Hence, Dn /Cn = C2 . Problem 1.1.19. What is the quotient group O(n)/SO(n) isomorphic to? Remark 1. The set of right cosets of a normal subgroup H ⊂ G is also a group. It follows from Proposition 1.1.1 that the group H\G is isomorphic to G/H. Remark 2. If a subgroup H is not normal in G, then the product of cosets defined in (1.1.23) is not, generally speaking, a coset. Therefore, in that case the multiplication in the coset space G/H cannot be defined.

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Definition 1.1.11. Finite group G is simple if it does not have nontrivial normal subgroups. Finite group G is semisimple if it does not have nontrivial Abelian normal subgroups. Note that the notion of simple group generalizes, in a certain sense, the notion of prime number (simple groups do not have nontrivial “divisors”). Examples 1. Group Cp = Zp is simple if p is a prime number. Indeed, its order equals ord(Cp ) = p, and it follows from the Lagrange theorem that Cp does not have any nontrivial subgroup whatsoever. 2. Group Dn is neither simple (Dn has a normal subgroup Cn ) nor semisimple (normal subgroup Cn is Abelian). Remark 3. In the case of infinite groups, the definition of simple group is different from that given in Definition 1.1.11. As an example, group SO(2n) is considered simple, even though it has a normal subgroup consisting of two elements I2n and (−I2n ). We give the definition of simple group for a certain class of infinite groups in Section 3.2.2. Remark 4. Let G be a finite group, and a ∈ G. Let us consider an infinite sequence of elements ak , k = 1, 2, 3, . . . . This sequence definitely has coincident elements, otherwise the group G would have infinite order. This shows that for a given element a ∈ G there exists the smallest integer n such that an = e. The set of elements {e, ak } (k = 1, 2, . . . , n − 1) is a subgroup Cn = Zn in G. Thus, every element a ∈ G is associated with some cyclic subgroup of G, whose order, through the Lagrange theorem, is a divisor of the order of G. The latter property is used for classifying finite groups of a given order. Problem 1.1.20. Show that finite group G whose order p is a prime number is unique: G = Zp = Cp . Problem 1.1.21. Show that if the order of a group G equals 2n, and H is a subgroup of G whose order is ord H = n, then H is a normal subgroup of G. Problem 1.1.22. Find the quotient group G/H where G is an additive group of integer multiples of 3, and H is a group of multiples of 15.

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1.1.3.

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Direct products of groups, conjugacy classes, center

Having two groups G1 and G2 , one constructs a new group G1 × G2 called direct product of G1 and G2 . The direct product is an important notion, since it enables one to construct new groups and, conversely, reduce the study of more complicated groups to that of simpler ones. Definition 1.1.12. Let G1 and G2 be two groups. The set of pairs (g1 , g2 ), where g1 ∈ G1 and g2 ∈ G2 , with component-wise multiplication (g1 , g2 ) · (h1 , h2 ) = (g1 · h1 , g2 · h2 ),

(1.1.25)

is a group G1 × G2 called direct product of groups G1 and G2 . Unit element in this new group is (e1 , e2 ) where e1 and e2 are unit elements in G1 and G2 , while the inverse element to (g1 , g2 ) is (g1−1 , g2−1 ). Clearly, the group G1 × G2 is isomorphic to G2 × G1 . The group G1 × G2 has two subgroups isomorphic to G1 and G2 , respectively, whose elements are {(g1 , e2 ) | g1 ∈ G1 } and {(e1 , g2 ) | g2 ∈ G2 }. These two subgroups are normal in G1 × G2 . If the groups G1 and G2 are simple and non-Abelian, then the group G1 × G2 is semisimple. Problem 1.1.23. Show that G1 and G2 are normal subgroups of G1 × G2 and establish the isomorphisms (G1 ×G2 )/G1 = G2 and (G1 ×G2 )/G2 = G1 . It is worth noting that the inverse relationship between “division” and “multiplication” is absent for groups. Namely, if H is a normal subgroup of G, then G/H is a group, but, generally speaking, (G/H) × H = G. Problem 1.1.24. Making use of the presentation (1.1.9), establish isomorphisms D2 = Z2 × Z2 and D6 = Z2 × D3 . Establish isomorphism Zp × Zq = Zpq , where p and q are coprime numbers. Show that Z4 /Z2 = Z2 , but Z4 = Z2 × Z2 . Check that the groups Z2 × Z4 and Z8 are not isomorphic. We give in Section 1.4.2 the definition of semidirect product of groups which generalizes the direct product. Nevertheless, the usual relationship between “division” and “multiplication” is absent, generally speaking, for the semidirect product as well. Let us introduce yet another concept of group theory. Definition 1.1.13. Two elements g1 and g2 of a group G are conjugate if there exists g ∈ G such that g1 = g · g2 · g −1 . A subset g˜0 = {g · g0 · g −1 |g ∈

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G} ⊂ G, where g0 is a given element of G, is called conjugacy class of the element g0 . Note that an element g ∈ G is conjugate to itself. If g1 ∈ G is conjugate to g2 ∈ G then g2 is conjugate to g1 , and if g2 is in turn conjugate to g3 ∈ G, then g1 is also conjugate to g3 . Thus, the set of elements of a group G is endowed with the equivalence relation (elements g1 and g2 are equivalent iff they are conjugate), and the equivalence classes are conjugacy classes. Any group G is partitioned into conjugacy classes which either coincide or do not overlap. Indeed, let two conjugacy classes g˜0 and g˜1 have nontrivial overlap, i.e., there exists an element g ∈ G which is conjugate to all elements of g˜0 and all elements of g˜1 . Then, due to the equivalence relation, all elements of g˜0 are conjugate to all elements of g˜1 , and hence the two conjugacy classes coincide. Examples 1. Unit element e of group G makes a conjugacy class which consists of one element. 2. Group Cn is decomposed into n conjugacy classes {e}, {g1}, {g2 }, . . . , {gn−1 }, each of which has one element. Indeed, the group Cn is Abelian, so ∀g ∈ Cn one has g · gn · g −1 = gn . This decomposition into conjugacy classes consisting of one element each is characteristic of all Abelian groups. 3. Group D2n+1 is decomposed into n+ 2 equivalence classes {e}, {gk , g−k } (k = 1, . . . , n), {r, r · g1 , . . . , r · g2n }. The fact that all improper elements of D2n+1 belong to one and the same conjugacy class follows from the relation r · gk · (r · gm ) · g−k · r = r · g2k−m ; it is important that the number of vertices of the polygon is odd. 4. Group D2n is decomposed into n + 3 conjugacy classes {e}, {gn }, {gk , g−k }, (k = 1, . . . , n − 1), {r, r · g2 , r · g4 , . . . }, {r · g1 , r · g3 , . . .}. Problem 1.1.25. Describe the conjugacy classes of the group O(2). Definition 1.1.14. An element g0 of a group G is self-conjugate, or central, if g · g0 · g −1 = g0 , ∀g ∈ G. A subset Z of all self-conjugate elements of a group G is the center of G. In other words, the center Z of a group G contains all elements which commute with every element of G: g · g0 = g0 · g,

∀g0 ∈ Z ⊂ G, ∀g ∈ G.

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Problem 1.1.26. Show that the center Z of group a G is an Abelian normal subgroup of G. Examples 1. 2. 3. 4.

Center of an Abelian group coincides with this group. Center of group D2n for n ≥ 2 consists of two elements {e, gn}. Center of group O(n), n > 2 consists of two elements {In , −In }. Center of group SO(2n), n > 1 also consists of two elements {I2n , −I2n }, while center of SO(2n + 1) is trivial, i.e., it consists of a single unit element I2n+1 .

1.1.4.

Example. Group of permutations (symmetric group) Sn

We consider in this subsection the group of permutations (symmetric group) Sn in some detail. Among other things, this group is important for applications. Here, using this group as an example, we illustrate concepts of the group theory which we have introduced so far. The group of permutations (symmetric group) Sn has been defined in Example 7 of Section 1.1.1. Elements of Sn are conveniently written in the following form:   1 2 3 ... n− 1 n , (1.1.26) A= a1 a2 a3 . . . an−1 an where {a1 , a2 , . . . , an } is a new arrangement of numbers {1, 2, . . . , n}. Permutations are also conveniently represented graphically. As an example, the permutation   1 2 3 4 5 6 7 ∈ S7 , (1.1.27) 3 6 4 1 2 5 7 can be drawn as a braid with seven strands: 1

2

3

4

5

6

7

1

2

3

4

5

6

7

(1.1.28)

The braid for the product A · B of two permutations A, B ∈ Sn is obtained as follows: one first draws the n-strand braid for the first permutation B

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and then connects to it the n-strand braid for the second permutation A: A·B

B A

=

(1.1.29)

(the braids here are schematically shown as boxes). This procedure is equivalent to multiplication of permutations defined in (1.1.12). Any permutation from the group Sn can be decomposed into a product of neighboring (elementary) transpositions σk = (k, k+1) (k = 1, . . . , n−1), which permute neighboring objects. This is a consequence of the fact that any permutation is a product of cycles (see Problem 1.1.9) and the results of the following problem.  Problem 1.1.27. Show that any cycle can be decomposed into a product of transpositions, namely, the following identity holds: (a1 , a2 , . . . , ak ) = (a1 , ak )(a1 , ak−1 ) · · · (a1 , a2 ).

(1.1.30)

Show that any transposition can be decomposed into a product of neighboring transpositions σi = (i, i + 1): (i, j) = σi · σi+1 · · · σj−2 · σj−1 · σj−2 · · · σi+1 · σi .

(1.1.31)

Problem 1.1.28. Derive (1.1.31) by employing the identities for transpositions (i, j) · (j, k) = (j, k) · (i, k) = (i, k) · (i, j), (i, j) · (k, m) = (k, m) · (i, j)

(i, j)2 = e

(∀ i, j, k),

(i = k = j, i = m = j).

(1.1.32) Check these identities by making use of the permutation multiplication rule (1.1.12). Write the permutation (1.1.27) as a product of neighboring transpositions σi ∈ S7 . Thus, the neighboring transpositions make the complete set of generators (see Definition 1.1.5) of the symmetric group. In fact, there are two convenient sets of generators of Sn . 1. The set of all neighboring transpositions σi = (i, i + 1) (i = 1, . . . , n − 1) which obey σi · σj = σj · σi ,

(|i − j| > 1),

(1.1.33)

σi · σi+1 · σi = σi+1 · σi · σi+1 ,

(1.1.34)

σi2 = e,

(1.1.35)

(i = 1, . . . , n − 1),

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where e is the identical permutation. The inverse elements of the generators are obvious from (1.1.35): σi−1 = σi . The latter formula enables one to find an inverse of any element of Sn written as a product of σi . Problem 1.1.29. Prove the relations (1.1.34) for neighboring transpositions σi . The presentation of the group Sn based on (1.1.33)–(1.1.35) is written as follows: σ1 , . . . , σn−1 | σi · σi+1 · σi = σi+1 · σi · σi+1 , σi · σj = σj · σi (|i − j| > 1),

σi2 = e.

(1.1.36)

It is called Moore–Coxeter presentation, and elements σi ∈ Sn are called Coxeter generators. 2. The set of two generators, namely, the first transposition σ1 = (1, 2) and the longest cycle C = (1, 2, . . . , n). Problem 1.1.30. Show that the set of two elements, the first transposition σ1 = (1, 2) and the longest cycle C = (1, 2, . . . , n), is the complete set of generators of Sn . Hint: make use of the relation C · σi · C −1 = σi+1 . Let us make a digression here. If one does not insist on the validity of the identity (1.1.35) and requires only that the generators σi are invertible and obey the relations (1.1.33) and (1.1.34), then σi generate a group Bn called braid group. The identities (1.1.33) and (1.1.34) are dubbed locality relation and braid group relation, respectively. This nomenclature is justified, e.g., by a graphic representation of the relation (1.1.34), which is an identity of two braids with three strands:

Here we make use of the representation

(1.1.37) We note a subtlety in this graphic representation in the case of Bn : when strands cross, it is important which one is above the other, as indicated in (1.1.37). In the case of group

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2

1

2

Fig. 1.1.3.

21

Braid closure (left) resulting in a knot known as trefoil.

Sn this is irrelevant, since, due to (1.1.35), we have

The braid group Bn , unlike permutation group Sn , has infinite order. This follows from the fact that all powers of generators σik ∈ Bn (k = 1, 2, . . . ) are nontrivial and independent elements of Bn . As an example,

i.e., the braid σi2 ∈ Bn has a link that cannot be unraveled. The braid group Bn and its matrix representations are of importance, since they enable one to describe in algebraic way any braid of n strands. By closing the braids with additional strands having the same initial and final positions (see example in Fig. 1.1.3), one obtains graphic representations of knots and links.c A theorem due to W. Alexander of 1923 states that any knot, and, indeed, any link can be obtained by closing some braid. Therefore, the braid group is of importance in the theory of knots and links.

Let us recall (see Definition 1.1.6) that a product of even (odd) number of transpositions is even (odd) permutation. In fact, the representation of c Knot

is one entangled strand, while link is made of several entangled and linked strands.

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a permutation as a product of transpositions is not unique. Nevertheless, different representations are related to each other by (1.1.32), and it is clear that manipulations based on (1.1.32) preserve parity of permutation, so parity is well defined. Even permutations make a subgroup of the symmetric group Sn . Problem 1.1.31. Check that the group axioms are satisfied for a subset of even permutations in Sn . The subgroup of even permutations in Sn is denoted by An and called alternating subgroup. The subgroup An is a normal subgroup of Sn , since the conjugation h → g · h · g −1 preserves the parity of h. Problem 1.1.32. Show that the quotient group Sn /An is isomorphic to Z2 . Let us now split the symmetry group into conjugacy classes (see Definition 1.1.13). We recall (see Problem 1.1.9) that any element A of Sn can be represented as a product of cycles consisting of different objects: A = (a1 , a2 , . . . , aλ1 )(aλ1 +1 , aλ1 +2 . . . , aλ1 +λ2 )

 

  λ1

λ2

· · · (aλ1 +···+λm−1 +1 , . . . , aλ1 +···+λm ),

 

(1.1.38)

λm

where λi is the length of ith cycle, and all objects on the right-hand side are different: aα = aβ for α = β. Clearly, one has λ1 + λ2 + · · · + λm = n. Since all these cycles commute with each other, we can arrange them in (1.1.38) in such a way that λ1 ≥ λ2 ≥ · · · ≥ λm . A nonincreasing sequence of positive integers [λ1 , λ2 , . . . , λm ] ≡ λ, whose sum equals n, is called partition of number n and is denoted by λ n. Thus, there is partition λ n associated with any given permutation from Sn . Proposition 1.1.3. Two permutations with one and the same partition λ n are conjugate to each other. Conjugacy classes in Sn are in one-to-one correspondence with partitions λ n.

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Proof. Let permutation B ∈ Sn be decomposed into a product of cycles whose lengths (λ1 , λ2 , . . . , λm ) are the same as for permutation A ∈ Sn in (1.1.38): B = (b1 , b2 , . . . , bλ1 ) (bλ1 +1 , bλ1 +2 . . . , bλ1 +λ2 )       λ1

λ2

· · · (bλ1 +···+λm−1 +1 , . . . , bλ1 +···+λm ),   

(1.1.39)

λm

i.e., the two permutations A and B have one and the same partition λ  n. Then the permutation (1.1.38) is related to the permutation (1.1.39) by conjugation B = T · A · T −1 , where the permutation T ∈ Sn is   a1 a2 . . . aλ1 aλ1 +1 . . . aλ1 +λ2 · · · aλ1 +···+λm−1 +1 . . . aλ1 +···+λm . T = b1 b2 . . . bλ1 bλ1 +1 . . . bλ1 +λ2 · · · bλ1 +···+λm−1 +1 . . . bλ1 +···+λm (1.1.40) To prove the latter statement, we note that T · A · T −1 = T (a1 , . . . , aλ1 ) T −1 · T (aλ1 +1 , . . . , aλ1 +λ2 ) T −1 · · · ,       λ1

λ2

so it suffices to check the equality T (ak , . . . , ak+λi ) T −1 = (bk , . . . , bk+λi ) for a single cycle. This is done straightforwardly by making use of the rule (1.1.12) for the product in Sn . So, the permutations A and B belong to the same conjugacy class. On the other hand, let T be an arbitrary permutation (1.1.40), then the conjugation operation A → T · A · T −1 = B transforms the permutation A with partition λ  n into permutation B (1.1.39) with the same partition. Thus, there is one-to-one correspondence between conjugacy classes in Sn and partitions of number n.

To figure out whether two permutations A and B belong to the same conjugacy class, one decomposes them into products of independent cycles, and places the cycles from top to bottom (from left to right in formulas) according to their lengths, with the longest cycle on the top, until the shortest cycle takes its place in the very bottom. If the number of the cycles and lengths of all cycles λ1 , λ2 , . . . , λm are the same for permutations A and B, then these permutations belong to the same conjugacy class, otherwise they do not. A partition λ n is conveniently represented by Young diagram m1

λ=

λ(1)

m2 m3 mk

λ(2)

...

⇔

mk m2 1 λ = [λm (1) , λ(2) , . . . , λ(k) ].

λ(3)

λ(k)

(1.1.41)

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In our context, this diagram corresponds to a conjugacy class of permutations with m1 cycles of length λ(1) (lengths of the first m1 cycles coincide, λ1 = · · · = λm1 , and are equal to λ(1) ), m2 cycles of length λ(2) , etc. A concrete permutation, e.g., (1.1.38), is represented by a tableau with cells filled with appropriate elements: a1 a2 ak1 +1 ak1 +2 ... ... akm−1 +1 . . .

a3 . . . . . . ak1 . . . . . . ak2 ... ... akm

(1.1.42)

Here we set for brevity k1 = λ1 , k2 = λ1 + λ2 , . . . , km = λ1 + · · · + λm = n.

Examples 1. Symmetric group S3 . There are the following partitions of 3! = 6 permutations of three elements into three different conjugacy classes: 1. e = (1)(2)(3), 2. (1, 2)(3), (1, 3)(2), (2, 3)(1), 3. (1, 2, 3), (1, 3, 2).

(1.1.43)

Their Young diagrams are

3 =• • • = [1 ],

= •• • = [2, 1],

= ••• = [3].

It follows from (1.1.43) that the number of elements in classes [13 ], [2, 1], [3] equals 1, 3, 2, respectively. In what follows we do not write trivial cycles in expressions like (1.1.43), with the exception of the first partition with trivial cycles only; as an example, the permutation (1, 2)(3) is denoted merely by (1, 2). 2. Symmetric group S4 . One has the following partition of 24 permutations of four objects into five different classes (# denotes the number of

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elements in each class) • 1. e = (1)(2)(3)(4) ∈ •• = [14 ] •

(#1),

2. (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4) ∈ •• •• = [2, 12 ]

(#6),

3. (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3) ∈ •• •• = [2, 2]

(#3),

4.

5.

(1, 2, 3), (1, 3, 2), (1, 2, 4), (1, 4, 2), (1, 3, 4), (1, 4, 3), (2, 3, 4), (2, 4, 3) ∈ •••• = [3, 1]

(#8),

(1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 2, 4), (1, 3, 4, 2), (1, 4, 2, 3), (1, 4, 3, 2) ∈ •••• = [4] (#6), (1.1.44)  # = 1 + 6 + 3 + 8 + 6 = 24 = 4!.

Problem 1.1.33. Split the group S5 into conjugacy classes and find the number of elements in each class. There is a general combinatoric formula for the number of conjugate m2 mk 1 elements in a class Kλ ⊂ Sn with λ = [λm (1) , λ(2) , . . . , λ(k) ]. Proposition 1.1.4. Consider conjugacy class Kλ in Sn whose Young mk m2 1 diagram λ = [λm (1) , λ(2) , . . . , λ(k) ] is shown in (1.1.41). Let Zλ be a set of permutations T rendering a given element A ∈ Kλ stable (i.e., T · A · T −1 = A). The set Zλ is a subgroup of Sn whose order is mk m2 1 ord Zλ = m1 ! λm (1) m2 ! λ(2) · · · mk ! λ(k) .

(1.1.45)

The number of elements in class Kλ is given by |Kλ | =

n! n! = m1 mk . 2 ord Zλ m1 ! λ(1) m2 ! λm (2) · · · mk ! λ(k)

(1.1.46)

Proof. We have seen that any element A ∈ Kλ is related to any other element B ∈ Kλ by the transformation B = T · A · T −1 , where T is given by (1.1.40), and elements A, B ∈ Kλ are given by (1.1.38), (1.1.39). Equality B = A holds iff (1.1.39) differs from (1.1.38) only by (a) permutation of cycles of equal length, and (b) cyclic permutations inside each cycle. These operations are independent, i.e., the corresponding elements T ∈ Zλ are different, as seen from (1.1.40). There are mi ! operations of the type (a) for m cycles of length λ(i) (whose number is mi ), and λ(i)i operations of the type (b). The product of these factors gives (1.1.45). Clearly, the set Zλ is a subgroup of Sn , and the stability subgroups Zλ of different elements from one and the same class Kλ are isomorphic. Now, let us consider an element

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A ∈ Kλ whose stability subgroup is Zλ . Then ∀B ∈ Kλ there exists T ∈ Sn such that B = T · A · T −1 and the conjugation operation moves elements of Sn within the class Kλ , i.e., ∀T ∈ Sn we have T · A · T −1 ∈ Kλ . All elements from the left coset T˜ · Zλ for a given T˜ ∈ Sn move A to one and the same element T˜ · A · T˜−1 ∈ Kλ , which, therefore, is in correspondence with this left coset T˜ · Zλ . Furthermore, different elements of Kλ correspond to different left cosets of the subgroup Zλ in Sn . Indeed, the / Zλ . Hence, the number condition T˜ · A · T˜ −1 = T˜ · A · (T˜ )−1 is equivalent to T˜ −1 · T˜  ∈ of elements in the class Kλ is equal to the number of elements in the coset space Sn /Zλ . Making use of the Lagrange theorem (see Problem 1.1.18) and the formula (1.1.45), we arrive at (1.1.46).

Corollary 1.1.1. According to (1.1.46), the only conjugacy class in Sn with one element has Young diagram [1n ] (this is the class of the unit element e ∈ Sn ). Therefore, the center of the group Sn is trivial. Note that the formula (1.1.46) is nontrivial from the combinatorics viewpoint, since all numbers |Kλ | are integer, while their sum over partitions λ n equals ord(Sn ) = n!. This implies a remarkable identity  −1 = 1. λn (ord Zλ ) Problem 1.1.34. Making use of (1.1.46), find the numbers of elements in conjugacy classes in S4 which correspond to diagrams [14 ], [2, 12 ], [22 ], [3, 1], [4] and compare them with the numbers in (1.1.44). Find the numbers of elements in conjugacy classes in S5 described by diagrams [15 ], [2, 13 ], [22 , 1], [3, 12 ], [3, 2], [4, 1], [5] and check that the sum of these numbers equals 5!. One of important properties of symmetric groups is that any finite group G of order n is a subgroup of Sn . This follows from Theorem 1.3.1 which we prove below in Section 1.3.2. In this sense the study of finite groups reduces to the study of various subgroups of symmetric groups. Also, as we discuss in the accompanying book, the symmetric groups are extremely relevant in the theory of finite-dimensional representations of groups SL(n, C) and SU (n), which have numerous applications in physics. 1.2.

1.2.1.

Matrix Groups. Linear, Unitary, Orthogonal and Symplectic Groups Vector spaces and algebras

In what follows, K denotes either field of complex numbers or field of real numbers. Definition 1.2.1. Vector (or linear) space V(K) over field K is a set of objects (vectors) x, y, . . . endowed with multiplication by numbers

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α, β, . . . ∈ K and addition, such that αx + β y ∈ V(K). Furthermore, (V(K), +) must be an Abelian group with unit element 0 ∈ V(K), called zero vector, 0 = αx| α=0 ,

∀x ∈ V(K).

Finally, the following properties must hold: (α + β)x = αx + βx,

α(x + y) = αx + α y,

(αβ)y = α(β y ).

Vector spaces V(R) and V(C) are called real and complex, respectively. Definition 1.2.2. Vector space Vn (K) is n-dimensional if there exist n linearly independent vectors ek ∈ Vn (K) (k = 1, 2, . . . , n) such that any x ∈ Vn (K) can be cast in the form x = ek xk .

(1.2.1)

The numbers xk ∈ K are called coordinates (or components) of the vector x in basis {ek }. In the case of complex vector space Vn (C), one can define complex conjugation operation. Namely, vector (z )∗ ∈ Vn (C) is complex conjugate to vector z = ek zk if its decomposition in basis in Vn (C) is (z )∗ = ek zk∗ , where coordinates zk∗ are related to coordinates zk by complex conjugation. Note that, generally speaking, this definition is not universal, in the sense that it depends on the choice of basis. Definition 1.2.3. Linear space A over field K is algebra if, besides addition of vectors and their multiplication by numbers from K, it is endowed with an operation of multiplication of vectors, which takes two vectors a, b ∈ A to the third vector a · b ∈ A and obeys the distributivity axioms a · (αb + βc) = α(a · b) + β(a · c), (αa + βb) · c = α(a · c) + β(b · c), ∀a, b, c ∈ A, ∀α, β ∈ K. (1.2.2) If multiplication obeys also the associativity axiom, a · (b · c) = (a · b) · c,

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then algebra A is associative. For K = R and K = C, algebra is called real and complex, respectively. Algebra A is n-dimensional if it is n-dimensional as vector space. 1.2.2.

Matrices. Determinant and Pfaffian, direct product

Let A = ||Aij || (i, j = 1, 2, . . . , n) be a complex (n × n) matrix, where indices i and j label rows and columns, respectively. The space Matn (C) of all these matrices is endowed with standard operations of matrix addition and multiplication, as well as multiplication of matrix by a number: A + B = ||Aij + Bij ||,

A · B = ||Aik Bkj ||,

∀A, B ∈ Matn (C),

α A = ||α Aij ||,

∀α ∈ C

(1.2.3)

(summation over repeated index k is implied). Real algebra Matn (R) of real n × n matrices is defined in a similar way. Problem 1.2.1. Show that the set of matrices Matn (C) is n2 -dimensional complex associative algebra with algebraic operations (1.2.3). In what follows we use the following standard notations: T AT = ||AT ij ||, Aij = Aji ,

A∗ = ||A∗ij ||, A† = (AT )∗ = ||A†ij || , A†ij = A∗ji , In = ||δij ||, where AT , A∗ , A† and In are transposed, complex conjugate, Hermitian conjugate and unit n×n matrices, respectively. Matrix A is called symmetric if AT = A, antisymmetric if AT = −A, Hermitian if A† = A, anti-Hermitian if A† = −A, orthogonal if AT · A = In and unitary if A† · A = In . Matrix −1 = A−1 · A = In . Matrix A is A−1 = ||A−1 ij || is inverse to A if A · A degenerate if it does not have an inverse, otherwise it is nondegenerate. n Important characteristics of a matrix A is its trace Tr(A) = i=1 Aii and determinant det(A) =

n  i1 ,...,in =1

εi1 i2 ...in Ai1 1 Ai2 2 · · · Ain n ,

(1.2.4)

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where {εi1 i2 ...in } are components of antisymmetric tensor (ε-tensor) of nth rank.d All components of εi1 ...in are uniquely determined by two conditions: 1. ε12...n = 1, 2. εi1 i2 ...ik ...im ...in = − εi1 i2 ...im ...ik ...in ;

(1.2.5)

the right- and left-hand sides of the condition 2 differ by transposition of two indices ik and im . It follows from the definition (1.2.5) that all components which carry at least two equal indices vanish, while if all indices are different, then εi1 i2 ...in = (−1)P (σ) ε12...n = (−1)P (σ) , where P (σ) is parity of permutation  1 2 3 ... n− 1 σ= i1 i2 i3 . . . in−1

 n , in

(1.2.6)

(1.2.7)

i.e., P (σ) = 0 and P (σ) = 1 for even and odd permutation σ, respectively. The number of nonvanishing components of ε-tensor is n!. Making use of (1.2.6), one writes the definition of determinant (1.2.4) as follows:  (−1)P (σ) Aσ(1) 1 Aσ(2) 2 · · · Aσ(n) n , (1.2.8) det(A) = σ∈Sn

where the sum runs over all permutations (1.2.7) from symmetric group Sn . Problem 1.2.2. Making use of the definitions (1.2.4), (1.2.6) and (1.2.8), prove the following identities: εi1 i2 ...in Ai1 j1 Ai2 j2 · · · Ain jn = det(A) εj1 j2 ...jn , εj1 j2 ...jn · εj1 j2 ...jn = n! ,

(1.2.9)

εij2 ...jn · εkj2 ...jn = (n − 1)! δik ,

(1.2.10) εi1 i2 j3 ...jn · εk1 k2 j3 ...jn = (n − 2)! (δi1 k1 δi2 k2 − δi1 k2 δi2 k1 ), ⎛ ⎞ δi1 k1 δi1 k2 . . . δi1 kr ⎜δi2 k1 δi2 k2 . . . δi2 kr ⎟ , εi1 ...ir jr+1 ,...jn · εk1 ...kr jr+1 ...jn = (n − r)! det ⎜ .. .. ⎟ ⎝ ... . ... . ⎠ δir k1 δir k2 . . . δir kr object {εi1 i2 ...in } is nth rank tensor in the sense that its components εi1 i2 ...in carry n indices, and it can be interpreted as an element of tensor product Vn⊗n of n vector spaces Vn , cf. Section 4.3.1. Tensor properties of {εi1 i2 ...in } are irrelevant in our context here. d The

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det(A) =

1 εi i ...i Ai j Ai j · · · Ain jn εj1 j2 ...jn , n! 1 2 n 1 1 2 2

(1.2.11)

det(A) = A1j1 A2j2 · · · Anjn εj1 j2 ...jn .

(1.2.12)

Problem 1.2.3. Let A and B be n × n matrices. Prove the identities det(A · B) = det(A) det(B), (A−1 )ji =

det(A) = det(AT ),

1 1 εii ...i Ai j · · · Ain jn εjj2 ...jn , (n − 1)! det(A) 2 n 2 2

(1.2.13) (1.2.14)

(note peculiar orders of indices in the left- and right-hand sides). Hint: make use of (1.2.9), (1.2.11) for proving (1.2.13); to prove (1.2.14), make a convolution of both sides with Aik and use (1.2.10). It follows from (1.2.14) that the matrix A is not degenerate iff det(A) = 0. Problem 1.2.4. Let A and B be n × n matrices, det(B) = 0, and be a small parameter. Show that εki2 ...in Aki1 + εi1 ki3 ...in Aki2 + · · · + εi1 i2 ...in−1 k Akin = Tr(A) εi1 i2 ...in , (1.2.15) det(B + A) = det(B) · [1 + Tr(B −1 A) + O( 2 )],

(1.2.16)

∂ det(etA ) = Tr(A) · det(etA ) ⇔ det(etA ) = etTr(A) . ∂t

(1.2.17)

Problem 1.2.5. Let g = ||gij || be n × n matrix. Making use of (1.2.11) and (1.2.14), prove the identity ∂ det(g) = det(g) · (g −1 )ji ∂gij

(1.2.18)

(note peculiar order of indices on the right-hand side). Assuming that g depends on parameter t, derive the formula (det(g))−1 · ∂t det(g) = (g −1 )ji ∂t gij , i.e., ∂t ln det(g) = Tr(g −1 ∂t g).

(1.2.19)

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Let us derive a convenient decomposition of a nondegenerate matrix A ∈ Matn (C) which is used in various applications. Proposition 1.2.1. Any nondegenerate complex n × n matrix A can be decomposed into a product A = H · U,

(1.2.20)

where U is a unitary matrix and H is a Hermitian matrix, U † U = In and H † = H. Proof. Consider matrix X = A · A† . It is Hermitian and has positive eigenvalues. The latter fact follows from the relation for the scalar product, zi Xij zj∗ = (zi Aik )(zj Ajk )∗ > 0,

(1.2.21)

where (z1 , . . . , zn ) is an arbitrary vector in Cn . Note that the right-hand side of (1.2.21) is nonzero for any nonzero vector z, since zi Aik = 0 would contradict nondegeneracy of A. Since X is Hermitian, it can be diagonalized by a unitary matrix V , so one can write X = V · D2 · V † = (V · D · V † )2 = H 2 ,

(1.2.22)

where D is diagonal real matrix whose diagonal elements are square roots of eigenvalues of X. Clearly, H = V · D · V † is a Hermitian matrix. The relation (1.2.22) can also be written as (H −1 · A) · (A† H −1 ) = In , i.e., matrix H −1 · A = U is unitary. This is equivalent to (1.2.20). Clearly, in Proposition 1.2.1, the unitary matrix U could be placed on the left of the Hermitian factor H. Also, the proof of Proposition 1.2.1 shows that the matrix H can be chosen positive-definite (i.e., its eigenvalues are positive). The decomposition (1.2.20) is called polar, since it generalizes the polar representation of complex numbers, z = ρ eiφ (indeed, the latter is precisely (1.2.20) for n = 1). Let A = ||Aik || be n × n matrix and B = ||Bab || be m × m matrix. Their direct product (A ⊗ B) is a composite (n · m) × (n · m) matrix with elements (A ⊗ B)ia,kb = Aik Bab .

(1.2.23)

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The matrix (A ⊗ B) has dimension (n · m) × (n · m) and it carries double indices ia and kb. It can be written as (n × n) block matrix ⎞ ⎛ A11 ||Bab || A12 ||Bab || . . . A1n ||Bab || ⎟ ⎜ ⎜ A21 ||Bab || A22 ||Bab || . . . A2n ||Bab || ⎟ ⎟ ⎜ ⎟, (1.2.24) (A ⊗ B) = ⎜ ⎟ ⎜ . . . .. .. . ⎟ ⎜ . . . . ⎠ ⎝ An1 ||Bab || An2 ||Bab || . . . Ann ||Bab || where each block is m×m. According to the definition (1.2.23), the product of matrices of this form reads (A ⊗ B)ia,kb · (C ⊗ D)kb,jc = Aik Bab Ckj Dbc = (A C)ij (B D)ac = (A · C ⊗ B · D)ia,jc , or, omitting indices, (A ⊗ B) · (C ⊗ D) = (A · C ⊗ B · D).

(1.2.25)

This agrees with the standard multiplication of matrices of the form (1.2.24), see the problem. Problem 1.2.6. Show that the product (1.2.25) of matrices of the form (1.2.24) is the standard product of (n · m) × (n · m) matrices. Show that Tr(A ⊗ B) = Tr(A) · Tr(B). Even-dimensional (2n × 2n) antisymmetric matrix B = −B T is characterized by yet another quantity, called Pfaffian: Pf(B) =

1 2n n!



εi1 i2 i3 i4 ...i2n−1 i2n Bi1 i2 Bi3 i4 . . . Bi2n−1 i2n . (1.2.26)

i1 ,...,i2n

Since the matrix B is antisymmetric, Pfaffian has an obvious property Pf(B T ) = (−1)n Pf(B).

(1.2.27)

Proposition 1.2.2. Let A be an arbitrary (2n × 2n) matrix, and B be an antisymmetric (2n × 2n) matrix. Then the following identities hold: det(A) · Pf(B) = Pf(A · B · AT ),

(1.2.28)

det(B) = [Pf(B)]2 ,

(1.2.29)

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and, if det(B) = 0, then Pf(B −1 ) = [Pf(B T )]−1 .

(1.2.30)

Proof. We make use of (1.2.9) and (1.2.26), and derive the identity (1.2.28): det(A) · Pf(B) = =

1 2n n! 1 2n n!

εi1 i2 ...i2n (Ai1 j1 Ai2 j2 · · · Ai2n j2n )Bj1 j2 Bj3 j4 · · · Bj2n−1 j2n εi1 i2 ...i2n (Ai1 j1 Bj1 j2 Ai2 j2 ) · · · (Ai2n−1 j2n−1 Bj2n−1 j2n Ai2n j2n )

= Pf(A · B · AT ).

(1.2.31)

Let us now prove the identity (1.2.29). This identity is obvious for a degenerate matrix B, since in that case Pf(B) = 0 (which follows directly from the definition (1.2.26)). Now, every even-dimensional antisymmetric, nondegenerate matrix B has the following representation: B = Q · J · QT ,

(1.2.32)

where Q is some nondegenerate (2n × 2n) matrix, and J is the standard antisymmetric matrix  J=

0 −In

 In . 0

(1.2.33)

Here 0 denotes zero (n × n) block. The representation (1.2.32) gives det(B) = det2 (Q) det(J),

(1.2.34)

and making use of (1.2.28) we have Pf(B) = Pf(Q · J · QT ) = det(Q)Pf(J). It is straightforward to see that det(J) = 1,

Pf(J) = (−1)n(n−1)/2 .

Then (1.2.29) follows from (1.2.34) and (1.2.35).

(1.2.35)

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Let us make a replacement B → B −1 , A → B in (1.2.28). Then, in view of (1.2.27), we get det(B) · Pf(B −1 ) = (−1)n Pf(B).

(1.2.36)

We now use (1.2.29) for det(B) and again recall (1.2.27) to obtain (1.2.30).

Consider n × n matrix

⎛

1 ⎜ x1 ⎜ ⎜ x2 1 X =⎜ ⎜ . ⎜ . ⎝ . xn−1 1

1 x2 x22 . ..

xn−1 2

⎞ 1 xn ⎟ ⎟ x2n ⎟ ⎟, . ⎟ .. ⎟ ⎠ xn−1 n

... ... ... ... ...

(1.2.37)

called Vandermonde matrix. det(X) is called Vandermonde determinant. Problem 1.2.7. Show that Vandermonde determinant equals  (xi − xj ). Δ(x1 , x2 , . . . , xn ) ≡ det(X) =

(1.2.38)

j 2 ∂k ∂m (∂l l ) = 0. We then apply operation ∂i ∂j to the original Eq. (3.4.25) and find ∂i ∂j (∂k m + ∂m k ) = 0. Upon permuting indices m ↔ j, k ↔ j and comparing the results, we finally derive ∂k ∂j ∂i m = 0,

∀i, j, k, m.

Thus, we see that for n > 2, the function m (x) is a polynomial in second order: m (x) = am + Tmk xk + bm,pk xp xk ,

(3.4.29) {xk }

of at most (3.4.30)

where am , Tmk , bm,pk are some constants, and bm,pk = bm,kp . Constraints on these constants are obtained by inserting the solution (3.4.30) back into Eq. (3.4.25). As a

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result, we find that coefficients am are arbitrary, while coefficients Tmk and bm,pk obey 2 Tmk + Tkm = ηkm (ηij Tij ), n (3.4.31) 2 bm,kp + bk,mp = ηkm (ηij bi,jp ). n The first of these equations has the general solution Tmk = 2 ωmk + λ ηmk ,

(3.4.32)

where ωmk = −ωkm and the coefficient 2 is introduced for convenience. To obtain the general form of coefficients bm,pk = bm,kp , we denote ηij bi,jp = n · bp

(3.4.33)

and substitute k ↔ p and m ↔ p in the second equation in (3.4.31): bm,pk + bp,mk = 2bk , bp,km + bk,pm = 2bm . We combine the latter with the second equation in (3.4.31) and find bk,mp = bm ηkp + bp ηkm − bk ηmp .

(3.4.34)

Then Eq. (3.4.33) becomes an identity, i.e., coefficients bm are arbitrary. Finally, we insert (3.4.32) and (3.4.34) in (3.4.30) and obtain the general solution for conformal Killing vector: i (x) = ai + 2 ηim ωmk xk + λxi + 2(bm xm )xi − bi (xm xm ).

(3.4.35)

It coincides with the vector (3.4.3). Therefore, the composition of transformations (3.4.1) is the most general (infinitesimal) conformal transformation in Rp,q , so transformations (2.1.36), (2.1.37), (2.1.38), (2.1.40) exhaust the connected component of unit element in group Conf(Rp,q ).

The proof of Proposition 3.4.1 shows that the case p + q ≡ n = 2 is special and needs separate study. In 2-dimensional case, Eq. (3.4.25) for conformal Killing vector is written as follows: ∂1 2 + ∂2 1 = 0, ∂1 1 = ±∂2 2 ,

(3.4.36)

where the sign “+” in the second equation refers to R2 , while “−” corresponds to R1,1 . Equations (3.4.36) simplify further in terms of variables x± = x1 ± x2 = x1 ∓ x2 , ± = 1 ± 2 for R1,1 ; x± = x1 ± ix2 , ± = 1 ± i2 for R2 . One has ∂ + − (x) = 0, ∂ − + (x) = 0,

(3.4.37)

∂ where ∂ ± = ∂x . Equations (3.4.37) imply that + (x) depends only on x+ , ± whereas − (x) depends only on x− . This solution to Eq. (3.4.36) describes conformal transformations

x+ → F+ (x+ ), x− → F− (x− ), in agreement with the analysis in Section 1.4.3.

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3.4.3.

Isomorphism conf(Rp,q ) = so(p + 1, q + 1), for p+q >2

We have seen in Section 2.1.3 that conformal group Conf(Rp,q ) with p + q > 2 is isomorphic to group O(p + 1, q + 1). Accordingly, their Lie algebras are isomorphic too. It is instructive, nevertheless, to establish the latter isomorphism explicitly. Proposition 3.4.2. Lie algebra conf(Rp,q ) of conformal group Conf(Rp,q ) (with p + q > 2) is isomorphic to Lie algebra so(p + 1, q + 1). Proof. Lie algebra conf(Rp,q ) has generators {Pm , Lmk , D, Km } (m, k = 1, . . . , p + q) and defining commutation relations (3.4.6). We set p + q = n. Elements Lij (1 ≤ i < j ≤ n), obey commutation relations given in (3.4.6) and generate subalgebra so(p, q) in Lie algebra conf(Rp,q ). So, one can view Lie algebra conf(Rp,q ) as an extension of so(p, q), obtained by adding extra generators {Pm , D, Km } with commutation relations (3.4.6). The generators {Pm , D, Km } can be realized as differential operators, see (3.4.5): Pm = −∂m , D = xm ∂m , Km = (xk xk ) ∂m − 2 xm D.

(3.4.38)

Dimension of this extended algebra is, clearly, n(n − 1) (n + 1)(n + 2) +2n+1 = , 2 2 which is the same as dimension of so(p + 1, q + 1). Let us show that the extended algebra with defining commutation relations (3.4.6) is isomorphic to Lie algebra so(p + 1, q + 1). Defining commutation relations of so(p + 1, q + 1) are, see (3.3.41): [Lab , Lcd ] = ηbc Lad + ηda Lbc + ηca Ldb + ηbd Lca ,

(3.4.39)

where we use the convention that a, b, c, d = 0, 1, . . . , n + 1, while the diagonal metric ηab has signature (p + 1, q + 1). We obtain from (3.4.39) that [Lij , Lk ] [L0j , L0k ] [L0 j , Lkm ] [Ln+1 j , Lkm ] [L0j , Ln+1,k ]

= ηjk Li + ηi Ljk + ηki Lj + ηj Lkj , = η00 Lkj , [Ln+1,j , Ln+1,k ] = ηn+1,n+1 Lkj , = ηjk L0 m − ηjm L0 k , = ηjk Ln+1 m − ηjm Ln+1 k , = ηjk Ln+1,0 ,

(3.4.40)

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where i, j, k,  = 1, . . . , n. We notice that η00 = −ηn+1,n+1 = 1,

(3.4.41)

and that ηij with i, j = 1, . . . , n is metric of space R(p,q) (so, ηij has signature (p, q)). We now introduce combinations Lk () = (L0k + Ln+1,k ), where  = ±1. We make use of antisymmetry Lij = −Lji and Eqs. (3.4.40), (3.4.41) and obtain [Lj (), Lk ()] = 0, [L0,n+1 , Lk ()] = − Lk (), [Lj (), Lkm ] = ηjk Lm () − ηjm Lk (), where k, j, m = 1, . . . , n. We identify Pk := Lk ()| =+1 = L0k + Ln+1,k , Kk := Lk ()| =−1 = L0k − Ln+1,k , D := L0,n+1 . (3.4.42) Making use of (3.4.40) it is straightforward to check that operators (3.4.42) together with Lij obey commutation relations (3.4.6). Therefore, Eq. (3.4.42) establishes isomorphism so(p + 1, q + 1) = conf(R(p,q) ). Corollary 3.4.1. Operators {Lkj , Pj = Ln+1,j } (k, j = 1, . . . , n) generate, in accordance with (3.4.40), a subalgebra of conf(Rp,q ) with defining relations [Lij , Lkm ] = ηjk Lim + ηmi Ljk + ηki Lmj + ηjm Lki , [Pj , Lkm ] = ηjk Pm − ηjm Pk , [Pj , Pk ] = α Ljk ,

(3.4.43)

where α = −ηn+1,n+1 . This subalgebra is isomorphic to so(p + 1, q) for α = −1 and isomorphic to so(p, q + 1) for α = +1. This result is used in the accompanying book. Remark. As we have seen in Section 1.4.3, 2-dimensional conformal groups Conf(R1,1 ) and Conf(R2 ) are infinite-dimensional, so their Lie algebras are not isomorphic to so(2, 2) and so(1, 3). However, in accordance with the result of Problem 1.4.4, 2-dimensional conformal groups have finitedimensional subgroups SL(2, R)/Z2 × SL(2, R)/Z2 ⊂ Conf(R1,1 ),

SL(2, C)/Z2 ⊂ Conf(R2 ),

whose Lie algebras are s(2, R)+s(2, R) and A(SL(2, C)), respectively. The latter, in turn, are isomorphic to so(2, 2) and so(1, 3), see Proposition 3.3.5.

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Locally Isomorphic Lie Groups. Universal Covering

We have seen in previous sections that different Lie groups may have identical (isomorphic) Lie algebras. These Lie groups are called locally isomorphic. Local isomorphism implies that vicinities of unit elements are isomorphic: all elements there can be written in exponential form (3.2.27), so the fact that Lie algebras coincide is equivalent to isomorphism of vicinities of unity. Let us state several propositions without proving them. (1) For any real Lie algebra A, there is one and only one simply connected ˆ is called universal ˆ whose algebra isd A. G (see Definition 2.1.7) Lie group G covering group. ˆ is (2) Any connected Lie group G, which is locally isomorphic to G, ˆ covered by the universal covering group G (i.e., there exists homomorphism ˆ → G). G (3) Universal covering of simple (semisimple) compact Lie group is compact. Recall in this regard that group U (1) is considered neither simple nor semisimple. Any nonsimply connected Lie group G which has m classes of homotopiˆ such that there cally inequivalent closed curves, has universal covering G ˆ are m elements of G which map to one and the same element of G under ˆ → G. In other words, group G ˆ covers group G precisely homomorphism G m times. ˆ We show in the end of this subsection that if the universal covering G ˆ covers m times group G, then group G always has normal subgroup Z of ˆ order m, such that quotient group G/Z is isomorphic to G. As an illustration, consider simply connected group SU (2) and nonsimply connected group SO(3) with m = 2 (see Examples 7 and 9 in Section 2.1.2). Recall that groups SU (2) and SO(3) are locally isomorphic, as their Lie algebras su(2) and so(3) are isomorphic (see Example 2 in Section 3.3.3). Proposition 3.5.1. Group SU (2) covers twice group SO(3), and SO(3) = SU (2)/Z2 , where Z2 is the center of SU (2). Homomorphism SU (2) → d The fact that for any Lie algebra A there exists Lie group G whose algebra is A, is proven in Section 5.3 for compact Lie algebras. However, we do not prove there the existence of simply connected Lie group with this property. Neither we show that such a group is unique.

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SO(3) is given by Oij =

1 Tr(σi U σj U † ), 2

where ||Oij || ∈ SO(3), U ∈ SU (2) and σi are Pauli matrices (2.1.18). Proof. Let us give explicit construction of homomorphism SU (2) → SO(3) which maps two elements of SU (2) to one element of SO(3). Let X be real vector space of traceless Hermitian (2 × 2) matrices X = X † , Tr(X) = 0. Any matrix X ∈ X can be written as   x3 x1 − ix2 = x1 σ1 + x2 σ2 + x3 σ3 , X= x1 + ix2 −x3

(3.5.1)

x1 , x2 , x3 ∈ R. (3.5.2)

Matrices σi make basis in 3-dimensional space X . One introduces the scalar product in X : (X, Y ) = Tr(X · Y ), ∀X, Y ∈ X , so that (σi , σk ) = 2δik and xi = 12 (σi , X).

(3.5.3)

Equations (3.5.2) and (3.5.3) define one-to-one mapping which maps a vector x ∈ R3 with real coordinates (x1 , x2 , x3 ) to 2 × 2 matrix X ∈ X . It is sometimes called Pauli map. Matrices (3.5.2) obey the identities − det(X) = x21 + x22 + x23 ,

(3.5.4)

X 2 = (x21 + x22 + x23 ) I2 .

(3.5.5)

In view of (3.5.4), the latter identity is in fact equivalent to Tr(X) = 0. Problem 3.5.1. Show that any 2 × 2 matrix X obeys characteristic identity X 2 − Tr(X) X + det(X) I2 = 0.

(3.5.6)

Together with (3.5.4), this identity shows equivalence of Tr(X) = 0 to (3.5.5). Consider now linear transformation in space X : X → X  = U X V,

(3.5.7)

where U and V are nondegenerate complex 2 × 2 matrices. They obey constraints ensuring that the new matrix X  again belongs to X , i.e., it

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obeys (3.5.1) and has representation (3.5.2). Furthermore, we require that the transformation (3.5.7) preserves matrix square: (X  )2 = X 2 = xi xi · I2 .

(3.5.8)

Then, in accordance with (3.5.5), linear transformation xi → xi given by (3.5.7) is orthogonal in R3 , i.e., it is transformation from group O(3). The pertinent constraint on matrices U, V is obtained by making use of (3.5.5) and (3.5.8), which give U X = X  V −1 ⇒ X  (U X) X = X  (X  V −1 ) X ⇒ X  U = V −1 X ⇒ X  = V −1 XU −1 ⇒ XV U = (V U )−1 X,

(3.5.9)

where we use (3.5.7) to obtain the last equality. Since matrix X is Hermitian, traceless but otherwise arbitrary, Eq. (3.5.9) gives V U = ± I2 ⇒ V = ± U −1 .

(3.5.10)

Problem 3.5.2. Obtain the constraint (3.5.10) from (3.5.9). Thus, the transformation (3.5.7) involves matrix U only, and is of two possible forms,

†

X → X  = − U X U −1 ,

(3.5.11)

X → X  = U X U −1 .

(3.5.12)

Now, since X  = X  , we find from (3.5.11) and (3.5.12) that [U † U, X] = 0. Since this property must hold for any matrix X given by (3.5.2), it is equivalent to U † U = αI, where α is an arbitrary real and positive parameter. We note that transformations (3.5.11) and (3.5.12) are invariant under dilatation U → λU , λ ∈ C. This enables us to normalize U in such a way that det(U ) = 1 and U † U = I, i.e., U ∈ SU (2). Thus, we have shown that all transformations of space X that preserve matrix square (3.5.8), are given by (3.5.11), (3.5.12) with U ∈ SU (2). Transformations (3.5.11) and (3.5.12) with U ∈ SU (2) are orthogonal transformations of vector x = (x1 , x2 , x3 ). Transformations (3.5.11) include reflection x → −x, so they belong to subset O− (3) ⊂ O(3). We consider from now on the case of orthogonal transformations (3.5.12), which belong to connected component of identity transformation x → x, and thus make a group of proper rotations O+ (3) = SO(3). Making use of (3.5.2),

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(3.5.3), one writes transformation (3.5.12) with U ∈ SU (2) as orthogonal transformation xi → xi = Oij xj where elements of matrix ||Oij || ∈ SO(3) are expressed through elements of matrix U : Oij = 12 Tr(σi U σj U † ) ≡ Tij (U ),

U ∈ SU (2).

(3.5.13)

This defines mapping T of SU (2) to SO(3). This mapping is homomorphism. Indeed, one writes the formula (3.5.13) as U σj U † = σi Tij (U ), and obtains  σk Tkj (U1 · U2 ) = (U1 U2 )σj (U1 U2 )† = U1 σi Tij (U2 ) U1† = σk Tki (U1 ) Tij (U2 ), i.e., Tkj (U1 · U2 ) = Tki (U1 ) Tij (U2 ). The above analysis shows that any orthogonal transformation x → x = O ·x can be cast into the form (3.5.12) with U ∈ SU (2); in other words, any matrix O ∈ SO(3) can be cast into the form (3.5.13). Problem 3.5.3. Give explicit proof of the last statement. Hint: write proper orthogonal matrix O as product O1 O2 O3 of rotations around three axes in R3 and construct matrix Ui ∈ SU (2) for each of matrices Oi ∈ SO(3). Finally, mapping (3.5.13) maps two elements ± U ∈ SU (2) to one and the same element of O ∈ SO(3). Therefore, group SU (2) covers twice group SO(3). Since the kernel of this homomorphism has two elements ±I2 ∈ SU (2) and is normal subgroup Z2 ⊂ SU (2), the correspondence between the groups SU (2) and SO(3) is given by isomorphism SU (2)/Z2 = SO(3). Problem 3.5.4. Show that there are precisely two solutions ±I2 to the following equation for matrix U ∈ SU (2): Tr(σi U σj U † ) = δij . This means that kernel of homomorphism SU (2) → SO(3) is indeed Z2 . Problem 3.5.5. Show that group SL(2, R) covers twice group SO↑ (1, 2), and there is isomorphism P SL(2, R) ≡ SL(2, R)/Z2 = SO↑ (1, 2). Hint: the proof is analogous to the proof of Proposition 3.5.1. ˆ be universal covering of Lie group G. There Proposition 3.5.2. Let G ˆ is isomorphism G/Z = G, where Z is some discrete subgroup of center

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ˆ If Z has order m, then there are exactly m classes of nonof group G. homotopic closed curves in G. ˆ are Proof. Group G and its simply connected universal covering group G ˆ locally isomorphic. Consider homomorphism ρ : G → G, and let Ker(ρ) be its kernel. In the first place, let us show that Ker(ρ) is a discrete subgroup ˆ Since vicinities of unit elements in G and G ˆ are isomorphic, there are of G. no other elements from Ker(ρ) in small neighborhood of eˆ. This property holds for any element z ∈ Ker(ρ): were the set Ker(ρ) continuous near z, one could make a combination z −1 ·z  ∈ Ker(ρ) out of two nearby elements z and z  from Ker(ρ), which would be close to unit element, which is impossible. Hence Ker(ρ) is discrete subgroup. ˆ one has gˆ · z · gˆ−1 ∈ Now, let z belong to Ker(ρ). Then for all gˆ ∈ G Ker(ρ) (since kernel of homomorphism is always normal subgroup, see Proposition 1.3.1) The set of elements gˆ · z · gˆ−1 is, on the one hand, ˆ is connected, and, on the other hand, connected due to the fact that G discrete since Ker(ρ) is discrete. Therefore, this set consists of one element ˆ and z ∈ Ker(ρ) one has gˆ · z · gˆ−1 = z, i.e., which is z itself. So, for all gˆ ∈ G ˆ and since G = G/Ker(ρ), ˆ Z = Ker(ρ) is a subgroup of center of group G, ˆ we conclude that G = G/Z. ˆ Finally, if Z has order m, then group manifold G = G/Z can be viewed ˆ as manifold G whose m points in every coset are identified. Besides the usual closed curve, such a manifold has (m−1) nonhomotopic closed curves which connect different identified points. Namely, one singles out one point out of m and connects it to itself (contractible curve in G) or to any other (m − 1) ˆ which are identified to form G = G/Z. ˆ points in G This shows that there ˆ are precisely m nonhomotopic closed curves in G = G/Z. An example of this situation is the relation SO(3) = SU (2)/Z2 which we know from Proposition 3.5.1. The two nonhomotopic closed curves in SO(3) are curves in SU (2) starting from some element U ∈ SU (2), one of which ends at U (contractible curve in SO(3)) and the other at element (−U ) ∈ SU (2) (noncontractible closed curve in SO(3)). Another example is covering of U (1). Group U (1) has infinitely many closed nonhomotopic curves. Universal covering of U (1) is group R of real numbers with addition as group operation. Likewise, group U (1) × U (1) × · · · × U (1), whose manifold is k-dimensional torus, is covered    k

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by group of translations in Rk . In this regard, the following property holds, which we leave the reader to prove. Problem 3.5.6. Let T (Rn ) be group of translations in Rn . Prove that any discrete subgroup of T (Rn ) is the group of translations by vectors a1 · n1 + a2 · n2 + · · · + ak · nk ,

k ≤ n,

where vectors a1 , . . . , ak are linearly independent, and n1 , . . . , nk are integers. Using this fact, show that any Abelian Lie group of dimension n is direct product U (1) × U (1) × · · · × U (1) ×T (Rn−k ) ≡ [U (1)]k × T (Rn−k ).    k

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Chapter 4

Representations of Groups and Lie Algebras

4.1.

Linear (Matrix) Representations of Lie Groups

In general, representation of group G in space M is the action (1.4.1) with properties (1.4.2). In what follows we mostly consider linear representations. In this case group G acts in vector space M = V, and action (1.4.1) reads η = F (g, ξ) = T (g) · ξ, where η, ξ ∈ V, g ∈ G, and T (g) is an invertible linear operator in V. The properties (1.4.2) imply that T (e) is an identity operator in V and that T (g1 · g2 ) = T (g1 ) · T (g2 ). 4.1.1.

Definition of representation of a group. Examples

Definition 4.1.1. Linear representation (or simply representation) T of group G in linear (vector) space V is homomorphic mapping T : G → Γ of G to group Γ of nondegenerate linear operators in V. In other words, representation T maps an element g ∈ G to an invertible linear operator T (g) ∈ Γ, and this mapping is consistent with group multiplication in G: T (g1 · g2 ) = T (g1 ) · T (g2 ),

∀g1 , g2 ∈ G.

(4.1.1)

Space V, in which the group Γ acts, is the space of representation. If V ≡ Vn is real (complex) n-dimensional space, then T is called n-dimensional real (complex) representation. It follows from (4.1.1) that −1

T (e) = I, T (g −1 ) = (T (g))

,

∀g ∈ G,

(4.1.2)

where e is unit element in G and I is unit operator (identity) in Γ. Note that Vn (C) can be viewed also as V2n (R), so that any n-dimensional complex 179

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representation can be viewed as 2n-dimensional real representation; the opposite is not, generally speaking, true. Once basis ei ∈ Vn (K) is chosen, there is one-to-one mapping between linear operators and (n × n) matrices, see (1.2.42). Therefore, n-dimensional representation T is at the same time mapping of G to group of (n × n) matrices with elements from K. An element g ∈ G is mapped to matrix ||Tij (g)|| ∈ Γ such that T (g) · ei = ej Tji (g).

(4.1.3)

Each of n2 elements of matrix ||Tij (g)|| is a function on group G, and (4.1.1), (4.1.2) and (4.1.3) give Tik (g1 )Tkj (g2 ) = Tij (g1 · g2 ),

(4.1.4)

Tij (g −1 ) = (T (g)−1 )ij , Tij (e) = δij .

(4.1.5)

So, representation T can be thought of as isomorphism T : G → Γ of group G to matrix subgroup Γ ⊂ GL(n, K).  = ψiei . We decompose vector T (g)ψ  Let ψi be coordinates of vector ψ in basis ei and obtain T (g)(ψiei ) = ψiej Tji (g). Thus, coordinates of vector transform under T (g) as follows: ψi → Tij (g)ψj . Definition 4.1.2. Let complex space V(C) be endowed with positivedefinite Hermitian form fH . Representation T in V(C) is unitary if operators T (g) are unitary with respect to fH : fH (T (g) · x, T (g) · y ) = fH (x, y ), ∀x, y ∈ V(C).

(4.1.6)

Let ei ∈ V(C) be basis, orthonormal with respect to fH , i.e., fH (ei , ek ) = δik . Then the property (4.1.6) is equivalent to unitarity of matrices Tij (g): Tmi (g)∗ Tmk (g) = δik ⇒ (T (g)† )im Tmk (g) = δik ⇒ T † T = I. We now give a few basic examples of representations of Lie groups. 1. Trivial representation Any group G has 1-dimensional representation T (g) = 1, ∀g ∈ G.

(4.1.7)

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2. Defining representation Let G be matrix Lie group whose elements are (n × n) matrices g = ||gij ||, like SU (n) or SO(n), and let Vn be n-dimensional vector space with basis {ei } (i = 1, . . . , n). Defining (fundamental in physics literature) representation T is mapping from G to space of linear operators in Vn given by T (g) · ei = ej gji .

(4.1.8)

Accordingly, defining representation acts on coordinates of a vector as follows:  )i = gij ψj . (T (g) · ψ

(4.1.9)

If all matrices in G are real, then defining representation is real. It is often useful, though, to consider representation defined by (4.1.8), (4.1.9) with real matrices, as representation in complex vector space; we call this complex representation defining as well, with qualifications when necessary. If matrices in G are complex, then defining representation is complex. 3. Contragredient representation (co-representation) T Let T be n-dimensional representation in complex space Vn . One introduces n of linear functionals on Vn . By definition, every dual to Vn vector space V  ∈ Vn is denoted ∈V n defines linear function Vn → C, whose value on ψ φ    by φ, ψ ∈ C. Representation T , contragredient to T , is mapping of G to n in such a way that space of linear operators, acting in V  T (g) · ψ  = φ,  ψ ,  T(g) · φ,

∀g ∈ G,

∈V n , ∀φ

 ∈ Vn . ∀ψ

(4.1.10)

 = εi n such that εi , ej = δij . We set φ Let {εi } (i = 1, . . . , n) be basis in V  and ψ = ej in (4.1.10) and obtain δij = T(g) · εi , T (g) · ej = εk , er Tki (g) Trj (g) = Tki (g) Tkj (g). Therefore, we have Tki (g) = Tik (g −1 ) and T(g) · εi = εj Tji (g) = εj (T (g)−1 )T ji ,

(4.1.11)

i.e., co-representation has matrices T(g) = (T (g)−1 )T . Mapping T is homomorphic: T(g1 ) · T(g2 ) = (T (g1 )−1 )T · (T (g2 )−1 )T = (T (g2 )−1 · T (g1 )−1 )T T  = T (g1 · g2 )−1 = T(g1 · g2 ).

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In accordance with (4.1.11), co-representation T acts on coordinates of  = φi εi ∈ V n as follows: vector φ  i = φj (T (g)−1 )ji = (T (g)−1 )T φj . (T(g) · φ) ij

(4.1.12)

This construction is literally valid also in the case of real representation and its real co-representation. It is worth noting that if Vn is endowed with n scalar product invariant under the group action (4.1.3), then the space V can be identified with the space Vn . 4. Conjugate representation T ∗ Let T be n-dimensional complex representation given by (4.1.3). Its conjugate representation T ∗ acts in the same space Vn (C) and is given by  )i = T ∗ (g) ψj , T ∗ (g) · ei = ej Tji∗ (g), (T ∗ (g) · ψ ij

(4.1.13)

where ||Tij∗ (g)|| are complex conjugate matrices. Mapping T ∗ is obviously homomorphic. In particular, representation conjugate to defining one is given by ∗  )i = g ∗ ψj . T ∗ (g) · ei = ej gji , (T ∗ (g) · ψ ij

(4.1.14)

This representation is dubbed antifundamental in physics literature. We note that this construction does not yield anything new in the case of real representation T : the latter has real matrices, so T ∗ = T . Problem 4.1.1. Let the space Vn of representation T be endowed with positive-definite Hermitian form, and representation T be unitary with respect to this form. Show that representation T ∗ conjugate to T coincides with co-representation T. Demonstrate explicitly that representation T ∗ of unitary group U (n) (or group SU (n)), which is conjugate to defining representation T , coincides with co-representation T. 5. Adjoint representation ad(G) of Lie group G Let G be matrix Lie group, and A(G) be its matrix Lie algebra. For every element g = ||gij || ∈ G one assigns linear operator ad(g) (“ad” from adjoint), which acts on vector A (matrix) belonging to Lie algebra, i.e., tangent space A(G), as follows: ad(g) A = g · A · g −1 .

(4.1.15)

Matrix g · A · g −1 is again an element of tangent space A(G), since it is tangent vector at point In ∈ G to a curve g · gA (t) · g −1 ∈ G where gA (t) is

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a curve with tangent vector A. Thus, formula (4.1.15) defines mapping ad of group G to set of linear operators acting in Lie algebra (vector space) A(G). To see that this mapping is homomorphic, we write ad(g1 · g2 ) A = (g1 · g2 ) · A · (g1 · g2 )−1 = g1 · (g2 · A · g2−1 ) · g1−1 = ad(g1 ) (ad(g2 ) A). The general definition of adjoint representation which goes beyond matrix groups makes use, instead of (4.1.15), of the following relation: gad(g) A (t) = g · gA (t) · g −1 ,

∀g ∈ G,

(4.1.16)

where gA (t) is again a curve in G emanating from unit element along the tangent vector A. This defines mapping ad of group G to algebra of linear operators acting in A(G). The fact that it is homomorphism follows from the identity (g1 · g2 ) · gA (t) · (g1 · g2 )−1 = g1 · (g2 · gA (t) · g2−1 ) · g1−1 . For given g, mapping ad(g): A(G) → A(G) is consistent with commutation: [ad(g)A, ad(g)B] = ad(g)[A, B].

(4.1.17)

This follows from the properties gad(g) A (t) · gad(g) B (t) = g · gA (t) · gB (t) · g −1 , [gad(g) A (t)]−1 = g · [gA (t)]−1 · g −1 and the general definition of commutator in A(G) given in Section 3.1.4. Therefore, ad(g) is inner automorphism of algebra A(G), see also Example 2 in Section 3.1.4. Adjoint representation of Lie group G, acting in space A(G), has matrices ||(ad g)ab || in a given basis Xa ∈ A(G): ad(g) · Xa = Xb (ad g)b a .

(4.1.18)

Due to the property (4.1.17), adjoint representation preserves structure constants, d [ad(g)Xa , ad(g)Xb ] = Cab · ad(g)Xd .

Therefore, adjoint representation preserves Killing metric (3.2.62), (3.2.63) in A(G), gab = (Xa , Xb ) = (ad(g) · Xa , ad(g) · Xb ) = gcd (ad g)c a (ad g)d b . (4.1.19)

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The latter relation, applied near unit element, yields the symmetry property (3.2.67). Problem 4.1.2. Prove the latter statement. Definition 4.1.3. Representation of Lie group, which is contragredient to adjoint representation, is co-adjoint. 4.1.2.

Regular and induced representations of finite groups. Faithful representations

Theorem 1.3.1 (Cayley) says that any finite group G of order n can be embedded in permutation (symmetric) group Sn . This property is closely connected to the fact that, as we now see, G always has n-dimensional real representation. It is constructed as follows. Let’s write n elements of G as a row (g1 , g2 , . . . , gn ). Action on the left (right) of a given element gi ∈ G on this row defines permutation of its components (see (1.3.5)): gi · (g1 , g2 , . . . , gn ) = (gi · g1 , gi · g2 , . . . , gi · gn ) = (gk1 , gk2 , . . . , gkn ), (g1 , g2 , . . . , gn ) · gi = (g1 · gi , g2 · gi , . . . , gn · gi ) = (gr1 , gr2 , . . . , grn ). These permutations can be expressed in terms of (n × n)-matrices (R) (R) ||Tjk (gi )|| and ||T˜jk (gi )||: (R)

(R)

gi · gj = gk Tkj (gi ), gj · gi = T˜jk (gi ) gk .

(4.1.20)

(R) (R) Matrices ||Tjk (gi )|| and ||T˜jk (gi )|| define permutations, so their elements are equal to either 0 or 1; each row and each column contain exactly one element 1 and other elements are zeros; if gi = e, then main diagonal has only zeros. Mappings T (R) and T˜ (R) of G to GL(n, R) are homomorphisms: (R) (R) (R) (R) (R) (R) Tlk (gm ) Tkj (gi ) = Tlj (gm · gi ), T˜jk (gi ) T˜kl (gm ) = T˜jl (gi · gm ).

(4.1.21) This follows from associativity of group operation in G. For example, we have for T (R) : (R)

(R)

(R)

gm · gi · gj = gm · (gi · gj ) = gm · gk Tkj (gi ) = gl Tlk (gm ) Tkj (gi ), (R)

gm · gi · gj = (gm · gi ) · gj = gl Tlj (gm · gi ),

(4.1.22)

The first of Eq. (4.1.21) is obtained by comparing the right-hand sides of these formulas. Homomorphisms T (R) and T˜ (R) define n-dimensional real

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matrix representations of group G, where n is order of G. Representations T (R) and T˜ (R) , constructed using left and right multiplication in G, are left and right regular representations of G, respectively. Problem 4.1.3. Given an arbitrary permutation σ ∈ Sn , one constructs matrix T (σ) = ||Tij (σ)|| (i, j = 1, . . . , n) such that σ · (g1 , . . . , gn ) = (gσ(1) , . . . , gσ(n) ) = (g1 , . . . , gn ) T (σ). Show that mapping T : σ → T (σ) is representation of symmetric group Sn . Show that this representation coincides with its co-representation, T (σ) = (T (σ)−1 )T . Hint: any permutation is a product of transpositions, (ij) = (ij)−1 , for which T (ij) = (T (ij))T . Problem 4.1.4. Show that matrices of regular representations T (R) and T˜ (R) obey the identities (T (R) (g))T = T (R) (g −1 ),

(T˜ (R) (g))T = T˜ (R) (g −1 ),

∀g ∈ G.

Hint: make use of the second result of Problem 4.1.3. As an example, we give explicit form of 3-dimensional left regular representation T (R) of cyclic group C3 . In that case the rule (4.1.20) reads e · (e, g1 , g2 ) = (e, g1 , g2 ) = (e, g1 , g2 ) T (R) (e), g1 · (e, g1 , g2 ) = (g1 , g2 , e) = (e, g1 , g2 ) T (R) (g1 ), g2 · (e, g1 , g2 ) = (g2 , e, g1 ) = (e, g1 , g2 ) T (R) (g2 ). We see that

⎛

1 T (R) (e) = ⎝0 0 ⎛ 0 (R) ⎝ T (g2 ) = 0 1

0 1 0 1 0 0

⎞ ⎛ 0 0 0 0⎠ , T (R) (g1 ) = ⎝1 0 1 0 1 ⎞ 0 1⎠ . 0

⎞ 1 0⎠ , 0 (4.1.23)

Problem 4.1.5. Elements of dihedral groups D3 and D4 are permutations of vertices of triangle and square, respectively. Construct isomorphism T1 : D3 → S3 and monomorphism (see Definition 1.3.2) T2 : D4 → S4 . Problem 4.1.6. Find explicit matrix form of left and right regular representations of groups D3 = S3 and D4 .

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The definition of regular representation has to be modified for infinite groups. This has to do with difficulties in defining infinite-dimensional analogs of (4.1.20). Regular representations of Lie groups are defined below in Section 4.5. Definition 4.1.4. Representation T : G → Γ is faithful if T is monomorphism of G to Γ (where Γ is the same as in Definition 4.1.1), i.e., T is isomorphism of G to image Im(T ) ⊂ Γ (see Definition 1.3.2). Representation is not faithful if more than one element of G is mapped to one and the same operator in Γ. If T is faithful, then kernel of homomorphism T : G → Γ is trivial. Conversely, let representation T of group G be not faithful, and let H = Ker(T ) be kernel of homomorphism T : G → Γ. Then T is faithful representation of quotient group G/H. Problem 4.1.7. Prove the above two statements. Show that representation T of group Sn described in Problem 4.1.3 is faithful. As an example, consider homomorphism T of group O(n) to group of two numbers Γ = {1, −1} (Γ can be viewed as a subgroup of operators in 1-dimensional real vector space), such that T (g) = det(g), ∀g ∈ O(n). This homomorphism is 1-dimensional representation which is not faithful. Kernel Ker(T ) = SO(n) is nontrivial, and, as we know, O(n)/SO(n) = Z2 . At the same time, isomorphism T : O(n)/SO(n) = Z2 → Γ , given by Z2 = {In · SO(n), R · SO(n)} ←→ Γ = {1, −1}, T

is faithful representation. Regular representations defined by (4.1.20) can be generalized in the following way. Let G be finite group, H its subgroup and ord(G) = n, ord(H) = m. Consider quotient set G/H of left cosets of H in G. The number of these cosets, and hence order of G/H is n/m = k. Let us choose one representative in each coset, kα ∈ G (α = 1, . . . , k), and put them into a row (k1 , k2 , . . . , kk ). Then the left action of an element g ∈ G on the row (k1 , k2 , . . . , kk ) gives permutation of its components modulo their

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multiplication on the right by elements of subgroup H: g · (k1 , k2 , . . . , kk ) = (g · k1 , g · k2 , . . . , g · kk ) = (kα1 · h(g,k1 ) , kα2 · h(g,k2 ) , . . . , kαk · h(g,kk ) ), (4.1.24) where (α1 , . . . , αk ) is some permutation of numbers (1, 2, . . . , k) and h(g,kα ) ∈ H. Thus, left action (4.1.24) can be written in terms of (k × k) matrix ||Tαβ (h(g,kβ ) )||, whose nonzero entries are elements of subgroup H: g · kβ = kα Tαβ (h(g,kβ ) ).

(4.1.25)

Each row and each column of the matrix ||Tαβ (h(g,kβ ) )|| contain exactly one nonzero element h(g,kβ ) ∈ H. Assume now that there is some d-dimensional representation Δ of subgroup H, which maps each element h ∈ H to matrix Δ(h) ∈ GL(d, K). Then, in accordance with (4.1.25), each element g ∈ G is mapped to block (k × k) matrix ||Tαβ (Δ(h(g,kβ ) ))|| whose blocks have size (d × d). Mapping T of group G to group GL(k · d, K): g → ||Tαβ (Δ(h(g,kβ ) ))||,

∀g ∈ G,

(4.1.26)

defines (k · d)-dimensional representation of group G. Problem 4.1.8. Show that mapping (4.1.26) is homomorphism. Hint: make use of associativity of group operation in G, (g · g  ) · kα = g · (g  · kα ). Formula (4.1.26) defines representation of G, induced from representation of subgroup H. In the particular case of trivial subgroup H, representation (4.1.25), (4.1.26) coincides with the left regular representation T (R) (see (4.1.20)). Instead of left action (4.1.24) of group G on left coset space G/H, one may consider right action of G on right coset space H\G and construct another version of induced representation. The latter coincides with right regular representation T˜ (R) in the particular case of trivial subgroup H. Problem 4.1.9. Show that representation T of dihedral group Dn , induced from representation Δ of its subgroup Cn , has the following form:



Δ(gi ) 0 Δ(g−i ) 0 , T (r · gi ) = . T (gi ) = 0 0 Δ(g−i ) Δ(gi ) where i = 0, 1, . . . , n − 1. Hint: choose, as representatives, elements (e = g0 , r) ∈ Dn .

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Generalization of induced representation (4.1.26) to Lie groups is given below in Section 7.3. 4.1.3.

Equivalent representations. Equivalence of defining representation and its conjugate for SU (2), characters

 are equivalent, Two representations T and T of group G in spaces V and V  → V such that if there is invertible operator S: V T(g) = S −1 · T (g) · S, ∀g ∈ G.

(4.1.27)

Note that if T is group representation, then any mapping T˜ defined by (4.1.27) is also representation (i.e., T˜ is homomorphism). This follows from the chain of identities T˜(g1 ) · T˜ (g2 ) = S −1 T (g1 )S S −1 T (g2 )S = S −1 · T (g1 ) · T (g2 ) · S = S −1 · T (g1 g2 ) · S = T˜(g1 · g2 ). Let representations T and T act in one and the same n-dimensional vector  In a given basis {ei } ∈ V the relation (4.1.27) is written in space V = V. matrix form (see (4.1.3)) −1 Tkm (g) Smj , ∀g ∈ G, Tij (g) = Sik

(4.1.28)

where ||Tij (g)|| are (n × n) matrix of operator T (g) in basis {ei }. Under the change of basis, ei  = Sjiej , matrix of operator T (g) is transformed in accordance with (1.2.48), i.e., it becomes equal to ||Tij (g)|| given by (4.1.28). In that case the equivalence transformation (4.1.28) can be interpreted simply as the change of basis. Example. Any element U of SU (2) in defining representation (4.1.8) has the form (2.1.14) (see Example 7 in Section 2.1.2), while the same element in conjugate representation (4.1.13) is given by matrix U ∗ :



α β α∗ β ∗ ∗ . U= , U = −β α −β ∗ α∗ It is now straightforward to check that these matrices are related as follows:

0 1 ∗ −1 U =SUS , S = . −1 0 Thus, defining representation of group SU (2) is equivalent to its conjugate representation (and hence equivalent to its contragredient representation,

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see Problem 4.1.1). It is worth emphasizing that similar property does not hold for groups SU (n) with n > 2. Equivalent representations T have T˜ equal traces, Tr(T˜ (g)) = Tr(S −1 · T (g) · S) = Tr(T (g)) (∀g ∈ G).

(4.1.29)

Thus, for given representation T , it is natural to define a function on G, χT (g) := Tr(T (g)) (∀g ∈ G). This function is called character of representation T and is an important characteristic of a representation. It follows from (4.1.29) that equivalent representations T and T have equal characters, χT = χT . There is an inverse statement valid for large class of groups and their representations: if characters of two representations of a given group coincide, then these representations are equivalent (see Section 4.6.2 in this regard). This simplifies the study of equivalence of representations. Note that elements g1 and g2 from the same conjugacy class (i.e., related by g1 = g · g2 · g −1 , g ∈ G) have the same value of character χT (g1 ) = χT (g2 ), so character is actually a function on conjugacy classes of group G. For unit element e ∈ G we have χT (e) = n, where n is the dimension of representation T (dimension of space V where operators T act). Example. Consider defining representation (4.1.8) of group SL(2, C) with 2 × 2 complex matrices

α11 α12 , det(||gij ||) = α11 α22 − α12 α21 = 1, αij ∈ C. ||gij || = α21 α22 Matrices of its conjugate representation, according to (4.1.13), are

∗ ∗ α α 11 12 ∗ ||gij || = . α∗21 α∗22 So, there are elements g ∈ SL(2, C) such that characters of defining and conjugate representations are different: χ(g) = α11 + α22 = α∗11 + α∗22 = χ(g ∗ ). iφ As an example, one chooses α12 = α21 = 0 and α11 = α−1 22 = ρe , where 2 iφ −1 −iφ

= χ(g ∗ ) = ρ = 1, φ = πn, then χ(g) = α11 + α22 = ρe + ρ e ∗ ∗ −iφ −1 iφ + ρ e . Therefore, these representations of SL(2, C) α11 + α22 = ρe are inequivalent.

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Problem 4.1.10. Compare characters of defining representation of SU (2) and its conjugate representation. 4.2. 4.2.1.

Representations of Lie Algebras Definition of Lie algebra representation

Definition 4.2.1. Representation T of Lie algebra A over field K in vector space V(K ), K ⊂ K , is homomorphism T of A to algebra of linear operators acting in V(K ), i.e., T (α A + β B) = α T (A) + β T (B) , α, β ∈ K,

(4.2.1)

T ([A, B]) = [T (A), T (B)],

(4.2.2)

where A, B ∈ A and [T (A), T (B)] ≡ T (A) · T (B) − T (B) · T (A). Definition 4.2.2. Representation is faithful if T is isomorphism of A to Im(T ); in other words, Ker(T ) = ∅. Note that real Lie algebras with K = R can have both real and complex representations with K = R and K = C, respectively. Representations of complex Lie algebras are always complex. Let representation T of Lie algebra A act in n-dimensional vector space Vn with basis {ei } (i = 1, . . . , n). In that case representation is called n-dimensional. Representation T maps each element of A ∈ A to linear operator T (A) acting in Vn , and also to matrix of operator T (A), defined, as usual, by T (A) · ei = ej Tji (A).

(4.2.3)

 = ψiei in the stanRepresentation T acts on components of vector ψ dard way: ψi → Tij (A)ψj . Definition 4.2.3. Representations T and T of Lie algebra A in spaces V  respectively, are equivalent if there exists invertible operator S : and V,  such that V → V, T(A) = S T (A) S −1 , ∀A ∈ A.

(4.2.4)

Let T : G → Γ be representation of Lie group G, where Γ is Lie group of linear operators in V. This representation induces representation T  of its Lie algebra A(G) in space V. To construct it, consider smooth curve

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g(t) near unit element e ∈ G with tangent vector A. This curve generates smooth curve T (g(t)) in Γ, whose expansion in t is T (g(t)) = I + t T  (A) + O(t2 ).

(4.2.5)

Tangent vector T  (A) to curve T (g(t)) is a linear operator in V. For smooth enough mappings, T (A) it is uniquely determined by vector A ∈ A(G) tangent to curve g(t) (i.e., it does not depend on particular choice of curve g(t), provided that it has tangent vector A). Thus, Eq. (4.2.5) defines mapping T  of Lie algebra A(G) to the space of operators in V. This mapping obeys all properties (4.2.1), (4.2.2), and hence it is representation of Lie algebra A(G) in space V. Problem 4.2.1. Check that mapping T  of Lie algebra A(G) to the space of operators in V defined by (4.2.5) obeys (4.2.1), (4.2.2). In accordance with the general definition (see Remark 2 in Section 2.2.1), mapping T  is derivative of mapping T : G → Γ at point e ∈ G. We denote in what follows the representation T of Lie group and its counterpart, representation T  of its Lie algebra, by the same symbol T , unless this leads to confusion. Let T be unitary representation of Lie group G (see Definition 4.1.2) in vector space V(C) with Hermitian form fH . Then Eqs. (4.1.6) and (4.2.5) show that operators T (A) of Lie algebra representation T are antiHermitian with respect to fH : fH (T (A) · x, y ) + fH (x, T (A) · y ) = 0, ∀x, y ∈ V(C). Remark 1. Let T be finite-dimensional representation of Lie algebra A. Then one can define invariant scalar product in A: (A, B)T = α Tr(T (A) T (B)),

A, B ∈ A,

(4.2.6)

where α = 0 is a numerical factor. Invariance of the scalar product (4.2.6) means that (cf. (3.2.67)) ([Y, A], B)T = −(A, [Y, B])T ,

∀A, B, Y ∈ A,

(4.2.7)

which is valid due to the cyclic property of trace, ([Y, A], B)T = α Tr([T (Y ), T (A)] T (B)) = −α Tr(T (A) [T (Y ), T (B)]) = −(A, [Y, B])T .

(4.2.8)

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For given basis {Xa } in A, the invariant metric reads (T )

gab = (Xa , Xb )T = α Tr(T (Xa ) T (Xb )).

(4.2.9)

Invariant scalar product defined in (4.2.6) depends on the choice of representation T . Nevertheless, this definition is quite general. In particular, the scalar product (3.2.64) based on Killing metric is obtained from (4.2.6), if T is chosen as adjoint representation (see formula (4.2.18) below, item 5 in Section 4.2.2). Scalar product (4.2.6) is used in Section 4.7.2 where we consider Casimir operators, and in Section 5.1 where we construct invariant metric in compact Lie algebras. ˆ be universal covering of group G (see Remark 2. Let Lie group G ˆ Section 3.5). Then G and G are locally isomorphic, i.e., they have ˆ The groups G ˆ and G are in general isomorphic Lie algebras A(G) = A(G). different, as their manifolds have different global properties. An example is again given by groups SU (2) and SO(3), where the former is double ˆ may covering of the latter (see Section 3.5). The universal covering group G have representations which are not representations of G. Therefore, not all representations of Lie algebra A(G) can be constructed as derivatives of representations of group G by making use of the procedure (4.2.5). Let us discuss this point in more detail. Proposition 4.2.1. Any representation of Lie group G is at the same ˆ time representation of its universal covering group G. ˆ covers G, there exists homomorphism T˜ : G ˆ → G (see Proof. Since group G Proposition (2) in the beginning of Section 3.5). Let T be a representation of G, which is homomorphism of G to matrix group Γ. Then the composition ˆ to Γ, which is precisely the desired of mappings T · T˜ is homomorphism of G ˆ representation of G. A statement inverse to Proposition 4.2.1 is not valid, generally speaking. ˆ may have representations which This implies that Lie algebra A(G) = A(G) ˆ are obtained via (4.2.5) from representations of universal covering group G, but cannot be obtained in the same way from representations of group G. ˆ cover group G in a nontrivial way, i.e., G is not simply Let group G ˆ is). In that case faithful irreducible representation T of connected (while G ˆ covering group G cannot be representation of group G. If the number of ˆ covers group G is m, such a representation is sometimes times the group G loosely called m-valued representation of group G.

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ˆ Proposition 4.2.2. Let G be Lie group, A(G) its Lie algebra and G universal covering of G. Then any finite-dimensional representation T of ˆ is generated from representation of the universal algebra A(G) = A(G) ˆ such that T¯ (ˆ covering group, i.e., there exists representation T¯(G), gA (t)) = 2 ˆ with tangent vector 1 + t · T (A) + O(t ), where gˆA (t) is a curve in G A ∈ A(G). Without proving, we substantiate this proposition with the following consideration. By making use of the exponential map, for any finitedimensional representation T of Lie algebra A(G) one constructs matrix group G whose elements are products of exponentials exp(T (A)). This ˆ by construction, so it is covered matrix group is locally isomorphic to G ˆ ˆ → G , which can be by G. Therefore, there exists homomorphism T¯ : G ˆ This representation T¯ determines, viewed as a representation of group G. in accordance with procedure (4.2.5), representation of Lie algebra of group ˆ which, again by construction, coincides with the original representation G, T of Lie algebra A(G), as desired. 4.2.2.

Examples of Lie algebra representations

Let us introduce a few important Lie algebra representations. These are related to Lie group representations introduced in Section 4.1.1. 1. Trivial representation Every Lie algebra A has trivial representation T (A) = 0, ∀A ∈ A. If A is Lie algebra of Lie group G, then this representation, in accordance with (4.2.5), corresponds to trivial representation (4.1.7) of Lie group G. 2. Defining representation Let A be matrix Lie algebra whose elements are (n×n) matrices (e.g., su(n), sl(n, C) or so(n)). Let Vn be n-dimensional vector space with basis {ei } (i = 1, . . . , n) and A = ||Aij || is Lie algebra element. Defining representation T in space Vn is given by T (A) · ei = ej Aji . This representation is called fundamental in physics literature. For Lie algebra A = A(G) of Lie group G, this representation corresponds to defining representation (4.1.8) of matrix group G. If matrices A are real,

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then defining representation can be considered as either real or complex; if matrices of A are complex, then the defining representation is necessarily complex. We find it convenient to deviate here from nomenclatures accepted in physics and mathematics. One of the reasons is that the two nomenclatures are not completely consistent with each other. One point is that despite the original definition of su(n), sl(n, C), so(n) and alike as matrix algebras, they are fully determined by their defining commutation relations. Therefore, they can be thought of abstract algebras without any reference to matrices. Then the notion of defining representation of these and similar algebras does not make so much sense. For this reason this notion is no longer widely used in mathematical literature. Nevertheless, we find this notion convenient, since it immediately tells what representation is considered, and also makes reference to the relevant group, see Section 4.1.1. Another point is that the term “fundamental representation” has different meaning in mathematical vs. physical literature. Yet the notion of fundamental representation (which we prefer to call defining) is common in theoretical physics, especially when one talks about algebras su(n), sl(n, C) and usp(2k).

3. Contragredient representation (co-representation) T Let T be n-dimensional representation of Lie algebra A in space Vn , which  ψ  is defined is given by (4.2.3). Let V˜n be dual vector space to Vn and φ, as in item 3 in Section 4.1.1. Contragredient representation T maps an element A ∈ A to linear operator T(A), acting in V˜n in accordance with the rule  ψ  = −φ,  T (A) · ψ .  T(A) · φ,

(4.2.10)

n be such that εi , ej = δij . Upon Let the basis {εi } (i = 1, . . . , n) in V  = ej in (4.2.10), we get  = εi and ψ setting φ T(A) · εi , ej = −εi , T (A) · ej = −Tji (A). On the other hand, the left-hand side here is T(A) · εi , ej = Tji (A). As a result, we have Tji (A) = −Tij (A), so co-representation in a given basis is given by T(A) · εi = εj Tji (A) = −Tij (A) εj ,

(4.2.11)

and its matrices are T(A) = −T (A)T . The fact that representation T is linear, is straightforward, while the fact that it is homomorphism is checked as follows: T [T(A1 ), T(A2 )] = [T (A1 )T , T (A2 )T ] = − ([T (A1 ), T (A2 )]) T = − (T ([A1 , A2 ])) = T ([A1 , A2 ]) , ∀A1 , A2 ∈ A.

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If A = A(G), and T (A) is generated from representation T (G) of Lie group G, then co-representation of Lie algebra A(G) is generated from co-representation of group G. 4. Representation T

∗

conjugate to T

Let A be real Lie algebra, and T its n-dimensional complex representation. Representation T ∗ is mapping of A to space of linear operators in Vn (C) given by T ∗ (A) ei = ej (Tji (A))∗ , ∀A ∈ A.

(4.2.12)

In the case of defining representation of matrix Lie algebra A(G), its conjugate representation is simply T ∗ (A) ei = ej A∗ji . It is worth emphasizing that an attempt to extend the definition (4.2.12) to complex Lie algebras would encounter, instead of linearity condition in (4.2.1), anti-linearity T ∗ (αA + βB) = α∗ T ∗ (A) + β ∗ T ∗ (B). This would be inconsistent with the definition of representation. 5. Adjoint representation ad of Lie algebra Adjoint representation of Lie algebra A is linear homomorphism of A to algebra of linear operators acting in A, cf. (3.2.66), with ad(Y ) · X = [Y, X],

∀X, Y ∈ A.

(4.2.13)

Mapping ad is indeed homomorphism: ad([X, Y ]) = [ad(X), ad(Y )].

(4.2.14)

This follows from the Jacobi identity (3.2.32), which gives ∀Z ∈ A ad([X, Y ]) · Z = [[X, Y ], Z] = [X, [Y, Z]]−[Y, [X, Z]] = [ad(X), ad(Y )] · Z. Problem 4.2.2. Prove the identity ad(X) · [A, B] = [ad(X) · A, B] + [A, ad(X) · B],

∀X, A, B ∈ A, (4.2.15)

which shows that operator ad(X) acts in Lie algebra A as differentiation (satisfies the Leibnitz rule).

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If A = A(G), then adjoint representation of Lie algebra is generated from adjoint representation of Lie group G given by (4.1.15). Indeed, in the case of matrix group G, one has near unit element gX = In + tX + O(t2 ). One inserts this into (4.1.15) and obtains ad(gX ) · Y = (In + tX + O(t2 )) Y (In − tX + O(t2 )) = Y + t [X, Y ] + O(t2 ), which, together with (4.2.13), gives ad(gX ) · Y = Y + t ad(X) · Y + O(t2 ).

(4.2.16)

In the general situation one makes use of Eq. (4.1.16) and definition of Lie algebra given in Section 3.2.1. The result (4.2.16) remains valid, provided that t is a parameter along the curve gX (t) defining tangent vector X. For a given basis {Xa } in Lie algebra A, Eq. (4.2.13) defines linear homomorphism ad of A to algebra of matrices of dimension (dim(A) × dim(A)): d a )db = Cab ≡ (ad(Xa ))db , ad : Xa → (X

(4.2.17)

a have been introduced in (3.2.68). Homomorphism where matrices X property (4.2.14) of mapping (4.2.17) is ensured by (3.2.70). Let us recall that the representation (4.2.17) can be used to write Killing metric (3.2.62), (3.2.69) and scalar product (3.2.64) in algebra A as follows: gab = Tr(ad(Xa ) · ad(Xb )),

(A, B) = Tr(ad(A) · ad(B)),

A, B ∈ A. (4.2.18)

Finally, we note that adjoint representation of real Lie algebra is real. Definition 4.2.4. Lie algebra representation, contragredient to adjoint representation, is called co-adjoint. Problem 4.2.3. Show that adjoint and co-adjoint representations of semisimple Lie algebra are equivalent. Hint: make use of the fact that Killing metric is not degenerate, and that the structure constants are antisymmetric, see (3.2.71). 4.3.

Direct Product and Direct Sum of Representations

Let T (1) and T (2) be two representations of Lie group G (or Lie algebra A). There are two procedures which enable one to construct new representations of G (or A). These are direct sum and direct product of representations.

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Direct (tensor) product of representations: Tensors

Consider n- and m-dimensional vector spaces Vn (K) and Vm (K) with bases {ei } (i = 1, . . . , n) and {a } (a = 1, . . . , m), respectively. Given two vectors x = xiei ∈ Vn (K) and y = yaa ∈ Vm (K), let us define an (n · m)dimensional vector z with coordinates (z1 , z2 , . . . , zn · m ) = (x1 y1 , . . . , x1 ym , x2 y1 , . . . , x2 ym , . . . , xn y1 , . . . , xn ym ). (4.3.1) This vector is an element of (n · m)-dimensional space Vn·m (K). Vector z ∈ Vn · m (K) is called direct product of vectors x and y and is denoted by (x ⊗y). Coordinates of vectors ei and a follow from their “decompositions” ei = δij ej and a = δabb . Therefore, the definition (4.3.1) implies that coordinates of vector (ei ⊗ a ) are (ei ⊗ a ) :

(z1 , z2 , . . . , zn·m ) = ( 0, 0, . . . , 0 , 1, 0, . . . , 0).   (i−1)m+a−1

So, the set of vectors {(ei ⊗ a )} makes basis in Vn · m (K). Consider now vectors of the form (summation over repeated indices is assumed) v = via (ei ⊗ a ) ∈ Vn · m (K), via ∈ K.

(4.3.2)

Addition of these vectors and their multiplication by number is defined in the standard way, so vectors (4.3.2) form linear (vector) space called direct (tensor) product of spaces Vn (K) and Vm (K) and denoted by Vn (K) ⊗ Vm (K). The space Vn (K) ⊗ Vm (K), as vector space, coincides with Vn·m (K), however, it has specific properties related to transformation of bases separately in Vn (K) and Vm (K). We emphasize again that dimension of space Vn (K) ⊗ Vm (K) is equal to product of dimensions of Vn (K) and Vm (K). Definition 4.3.1. Elements of tensor product of r vector spaces v ∈ Vn1 (K) ⊗ Vn2 (K) ⊗ · · · ⊗ Vnr (K), (k)

are tensors of rank r. Let eik Vnk (K). Then

(4.3.3)

(ik = 1, . . . , nk ) be basis vectors in spaces (1)

(r)

v = vi1 ,i2 ,...,ir ei1 ⊗ · · · ⊗ eir , and vi1 ,i2 ,...,ir ∈ K are coefficients of tensor v.

(4.3.4)

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With this definition, vector x ∈ Vn (K) is tensor of rank 1, while vector (4.3.2) is tensor of rank 2. Let T (1) and T (2) be two representations of group G in spaces Vn (K) and Vm (K), respectively. Then one defines representation T (p) = T (1) ⊗T (2) in Vn (K) ⊗ Vm (K) as (1)

(2)

T (p) (g) · (ek ⊗ b ) = (T (1) (g) · ek ⊗ T (2) (g) · b ) = (ei ⊗ a ) Tik (g) Tab (g), (4.3.5) (1)

(2)

where g ∈ G, and ||Tik (g)|| and ||Tab (g)|| are (n×n) and (m×m) matrices of representations T (1) and T (2) . The definition (4.3.5) of representation T (p) implies the following transformation property of coordinates of vector v given by (4.3.2): (T (p) (g) · v )ia = (T (1) (g) ⊗ T (2) (g))ia,kb vkb (1)

(2)

= Tik (g) Tab (g) vkb = (T (1) (g) · v · (T (2) (g))T )ia .

(4.3.6)

Expressions (4.3.5) and (4.3.6) involve composite (n · m × n · m) matrix   (p) (1) (2) || = ||Tik (g) Tab (g)||, ||Tia,kb (g)|| = || T (1) (g) ⊗ T (2) (g) ia,kb

(4.3.7) (1)

(2)

which is precisely direct product of matrices ||Tik (g)|| and ||Tab (g)||, introduced in (1.2.23), ⎛ (1) ⎞ (2) (1) (2) T11 (g)||Tab (g)|| ... T1n (g)||Tab (g)|| ⎜ ⎟ ⎜ (1) ⎟ (2) (1) (2) ⎜T21 (g)||Tab (g)|| ... T2n (g)||Tab (g)||⎟ ⎜ ⎟ (4.3.8) T (p) (g) = ⎜ ⎟, .. .. ⎜ ⎟ ⎜ ⎟ ... . . ⎝ ⎠ (1) (2) (1) (2) Tn1 (g)||Tab (g)|| ... Tnn (g)||Tab (g)|| where each block has dimension (m × m). The homomorphism property of mapping T (p) follows from the fact that T (1) , T (2) are homomorphisms; it is straightforwardly checked in operator language: T (p) (g1 ) · T (p) (g2 ) = (T (1) (g1 ) ⊗ T (2) (g1 )) · (T (1) (g2 ) ⊗ T (2) (g2 )) = T (1) (g1 )T (1) (g2 ) ⊗ T (2) (g1 )T (2) (g2 ) = T (1) (g1 g2 ) ⊗ T (2) (g1 g2 ) = T (p) (g1 g2 ). The proof in the matrix language makes use of the multiplication rule (1.2.25) for matrices (A ⊗ B) and (C ⊗ D). Thus, mapping T (p) = T (1) ⊗ T (2) is indeed a new representation of group G, which is called direct

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(or tensor ) product of representations T (1) and T (2) . Note that Eq. (4.3.8) shows that character of representation T (p) = T (1) ⊗T (2) is equal to product of characters of representations T (1) and T (2) : χT (p) (g) = χT (1) (g) · χT (2) (g). Example. Consider two representations of group C3 = Z3 , namely, 1-dimensional representation T (1) : T (1) (e) = 1, T (1) (g1 ) = q, T (1) (g2 ) = q 2 ,

q ≡ exp(2πi/3),

(4.3.9)

and 3-dimensional regular representation T (2) = T (R) given by (4.1.23). New representation T (p) = T (1) ⊗ T (R) is 3-dimensional, and, in accordance with (4.3.8), it is given by ⎛ ⎞ ⎛ ⎞ 2πi 0 0 e 3 1 0 0 ⎜ 2πi ⎟ ⎜ ⎟ T (p) (e) = ⎝0 1 0⎠, T (p) (g1 ) = ⎜ 0 0 ⎟ ⎝e 3 ⎠, 2πi 0 0 1 0 0 e 3 ⎛ ⎞ 4πi 0 e 3 0 ⎜ ⎟ 4πi (p) ⎜ T (g2 ) = ⎝ 0 0 e 3⎟ ⎠. 4πi e 3 0 0 The prescription (4.3.5), (4.3.6) for tensor product of two representations is generalized in an obvious way to tensor product of any number of representations. Namely, let T (α) (α = 1, . . . , r) be representations of (p) group G in spaces Vnα (K). Then one defines the representation Tr = T (1) ⊗ · · · ⊗ T (r) of G in space Vn1 (K) ⊗ · · · ⊗ Vnr (K) (space of rank-r tensors, see Definition 4.3.1): (1)

(r)

Tr(p) (g) · (ei1 ⊗ · · · ⊗ eir ) = (ej1 ⊗ · · · ⊗ ejr ) Tj1 i1 (g) · · · Tjr ir (g),

∀g ∈ G. (4.3.10)

Accordingly, transformation of coordinates vi1 ,...,ir of tensor (4.3.4) is (1)

(r)

vi1 ,...,ir → (Tr(p) (g) · v)i1 ,...,ir = Ti1 j1 (g) · · · Tir jr (g) vj1 ,...,jr . (4.3.11) We use the following definition later on. Definition 4.3.2. Tensor v ∈ Vn1 (K) ⊗ · · · ⊗ Vnr (K) is invariant under group G, if for all g transformations (4.3.10), (4.3.11) of its coordinates give (p) trivial results: (Tr (g) · v)i1 ,...,ir = vi1 ,...,ir .

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Let G be Lie group, and gA (t) be curve in G emanating from unit element, gA (0) = e, along tangent vector A ∈ A(G). Then the representation T (p) = T (1) ⊗ T (2) is expanded near unity as follows (see (4.2.5)): T (p) (gA (t)) = T (1) (gA (t))) ⊗ T (2) (gA (t))) = (In + t T (1) (A) + O(t2 )) ⊗ (Im + t T (2) (A) + O(t2 )) = In ⊗ Im + t(T (1) (A) ⊗ Im + In ⊗ T (2) (A)) + O(t2 ). This shows that tensor product of representations T (1) and T (2) of Lie algebra A(G) is given by T (p) (A) = (T (1) (A) ⊗ Im + In ⊗ T (2) (A)).

(4.3.12)

It acts in space Vn (K) ⊗ Vm (K) and transforms coordinates of vectors there as follows:   (p) T (A) · v ia = (T (1) (A) ⊗ Im + In ⊗ T (2) (A))ia,kb vkb (4.3.13) (1) (2) = Tik (A) vka + Tab (A) vib . (p)

Similarly, the action of representation Tr tensors (4.3.3) is Tr(p) (A) =

r 

= T (1) ⊗ · · · ⊗ T (r) in space of

I(1) ⊗ · · · ⊗ I(m−1) ⊗ T (m) (A) ⊗ I(m+1) ⊗ · · · ⊗ I(r) ,

m=1

(4.3.14) where I(k) := Ink . Definitions (4.3.12), (4.3.14) of direct product of representations are directly extended to general Lie algebras, both real and complex, irrespectively of Lie groups. Problem 4.3.1. Write the analog of formula (4.3.13) for tensor (4.3.3) of rank r > 2. In quantum mechanics, direct product T (1) ⊗ T (2) of two representations of group G describes a system consisting of two independent subsystems (e.g., spins) whose wave functions transform under representations T (1) and T (2) . The formula (4.3.12) corresponds to what is called spin addition rule in quantum mechanics.

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4.3.2.

201

Direct sum of representations

Given two vectors x = xiei ∈ Vn (K) and y = yaa ∈ Vm (K), one can construct a new (n + m)-dimensional vector w  with coordinates {wA } (A = 1, . . . , n + m): (w1 , w2 , . . . , wn+m ) = (x1 , x2 , . . . , xn , y1 , y2 , . . . , ym ).

(4.3.15)

In other words, vector w belongs to (n + m)-dimensional space with basis vectors ei and a , and equals to w  = ei xi + a ya .

(4.3.16)

This (n + m)-dimensional vector space is called direct sum of spaces Vn (K) and Vm (K) and is denoted by Vn (K) ⊕ Vm (K). In analogy to Section 4.3.1, one considers two representations T (1) and T (2) of group G, which act in spaces Vn (K) and Vm (K). For all g ∈ G one  given by (4.3.16) defines linear operators T (s) (g) which act on vectors w, and belonging to space Vn (K) ⊕ Vm (K), as follows: (1)

(2)

 = T (1) (g) · x + T (2) (g) · y = ej Tji (g) xi + b Tba (g) ya . T (s) (g) · w (4.3.17) This shows that the transformation rule for coordinates (4.3.15) of vector w  is (s)

 A = TAB (g) wB , (T (s) (g) · w) where (n + m) × (n + m) matrix of linear operator T (s) (g) is block-diagonal: (1)

||Tij (g)|| 0 (s) ||TAB (g)|| = . (4.3.18) (2) 0 ||Tab (g)|| It is sometimes convenient to represent operators T (s) (g) themselves as block matrices with operator coefficients,

0 T (1) (g) (s) , (4.3.19) T (g) = 0 T (2) (g)   represent vector w  by column xy , and write (4.3.17) as follows:





T (1) (g) x 0 T (1) (g) · x (s) = · . T (g) · w  = (2) (2) y T (g) · y 0 T (g)

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In this way one constructs a new representation T (s) of group G, acting in (n + m)-dimensional space Vn (K) ⊕ Vm (K). The fact that T (s) is homomorphism is straightforward to check:



T (1) (g1 ) T (1) (g2 ) 0 0 (s) (s) T (g1 )T (g2 ) = 0 T (2) (g1 ) 0 T (2) (g2 )

T (1) (g1 g2 ) 0 = = T (s) (g1 g2 ). 0 T (2) (g1 g2 ) Operator T (s) (g) given by (4.3.17), (4.3.19) is called direct sum of operators T (1) (g) and T (2) (g) and is denoted by T (s) (g) = T (1) (g) ⊕ T (2) (g). The new representation T (s) is direct sum of representations T (1) and T (2) and is denoted by T (1) ⊕ T (2) . Note that the formula (4.3.18) shows that character of representation T (s) = T (1) ⊕ T (2) is equal to the sum of characters: χT (s) (g) = χT (1) (g) + χT (2) (g). Example. Let us again consider two representations of group C3 , namely, 1-dimensional representation T (1) given by (4.3.9), and 3-dimensional regular representation (4.1.23). The new representation T (s) = T (1) ⊕ T (R) is 4-dimensional, and its matrices are ⎛

1 0 ⎜ 0 1 T (s) (e) = ⎜ ⎝0 0 0 0 ⎛ 2πi e 3 ⎜ 0 T (s) (g1 ) = ⎜ ⎝ 0 0 ⎛ 4πi e 3 ⎜ 0 T (s) (g2 ) = ⎜ ⎝ 0 0

0 0 1 0 0 0 1 0 0 0 0 1

⎞ 0 0⎟ ⎟, 0⎠ 1 ⎞ 0 1⎟ ⎟, 0⎠ 0 ⎞ 0 0 1 0⎟ ⎟. 0 1⎠ 0 0 0 0 0 1

Let G be Lie group and gA (t) be curve in G defining tangent vector A ∈ A(G) at unit element. Expansion of this curve in representation T (s)

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given by (4.3.19) reads:

(1) (g (t))) 0 T A T (s) (gA (t))) = 0 T (2) (gA (t))



0 In 0 T (1) (A) + O(t2 ), = +t 0 Im 0 T (2) (A)

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(4.3.20)

where T (1) (A) and T (2) (A) are representations of element A ∈ A(G), corresponding to representations T (1) and T (2) of group G. Expansion (4.3.20) shows that representation T (s) of Lie algebra A(G), which corresponds to Lie group representation T (1) ⊕ T (2) , is

(1) (A) 0 T , (4.3.21) T (s) (A) = 0 T (2) (A) i.e., it has block-diagonal form. The latter definition of direct sum T (1) ⊕ T (2) of Lie algebra representations T (1) and T (2) is directly extended to general Lie algebras, both real and complex, irrespectively of Lie groups. 4.4. 4.4.1.

Reducible and Irreducible Representations Definition of reducible and irreducible representations

Direct sum of representations, given by (4.3.18), is constructed out of representations T (1) and T (2) which have lower dimensions. The study of representation T (s) is thus reduced to that of individual representations T (1) and T (2) . It is worth noting that the similarity transformation (4.1.28) with (n + m) × (n + m) matrix S, applied to matrices T (s) (g), gives equivalent (n + m)-dimensional representation, which, however, no longer has blockdiagonal form (4.3.18). Conversely, for a given representation we can try to find similarity transformation that brings all its matrices to block-diagonal form and in this way partitions this representation into lower dimensional ones. Clearly, not all representations admit such a partition. So, we obtain the following definition. Definition 4.4.1. Representation T of group G which can be cast, by similarity transformation (4.1.28), to block-diagonal form (4.3.18) for all g ∈ G is decomposable. If representation T for all g ∈ G can be cast, by

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similarity transformation (4.1.28), to block-triangle form (1)



||Tij (g)|| ||X(g)ib || T (1) (g) X(g) T (g) = = , X(g) = 0, (2) 0 T (2) (g) 0 ||Tab (g)|| (4.4.1) then the representation is reducible.a If such a similarity transformation does not exist, i.e., matrices of representation T cannot simultaneously (i.e., for all g) be cast to block-triangle form (4.4.1), then representation T is irreducible. Homomorphism property T (g1 )T (g2 ) = T (g1 g2 ) of reducible representation (4.4.1) gives (1)

(1)

T (g1 ) X(g1 ) T (g2 ) X(g2 ) T (2) (g1 )

0 ≡

T (2) (g2 )

0

T (1) (g1 )T (1) (g2 ) T (1) (g1 )X(g2 ) + X(g1 )T (2) (g2 ) 0

=

T (2) (g1 )T (2) (g2 )

T (1) (g1 · g2 )

X(g1 · g2 )

0

T (2) (g1 · g2 )

.

(4.4.2)

This shows that T (1) (g1 ) T (1) (g2 ) = T (1) (g1 · g2 ),

T (2) (g1 ) T (2) (g2 ) = T (2) (g1 · g2 ),

and, therefore, blocks T (1) (g) and T (2) (g) are, in fact, representations of group G. So, if representation T is reducible (not necessarily decomposable), it still involves representations of lower dimensions, T (1) and T (2) . The main goal of representation theory is to find all irreducible representations, which are building blocks of other representations constructed in accordance with Eqs. (4.3.18) and (4.4.1). The fact that matrices T (1) (g) make representation of G can be understood as follows. Representation (4.4.1) transforms (n + m)-dimensional “composite” vectors (x, y) → (x , y  ), where x, x  ∈ Vn and y, y  ∈ Vm . a Clearly, representation is reducible also when the upper right block is zero instead. The latter representation is equivalent to (4.4.1), as these representations are related by similarity transformation which permutes rows and columns.

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In terms of coordinates, this transformation is given by (1)

(2)

xi = Tij (g) xj + Xia (g) ya , ya = Tab (g) yb .

(4.4.3)

This shows that the subspace with ya = 0 for all a = 1, . . . , m is transformed into itselfb : for any vector (x, 0) ∈ Vn ⊂ Vn ⊕ Vm we have T (g) · (x, 0) = (T (1) (g) · x, 0) ∈ Vn . This observation suggests the following definition. Definition 4.4.2. Let V be space of representation T of Lie group G and W linear subspace of V. The subspace W is invariant subspace of representation T , if for all v ∈ W and g ∈ G one has T (g) · v ∈ W, i.e., operator T (g) does not take vectors away from W. Trivial invariant subspaces are the space V itself and subspace with zero vector only. Definition 4.4.3 (equivalent to Definition 4.4.1). Representation T of group G in space V is irreducible if there are no invariant subspaces in V. Conversely, if there are nontrivial invariant subspaces in V, then representation T is reducible. Definition 4.4.4 (also equivalent to Definition 4.4.1). Reducible representation T of group G in space V is decomposable if there are two invariant subspaces W1 and W2 in V such that V = W1 ⊕ W2 . Decomposable representation T in space V = W1 ⊕ W2 is direct sum T = T (1) ⊕ T (2) of its two subrepresentations T (1) and T (2) acting in invariant subspaces W1 and W2 , respectively. If reducible representation T of group G is decomposed into direct sum of irreducible representations, then the representation T is completely reducible. Let us give a useful criterion of irreducibility of representation. Proposition 4.4.1. Finite-dimensional representation T of Lie group G in space V is irreducible if and only if for any v ∈ V, linear span of elements of group G in subspace Vn is determined by matrices T (1) (g). For this reason representation T (1) is called restriction of representation T to subspace Vn ⊂ Vn ⊕ Vm . The second representation T (2) acts in quotient space (Vn ⊕Vm )/Vn , whose elements are cosets (0, y  )+Vn . Formula (4.4.3) shows that T (g) acts in the coset space, as it transforms y ) + Vn . For this reason the representation T (2) is coset (0,  y ) + Vn to coset (0, T (2) (g) ·  sometimes called quotient representation of group G in space (Vn ⊕ Vm )/Vn .

b Action

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T (g) · v , where g runs over the whole group  G, coincides with V. Linear span here is a linear subspace of vectors g dμ(g) ag T (g) · v , where ag are numerical functions on the groupc and μ(g) is a left-invariant measure on the group. Proof. We begin with the first part of this proposition. Let for any v ∈ V linear span of elements T (g) · v coincide with V. Then no vector v can belong to a nontrivial invariant subspace, so there are no invariant subspaces, and T is irreducible.  To prove the second part, let us consider linear space Ta · v , where Ta = g dμ(g) ag T (g) and ag ≡ a runs over all functions on group. This space is invariant, since   T (˜ g)Tav = dμ(g) ag T (˜ g · g)v = dμ(g) ag˜−1 ·g T (g)v g

g

again belongs to space Ta · v . Since there are no nontrivial invariant subspaces of irreducible representation T , the space Ta · v coincides with V. The above proposition can be viewed as a consequence of an important theorem of representation theory. Proposition 4.4.2 (Burnside theorem). Let G be a group, T its finitedimensional irreducible representation in space V. Then linear span of operators T (g), where g runs over the whole group G, coincides with algebra of all linear operators in V. We do not give the proof of this theorem, it can be found in books [4, 8]. Corollary 4.4.1. Matrix elements Tij (g) of finite-dimensional irreducible representation T of group G make a system of linearly independent functions on G. Indeed, linear dependence means that Tr(C T (g)) = 0, where Tr is trace in the space of representation V and C is a constant nonzero matrix. Since any element of algebra of operators in V can be constructed as linear combination of operators T (g), then one can replace matrix ||Tij (g)|| in Tr(C T (g)) = 0 by matrix units, and obtain C = 0. Definitions 4.4.1, 4.4.3 and 4.4.4 for Lie group representations have natural analogs in the theory of Lie algebra representations. c More

accurately, one considers functions ag with compact support, and then defines the closure of this set using natural topology in V.

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Definition 4.4.5. Matrix representation T of Lie algebra A which can be cast, by similarity transformation (4.2.4), to block-diagonal form (4.3.21) for all A ∈ A is decomposable. If matrix representation of Lie algebra A can be cast, by similarity transformation (4.1.28), to block-triangle form

T (1) (A) X(A) , X(A) = 0 (∀A ∈ A), (4.4.4) T (A) = 0 T (2) (A) then this representation is reducible. If there are no similarity transformations with the latter property, then the representation is irreducible. We have the following expansion for reducible representation (4.4.1) of group G near unit element, see (4.2.5):

T (1) (gA (t))) X(gA (t))) T (gA (t)) = 0 T (2) (gA (t))



T (1) (A) X(A) In 0 +t + O(t2 ), = 0 Im 0 T (2) (A) (4.4.5) and for decomposable representation we have X(A) = 0. According to the general prescription, the second term on the right-hand side here involves the representation of Lie algebra A(G). This shows that the standard procedure (4.4.5) of constructing Lie algebra representation out of representation of its Lie group relates irreducible, reducible and decomposable representations of Lie group to representations of its Lie algebra falling into the same categories. Lie algebra analogs of Definitions 4.4.2–4.4.4 are also obvious. Definition 4.4.6. Let V be space of representation T of Lie algebra A, W linear subspace in V. It is called invariant subspace if for all v ∈ W and A ∈ A one has T (A) v ∈ W. Representation T of Lie algebra A in V is irreducible if there are no nontrivial invariant subspaces in V, and reducible, if invariant subspace(s) exist. Representation T of Lie algebra A in V is decomposable, if there are two invariant subspaces W1 , W2 such that V = W1 ⊕ W2 . If representation of Lie algebra A is partitioned into direct sum of its irreducible representations, then this representation is completely reducible. As an illustration, let us consider adjoint representation ad of Lie algebra, defined in item 4 of Section 4.2.2.

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Proposition 4.4.3. Adjoint representation of Lie algebra A is irreducible iff A is simple. Proof. Let us first prove, ad absurdum, that adjoint representation of simple algebra is irreducible. If adjoint representation ad of simple algebra A is reducible, then, in accordance with Definition 4.4.6, there exists a nontrivial invariant subspace V ⊂ A, such that for all Y ∈ V and all X ∈ A one has ad(X) · Y = [X, Y ] ∈ V. This shows that V is ideal in A, which contradicts the assumption of simplicity of A. On the other hand, let adjoint representation of A be irreducible. Then there are no invariant subspaces in the space of this representation — Lie algebra A itself — and hence A is simple. Note that Proposition 4.4.3 is valid for both real and complex Lie algebras. Remark 1. Complex Lie algebras and their real forms have basically the same complex representations. Indeed, let T (AC ) be representation of complex Lie algebra AC . Let the generators Xi of AC be such that the structure constants of AC are real. Then Xi are at the same time generators of real form AR and real linear combinations of T (Xi ) make representation T (AR ) (the latter can be viewed as induced from T (AC )). Conversely, complex representation T (AR ) of real algebra AR defines the set of operators T (Xi ), where now Xi are generators of AR . Complex linear combinations of T (Xi ) make representation T (AC ) of algebra AC , which is complexification of AR . Proposition 4.4.4. Let AC be complex Lie algebra, AR its real form. Representation T (AC ) is irreducible iff complex representation T (AR ) induced from T (AC ) is irreducible. Proof. Let us give the proof ad absurdum. Let representation T (AC ) in complex space V be irreducible, and T (AR ) be reducible. Since T (AR ) is reducible, there exists nontrivial invariant subspace V1 ⊂ V, such  that T (Xi )V1 ⊂ V1 for any generator Xi ∈ AR . Then T ( i ai Xi )V1 =  i ai T (Xi )V1 ⊂ V1 for all complex ai , which contradicts irreducibility of T (AC ). The proof of the rest of the proposition is analogous. Remark 2. Relationship between representations of complex Lie algebra AnC and representations of its realification A2n R is a lot less trivial as

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compared to the situation in Remark 1. Indeed, according to Remark 1, real 2n algebra A2n R and its complexification AC have the same representations. 2n Proposition 3.2.3 tells that algebra AC is partitioned into direct sum A(+) + A(−) of its subalgebras, and A(+) is isomorphic to AnC , while A(−) is related to AnC by a certain antilinear isomorphism ρ. Therefore, 2n complex representations of A2n C , and hence representations of AR may be constructed as direct products of representations of algebras AnC and ρ(AnC ). So, the variety of representations of A2n R is larger than the variety of n representations of AC . Let us illustrate this point by considering complex algebra s(2, C) and its realification A(SL(2, C)) = A6R = so(1, 3). We know from Proposition 3.3.5 that complexification of algebra A6R is algebra so(4, C) = s(2, C)+s(2, C), so complex representations of A6R are the same as representations of s(2, C)+s(2, C). The latter representations are direct products of representations of the first and second subalgebras s(2, C). Finite-dimensional irreducible representations of A6R are thus determined by two integer or half-integer numbers (two spins) relating to the first and second s(2, C) subalgebra (see Section 4.7.3 below), while representations of the original complex algebra s(2, C) are determined by a single spin. Thus, the variety of representations of s(2, C) is smaller than that of its realification A6R . This example is of importance from the viewpoint of relativistic physics; it is studied in detail in accompanying book. 4.4.2.

Schur’s lemma

It is of importance to have a working criterion for irreducibility of a representation. Such a criterion is given by Schur’s lemma. Lemma 4.4.1 (Schur’s lemma). (1) Operator A = 0 that commutes with all elements of group G in its irreducible complex representation T, T (g) · A = A · T (g) (∀g ∈ G), is proportional to unit operator, A = λI. (2) Let T (1) and T (2) be two inequivalent complex irreducible representations of group G in vector spaces Vn , Vm , and A be linear map Vm → Vn such that ∀g ∈ G one has T (1) (g) · A = A · T (2) (g). Then A = 0. We note in passing that if an operator A has the property T · A = A · T , where T and T  are two operators acting in vector spaces V and V  , then this operator A is called intertwining operator. Proof. (1) Operator A = 0 that commutes with all operators T (g) (∀g ∈ G) of complex irreducible representation in vector space V is not degenerate. Let us see this ad absurdum. Let there be nonzero vectors x ∈ V

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such that Ax = 0. These vectors make linear subspace Ker(A) in V. It follows from T (g) · A = A · T (g) that T (g) · Ax = A · T (g)x = 0. Therefore, if x ∈ Ker(A), then T (g)x ∈ Ker(A) (∀g ∈ G), i.e., Ker(A) is invariant subspace of V (Ker(A) = V is impossible, since A = 0 in that case). Thus, representation T is reducible, in contradiction to the original assumption. So, A does not have zero vectors, i.e., it is not degenerate. Since A is not degenerate, there exists eigenvalue λ = 0 of operator A with eigenvectors v making subspace Ker(A − λI) ⊂ V. Then one has T (g)(A − λI)v = (A − λI)T (g)v = 0 (∀g ∈ G). Thus, either Ker(A − λI) is a nontrivial invariant subspace and the representation is reducible (which is in contradiction to the assumption of Lemma) or Ker(A − λI) coincides with the whole space V, and hence (A − λI) = 0, as promised. (2) Let A = 0. Then, in complete similarity to above, Ker(A) is invariant subspace in Vm . Furthermore, Im(A) is invariant subspace in Vn . Indeed, ∀x ∈ Vm , and ∀g ∈ G we have T (1) (g) (Ax) = AT (2) (g) x, i.e., T (1) (g) Im(A) ⊂ Im(A). Since representations T (1) and T (2) are irreducible, and A = 0, we have Ker(A) = ∅ and Im(A) = Vn . Thus, the map A : Vm → Vn is isomorphism, i.e., representations T (1) and T (2) are equivalent, which contradicts the assumption of lemma. Hence, A = 0. Corollary 4.4.2. If T is complex representation, and there exists a nontrivial operator A = λI, such that T (g) A = A T (g) (∀g ∈ G), then representation T is reducible. Corollary 4.4.3. All finite-dimensional complex representations of an Abelian group G are 1-dimensional. Indeed, for any representation T of this group and ∀g, h ∈ G we have T (g)T (h) = T (h)T (g). Let T be complex and irreducible. For given h, operator T (h) commutes with all T (g), and due to irreducibility of T we have T (h) = λ(h)I, where λ(h) is a numerical function on G. This is valid for any h, so I = 1 is unit 1 × 1 “matrix” (otherwise T (h) would be direct sum of 1 × 1 matrices λ(h), and representation would be reducible). So, representation T is 1-dimensional. Remark 1. Schur’s lemma and its proof are literally valid also for complex representations of Lie algebras. They are valid for representations of all associative algebras as well. Remark 2. The assumption that representation T is complex is important for Schur’s lemma. The reason is that its proof is based, in particular, on the property that there exist eigenvectors and eigenvalues of any operator

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A, i.e., subspaces Ker(A − λI); the eigenvalues λ may be complex even for real matrices of operators A. Let us consider an example that illustrates Remark 2. Real 2-dimensional representation T of Abelian group U (1) = SO(2) is given in terms of matrices Oφ , defined in (3.1.15), and describes rotations in R2 . It is irreducible as real representation, even though all matrices Oφ commute with antisymmetric matrix A = i given by (3.1.14). If, however, the matrices (3.1.15) of representation T are considered as acting in complex 2-dimensional space C2 , then the representation T is reducible, in complete accordance with the Corollary 4.4.3: all real matrices Oφ given by (3.1.15) are simultaneously diagonalized by (complex) similarity transformation (4.1.28), and the complex representation T is decomposed into direct sum of two 1-dimensional irreducible complex representations of U (1) = SO(2). These 1-dimensional representations are T (1) (Oφ ) = eiφ , T (1)∗ (Oφ ) = e−iφ . Problem 4.4.1. Find complex 2 × 2 matrix S, such that



iφ 0 cos φ − sin φ e S −1 . S= 0 e−iφ sin φ cos φ

(4.4.6)

(4.4.7)

We note that any n-dimensional complex representation in Vn (C) can be considered as 2n-dimensional real representation in V2n (R). As an example, 1-dimensional complex representation T (1) (Oφ ) = eiφ of group U (1) = SO(2) is phase rotation in complex plane z → eiφ z. By setting z = x + iy, where x, y ∈ R, one rewrites this phase rotation as 2-dimensional real transformation



  cos φ − sin φ x x → . y sin φ cos φ y Thus, realification of irreducible complex representation T (1) (Oφ ) is precisely the 2-dimensional real representation T with matrices Oφ given by (3.1.15). Now, any n-dimensional real representation can be complexified, so that it can be viewed as n-dimensional complex representation. We see that complete classification of complex representations of a given Lie group would at the same time give information on real representations, which sometimes are important from physics prospective. The classification

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of complex irreducible representations of a Lie group (or Lie algebra) is simpler, in particular, due to Schur’s lemma. Corollary 4.4.4. Let A be simple complex Lie algebra or real form of simple complex Lie algebra. Then matrix A that commutes with all matrices of its adjoint representation, ad(Y ) · A = A · ad(Y ),

∀Y ∈ A,

(4.4.8)

is proportional to unit matrix, A = λ I. In the case of complex algebra A, this is a direct consequence of Schur’s Lemma, since, due to Proposition 4.4.3, complex representation ad is irreducible in that case. The claim that A = λ I may be wrong for simple real Lie algebra A whose adjoint representation is real, see Remark 2 above. Nevertheless, the statement of Corollary 4.4.4 is valid for real forms of simple complex Lie algebras. Indeed, let matrix A ∈ A obey (4.4.8), where A is real form of simple complex Lie algebra A(C). Let us choose basis {Xa } in real algebra A with defining commutation relations (3.2.30). In this basis, matrix ad(Y ) is equal to d , ad(Y )db = Y a Cab

where Y a are coordinates of element Y = Y a Xa ∈ A. The relation (4.4.8) then reads p d Adp = Abd Y a Cad Y a Cab

(4.4.9)

for all real Y a . Equation (4.4.9) gives p d (Y a + i Y a ) Cab Apd = Adb (Y a + i Y a ) Cad ,

∀ Y a , Y b ∈ R,

(4.4.10)

i.e., all matrices of adjoint representation of simple complex Lie algebra A(C) commute with A. So, matrix A must be proportional to unit matrix. Examples 1. Matrices T (g) of decomposable representation T of Lie group G can be cast into the form (4.3.18) by equivalence transformation (4.1.27). Matrices T (g) of this type commute, for all g ∈ G, with block matrices

0 λ1 In , (λ1 = λ2 ). A= 0 λ2 Im These matrices are not proportional to unit matrix In+m . This fact serves as an illustration to the first part of Schur’s lemma.

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2. Defining representations T of groups SL(n, C) and SU (n) are irreducible. However, their tensor products are reducible. Consider representation T ⊗ T of SU (n) (the reasoning that follows is valid also for SL(n, C)) in complex space Vn2 (C) = Vn (C) ⊗ Vn (C). Elements g ∈ SU (n) in representation T ⊗ T act in Vn2 (C) as given in (4.3.6). In accordance with Schur’s lemma, if T ⊗ T is reducible, then there must exist an operator P = λIn ⊗ In , acting in Vn2 (C) and commuting with all operations (4.3.6). Such an operator does exist: it is the permutation operator P · (v1 ⊗ v2 ) = (v2 ⊗ v1 ),

∀v1 , v2 ∈ Vn (C),

P 2 = In ⊗ In ≡ In2 . (4.4.11)

Let {ei } be basis in Vn (C) and {(ei ⊗ ej )} be basis in Vn2 (C). The matrices of operator P and unit operator in the latter basis have elements k r kr k r P kr ij = δj δi and I ij = δi δj , which is clear from the relations P · (ei ⊗ ej ) = (ej ⊗ ei ) = (ek ⊗ er ) P kr ij , In2 · (ei ⊗ ej ) = (ei ⊗ ej ) = (ek ⊗ er ) I kr ij .

(4.4.12)

Problem 4.4.2. Check that permutation operator P given by (4.4.11), (4.4.12) commutes with all operators of representation T ⊗ T which act as given in (4.3.6). Making use of operator P , one constructs two projection operators P + and P − : P + = 12 (In2 + P ), P − = 12 (In2 − P ),

(4.4.13)

P + P − = 0, (P ± )2 = P ± , P + + P − = In2 .

(4.4.14)

They also commute with operations (4.3.6) and single out two invariant (+) (−) subspaces Vm+ (C) and Vm− (C) of Vn2 (C). In other words, any vector  =w  (+) + w  (−) of a symmetric w  ∈ Vn2 (C) is partitioned into the sum w (+) (+) = w  and P · w  (−) = −w  (−) , and antisymmetric vectors, P · w  respectively:  w  (−) = P − · w.  w  (+) = P + · w,

(4.4.15)

Since P commutes with operations (4.3.6), the two subspaces V (±) are invariant under the representation T ⊗ T .

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Problem 4.4.3. Making use of the decomposition (4.3.2), show that components of vectors (4.4.15) are symmetric and antisymmetric tensors (+)

wij = 12 (wij + wji ),

(−)

wij = 12 (wij − wji ),

(4.4.16)

 ∈ Vn2 (C). Show that dimensions of these where wij are components of w . representations are m± = n(n±1) 2 In fact, V (±) are subspaces of two irreducible representations of group SU (n) (or SL(n, C)), since there is only one nontrivial operator that commutes with operators T ⊗ T , and this is the permutation operator P . We discuss this property in some detail in the accompanying book. Thus, tensor product of two defining representations T (we use also notation [n] for defining representation, according to its dimension) is partitioned into sum of symmetric and antisymmetric irreducible ] and [ n(n−1) ], respectively: representations, [ n(n+1) 2 2     n(n + 1) n(n − 1) [n] ⊗ [n] = ⊕ . (4.4.17) 2 2 The partitions of direct products of several defining representations of SU (n) and SL(n, C) into direct sums of irreducible representations are discussed in the accompanying book. 3. Let us again consider defining representation [n] of group SU (n). We denote complex conjugate representation to [n] by [¯ n]. In these representations, elements g ∈ SU (n) act on coordinates of vectors in accordance with (4.1.9) and (4.1.14), respectively. Consider now direct product [n] ⊗ [¯ n]. Vectors in n2 -dimensional space of representation [n]⊗ [¯ n] are second rank tensors with components wij whose transformations are dictated by (4.1.9) and (4.1.14):  ∗ = gik wkm gjm ⇒ w = g · w · g −1 . wij → wij

(4.4.18)

The second equality here uses the fact that g † = g −1 for SU (n) matrices. The decomposition   Tr(w) Tr(w) wij = wij − δij + δij n n then tells that there are two invariant subspaces in the space of representation [n] ⊗ [¯ n], namely (1) (n2 − 1)-dimensional subspace of traceless matrices (wij − δij Tr(w)/n), which transforms (see (4.4.18)) according to adjoint representation of SU (n); we denote the adjoint

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representation here by [n2 − 1]; (2) 1-dimensional subspace of matrices δij Tr(w) which is the space of trivial representation of SU (n); we denote this representation by [1]. It follows from Proposition 4.4.3 that representation [n2 − 1] is irreducible; the representation [1] is obviously irreducible too. So, representation [n] ⊗ [¯ n] is partitioned into direct sum of two irreducible representations [n] ⊗ [¯ n] = [n2 − 1] ⊕ [1].

(4.4.19)

Problem 4.4.4. Find an operator A = λI that acts in the space of representation [n] ⊗ [¯ n] and commutes with T (g) ⊗ T ∗ (g) for all g. 4.5.

Representations of Finite Groups and Compact Lie Groups. Group Algebra and Regular Representations

In this and next sections we repeatedly make use of the right- and leftinvariance of summation over finite group G    X(g) = X(g h) = X(h g), ∀h, h ∈ G, (4.5.1) g∈G

g∈G

g∈G

where X(g) is an arbitrary function on G. Since there exists invariant measure on compact Lie group G, and the volume of G is finite (see Section 2.2.2), the results we study here are valid not only for finite groups, but also for compact Lie groups. In the latter analysis, summation over finite group is to be replaced by invariant integration over compact Lie group, and order of finite group N by volume V of compact Lie group. In fact, finite groups can be viewed as compact Lie groups of zero dimension. For this reason, we use the term “compact group” when the results are valid for both finite groups and compact Lie groups. We prove most of the facts by considering finite groups only, but the proofs are straightforwardly generalized to compact Lie groups, as explained above. The following three propositions are valid for both finite groups and compact Lie groups. Proposition 4.5.1. Reducible representations of compact groups are completely reducible.

Proof. Let G be finite group of order N . Its reducible representation has matrices of the form (4.4.1). We make similarity transformation (change of

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basis) (4.1.28) with In S= 0



Y Im

, Y =

1  X(g) T (2)(g −1 ). N

(4.5.2)

g∈G

Then the representation (4.4.1) is cast into block-diagonal form (4.3.18). Indeed, let us prove that



0 T (1) (g) X(g) T (1) (g) −1 S= . (4.5.3) S 0 T (2) (g) 0 T (2) (g) We use explicit expression S −1 =

In

−Y

0

Im

.

Then (4.5.3) is equivalent to T (1) (g)Y − Y T (2) (g) + X(g) 1  (1) [T (g)X(h)T (2) (h−1 ) − X(h)T (2) (h−1 )T (2) (g)] + X(g) = 0. ≡ N h∈G

The latter is indeed an identity, which is proven by going from summation over h to summation over gh and using the formula X(g1 · g2 ) = T (1) (g1 ) X(g2 ) + X(g1 ) T (2) (g2 ), which follows from (4.4.2). The proof for compact Lie groups is completely analogous. This proposition says that any reducible representation is partitioned into direct sum of irreducible representations. This partition is, in fact, unique. We prove the latter property ad absurdum. Let completely reducible representation T be partitioned into irreducible representations in two different ways. Then these two partitions have different invariant subspaces which intersect in nontrivial ways. The intersections are invariant subspaces themselves, which contradicts complete reducibility of representation T . Proposition 4.5.2. Any representation of compact group is equivalent to unitary representation, i.e., there exists positive-definite Hermitian scalar product in the space of representation, with respect to which the operators of representation are unitary.

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Proof. Let V be space of representation T of compact group G. One can always define some positive-definite Hermitian scalar product (Hermitian form) x, y 1 = y , x ∗1 , x, y ∈ V. Recall that positive-definiteness means that x, x 1 ≥ 0 for all x ∈ V, and x, x 1 = 0 implies x = 0. One defines the new form 1  T (g) · x, T (g) · y 1 . (4.5.4) x, y := N g∈G

It is also positive-definite and Hermitian. Making use of the invariance of summation over group, it is straightforward to show that the form (4.5.4) is invariant under all operations T (h): x, y = T (h) · x, T (h) · y ,

∀h ∈ G.

(4.5.5)

Hence, T is unitary representation of G with respect to the scalar product (4.5.4). Remark 1. In the case of real representation, this proposition and the result (4.5.5) mean that there always exists positive-definite scalar product in the space of representation, which is invariant under the group action, and matrices of this representation are orthogonal in the orthonormal basis. Problem 4.5.1. Prove Proposition 4.5.1 on the basis of Proposition 4.5.2. Hint: make use of the fact that orthogonal complement of an invariant subspace in V with respect to the scalar product (4.5.5) is also invariant subspace. Remark 2. Propositions 4.5.1 and 4.5.2 are valid also for finitedimensional associative algebras. Proposition 4.5.3. Positive-definite Hermitian scalar product, under which an irreducible complex representation of compact group is unitary, is unique up to a constant numerical pre-factor. Proof. Let there be two scalar products  , 1,2 in the space V of irreducible complex representation T of group G, and this representation is unitary with respect to both of them. Let ea be basis in V, which is orthonormal in scalar product  , 1 . Then ea , eb 2 = ηab .

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Since the representation T is unitary in the scalar product  , 2 , we have ηab = T (g) · ea , T (g) · eb 2 = Tca ec , Tdb ed 2 ∗ = Tca Tdb ηcd = (T † · η · T )ab ,

where the last expression involves product of matrices. Now, since the representation T is unitary in the scalar product , 1 , the matrix Tab is unitary. Therefore, T −1 · η · T = η. Hence, matrix η commutes with all matrices of irreducible complex representation, and therefore it is proportional to unit matrix due to Schur’s lemma. We make use the definition of group algebra in following sections. Definition 4.5.1. Group algebra K[G] of finite group G over field K is vector space over field K, which is a linear span of elements of group G considered as basis in K[G]. Thus, any element of K[G] reads  αg g (αg ∈ K), (4.5.6) a= g∈G

where coefficients αg are functions on group taking values in K. Addition and multiplication by number β ∈ K of elements (4.5.6) are defined as follows:     αg g + βg g = (αg + βg ) g, β a = (βαg ) g, a+b= g∈G

g∈G

g∈G

g∈G

while multiplication of elements (4.5.6) is defined in accordance with group operation in G:     αh h · βg g = (αh βg ) (h · g) = γg g. (4.5.7) a·b = h∈G

g∈G

g,h∈G

g∈G

Here γg =



αh βh−1 g ,

(4.5.8)

h∈G

and we use invariance of summation over group when obtaining (4.5.7).

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One can give equivalent definition by making use of coordinate language, i.e., instead of formal vectors (4.5.6) considering functions αg = α(g) on group G. Namely, taking for definiteness K = C, the group algebra C[G] is equivalently defined as algebra of complex functions αg = α(g) on G with multiplication (4.5.8). This construction is defined not only for finite groups but also for compact Lie groups. In the latter case one requires that functions α(g) belong to the space L2 (G, dμ) of square integrable (with Haar measure dμ) complex functions on G. This trick (employing algebra of functions on G instead of group itself) can be used for giving equivalent definition of regular representations of both finite groups (see Eq. (4.1.20) in Section 4.1.2) and compact Lie groups. To this end, we note first that regular representation of finite group G induces representation in group algebra, which we also call regular. Namely, we write left and right action of an element gi ∈ G on vectors (4.5.6) in group algebra:   gi · a = α(gi−1 · gk ) gk , a · gi = α(gk · gi−1 ) gk , (4.5.9) gk ∈G

gk ∈G

where we again make use of invariance of summation over group. We now recall the first formula in (4.1.20) and write   (R) gi α(gm )gm = α(gm )gk Tkm (gi ). gm ∈G

gm ,gk ∈G

Comparing coefficients of gk here and in the first of (4.5.9), we get (R)

Tkm (gi ) α(gm ) = α(gi−1 · gk )

(4.5.10)

(summation over m is assumed on the left-hand side). Similarly, we have for representation T(R) that (R) α(gm ) Tmj (gi ) = α(gj · gi−1 ).

The latter equality and the property T(R) T (g) = T(R) (g −1 ), valid for finite groups (see Problem 4.1.4), imply that (R) Tjm (gi ) α(gm ) = α(gj · gi ).

(4.5.11)

The left and right regular representations in the space of functions on group G are now defined by Eqs. (4.5.10), (4.5.11): these are directly related to left and right translations in G. In the case of finite group of order n, functions α(gm ) can be thought of as n-dimensional columns with elements αm = α(gm ). Then Eq. (4.5.10) is a linear transformation in the space of these columns.

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Now we employ analogs of Eqs. (4.5.10), (4.5.11) to define left and right regular representations of compact Lie group G. Namely, we introduce linear operators T (R) (h) and T(R) (h), which act in the space of functions L2 (G, dμ) as left and right translations, in G: [T (R) (h)α](g) = α(h−1 · g), (R)

[T

(h)α](g) = α(g · h).

(4.5.12) (4.5.13)

Mappings T (R) and T(R) of G to the space of linear operators in space of functions on G are precisely left and right regular representations of group G. These representations are infinite-dimensional, since the space L2 (G, dμ) is infinite-dimensional. Problem 4.5.2. Show that mappings T (R) and T(R) , given by (4.5.12) and (4.5.13), are homomorphisms of G to group of linear operators in L2 (G, dμ). Problem 4.5.3. Show that regular representations T (R) and T(R) of compact group G are unitary with respect to scalar product in L2 (G, dμ) given by  (α, β) = dμ(g) α∗ (g) β(g). G

4.6.

4.6.1.

Elements of Character Theory for Finite Groups and Compact Lie Groups Examples. Irreducible representations and characters of C3 and S3

All representations in this subsection are complex. 1. Group C3 . Regular representation T (R) is given by matrices (4.1.23). The characters are (4.6.1) χR (e) = 3, χR (g1 ) = 0, χR (g2 ) = 0. Since C3 is Abelian group, its three-dimensional representation (4.1.23) is reducible by Schur’s lemma. To see this explicitly, we note that the eigenvectors of matrices (4.1.23) are v3 = (1, q 2 , q), (4.6.2)  v1 = (1, 1, 1), v2 = (1, q, q 2 ),  v12 = ( v2 ,  v3 ) = 3, ( v1 ,  v2 ) = ( v1 ,  v3 ) = v22 = where q = e2πi/3 . We use the equalities  2 2 v3 = 0, which follow from the identity q + q + 1 = 0. We compose columns and rows of matrices A and A−1 out of vectors (4.6.2): ⎞ ⎞ ⎛ ⎛ 1 1 1 1 1 1 1 ⎟ ⎟ ⎜ ⎜ A = ⎝1 q q 2 ⎠ , A−1 = ⎝1 q 2 q ⎠ . (4.6.3) 3 2 2 1 q q 1 q q

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Then Aij Tjk = λi Aik , and similarity transformation with S = A−1 diagonalizes matrices (4.1.23): (R)

⎛

1 ⎜ T˜(R) (e) = ⎝0

0 1

0

0

0

⎞

⎟ 0⎠,

⎛

1 ⎜ T˜ (R) (g1 ) = ⎝0

1

0

0

0

⎞

q

⎟ 0 ⎠,

0

q2

⎛

1 ⎜ T˜ (R) (g2 ) = ⎝0

q2

⎞ 0 ⎟ 0⎠.

0

0

q

0

(4.6.4) We see explicitly that the regular representation (4.1.23) is completely reducible and is partitioned into direct sum of 1-dimensional irreducible representations T (R) = I ⊕ Γ ⊕ Γ∗ ,

(4.6.5)

where I(C3 ) is trivial representation, Γ(C3 ) is faithful representation Γ(e) = 1, and

Γ∗ (C3 )

Γ(g1 ) = e2πi/3 = q,

Γ(g2 ) = e4πi/3 = q 2 (q 3 = 1),

(4.6.6)

is complex conjugate of the latter,

Γ∗ (e) = 1,

Γ∗ (g1 ) = e−2πi/3 = q 2 ,

Γ∗ (g2 ) = e−4πi/3 = q.

(4.6.7)

We show in Section 4.6.2 that regular representation of finite group G contains all irreducible representations of G. So, representations I, Γ, Γ∗ exhaust the class of inequivalent irreducible representations of C3 . 1-dimensional representations T have χT (g) = T (g), so the table of characters χT (g) of irreducible representations of group C3 is —

(e)

(g1 )

(g2 )

I(C3 )

χI = 1

χI = 1

χI = 1

Γ(C3 )

χΓ = 1

χΓ = q

χΓ = q 2

Γ(C3 )∗

χΓ ∗ = 1

χΓ ∗ = q 2

χΓ ∗ = q

T (R) (C3 )

χR = 3

χR = 0

χR = 0

The first three rows of this table determine the characters (4.6.1) of regular representation T (R) (C3 ) which, according to (4.6.5) are equal to the sum of characters of irreducible representations, χR = χI + χΓ + χΓ∗ (the fourth row of the table). We note that direct products of representations I(C3 ), Γ(C3 ) and Γ∗ (C3 ) are again 1-dimensional irreducible representations: I(C3 ) ⊗ Γ(C3 ) = Γ(C3 ), Γ(C3 ) ⊗ Γ(C3 ) = Γ∗ (C3 ), Γ∗ (C3 )

⊗

I(C3 ) ⊗ Γ∗ (C3 ) = Γ∗ (C3 ), Γ(C3 ) ⊗ Γ∗ (C3 ) = I(C3 ),

Γ∗ (C3 )

= Γ(C3 ).

This follows, in particular, from multiplication table of characters: χΓ χΓ = χΓ∗ ,

χΓ χΓ∗ = χI ,

χΓ∗ χΓ∗ = χΓ .

We see that the set of inequivalent irreducible representations {I, Γ, Γ∗ } of C3 is an Abelian group C3∗ isomorphic to C3 , where the group operation is direct product ⊗, trivial representation I(C3 ) is unit element and representations Γ(C3 ) and Γ∗ (C3 )

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are inverse to each other. Group C3∗ of irreducible representations of C3 is called dual group of C3 . Dual group is always isomorphic to the original group for Abelian finite groups (remarkable result by Pontryagin [3]). This property does not hold for non-Abelian groups. Problem 4.6.1. Construct table of characters for group C4 and show that C4∗ = C4 . 2. Group S3 = D3 . Let us construct the table of characters for non-Abelian group D3 = S3 , whose elements are denoted by (e, g1 , g2 , r, rg1 , rg2 ). Left regular representation T (R) is defined by the relations g1 · (e, g1 , g2 , r, rg1 , rg2 ) = (g1 , g2 , e, rg2 , r, rg1 ) = (e, g1 , g2 , r, rg1 , rg2 ) · T (R) (g1 ), r · (e, g1 , g2 , r, rg1 , rg2 ) = (r, rg1 , rg2 , e, g1 , g2 ) = (e, g1 , g2 , r, rg1 , rg2 ) · T (R) (r). Accordingly, regular representation (dots stay for zeros) ⎛ . . 0 0 1 ⎜1 0 0 . . ⎜ ⎜ ⎜0 1 0 . . ⎜ T (R) (g1 ) = ⎜ ⎜. . . 0 1 ⎜ ⎜ 0 0 ⎝. . . .

.

.

1

0

of the two generators g1 and r of group D3 is .

⎞

.⎟ ⎟ ⎟ .⎟ ⎟ ⎟, 0⎟ ⎟ ⎟ 1⎠

⎛

.

.

1

0

.

.

0

1

.

.

0

0

0

0

.

.

1

0

.

.

⎞ 0 0⎟ ⎟ ⎟ 1⎟ ⎟ ⎟ .⎟ ⎟ ⎟ .⎠

1

.

.

.

. ⎜. ⎜ ⎜ ⎜. ⎜ (R) (r) = ⎜ T ⎜1 ⎜ ⎜ ⎝0 0

0

0

(4.6.8) We see that matrices of this representation are constructed out of blocks of matrices of regular representation of group C3 given by (4.1.23). We already know eigenvectors (4.6.2), so we immediately find 1-dimensional invariant subspaces of representation (4.6.8). These are vectors w  1 = (1, 1, 1, 1, 1, 1) (trivial representation T (1) (gk ) =  2 = (1, 1, 1, −1, −1, −1) (1-dimensional representation distinT (1) (rgk ) = 1) and w guishing even and odd transpositions, T (2) (gk ) = 1, T (2) (rgk ) = −1). We can also extract two 2-dimensional invariant subspaces with basis vectors v2 , ± v3 ), w  3± = (1, q, q 2 , ±1, ±q 2 , ±q) = ( v3 ,  v2 ) w  4± = (±1, ±q 2 , ±q, 1, q, q 2 ) = (± such that ⇒w  3± T (R) (g1 ) = q w  3± ,

w  4± T (R) (g1 ) = q −1 w  4± ,

w  3± T (R) (r) = ± w  4± .

So, besides the two 1-dimensional representations T (1) and T (2) we have two irreducible representations T (+) and T (−) :   q 0 0 ±1 (±) (r) = . (4.6.9) T (±) (g1 ) = , T 0 q −1 ±1 0 The latter are equivalent to each other, since they are related by similarity transformation (4.1.28) with S = diag(1, −1). Let us now choose the matrix S −1

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as follows:

⎛

S −1

⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝

1

1

1

1

1

1

1

1

−1

−1

1

q

q2

1

q2

1

q2

q

1

q

1

q

q2

−1

−q 2

−1

−q 2

−q

1

q

1

223

⎞

−1⎟ ⎟ ⎟ q ⎟ ⎟ ⎟, q2 ⎟ ⎟ ⎟ −q ⎠

(4.6.10)

q2

where rows contain coordinates of vectors w  1, w  2, w  3+ , w  4+ , w  3− , w  4− . Then the similarity transformation (4.1.28) with matrix (4.6.10) gives for (4.6.8) ⎛

1

⎜0 ⎜ ⎜ ⎜. ⎜ −1 (R) (g1 ) S = ⎜ S T ⎜. ⎜ ⎜ ⎝. ⎛

0

.

.

.

.

1

.

.

.

.

.

q

0

.

.

.

0

q −1

.

. 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

.

.

.

q

.

.

.

.

0 q −1

1

0

.

.

.

.

.

.

0

1

.

1

0

.

.

.

0

.

.

−1

⎜0 −1 ⎜ ⎜ ⎜. . ⎜ S −1 T (R) (r) S = ⎜ ⎜. . ⎜ ⎜ . ⎝. .

.

⎞

.

. ⎟ ⎟ ⎟ . ⎟ ⎟ ⎟. . ⎟ ⎟ ⎟ −1⎠

(4.6.11)

0

So, regular representation (4.6.8) contains two inequivalent 1-dimensional representations T (1) and T (2) and two 2-dimensional equivalent representations T (±) given by (4.6.9). Regular representation of group D3 = S3 contains all inequivalent irreducible representations (see Section 4.6.2). It is now straightforward to find the table of characters of all irreducible representations of group D3 = S3 : classes repres. T (1) (S3 )

(1, 1, 1) = e

T (2) (S3 ) T (+) (S

(2, 1)

(3)

χT (1) = 1

χT (1) = 1

χT (1) = 1

χT (2) = 1

χT (2) = −1

χT (2) = 1

χT (+) = 2

χT (+) = 0

χT (+) = −1

3)

χT (−) = 2

χT (−) = 0

χT (−) = −1

3)

χR = 6

χR = 0

χR = 0

3)

T (−) (S T (R) (S

where (1, 1, 1), (2, 1), (3) denote conjugacy classes in S3 . The numbers in parenthesis refer to lengths of cycles in permutations, i.e., (1, 1, 1) is identical permutation, (2, 1) stands for odd permutations (involving r) and (3) for cyclic permutations g1 and g2 (see (1.1.1), (1.1.3) and Example 7 in Section 1.1.1):  1 2 3 r= = (1)(23), g1 = (123) , g2 = (132) . (4.6.12) 1 3 2

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Symmetric groups Sn , their representations and conjugacy classes will be studied in details in the accompanying book. Let us highlight the fact seen for group D3 = S3 , which is actually valid for all finite groups. Namely, every irreducible m-dimensional representation of finite group G enters its regular representation exactly m times. We prove this property in the next subsection.

4.6.2.

Characters of finite groups and compact Lie groups

All functions in this subsection are complex valued, and all representations of Lie groups are complex. Most of the results studied in this subsection, like the results of Section 4.5, are valid for both finite groups and compact Lie groups. We give proofs for finite groups with understanding that these proofs are extended to compact Lie groups, as described in the beginning of Section 4.5. We still use the term “compact group” when the result is valid for both finite groups and compact Lie groups; we make explicit emphasis on the cases when the result is valid for finite groups only. Presentation of the theory of characters for finite groups and compact Lie groups, which is fairly similar to ours, can be found in book [9]. Let T (ν) : G → GL(Nν , C) be complete set of inequivalent and irreducible representations of finite (or compact) group G, which has order N (volume V ). They act in complex vector spaces Vν of dimensions Nν < ∞; here ν labels inequivalent and irreducible representations. Proposition 4.6.1. The following identity holds for finite groups: 1  (ν) −1 (μ) Tiν ,jν (g ) Tkμ ,mμ (g) N g∈G

=

1 μν δ δiν ,mμ δkμ ,jν , Nν

(kμ , mμ = 1, . . . , Nμ ; iν , jν = 1, . . . , Nν ). (4.6.13)

Similar identity for compact Lie groups reads  1 (ν) (μ) dμ(g) Tiν ,jν (g −1 ) Tkμ ,mμ (g) V g∈G =

1 μν δ δiν ,mμ δkμ ,jν , Nν

(kμ , mμ = 1, . . . , Nμ ; iν , jν = 1, . . . , Nν ), (4.6.14)

where dμ(g) is Haar measure.

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Proof. In the case of finite group, consider two sets of rectangular matrices Akμ ,jν and Biν ,mμ with elements (Akμ ,jν )iν ,mμ =

1  (ν) −1 (μ) Tiν ,jν (g ) Tkμ ,mμ (g) = (Biν ,mμ )kμ ,jν . N g∈G

(4.6.15) Let us prove that for all h ∈ G, these matrices obey T (ν) (h) Akμ ,jν = Akμ ,jν T (μ) (h),

T (μ) (h) Biν ,mμ = Biν ,mμ T (ν) (h). (4.6.16)

To this end, we make use of (4.5.1) and find for all h ∈ G that (ν)

Tkν ,iν (h) (Akμ ,jν )iν ,mμ =

1  (ν) (ν) (μ) Tkν ,iν (h) Tiν ,jν (g −1 ) Tkμ ,mμ (g) N g∈G

=

1  (ν) (μ) Tkν ,jν (hg −1 ) Tkμ ,mμ (gh−1 h) N g∈G

=

1 N



(ν)

(μ)

(μ)

Tkν ,jν (f −1 ) Tkμ ,jμ (f ) Tjμ ,mμ (h)

f =gh−1 ∈G (μ)

= (Akμ ,jν )kν ,jμ Tjμ ,mμ (h).

(4.6.17)

Similarly, (μ)

Tiμ ,kμ (h) (Biν ,mμ )kμ ,jν =

1  (μ) (ν) Tiμ ,mμ (hg) Tiν ,jν ((hg)−1 h) N g∈G

=

1 N

 f =hg∈G

(μ)

(ν)

Tiμ ,mμ (f ) Tiν ,jν (f −1 h) (ν)

= (Biν ,mμ )iμ ,kν Tkν ,jν (h).

(4.6.18)

The matrix forms of these equalities are precisely (4.6.16). Hence, Akμ ,jν and Biν ,mμ are intertwining operators for irreducible representations T (μ) and T (ν) . Then, Schur’s lemma (Lemma 4.4.1) says that if these representations are inequivalent, i.e., μ = ν, then Akμ ,jν = Biν ,mμ = 0; if they are equivalent, μ = ν, then Akμ ,jν and Biν ,mμ are proportional to unit matrices. So, we have (Akμ ,jν )iν ,mμ = (Biν ,mμ )kμ ,jν = λδ μν δiν ,mμ δkμ ,jν ,

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or

9in x 6in

1  (ν) −1 (μ) Tiν ,jν (g ) Tkμ ,mμ (g) = λδ μν δiν ,mμ δkμ ,jν , N

(4.6.19)

g∈G

where λ is a constant which can be found by setting μ = ν, kμ = jν and evaluating the sums of the two sides in (4.6.19) over jν . As a result, making use of the fact that (ν)

(ν)

Tiν ,jν (g −1 ) Tjν ,mν (g) = δiν mν , we find λ = 1/Nν , where Nν is dimension of representation T (ν) . The proof for compact Lie groups is completely analogous. Since we can treat any representation as unitary, we choose orthonormal basis in Vν such that Tiν ,jν (g −1 ) = (Tjν ,iν (g))∗ . Then the result (4.6.13) takes the form of orthogonality relation 1 μν 1  (ν)∗ (μ) Tjν ,iν (g) Tkμ ,mμ (g) = δ δiν ,mμ δkμ ,jν . (4.6.20) N Nν g∈G

Let us introduce scalar product in the space F of functions on compact Lie group 1  ∗ φ (g) ψ(g), ∀φ, ψ ∈ F . (4.6.21) φ, ψ = N g∈G

For characters χν and χμ we have χν , χμ =

1  ∗ χν (g) χμ (g). N

(4.6.22)

g∈G

In the case of compact Lie group, these formulas read  1 φ, ψ = dμ(g) φ∗ (g) ψ(g), (4.6.23) V g∈G  1 dμ(g) χ∗ν (g) χμ (g). χν , χμ = V g∈G Nν (ν) Tii (g) and obtain from We recall the definition of character χν (g) = i=1 (4.6.20) that the characters of irreducible representations are orthonormal, χν , χμ = δ μν .

(4.6.24)

Let χ1 , . . . , χh be characters of all inequivalent and irreducible finitedimensional representations T (1) , . . . , T (h) of compact group G (h is finite

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for finite groups, as we see below, and h is infinite for compact Lie groups). In accordance with Proposition 4.5.1, any finite-dimensional representation T can be partitioned into direct sum of irreducible representations, T = m1 T (1) ⊕ · · · ⊕ mh T (h) , where multiplicities mi are positive integers. Then the character χ of representation T equals χ = m1 χ1 + · · · + mh χh , and orthonormality relation (4.6.24) gives χ, χν = mν , χ, χ =

h 

m2ν .

(4.6.25)

ν=1

The formula χ = m1 χ1 + · · · + mh χh , as well as the series (4.6.25) are well defined in the compact Lie group situation too, when h = ∞, since the representation T is finite-dimensional, and we actually deal with a finite sum. Let G be finite group of order N and T (R) its left regular representation, see Section 4.1.2: gi gk =

N 

(R)

gm Tmk (gi ),

(4.6.26)

m=1 (R)

where Tmk (gi ) = δm,ki for gi gk = gki . Clearly, its character χR is given by χR (e) = N,

χR (g) = 0

(∀g = e),

(4.6.27)

where the second equality is due to the fact that diagonal elements vanish, (R) Tkk (g) = 0, for all g = e. Making use of the first formula in (4.6.25), we calculate multiplicity of every irreducible representation T (ν) in the regular representation T (R) : χR , χν =

1  ∗ 1 1 N χν (e) = Nν . χR (g) χν (g) = χ∗R (e) χν (e) = N N N g∈G

(4.6.28) So, every irreducible Nν -dimensional representation T (ν) enters T (R) precisely Nν times (see discussion of Example 2 in Section 4.6.1). This shows, in particular, that all irreducible representations of finite group G are contained in its regular representation. Once one knows multiplicities (4.6.28), one can write the expansion of χR in the basis of characters χν of h inequivalent irreducible representations, χR = ν=1 Nν χν . These formulas yield a remarkable relation between dimensions Nν of all irreducible

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complex representations T (ν) of finite group G and its order N : ord(G) ≡ N = χR (e) =

h  ν=1

Nν χν (e) =

h 

Nν2 .

(4.6.29)

ν=1

An obvious consequence of this relation is that the number h of all inequivalent irreducible complex representations of finite group G is finite and cannot exceed N = ord(G). We note that the identity (4.6.29) is valid also for all N -dimensional associative algebras (for finite N ). Since Eq. (4.6.29) expresses an integer N in terms of squares of integers Nν , it often enables one to figure out the number and dimensions of inequivalent irreducible representations of finite groups. As an example, irreducible complex representations of Abelian group G are 1-dimensional, so their number is found from (4.6.29) to be equal to ord(G) = N . We emphasize that formulas (4.6.27)–(4.6.29) are valid for finite groups only. Problem 4.6.2. Making use of (4.6.29), find dimensions of all inequivalent irreducible representations of group S3 . Hint: recall that S3 has two 1-dimensional irreducible representations, one of which is trivial, and another assigns −1 and 1 to odd and even permutations, respectively. Definition 4.6.1. Function f on group G is central, if f (h g h−1 ) = f (g) (∀g, h ∈ G). In other words, central function on group G is a function on conjugacy classes in G. For central function f and representation T of compact group G, consider matrix 1  ∗ f (g)Tmk (g) = f , Tmk . (4.6.30) Amk (f ∗ , T ) = N g∈G

Proposition 4.6.2. If representation T = T (ν) is irreducible and has dimension Nν and character χν , then matrix Amk (f ∗ , T ) defined in (4.6.30) is proportional to unit matrix and is given by Amk (f ∗ , T (ν) ) =

1 f, χν δmk . Nν

(4.6.31)

Proof. We make replacement T → T (ν) in (4.6.30) and consider change of variables g → h−1 · g · h to show (since f is a central function)

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(ν)

that Trm (h−1 ) Amk Tkj (h) = Arj for all h. In accordance with Schur’s lemma, we have Amk = λ δmk . Evaluating trace of the latter equality, we obtain (4.6.31). The result (4.6.31) is valid also for compact Lie groups, once the scalar product in (4.6.30), (4.6.31) is understood as integral (4.6.23). Consider the space H of all central functions on G. Clearly, characters {χ1 , . . . , χh } belong to this space. Proposition 4.6.3. Characters {χ1 , . . . , χh } make orthonormal basis in the space H. Proof. In accordance with (4.6.24), functions {χ1 , . . . , χh } make orthonormal system in H. We have to show that this system is complete, i.e., any central function f ∈ H, orthogonal to all characters χν , vanishes. Let us form matrix Amk (f ∗ , T ), defined in Eq. (4.6.30), for such a function. It vanishes for any irreducible representation T in view of (4.6.31), since f is orthogonal to character χT . Since any representation, including regular, of finite group can be partitioned into direct sum of irreducible (R) representations, we conclude that Amk (f ∗ , T (R) ) = f , Tmk = 0 also for regular representation (4.6.26). This in turn means that for all gk ∈ G one has ⎛ ⎞ N  1 ⎝ 1  ∗ (R) f ∗ (g) g⎠ · gk = f (g) Tmk (g) gm N N m=1 g∈G

g∈G

=

N 

(R)

f, Tmk gm = 0.

(4.6.32)

m=1

 ∗ This shows that element g∈G f (g) g of group algebra C[G] vanishes, and since elements g ∈ G make basis in C[G], then f ∗ (g) = 0, and hence f (g) = 0 for all g ∈ G. This proposition is valid for compact Lie groups as well. Problem 4.6.3. Extend this proof to compact Lie groups. Hint: make use of definitions of group algebra and regular representation of compact Lie group given above in Section 4.5.

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To summarize, any central function f ∈ H can be expanded in a sum over characters of inequivalent and irreducible representations, f (g) =

h 

cν χν (g).

(4.6.33)

ν=1

Recall that group elements g and g  are conjugate if there is an element h of group G, such that g  = hgh−1 . Group G is partitioned into cosets C1 , . . . , Ck (conjugacy classes), each combining mutually conjugate elements. Proposition 4.6.4. The number k of all conjugacy classes of finite group G is equal to the number h of inequivalent irreducible representations of G. Proof. Consider an arbitrary central function f ∈ H. It is constant in each conjugacy class, Cm , i.e., f (Cm ) = λm . Thus, f is determined by k constants {λ1 , . . . , λk }. This means that the dimension of space H equals k. On the other hand, in accordance with Proposition 4.6.3, dimension of H equals the number of independent characters χ1 , . . . , χh , and this number coincides with the number of inequivalent irreducible representations of G. Hence, k = h. Problem 4.6.4. Order of subgroup of even permutations A4 ⊂ S4 equals 12. Making use of the fact that number of conjugacy classes in A4 is 4 (show that by decomposing even permutations into products of cycles, see Proposition 1.1.3) and formula (4.6.29), find the number of all inequivalent irreducible representations of group A4 . Let central function fg be equal to 1 on conjugacy class Cg of element g  ∈ G and equal to zero otherwise. This function can be decomposed into h sum over characters, fg = ν=1 λν χν , where 1  ∗ c(g  ) ∗  λν = χν , fg = χ (g ), χν (g) fg (g) = N N ν g∈Cg



and c(g ) = dim(Cg ). Hence, for any g ∈ G we have  h h  1 c(g  )  ∗  fg (g) = λν χν (g) = χν (g ) χν (g) = N ν=1 0 ν=1

for g ∈ Cg , for g ∈ / Cg . (4.6.34)

The latter identity is another form of orthonormality of characters χν (g) (cf. (4.6.24)).

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Problem 4.6.5. Construct all inequivalent irreducible representations of group Cn = Zn . Check explicitly the validity of (4.6.22) and (4.6.34) for characters of these representations and describe the space of central functions on Cn . Remark 1. Character χR of right (left) regular representation of Lie group G is defined as distribution on group with support at unit element e ∈ G,  1 (4.6.35) dμ(g) χR (g) f (g) = f (e), V where f (g) is an arbitrary test function on G. Formula (4.6.35) is a natural generalization of definition (4.6.27) of character χR for finite group. Then the relation (4.6.28), which we have proven for finite groups, is straightforwardly translated to compact Lie groups,  1 dμ(g) χR (g) χν (g) = χν (e) = Nν . χR , χν = V Thus, we come to the following proposition (see also [4]). Proposition 4.6.5. Regular representation T (R) of compact Lie group G (or finite group) contains all irreducible finite-dimensional representations T (ν) of group G, and multiplicity of T (ν) in T (R) equals the dimension of T (ν) . This proposition is closely related to the following proposition which we give here without proof. Proposition 4.6.6. Any irreducible representation of compact Lie group G in Hilbert spaced is finite-dimensional and equivalent to a subrepresentation of right (left) regular representation of G. In view of this proposition and since the dimension of regular representation of compact Lie group is infinite, Eq. (4.6.29) shows that compact Lie group has infinite number of finite-dimensional inequivalent irreducible representations. Proposition 4.6.4 and formula (4.6.34) were proven for finite groups. Since compact Lie groups have infinitely many irreducible representations, d A Hilbert space is a complex vector space H with a Hermitian product f, g (∀f, g ∈ H)

such that the norm defined by |f | = f, f  turns H into a complete metric space.

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Proposition 4.6.4 is pointless for compact Lie groups (except for nearly evident statement that Lie group has infinitely many conjugacy classes). Remark 2. Left and right regular representations (4.5.12) and (4.5.13) of compact Lie group G are reducible and unitary (see Problem 4.5.3). In accordance with Propositions 4.6.5 and 4.6.6 these representations are decomposed into direct sums of finite-dimensional inequivalent and irreducible representations of G. Due to this fact one can prove the important theorem by F. Peter and G. Weyl. Theorem 4.6.1 (Peter–Weyl). Let {T (ν) } be system of all finitedimensional inequivalent and irreducible representations of compact Lie group G (where ν labels the representations). Then any continuous function (ν) f ∈ L2 (G, dμ) on G can be expanded in a series in matrix elements Tjk of these representations:  (ν) Cνjk Tjk (g). (4.6.36) f (g) = ν,j,k (ν)

This theorem tells that matrix elements Tjk make complete system of functions in space L2 (G, dμ). We do not give exhaustive proof of Peter–Weyl theorem here. The idea of the proof is to make use of the generalization of orthonormality property (4.6.20) to compact Lie groups (see (4.6.14)):  1 μν 1 (ν)∗ (μ) δ δiν ,mμ δkμ ,jν . (4.6.37) dμ(g) Tjν ,iν (g) Tkμ ,mμ (g) = V Nν If function f admits expansion (4.6.36), then it is unambiguously determined by its expansion coefficients, since Cνjk are obtained from (4.6.37). It (ν) remains to prove that any function f (g) orthogonal to all vectors Tjk (g)  (ν) (i.e., dμ(g) f ∗ (g) Tjk (g) = 0, for all ν, j, k) vanishes. The latter fact is a consequence of the statement that regular representation of compact Lie group can be partitioned into direct sum of representations T (ν) (compare with proof of Proposition 4.6.3). 4.6.3.

Irreducible representations and characters of SO(2) = U (1)

As a simple application of character theory, we consider compact group SO(2) = U (1). Group U (1) acts on functions on this group by translation, ρ(Oθ ) · f (φ) = f (φ − θ). Hence, functions on U (1) are periodic, f (φ) = f (φ + 2π). Since compact group U (1) has infinite order and is Abelian, its

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regular representation is infinite-dimensional and is partitioned into infinite number of 1-dimensional inequivalent and irreducible representations ρ(n) labeled by integer n: ρ(n) (gφ ) = einφ ,

∀n ∈ Z.

(4.6.38)

Clearly, characters of representations (4.6.38) equal χn (φ) = einφ . Character theory says that χn (φ) = einφ make complete system of central functions on U (1). Since U (1) is Abelian, all (periodic) functions on it are central. Thus, any periodic function f (φ) can be expanded in a series in characters (see (4.6.33)): f (φ) =

∞ 

cn einφ .

(4.6.39)

n=−∞

This is nothing but Fourier series on a circle. The result (4.6.39) reiterates the obvious fact that representations (4.6.38) are the only complex inequivalent and irreducible representations of SO(2) = U (1). Orthonormality property of characters, Eq. (4.6.24) is nothing but the property known from the theory of Fourier series:  2π  2π 1 1 eimφ (einφ )∗ dφ = eimφ e−inφ dφ = δmn . 2π 0 2π 0 The latter formula involves invariant integral over group U (1) with measure dφ; 2π is interpreted as volume of U (1). Measure dφ is invariant, since it does not change under translation φ → φ − θ. To end up this section, we note that since irreducible representations (4.6.38) are 1-dimensional, expansion (4.6.39) can be viewed also as an example of the result of Peter–Weyl theorem (Theorem 4.6.1). 4.7.

4.7.1.

Universal Enveloping Algebra. Casimir Operators, Yangians Definition of universal enveloping algebra U(A) of Lie algebra A

Let T be representation of Lie algebra A. Consider operators Ta ≡ T (Xa ), where Xa are generators of A obeying defining commutation relations (3.2.30). Operators Ta are dubbed in physics literature as generators in

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representation T ; we sometimes follow this nomenclature. Any element A of algebra A is a linear combination of generators Xa , and any operator T (A) is a linear combination of operators Ta . Irrespectively of representation considered, the commutation relations for operators Ta have one and the same form: c Tc , [Ta , Tb ] = Cab

(4.7.1)

c where Cab are structure constants of algebra A. While the only multiplication operation in Lie algebra A is commutator [A, B], operators T (A) and T (B) can be multiplied, T (A) · T (B) (recall that product of operators is their consecutive action). Important objects in representation theory of Lie algebras are polynomials in operators Ta . However, these are not images of any objects existing in Lie algebra, as similar polynomials cannot be defined in Lie algebra itself. We see that representation T associates to algebra A an algebra of operators UT (A) composed of all products of operators Ta and their linear combinations. All algebras UT (A), for various representations T , are associative and have the property (4.7.1). The point is that they can be defined as representations of one and the same universal associative algebra U(A), which is unambiguously determined by Lie algebra A and called universal enveloping algebra. Common properties of representations T and their algebras UT (A) are encoded in the properties of enveloping algebra U(A). The definition of this universal algebra U(A) is the topic of this section. Given Lie algebra A over field K with generators Xa , one constructs new infinite-dimensional associative algebra U(A) over field K, whose basis elements are ordered formal combinations

X a1 · X a2 · X a3 · · · X ak

(a1 ≤ a2 ≤ · · · ≤ ak )

(4.7.2)

(property a1 ≤ a2 ≤ · · · ≤ ak is important!). These are have arbitrary lengths k = 0, 1, 2, 3, . . . . Zero length element is denoted by I. Any element of algebra U(A) is a linear combination, with coefficients from K, of basis elements {I, Xa , Xa · Xb , Xa · Xb · Xc , . . .} ,

(a ≤ b ≤ c ≤ . . . ).

(4.7.3)

Multiplication in algebra U(A) is defined as follows: I · (Xa1 · Xa2 · · · Xak ) = (Xa1 · Xa2 · · · Xak ) · I = Xa1 · Xa2 · · · Xak , (4.7.4)

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(Xa1 · Xa2 · · · Xak ) · (Xb1 · Xb2 · · · Xbr ) = Xa1 · Xa2 · · · Xak · Xb1 · Xb2 · · · Xbr .

(4.7.5)

This operation is evidently associative. However, to make sure that element on the right-hand side of (4.7.5) again belongs to U(A), it has to be cast in the ordered form (4.7.2). To this end one endows U(A) with the following relation: c Xc ≡ [Xa , Xb ], Xa · Xb − Xb · Xa = Cab

(4.7.6)

c where Cab are structure constants of Lie algebra A. The latter relation enables one to transpose neighboring elements on the right-hand side of (4.7.5), and in this way to express it as a linear combination of basis elements (4.7.3). This procedure defines the infinite-dimensional associative algebra U(A) for given Lie algebra A. We emphasize that multiplication in U(A) is not directly related to multiplication (commutation) in the original algebra A. The fact that algebra U(A) has to do with Lie algebra A is entirely due to relations (4.7.6).

Definition 4.7.1. Infinite-dimensional associative algebra U(A) with basis (4.7.3), multiplication (4.7.4), (4.7.5) and structure relations (4.7.6) is universal enveloping algebra of Lie algebra A. Thus, universal enveloping algebra U(A) is a linear space of all linear combinations of monomials Xa1 · Xa2 · Xa3 · · · Xak , where linear combinations that are related to each other by commutation (4.7.6) are identified. As an example, for A, B, C, D ∈ U(A) we have A · B · C · D = A · C · B · D + A · [B, C] · D = A · C · B · D + A · D · [B, C] + A · [[B, C], D],

(4.7.7)

etc. The multiplication operation (4.7.5) in this space simply glues monomials together, (A · B · C) · (D · E) = A · B · C · D · E,

(4.7.8)

while multiplication of their linear combinations amounts to opening parentheses, (αA · B + βC · D) · (γE · F + δG) = αγ A · B · E · F + αδ A · B · G + βγ C · D · E · F + βδ C · D · G, (4.7.9)

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where α, β, γ, δ ∈ K. These properties define the universal covering algebra completely. Let us repeat the reasoning behind introducing the universal covering algebra. Consider representation T of algebra A and all polynomials of operators Ta . As we have already discussed, these polynomials make an associative algebra UT (A). Properties of this algebra valid irrespectively of the concrete choice of representation T are the same as properties of algebra U(A). In other words, every representation T of Lie algebra A is a representation of associative algebra U(A), i.e., it is homomorphic mapping of U(A) to UT (A). Study of algebra U(A) enables one to figure out common properties of all representations of Lie algebra A, see Section 4.7.2 in this regard. A comment is in order. Elements like A · B of enveloping algebra must be understood in a formal way, assuming that nothing is known about the original Lie algebra besides its defining commutation relations. This is true, e.g., for matrix Lie algebras whose structure is explicit. As an example, Pauli matrices (2.1.18) that make basis in algebra su(2) (see Section 3.3.1) obey σ{i σj} =

1 1 (σi σj + σj σi ) = δij σk σk . 2 3

This relation, however, cannot be considered valid for generators of U(su(2)), since it is satisfied only in defining representation of su(2) and does not hold in other representations. Likewise, covering algebra does not respect the relation σ{i σj} = δij which is not valid for trivial representation of su(2). 4.7.2.

Representations of U(A). Center of U(A) and Casimir operators

Since there is homomorphic embedding of Lie algebra A to its universal covering algebra U(A) (A is subalgebra of U(A) consisting of monomials with algebraic operation in commutator form (4.7.6)), every representation T of covering algebra U(A) defines representation of Lie algebra A by mapping of its generators, Xa → T (Xa ). On the other hand, every representation T of Lie algebra A defines representation T of algebra U(A). This is homomorphic mapping U(A) → UT (A) defined on basis elements (4.7.3) of U(A) by T (Xa1 · Xa2 · · · Xar ) = T (Xa1 ) · T (Xa2 ) · · · T (Xar ) = Ta1 · Ta2 · · · Tar . (4.7.10)

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This is indeed representation of U(A), since operators Ta ≡ T (Xa ) obey (4.7.6) with substitution Xa → Ta , while algebra of operators UT (A) has properties (4.7.7), (4.7.8), (4.7.9) (with replacement A → T (A), B → T (B), etc.). Let C be an element of universal enveloping algebra U(A) which commutes with all generators Xa of Lie algebra A, and hence with all elements of algebra U(A). In other words, element C belongs to center Z of enveloping algebra. Then [T (C), T (A)] = 0 for all A ∈ U(A) and all representations T of algebra U(A). If T is complex and irreducible representation, then, by Schur’s lemma, operator T (C) is proportional to unit operator T (C) = λ · I,

(4.7.11)

where the value of λ depends on the choice of C and, for given C, is an important parameter of the representation T . If the representation T is reducible, then operator T (C) may have several eigenvalues λ, and all subspaces Ker(T (C) − λ · I) are invariant subspaces. Definition 4.7.2. Elements of center Z of universal enveloping algebra U(A), which comprise the minimal set of generators of Z, are called Casimir operators of Lie algebra A. We recall that a set of generators of an algebra Z consists of elements which are sufficient for constructing all other elements of Z by applying algebraic operations of multiplication, addition and multiplication by number, and the minimal set of generators contains minimum number of elements. So, Casimir operators make the minimal set of independent elements which commute with all elements of U(A). One can show that for simple Lie algebra, the number of Casimir operators is equal to the rank of this algebra, see Proposition 4.7.2 below and definition of rank of Lie algebra in Section 6.1. This fact is closely related to the statement that the set of eigenvalues {λ} of Casimir operators unambiguously characterizes complex irreducible finite-dimensional representation of simple Lie algebra. Therefore, the spectrum of Casimir operators is of considerable importance in Lie algebra representation theory. In the rest of this section we consider (unless the opposite is stated explicitly) simple Lie algebras A. Many propositions of this section can be translated to semisimple Lie algebras too. The simplest Casimir operator of algebra A is quadratic: C2 = gab Xa Xb ,

(4.7.12)

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where gab are elements of inverse matrix of Killing metric (3.2.62) in basis Xa . d Problem 4.7.1. Making use of antisymmetry of Cabc = gcd Cab in all indices, show that C2 = gab Xa Xb commutes with all operators of simple Lie algebra.

Problem 4.7.2. Show that quadratic Casimir operator (4.7.12) is independent of the choice of basis in A (in particular, independent of normalization of generators Xa ). Problem 4.7.3. Show that there are no first-order Casimir operators (operators linear in Xa ) for simple Lie algebra. This construction, for Lie algebra su(2), is well known in quantum mechanics: operator 2C2 is the operator of spin squared, 2C2 = −τα τα = J 2 ,

(4.7.13)

where τα are basis elements of algebra su(2), which obey (see (3.3.12)) [τα , τβ ] = εαβγ τγ .

(4.7.14)

Killing metric (3.2.62) is obtained from (4.7.14) and equals gαβ = εαδγ εβγδ = −2δαβ . Hermitian generators i τα are interpreted as components of spin. We already know three representations of su(2): trivial 1-dimensional representation T0 such that T0 (τα ) = 0; defining 2dimensional representation (3.3.11) with generators given by Pauli matrices, T1/2 (τα ) = −iσα /2;

(4.7.15)

and 3-dimensional adjoint representation T1 , which, according to (4.7.14), has the following matrices: [T1 (τα )]βγ = ad(τα )βγ = −εαβγ .

(4.7.16)

The result (4.7.11) implies that spin squared J 2 , defined in (4.7.13), is the same for all vectors (states) of irreducible representation of su(2) and is equal to λ. This parameter is conveniently written as λ = j(j + 1), where the real number j ≥ 0 is spin. Finite-dimensional irreducible representation of su(2) algebra is unambiguously characterized by integer or half-integer spin j (see Section 4.7.3 and accompanying book).

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Problem 4.7.4. Show that the defining and adjoint representations of su(2), given by Eqs. (4.7.15) and (4.7.16), obey T1/2 (J 2 ) = 34 I2 and T1 (J 2 ) = 2 I3 , i.e., j = 12 and j = 1, respectively. Let us come back to the general case and consider the construction of all Casimir operators of a simple Lie algebra A. In the first place, it is convenient to introduce basis in enveloping algebra U(A), different from basis (4.7.3). Namely, the new basis is made of symmetric homogeneous polynomials X{a1 Xa2 . . . Xak } , where curly brackets denote total symmetrization, 1  X{a1 Xa2 . . . Xak } = Xaσ(1) · Xaσ(2) · · · Xaσ(k) . (4.7.17) k! σ Here the sum runs over all permutations σ ∈ Sk . Symmetric polynomial of zeroth order is unit element I. The statement that symmetric polynomials (4.7.17) make basis in U (A) is proven by induction in k. For k = 1, elements (4.7.17) are original generators Xa , which make basis in the subspace of first-order polynomials in U (A). Consider now k > 1 and assume that any polynomial in U (A) of order (k − 1) can be symmetrized, i.e., can be written as a linear combination of symmetric polynomials (4.7.17) of order m ≤ k−1. Consider monomial Xa1 · Xa2 · · · Xak of order k. It can be symmetrized by employing the relation (4.7.6), so that this monomial becomes a sum of symmetric polynomial (4.7.17) and monomials of order (k − 1) and lower. The latter emerge when applying commutation operation (4.7.6); they can be symmetrized by assumption. So, any homogeneous polynomial of order k is a linear combination of symmetric polynomials (4.7.17) of order k and lower. Symmetric polynomials X{a1 Xa2 . . . Xak } with different sets of indices are linearly independent and cannot be reduced to each other and symmetric polynomials of lower order by applying operation (4.7.6). This completes the proof.

Commutation of symmetric polynomial X{a1 Xa2 . . . Xak } with generators Xb gives a linear combination of symmetric polynomials with the maximum order again equal to k. Therefore one conjectures that Casimir operators are linear combinations of symmetric polynomials X{a1 Xa2 . . . Xak } , all having the same order k. This is indeed the case. As an example, cubic Casimir operator reads C3 = dabc Xa · Xb · Xc ,

(4.7.18)

where numerical coefficients dabc are totally symmetric in their indices. Let us figure out the properties of dabc . Let Lie algebra A correspond to Lie group G, i.e., A = A(G) (we show in Section 4.7.4 that this assumption can be dropped). We define in Section 4.1.1, item 5, adjoint representation of group G. It acts in space A = A(G) as follows: ad(g) · Xa = Xb (ad g)b a ,

(4.7.19)

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where (ad g)ab is the matrix of adjoint representation in basis Xa . This representation is extended to the representation of group G in the entire enveloping algebra U(A); the key ingredient is ad(g) · (Xa1 · · · Xak ) = (Xb1 · · · Xbk ) (ad g)b1a1 · · · (ad g)bkak .

(4.7.20)

Element C ∈ U(A) is central, [C, A] = 0,

A ∈ A(G),

(4.7.21)

iff it is invariant: ad(g) · C = C,

(4.7.22)

since (4.7.21) is equivalent to (4.7.22) for g = et A , where t is small parameter. We note that the definition (4.7.19) and condition (4.7.22) are explicit for matrix groups, g · Xa · g −1 = Xb (ad g)ba ,

g · C · g −1 = C.

(4.7.23)

Until now we have not specified to cubic Casimir operator. Let us now come to this case. In accordance with (4.7.20), adjoint action of G on cubic Casimir operator is ad(g) · C3 = (ad g)da (ad g)eb (ad g)fc dabc Xd Xe Xf . So, invariance of C3 is equivalent to the requirement (ad g)da (ad g)eb (ad g)fc dabc = ddef .

(4.7.24)

Symmetric tensor dabc can be viewed as an element of tensor product of three co-adjointe representations. Then the requirement (4.7.24) means that this tensor is invariant under co-adjoint action of group G. This tensor can be constructed by making use of a faithful irreducible representation T of simple Lie algebra A(G) and its Killing metric gad :   (4.7.25) dabc = gad gbe gcf Tr T{d Te Tf } . The fact that tensor (4.7.25) obeys (4.7.24) is demonstrated below, when we consider more general tensors da1 a2 ...ak of rank k ≥ 2 (see Proposition 4.7.1). e Recall

that co-adjoint and adjoint representations are equivalent for simple and semisimple Lie groups and algebras, see Problem 4.2.3.

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Casimir operators of higher orders (if exist) are constructed in an analogous way. They are independent invariant elements of the form Ck = da1 ...ak Xa1 · · · Xak ,

(4.7.26)

where da1 ...ak are rank k symmetric tensors. The invariance of Ck is equivalent to invariance of tensor da1 ...ak under co-adjoint action of group G (Ad-invariance): (ad g)ba11 (ad g)ba22 · · · (ad g)bakk da1 a2 ...ak = db1 b2 ...bk , ∀g ∈ G.

(4.7.27)

This generalizes (4.7.24). Note that if tensor da1 ...ak in (4.7.26) were not symmetric, the requirement of Ad-invariance (4.7.27) would be sufficientonly condition for invariance of Ck . Problem 4.7.5. Let elements g ∈ G in (4.7.27) be close to unity, i.e., g = etA , where t is small and A ∈ A(G). Show that the requirement (4.7.27) means invariance of da1 a2 ...ak under co-adjoint action of Lie algebra A(G): (ad A)ba11 da1 b2 ...bk + (ad A)ba22 db1 a2 b3 ...bk + · · · + (ad A)bakk db1 ...bk−1 ak = 0. (4.7.28) Show directly that the requirements [A, Ck ] = 0 and (4.7.28) are equivalent. Symmetric Ad-invariant tensor da1 ...ak in (4.7.26) can be chosen as follows (cf. (4.7.25)): da1 ...ak = ga1 b1 · · · gak bk Tr(T{b1 · · · Tbk } ),

Ta = T (Xa ),

(4.7.29)

where T is faithful representation (see Proposition 4.7.1 below). One convenient choice here is adjoint representation, T = ad. Then operators Ck given by (4.7.26) are independent of the choice of basis Xa ; in particular, they are independent of the normalization of generators Xa . In the case of matrix algebra A(G), another choice is defining representation. In that case one has to keep track of the normalization of generators Ta when calculating the values of the Casimir operators Ck . Proposition 4.7.1. Tensor (4.7.29) satisfies Ad-invariance condition (4.7.27). Proof. We show that Ad-invariance (4.7.27) holds for nonsymmetrized tensor ga1 ...ak = ga1 b1 · · · gak bk Tr(Tb1 · · · Tbk ),

Tb ≡ T (Xb );

(4.7.30)

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this ensures the invariance of any linear combination of tensors (4.7.30) with arbitrarily ordered indices (a1 , . . . , ak ), including symmetric combination (4.7.29). To this end, we have (ad g)da11 · · · (ad g)dakk ga1 a2 ...ak = Tr (Tb1 · · · Tbk ) (ad g −1 )ba11 · · · (ad g −1 )bakk ga1 d1 · · · gak dk = Tr((T (g −1 )Ta1 T (g)) · (T (g −1 )Ta2 T (g)) · · · (T (g −1 )Tak T (g))) ×ga1 d1 · · · gak dk = Tr(T (g −1 )Ta1 · · · Tak T (g)) ga1 d1 · · · gak dk = gd1 d2 ...dk , where the first equality makes use of (4.1.19) written for inverse metric, the second equality employs (4.7.23), and the last equality uses the cyclic property of trace. Problem 4.7.6. Considering Lie algebra only (without reference to Lie group) and making use of (4.7.28), show that polynomial (4.7.26) with constants (4.7.29) is Casimir operator. A question that naturally arises is whether any Ad-invariant symmetric tensor da1 ...ak can be written as symmetric combination of tensors (4.7.30), like in (4.7.29). Affirmative answer is given by the following proposition. Proposition 4.7.2. For simple Lie algebra A of rank r, there are r elements (4.7.26), (4.7.29), which make basis in center Z ⊂ U(A). In other words, for simple algebra of rank r, the set of Casimir operators has one quadratic, one cubic, etc., one polynomial of order (r + 1). Each of them has the form (4.7.29). Generalization of this proposition to semisimple Lie algebras is evident. Its complete proof, however, is beyond the scope of this book, see, e.g., [10, 11]. We substantiate the claim of Proposition 4.7.2 by considering simple matrix Lie algebras. In accordance with the above discussion, any central element is a polynomial in invariants P (Xa ) = da1 ...ak Xa1 · · · Xak , where da1 ...ak are symmetric Ad-invariant tensors. Therefore, the description of the generators of center Z is reduced to classification of all independent symmetric invariant tensors da1 ...ak . These tensors can be thought of as independent symmetric polynomials P (ua ) = da1 ...ak ua1 · · · uak , where parameters ua transform under adjoint representation of group G. Polynomials P (ua ), in their turn, are conveniently treated as functions of matrices u = ua gab Xb ∈ A(G), so that these functions P (u) must be invariant under adjoint action, u → g · u · g −1 , ∀g ∈ G.

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Adjoint transformation of u ∈ A(G) can be chosen in such a way that this element takes the form r  vk Hk , (4.7.31) u= k=1

where Hk are diagonal generators of A(G) (generators of Cartan subalgebra, whose number is equal to the rank of A(G), see Section 6.1). There are still transformations u → g · u · g −1 , which keep diagonal form (4.7.31) but permute parameters vk (these transformations form Weyl group). Because of Ad-invariance, function P (u) is symmetric polynomial in variables {v1 , . . . , vr }. It is known that algebra of symmetric polynomials of r variables {v1 , . . . , vr } has r generators in the form of power sums Pα (v1 , . . . , vr ) =

r  k=1

vkα ,

α = 1, . . . , r.

Every power sum Pα (v1 , . . . , vr ) defines invariant polynomial Pα (ua ), which, in turn, defines invariant element Pα (Xa ) ∈ Z. These invariant elements generate center Z ⊂ U (A). Thus, the number of independent generators of center Z ⊂ U (A) is equal to rank r of Lie algebra A. Now, it is clear that the variables vk are related to eigenvalues of matrix u, while symmetric functions of eigenvalues are expressed through Tr(uk ). The latter correspond precisely to invariant elements (4.7.26), (4.7.29), where T is adjoint representation. One concludes that any generator of center Z in U (A) can be constructed out of central elements Ck given in (4.7.26), (4.7.29). We note that we have described here isomorphism of symmetric polynomial algebra and center Z of enveloping algebra U (A). This is known as Harish-Chandra isomorphism.

Algebra su(2) has only one, quadratic Casimir operator. This follows from the fact that in su(2) case, all Ad-invariant symmetric tensors da1 ...ak of odd rank k vanish, while they are expressed through gab for even rank. Lie algebra su(3) has not only quadratic, but also cubic Casimir operator (the number of Casimir operators for su(3) equals rank(su(3)) = 2). Expression for the third rank invariant symmetric tensor (4.7.25), used in physics, reads (see (3.3.29)) dijk = 12 Tr(λ{i λj λk} ),

(4.7.32)

where λi are Gell-Mann matrices (3.3.25), (3.3.26). Modulo factor i, the latter are generators in defining representation. The components dijk for su(3) are listed in (3.3.30). Since there is only one cubic Casimir operator in su(3), we have for any irreducible representation T of this algebra that Tr(T{i Tj Tk} ) = c3 · dijk ,

(4.7.33)

where numerical parameter c3 is a characteristic of representation T . The fact that there is only one quadratic Casimir operator for any simple Lie algebra, can be proven very directly.

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Proposition 4.7.3. There is only one quadratic Casimir operator for any simple complex Lie algebra, as well as for all its real forms. Proof. Let C2 = dab Xa Xb be a quadratic Casimir operator of algebra A. Then dab is invariant tensor, obeying (4.7.28) with k = 2: ad(A)ac dcb = −dac ad(A)bc ⇒ ad(A) · d = −d · ad(A)T .

(4.7.34)

Inverse of Killing metric, g−1 = ||gab ||, entering the definition of quadratic Casimir operator (4.7.12), also obeys ad(A) · g−1 = −g−1 · ad(A)T ⇒ g · ad(A) = −ad(A)T · g.

(4.7.35)

We multiply (4.7.34) by matrix g on the right and make use of the second formula of (4.7.35) to obtain ad(A) · (d · g) = (d · g) · ad(A).

(4.7.36)

Hence, matrix d · g = ||dac gcb || commutes with all matrices ||ad(A)ab || of adjoint representation. The latter representation is complex and irreducible (see Proposition 4.4.3), so, by virtue of Lemma 4.4.1, the product d · g is proportional to unit matrix, d · g = λ I, or dab = λ gab . This proves the uniqueness of the quadratic Casimir operator (4.7.12) for complex simple Lie algebra. Adjoint representation of any real form of simple complex Lie algebra A is also irreducible (see Proposition 4.4.3). Furthermore, this representation is irreducible as complex representation, otherwise A would not be simple, see Corollary 4.4.4. Therefore, we have d · g = λ I for any real form of algebra A, which ensures uniqueness of quadratic Casimir operator not only for simple complex algebra, but also for all its real forms. Corollary 4.7.1. For simple complex Lie algebra A and all its real forms, there is one and only one invariant (obeying (4.7.34)) second rank tensor dab , and this tensor is inverse of Killing metric, gab . Accordingly, there is one and only one invariant tensor (d−1 )ab , which is Killing metric gab of algebra A. Let us write the quadratic Casimir operator (4.7.11) in the following form: gab Ta Tb = C2T I(T ) ,

(4.7.37)

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where T is complex irreducible representation of simple Lie algebra A, I(T ) is unit matrix, and C2T is the value of the Casimir operator C2 in representation T . We are going to find C2T for defining and adjoint representations of su(n). We perform the calculation for defining representation with normalization (3.3.21) used in physics for anti-Hermitian generators (f ) Ta of su(n). In this basis, the Killing metric is gab = −nδab , so, using (f ) (f ) Hermitian generators Ta → iTa , we have gab = nδab , gab =

1 ab δ . n

(4.7.38)

Problem 4.7.7. Show that Killing metric for su(n) in basis (3.3.18) with normalization (3.3.21) is gab = −nδab . Hint: calculate coefficient λ in gab = p d Cbd = λ δab by setting b = a and choosing index a in such a way that the Cap d . generator Ta is diagonal; then make use of (3.3.23) to calculate Cap In accordance with (4.7.37), the value C2T of Casimir operator C2 for any irreducible representation T of su(n) is given by 1 Ta · Ta = C2T I(T ) n

(4.7.39)

(summation over a is assumed). We have for defining (fundamental) representation T (f ) ≡ [n] 1 (f ) (f ) [n] T · Ta = C2 · In , n a (f )

where generators Ta are given in (3.3.20). Since the dimension of (f ) (f ) [n] representation [n] equals n, we have Tr(Ta · Ta ) = C2 n2 , and (3.3.21) (f ) (f ) gives Tr(Ta · Ta ) = (n2 − 1)/2. We have finally [n]

C2 =

n2 − 1 . 2n2

(4.7.40)

[n]

Note that this formula gives 2C2 = 3/4 for n = 2, in agreement with the result of Problem 4.7.4. Quadratic Casimir operator takes the same value (4.7.40) in complex conjugate to defining representation (antifundamental representation) [¯ n]. The value of quadratic Casimir operator C2 in adjoint representation can be calculated for any simple Lie algebra. To this end, we write the formula (4.7.37) as (ad)

gab ad(Xa ) · ad(Xb ) = C2

I(ad) ,

(4.7.41)

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and evaluate the traces of both sides. We recall the definition of Killing metric (4.2.18) and find for any simple Lie algebra (ad)

C2

= 1.

(4.7.42)

This result is valid, in particular, for su(n); it agrees, in view of (4.7.13), with the value of T1 (J 2 ) for su(2) calculated in Problem 4.7.4. Problem 4.7.8. Give alternative derivation of the result (4.7.42) for su(n), which employs the fact (4.4.19) that tensor product of defining and antifundamental representations is partitioned into direct sum of (complexified) adjoint representation [n2 − 1] and singlet representation [1]. Recall that according to Proposition 4.7.3, there is a single quadratic Casimir operator for simple complex Lie algebra A and any of its real forms. Let us show that for any irreducible representation T of A one has Tr(Ta Tb ) = gab ·

C2T · dim(T ) , dim(A)

(4.7.43)

where dim(T ) is dimension of (the space of) representation T . Indeed, in addition to the relation (4.7.37) we have (invariant metric is unique, modulo numerical factor) Tr(Ta Tb ) = c2 (T ) · gab ,

(4.7.44)

where the constant c2 (T ) depends on the choice of representation T . We contract this with gab and make use of (4.7.37) to obtain C2T · dim(T ) = c2 (T ) · dim(A), so that c2 (T ) =

C2T · dim(T ) , dim(A)

(4.7.45)

as desired. For defining representation we recall (4.7.40) and get c2 ([n]) =

1 1 ⇒ Tr(Ta Tb ) = δab , 2n 2

which agrees with the standard normalization of operators Ta .

(4.7.46)

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Problem 4.7.9. Making use of defining relations (3.3.41) and (3.3.48), rs find structure constants Cij, k of Lie algebras so(p, q) and sp(2r, K) and calculate their Killing metrics. Show that quadratic Casimir operators (4.7.12) for so(p, q) and sp(2r, K) read (so)

C2

=−

1 Lab Lab , 4(p + q − 2) 

(sp)

C2

=−



1 Mab M ab , 4(2r + 2) 

(4.7.47)



where Lab = η aa η bb La b and M ab = J aa J bb Ma b . Show that their values in defining (fundamental) representations of so(p, q) and sp(2r, K) are [p+q]

C2

=

(p + q − 1) , 2(p + q − 2)

[2r]

C2

=

(2r + 1) . 4(r + 1)

Problem 4.7.10. Show that elements Ck = η bk a1 La1 b1 η b1 a2 La2 b2 η b2 a3 · · · Lak−1 bk−1 η bk−1 ak Lak bk ,

(4.7.48)

Ck

(4.7.49)

=J

bk a1

Ma1 b1 J

b1 a2

Ma2 b2 J

b2 a3

· · · Mak−1 bk−1 J

bk−1 ak

Mak bk

are central in enveloping algebras U(so(p, q)) and U(sp(2r, K)), respectively. Show that any permutation σ of generators Lap bp in Ck and Map bp in Ck leaves the elements σ · Ck and σ · Ck central (Hint: show that σ · Ck and σ · Ck are invariant under adjoint action of groups SO(p, q) and Sp(2r, K)). Show that the central elements σ · Ck and σ · Ck are linear combinations of the Casimir operators Cq and Cq with q ≤ k, i.e., the permutation of generators does not introduce new Casimir operators. Problem 4.7.11. Consider operators (4.7.48), (4.7.49). Show that elements Ck and Ck with odd k are expressed through Cr and Cr with even r < k (Hint: make use of defining commutation relations (3.3.41), (3.3.48) and symmetry properties of Mab , ηab , Lab and Jab ). Hence, Casimir operators for groups SO(p, q) and Sp(2r, K) are monomials of even order in their generators. We note that in physics nomenclature, the term “quadratic Casimir coefficient” (sometimes “quadratic Casimir operator” or simply “quadratic Casimir”) is used for the parameter c2 that enters (4.7.44) and is related to C2T by (4.7.45). Parameter c2 , like C2T , is independent of the choice of basis in Lie algebra, but does of course depend on the choice of representation. [n] We have for su(n) that c2 = 1/(2n), cad 2 = 1. Likewise, the term “cubic Casimir coefficient” is used in physics literature for parameter c3 in (4.7.33).

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Problem 4.7.12. Calculate the sum of parameters c2 for symmetric and antisymmetric representations [n(n+1)/2] and [n(n−1)/2] of su(n), which enter the partition (4.4.17) of tensor product of two defining representations [n]. Problem 4.7.13. Find cubic Casimir coefficient c3 and the value of C3T with normalization (4.7.32) for fundamental, antifundamental and adjoint representations of su(3). Problem 4.7.14. Making use of isomorphism conf(Rp,q ) = so(p + 1, q + 1) (see Proposition 3.4.2), show that quadratic Casimir operator of conformal algebra conf(Rp,q ) with defining relations (3.4.6) is proportional to operator C2 = Lij Lij + P i Ki + K i Pi − 2 D2 .

(4.7.50)

Show that in representation T defined in (3.4.7), operator (4.7.50) reads [T ] C2 = 2Δ(n − Δ) + Σij Σij , where n = p + q. Quadratic Casimir operators gab Ta Tb of Lie algebra A in representations where generators Ta are differential operators, are Laplace operators of algebra A. Differential operators of this sort are discussed in Chapter 7. 4.7.3.

Finite-dimensional representations of Lie algebras. su(2) and s(2, C)

Lie algebras su(2) and s(2, C) serve as examples of utilizing universal enveloping algebra for constructing Lie algebra representations. We show here that algebras su(2) and s(2, C) have infinite series of finitedimensional inequivalent irreducible representations T (j) , which are characterized by nonnegative integers and half-integers j = 0, 12 , 1, 32 , 2, . . . (spins) and have dimensions (2j + 1). We construct these representations explicitly and show that they comprise all finite-dimensional irreducible representations of su(2) and s(2, C). To begin with, we recall that Lie algebra su(2) is real form of s(2, C), and generators τα of su(2) are at the same time generators of s(2, C). Another basis {e± , h} in s(2, C) (see (3.3.1)) is constructed out of {τα } via simple linear relations (with complex coefficients) e± = i τ1 ∓ τ2 , h = i τ3 , i 1 τ1 = − (e− + e+ ), τ2 = (e− − e+ ), 2 2

τ3 = −i h,

(4.7.51)

so that commutation relations are (3.2.51) and (3.3.12). In accordance with Remark 1 in Section 4.4.1, there is one-to-one correspondence

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between finite-dimensional complex representations of Lie algebra s(2, C) and its real form su(2). Namely, given a representation of s(2, C), the representations of its generators {e± , h} are known, so, in view of (4.7.51), representation of generators τi is known as well. The latter defines complex representation of su(2), and this procedure works the other way around.f The construction of complex representations of algebra s(2, C) is somewhat more straightforward than for su(2), so it is s(2, C) that is studied in the rest of this section. Representation T of Lie algebra s(2, C) is homomorphism of s(2, C) to associative algebra UT (s(2, C)) of linear operators in vector space V. Algebra UT (s(2, C)) is a representation of universal enveloping algebra U(s(2, C)). Therefore, constructing representations of s(2, C) is equivalent to constructing representations of enveloping algebra U(s(2, C)); this is discussed in general terms in Sections 4.7.1 and 4.7.2. The defining relations (3.2.51) for enveloping algebra U(s(2, C)) can be written as follows: e+ e− = 2 h + e− e+ ,

h e± = e± (h ± 1).

(4.7.52)

The quadratic Casimir operator reads J 2 = −τα τα =

1 (e− e+ + e+ e− ) + h2 = e− e+ + h(h + 1) ∈ U(s(2, C)). 2 (4.7.53)

This operator commutes with all generators of U(s(2, C)). As we pointed out in Section 4.7.2, the element T (J 2 ) is proportional to unit operator in any irreducible representation of Lie algebra s(2, C). Problem 4.7.15. Show that the operator (4.7.53) obeys [J 2 , e± ] = 0, [J 2 , h] = 0.

(4.7.54)

Let V be vector space of a representation T of algebra s(2, C) (or su(2)). Group SU (2) is compact, so any of its representations is equivalent to unitary. Accordingly, any representation of Lie algebra su(2), including T , is anti-Hermitian, T (τα )† = −T (τα ). In particular, the operator T (h) = iT (τ3 ) is Hermitian. Therefore T (h) can be diagonalized, and the space V f Some complex representations of s(2, C) may be considered as real representations of su(2) (see, e.g., (4.7.16)).

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can be decomposed into direct sum of subspaces Vλ labeled by eigenvalue λ of generatorg h, V = ⊕λ Vλ , where subspaces Vλ are Vλ = {v (λ) ∈ V | h · v (λ) = λ v (λ) }. If v (λ) ∈ Vλ , then e+ v (λ) ∈ Vλ+1 and e− v (λ) ∈ Vλ−1 , since (4.7.52) gives he± v (λ) = e± (h ± 1)v (λ) = (λ ± 1)e± v (λ) . Therefore, operators e+ and e− , when acting on eigenvectors of operator h, give new eigenvectors of h with eigenvalues raised and lowered by 1. Generally, one generates infinitely many eigenvectors of operator h out of a vector v (λ) : ek− v (λ) ∈ Vλ−k and en+ v (λ) ∈ Vλ+n , where k and n are arbitrary nonnegative integers. If vectors ek− v (λ) and en+ v (λ) do not vanish for any n and/or k, then the space V is infinite-dimensional, since vectors ek− v (λ) and en+ v (λ) are linearly independent, being eigenvectors of h with different eigenvalues, see Proposition 1.2.3. Representation of algebra s(2, C) is finite-dimensional, only if the series of eigenvectors of h of the form ek± v (λ) terminates. The latter requirement (λ) (λ) = 0 = en+1 for some k and n. These equalities determine gives ek+1 − v + v the dimension of representation, and, as we see below, the eigenvalue of the Casimir operator (4.7.53). It is convenient to begin the construction with the eigenvector v0 = en+ v (λ) ∈ V which has the highest eigenvalue (λ + n). Acting by “lowering” operator e− , one then constructs all basis vectors in space V. Equivalently, one could start with the eigenvector v0 = ek− v (λ) with the lowest eigenvalue (λ − k) and construct all basis vectors by applying “raising” operator e+ to v0 . Definition 4.7.3. Nonzero vector v0 , such that e+ v0 = 0, h v0 = λ v0 ,

(4.7.55)

is the highest weight vector, and its eigenvalue λ is the highest weight. Representation obtained by multiple action of generators of s(2, C) on the highest weight vector v0 is called highest weight representation. Nonzero vector v, such that h v = μ v, e− v = 0,

(4.7.56)

is the lowest weight vector, and μ is the lowest weight. Accordingly, representation obtained by multiple action of generators of s(2, C) on the lowest weight vector v is called lowest weight representation. g Hereafter

we write e± and h instead of T (e± ) and T (h), unless this leads to confusion.

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It follows from the above analysis that finite-dimensional representations are combinations of representations which are simultaneously highest weight and lowest weight. In detail, let T be finite-dimensional irreducible representation of s(2, C). By above reasoning, it has highest weight vector v0 and highest weight λ; we will soon see that there is only one highest weight vector. We generate new vectors from v0 by applying lowering operators e− : vk =

1 k e v0 , k! −

k = 1, 2, 3, . . . .

(4.7.57)

Making use of (4.7.52), we obtain (a) h vk = (1/k!) h ek− v0 = (1/k!) ek− (h − k) v0 = (λ − k) vk , (b) e− vk = (1/k!) ek+1 − v0 = (k + 1) vk+1 , (c) e+ vk =

(1/k!) e+ ek−

(4.7.58)

v0 = (2λ − k + 1) vk−1 (k ≥ 0),

where we set v−1 = 0 in the last equality. Formulas (a) and (b) are obvious. Formula (c) follows from the following identity valid in enveloping algebra U(s(2, C)) and its representations: (2 h − k + 1). e+ ek− = ek− e+ + k ek−1 −

(4.7.59)

Problem 4.7.16. Derive the identity (4.7.59) by making use of (4.7.52). It follows from (4.7.58)(a) that all vk have different eigenvalues of the operator h and hence are linearly independent by Proposition 1.2.3. Let us follow tradition and change notations. Namely, we denote the highest weight by j (instead of λ). Then there is the smallest integer n ≥ 0, such that vn = 0 but vn+1 = 0. It follows from (4.7.58) that subspace spanned by vectors v0 , v1 , . . . , vn is invariant. Since the representation is irreducible, this subspace coincides with the whole space of representation V; in particular, there is only one highest weight vector. In other words, the basis in the space of representation V is formed by vectors (v0 , v1 , . . . , vn ) with the eigenvalues of h equal to (j, j − 1, j − 2, . . . , j − n). So, V is partitioned into the sum of 1-dimensional subspaces V = Vj ⊕ Vj−1 ⊕ · · · ⊕ Vj−n ⇒ dim(V) = n + 1.

(4.7.60)

Consider then formula (4.7.58)(c) for k = n + 1: e+ vn+1 = (2j−n) vn . Since vn+1 = 0, vn = 0, we obtain j = n2 . We conclude that for finite-dimensional

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irreducible representation T (j) , the highest weight is nonnegative integer or half-integer. Problem 4.7.17. Show that representation T (j) with highest weight j = n2  is irreducible. Hint: prove that any nonzero vector k αk vk ∈ V can be transformed into any basis vector vk ∈ V, k = 0, . . . , n by action of generators e+ and e− . To summarize, a representation of s(2, C) is finite-dimensional and irreducible, iff it is highest weight representation with j = n/2, where n is non-negative integer. According to (4.7.60), the space of this representation is partitioned into sum of 1-dimensional spaces, V = Vj ⊕ Vj−1 ⊕ · · · ⊕ V−j ,

(4.7.61)

and its dimension is 2j+1. Basis in V is made of vectors vk given by (4.7.57), where k = 0, 1, . . . , 2j. Vectors vk ∈ Vj−k are eigenvectors of operator h with eigenvalues n/2 − k = j − k ≡ m. The spectrum of operator h is Spec(h) : m = −j, −j + 1, . . . , j − 1, j.

(4.7.62)

Thus, eigenvalues m are integer for integer j and half-integer for halfinteger j. Irreducible representations with different spins j are inequivalent, as they have different dimensions. Quadratic Casimir operator J 2 given by (4.7.53) takes one and the same value on all vectors of irreducible representation. This value is straightforwardly obtained by considering the highest weight vector: J 2 v0 = (e− e+ + h(h + 1)) v0 = j(j + 1) v0 .

(4.7.63)

Let us construct matrices T of irreducible representation with highest weight j = n2 in space (4.7.61). They are given by analogs of formula (1.2.42), i.e., h · vk = v T k (h), e± · vk = v T k (e± ). In accordance with (4.7.58) we have n  T k (h) = − k δ k , T k (e+ ) = (n − k + 1)δ ,k−1 , 2  T k (e− ) = (k + 1)δ ,k+1 . (4.7.64) Here , k = 0, 1, . . . , n and δ ,−1 = 0 = δ ,n+1 . Note that matrices T(h) and T(e± ) are real, so they can be used as matrices of real representations of algebra s(2, R) (the representations of su(2) are complex for half-integer j and are equivalent to real representations for integer j).

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Problem 4.7.18. Making use of (4.7.64), write explicitly 1-, 2- and 3dimensional matrix representations (j = 0, 1/2, 1, 3/2) of generators h and e± . Find the corresponding representations of su(2) generators τα and compare them with representations (4.7.15) and (4.7.16). Thus, we have proven the following proposition. Proposition 4.7.4. All finite-dimensional irreducible representations T (j) of Lie algebra s(2, C) (and su(2)) are representations with highest weights j = n2 , where n is nonnegative integer. Dimensions of these representations are determined by the highest weight and are equal to 2j + 1 = n + 1. Representations T (j) are inequivalent for different j. 4.7.4.

Coproduct in universal enveloping algebra U(A). Yangians

1. Coproduct. Bialgebras and Hopf algebras As we pointed out in Section 4.3.1, definition (4.3.12) of direct product of two representations is naturally extended to general Lie algebras A irrespectively of Lie groups. Furthermore, this definition can be extended to universal enveloping algebras U (A). In this section we consider this construction in some detail. It is worth noting that not every algebra admits direct (tensor) product of its representations. Algebras which admit this procedure are endowed with a special operation called coproduct (or comultiplication). We show in this section that coproduct can be defined in universal enveloping algebra U (A) of any Lie algebra A, so the enveloping algebras admit direct products of their representations. Towards the end of this section we give examples of associative algebras with coproduct (Yangians) which are quite different from universal enveloping algebras of Lie algebras. As we discuss in the beginning of Section 4.7.1, every representation T of Lie algebra A over field K defines associative algebra UT (A) of operators T (X), where X ∈ U (A). Consider, in particular, direct product T (p) = T (1) ⊗ T (2) of two representations. Representation T (p) of Lie algebra A induces homomorphism of enveloping algebra U (A) to direct sum of associative algebras: T (p) : U (A) → UT (1) (A) ⊗ UT (2) (A) = UT (p) (A).

(4.7.65)

An element A ∈ A ⊂ U (A) is mapped in accordance with (4.3.12), and mapping of product B · A is dictated by the rule of sequential action of operators: T (p) (B · A) = T (p) (B) · T (p) (A) = T (p) (B) · (T (1) (A) ⊗ Im + In ⊗ T (2) (A)) = T (1) (B) T (1) (A) ⊗ Im + T (1) (A) ⊗ T (2) (B) + T (1) (B) ⊗ T (2) (A) + In ⊗ T (2) (B) T (2) (A), where n and m are dimensions of representations T (1) and T (2) . Similar formulas are valid for monomials of higher orders and their linear combinations. Since mapping (4.7.65) makes sense for any representations T (1) and T (2) , and both algebras UT (1) (A) and UT (2) (A) are representations of one and the same enveloping

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algebra U (A), one is naturally lead to define a universal analog of mapping (4.7.65), namely, ˆ U (A) Δ : U (A) → U (A) ⊗ (4.7.66) ˆ (hereafter we use the symbol ⊗ for direct product of algebra elements to distinguish it from direct product of operators or matrices; this distinction disappears once one comes to representations). It has to be homomorphic, Δ(A · B) = Δ(A) Δ(B), Δ(αA + βB) = αΔ(A) + βΔ(B),

∀A, B ∈ U (A), ∀α, β ∈ K,

(4.7.67)

and there has to be correspondence between mappings (4.7.65) and (4.7.66): T (p) (X) = (T (1) ⊗ T (2) ) Δ(X),

∀ X ∈ U (A).

(4.7.68)

ˆ U (A) On the right-hand side here, the first and second factors in Δ(X) ∈ U (A) ⊗ are taken in representations T (1) and T (2) , respectively. Below we explain the formula (4.7.68) further. ˆ U (A), as vector space, is direct product of two identical vector spaces Algebra U (A) ⊗ ˆ YB ), where αAB ∈ K and U (A). It is a set of linear combinations of the form αAB (YA ⊗ ˆ U (A) is defined in the YA are basis monomials (4.7.3) in U (A). Multiplication in U (A) ⊗ standard way, ˆ YB ) · (β CD YC ⊗ ˆ YD ) = αAB β CD (YA · YC ⊗ ˆ YB · YD ). (αAB YA ⊗ Mapping Δ in (4.7.66) is uniquely determined by its action on unit element and on generators Xa of Lie algebra A : ˆ I, Δ(I) = I ⊗

ˆ I +I⊗ ˆ Xa . Δ(Xa ) = Xa ⊗

(4.7.69)

The second formula here is the counterpart of the relation (4.3.12) that defines direct product of Lie algebra representations. Extension of Δ to the whole algebra U (A) is then uniquely constructed by making use of (4.7.67). As an example, Δ(αXa · Xb + βI) = αΔ(Xa ) · Δ(Xb ) + βΔ(I) ˆI ˆ I+I⊗ ˆ Xa ) · (Xb ⊗ ˆ I+I⊗ ˆ Xb ) + βI ⊗ = α(Xa ⊗ ˆ Xa · Xb ) + βI ⊗ ˆ I. ˆ I + Xb ⊗ ˆ Xa + Xa ⊗ ˆ Xb + I ⊗ = α(Xa · Xb ⊗ To see that Δ is homomorphism, it suffices to check that it preserves commutation relations (4.7.6): ˆ I +I⊗ ˆ Xa ), (Xb ⊗ ˆ I +I⊗ ˆ Xb )] [Δ(Xa ), Δ(Xb )] = [(Xa ⊗ ˆ I +I⊗ ˆ [Xa , Xb ] = C d Δ(Xd ) = Δ([Xa , Xb ]). = [Xa , Xb ] ⊗ ab This completes the construction of mapping Δ. Proposition 4.7.5. Homomorphism Δ in (4.7.66), defined by (4.7.67) and (4.7.69), obeys (Δ ⊗ 1)Δ(X) = (1 ⊗ Δ)Δ(X), ∀X ∈ U (A), (4.7.70) where mappings (Δ ⊗ 1)Δ and (1 ⊗ Δ)Δ are homomorphisms of U (A) to ˆ U (A)⊗ ˆ U (A). U (A)⊗ ˆ U (A) to U (A)⊗ ˆ U (A)⊗ ˆ U (A), which acts on an Here Δ ⊗ 1 is mapping of U (A)⊗ ˆ Y ∈ U (A)⊗ ˆ U (A) as follows: the first factor X is mapped to Δ(X) (i.e., it element X ⊗ is “split” into two factors), while the second factor remains intact, ˆ Y ) = Δ(X) ⊗ ˆ Y. (Δ ⊗ 1)(X ⊗

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AC ˆ In other words, for Δ(X) expanded in basis, Δ(X) = αAC X YA ⊗ YC , αX ∈ K, one has ˆ YC . Mapping (1 ⊗ Δ) in (4.7.70) is understood in an Δ(Y ) ⊗ (Δ ⊗ 1)Δ(X) = αAC A X analogous way. To prove the property (4.7.70), it is again sufficient to consider generators Xa of Lie algebra A :

ˆ I+I⊗ ˆI⊗ ˆ Xa ˆ I+I⊗ ˆ Xa ) = (Xa ⊗ ˆ I +I⊗ ˆ Xa ) ⊗ (Δ ⊗ 1)Δ(Xa ) = (Δ ⊗ 1)(Xa ⊗ ˆI⊗ ˆ I +I⊗ ˆ (Xa ⊗ ˆ I +I⊗ ˆ Xa ) = (1 ⊗ Δ)Δ(X), = Xa ⊗ as desired. Proof that mapping (Δ ⊗ 1)Δ = (1 ⊗ Δ)Δ is homomorphism of U (A) to ˆ U (A) ⊗ ˆ U (A) is left to the reader. U (A) ⊗  Definition 4.7.4. Mapping Δ in (4.7.66) obeying (4.7.67)–(4.7.70), is coproduct (comultiplication) in algebra U (A). The property (4.7.70) is called coassociativity of coproduct Δ. Coproduct is conveniently used when one writes an element of X ∈ U (A) in tensor product of several representations. As an example, consider tensor product of two representations. We would like to write an element X ∈ U (A) in this tensor product. To this end, we apply coproduct Δ to element X thus transforming it to element ˆ U (A), i.e., “split ” X into two factors. Then ˆ C of algebra U (A) ⊗ Δ(X) = αAC YA ⊗Y X

mapping (T (1) ⊗ T (2) ) is applied to element Δ(X), see (4.7.68): (1) (YA ) ⊗ T (2) (YC ), (T (1) ⊗ T (2) ) Δ(X) = αAC X T

(4.7.71)

i.e., elements of U (A) in the first and second factors in Δ(X) are taken in representations T (1) and T (2) , respectively. This is precisely the desired expression of element X in representation (T (1) ⊗ T (2) ). On the other hand, the representation T (p) = T (1) ⊗ T (2) (4.3.12) of Lie algebra A can be extended to enveloping algebra U (A) by making use of (4.7.10). The two constructions must agree, which is indeed the case. The latter fact boils down to the identity for basis monomials (4.7.2): T (p) (Xa1 · Xa2 · · · Xar ) = (T (1) ⊗ T (2) ) Δ(Xa1 · Xa2 · · · Xar ).

(4.7.72)

This identity is straightforwardly checked for r = 1: ˆ I+I⊗ ˆ Xa ) T (p) (Xa ) = (T (1) ⊗ T (2) )Δ(Xa ) = (T (1) ⊗ T (2) )(Xa ⊗ = T (1) (Xa ) ⊗ IT (2) + IT (1) ⊗ T (2) (Xa ),

(4.7.73)

which agrees with (4.3.12). The identity (4.7.72) for all r becomes obvious when written as follows: T (p) (Xa1 ) · T (p) (Xa2 ) · · · T (p) (Xar ) = (T (1) ⊗ T (2) ) Δ(Xa1 ) · (T (1) ⊗ T (2) ) Δ(Xa2 ) · · · (T (1) ⊗ T (2) ) Δ(Xar ), where we make use of the fact that mappings T (p) , T (1) , T (2) and Δ are homomorphisms. (p) Considering tensor product of three or more representations, Tk = T (1) ⊗ T (2) ⊗ · · · T (k) , one “splits ” an element X = Xa1 · Xa2 · · · Xar ∈ U (A) to k factors by

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consecutively applying coproduct mapping Δ. Due to coassociativity (4.7.70), it is unimportant which factor is “split ” at which step. As a result, formulas (4.7.68) and (4.7.72) are generalized: (p)

Tk

= (T (1) ⊗ T (2) ⊗ · · · ⊗ T (k) ) Δk−1 ,

(4.7.74)

where Δk−1 (X) = (Δ ⊗ 1 ⊗ · · · ⊗ 1) · · · (Δ ⊗ 1 ⊗ 1) (Δ ⊗ 1)Δ(X).

 

(4.7.75)

k−2

In particular, we have for basis element Xa ∈ A (cf. (4.3.14)) (p)

Tk (Xa ) =

k 

(I(1) ⊗ · · · ⊗ I(m−1) ⊗ T (m) (Xa ) ⊗ I(m+1) ⊗ · · · ⊗ I(k) ).

(4.7.76)

m=1

Coproduct Δ is in some sense an “inverse ” map to multiplication m : ˆ U (A) → U (A), where m maps two elements X, X  ∈ U (A) to their product U (A) ⊗ X · X  ; arrow in m is directed in opposite way as compared to Δ in (4.7.66). Importantly, mappings m and Δ are consistent with each other, see the first of Eq. (4.7.67). We emphasize again that tensor product of representations of an associative algebra can be defined only if this algebra possesses coassociative coproduct operation. Algebras with both multiplication and coproduct are bialgebras. Definition 4.7.5. Associative algebra B with unit element I and basis {YA } is ˆ B, bialgebra, if it is endowed with coassociative coproduct Δ: B → B ⊗ ˆ Δ(YA ) = ΔBC A YB ⊗YC ,

ΔBC ∈ C, A

and co-unit element  : B → C, which have the following properties (∀a, b ∈ B): DK = ΔKB ΔLC , (Δ ⊗ 1)Δ = (1 ⊗ Δ)Δ ⇒ ΔBC A ΔB A B

Δ(a · b) = Δ(a) · Δ(b),

ˆ Δ(I) = I ⊗I,

m(( ⊗ 1)Δ(a)) = m((1 ⊗ )Δ(a)) = a ⇒

(a · b) = (a) · (b), ΔBC A  B YC

=

(4.7.77) (I) = 1,

ΔBC A YB  C

= YA , (4.7.78)

where A := (YA ) ∈ C, and m is multiplication map: m(C ⊗ b) = m(b ⊗ C) = C · b. We have seen that universal enveloping algebra U (A) of any Lie algebra A has coassociative coproduct Δ given by (4.7.67) and (4.7.69). Furthermore, in this case one defines homomorphism  : U (A) → C by setting (Xa ) = 0 and (I) = 1. This homomorphism satisfies all axioms of co-unity  listed in Definition 4.7.5. Therefore, enveloping algebra U (A) of any Lie algebra A is bialgebra. When discussing a new class of infinite-dimensional algebras (Yangians) towards the end of this section, we encounter yet another important construct, Hopf algebra. Definition 4.7.6. Bialgebra B is Hopf algebra, if, besides coproduct Δ and co-unity , there is an extra map S : B → B, such that m((S ⊗ 1)Δ(a)) = m((1 ⊗ S)Δ(a)) = (a) I,

∀ a ∈ B,

(4.7.79)

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ˆ where m is multiplication map, m(a⊗b) = a · b. Map S is called antipode; it is antihomomorphism with respect to both multiplication and coproduct in B: S(a · b) = S(b) · S(a),

(S ⊗ S)Δ(a) = σ · Δ(S(a)),

∀ a, b ∈ B,

(4.7.80)

ˆ b) = (b ⊗ ˆ a). where σ is transposition operator: σ · (a ⊗ We note that the antipode S is required for defining contragredient representations of bialgebra B. Example. Consider universal covering algebra (bialgebra) U (A) and define mapping S : U (A) → U (A), S(Xa ) = −Xa , S(I) = I, (4.7.81) where Xa are basis elements in Lie algebra A. Mapping (4.7.81) can be extended to the whole bialgebra U (A) via the first of Eq. (4.7.80). Mapping S constructed in this way obeys axioms (4.7.79) and (4.7.80), so S is antipode in U (A) and bialgebra U (A) is Hopf algebra. 2. Split Casimir operators and central elements of algebra U (A) Here we make use of coproduct operation to construct in an alternative way the Casimir operators (4.7.26), (4.7.29) of simple Lie algebra A (without any reference to Lie groups). Consider element ˆ = gab Xa ⊗ ˆ A ⊂ U (A) ⊗ ˆ U (A), ˆ Xb ∈ A ⊗ (4.7.82) C called split (or polarized) Casimir operator of Lie algebra A. ˆ is independent of choice of basis in A. It has the Proposition 4.7.6. Element C property (Ad-invariance) ˆ = [(A⊗I ˆ = 0, ˆ + I ⊗A), ˆ [Δ(A), C] C]

∀A ∈ A,

(4.7.83)

where Δ is coproduct (4.7.69), and obeys ˆ 23 ] = 1 [C ˆ 13 − C ˆ 23 ], ˆ 12 , C ˆ 13 , C [C 2

(4.7.84)

where we use the notations ˆ 12 = gab Xa ⊗ ˆ Xb ⊗ ˆ I, C ˆ 23 = gab I ⊗ ˆ Xa ⊗ ˆ Xb , C

ˆ 13 = gab Xa ⊗ ˆ I⊗ ˆ Xb , C (4.7.85)

ˆ ij ∈ U (A) ⊗ ˆ U (A) ⊗ ˆ U (A). so that C ˆ does not depend on the choice of basis in A, since the change Proof. Element C of basis Xa → Ba b Xb leads to transformation of inverse Killing metric gab → gcd (B −1 )c a (B −1 )d b . To prove (4.7.83), it suffices to consider an arbitrary generator A = Xr of Lie algebra A and write ˆ = [Xr ⊗ ˆ ˆ I +I⊗ ˆ Xr , C] [Δ(Xr ), C] k X ⊗ k ˆ Xk ) = gab gkp (Crap + Crpa )Xk ⊗ ˆ Xb = 0, = gab (Cra k ˆ Xb + Crb Xa ⊗

(4.7.86)

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where we use antisymmetry (3.2.71) of structure constants. To derive (4.7.84), we first note that ˆ 13 , C ˆ 23 ] = 0. ˆ 12 + C (4.7.87) [C Indeed, making use of definitions (4.7.85), we obtain for the left-hand side of (4.7.87) ˆ = [gpr Xp ⊗ ˆ ˆ C] ˆ C] ˆ (Xr ⊗ ˆ I +I⊗ ˆ Xr ), I ⊗ ˆ Δ(Xr ), I ⊗ [gpr Xp ⊗ ˆ = 0, ˆ [Δ(Xr ), C] = gpr Xp ⊗ where the last equality follows from (4.7.83). Likewise, we have ˆ 12 , C ˆ 13 + C ˆ 23 ] = 0. [C

(4.7.88)

Combining (4.7.87) with (4.7.88), we obtain (4.7.84). Remark 1. Let us define Lie algebra with generators tij = tji (i, j = 1, . . . N , N > 2), which obey (4.7.89) [tij , tk ] = 0, [tij , tik + tjk ] = 0, where all indices take different values. This is known as Kohno–Drinfeld algebra. Consider elements (cf. (4.7.85)) ˆ ˆ ˆ ˆ ij := gab (I ⊗(i−1) ˆ a ⊗I ˆ ⊗(j−i−1) ˆ b ⊗I ˆ ⊗(N−i−j) ), C ⊗X ⊗X

ˆ ij ˆ ji = C C

(j > i).

(4.7.90)

ˆ ij form representation of generators tij of Kohno–Drinfeld algebra, since C ˆ ij Elements C evidently obey the first of Eq. (4.7.89), and the property (4.7.84) ensures that the second relation in (4.7.89) is also satisfied. Note that product of two Ad-invariant elements is again Ad-invariant. This leads to ˆ Ad-invariance of any power of split Casimir operator C: ˆ k ] = 0, [Δ(A), C

∀A ∈ A,

(4.7.91)

where ˆ k = ga1 b1 · · · gak bk Xa · · · Xa ⊗ ˆ U (A). ˆ Xb · · · Xb ∈ U (A) ⊗ C 1 k 1 k

(4.7.92)

ˆ  ∈ U (A) ⊗ ˆ U (A) be Ad-invariant, Proposition 4.7.7. Let element C ˆ  ] = 0, [Δ(A), C

∀A ∈ A,

(4.7.93)

ˆ  in the and let T be any representation of algebra A and hence U (A). Let us write C following form: ˆ  = YB ⊗ ˆ Y B = YB ⊗ ˆ D BC YC , (4.7.94) C where D BC ∈ K, and YB are basis elements in U (A). Then element C = YB D BA Tr(T (YA )) ∈ U (A)

(4.7.95)

belongs to center Z of algebra U (A). ˆ I +I ⊗ ˆ A, so Ad-invariance property Proof. For any element A ∈ A we have Δ(A) = A ⊗ ˆ  reads (4.7.93) for element C ˆ Y B = −YB ⊗ ˆ [A , Y B ]. [A, YB ] ⊗

(4.7.96)

We apply mapping T (representation of U (A)) to the second factors here, ˆ T (Y B ) = −YB ⊗ ˆ [T (A), T (Y B )], [A, YB ] ⊗

(4.7.97)

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calculate traces of these factors and recall that Tr[T (A), T (Y B )] = 0. In this way we obtain the identity [A, YB D BA Tr(T (YA ))] = 0 for all A ∈ A. Thus, element C = YB D BA Tr(T (YA )) is central in algebra U (A). Problem 4.7.19. Prove Ad-invariance of elements ˆ (σ,k) = ga1 b1 · · · gak bk (Xa · · · Xa ) ⊗ ˆ (Xb · Xbσ(2) · · · Xbσ(k) ), C 1 k σ(1)

(4.7.98)

where σ is arbitrary permutation of k elements. Hint: perform calculation similar to that done in (4.7.86). Let T be a nontrivial representation of simple Lie algebra A. Consider Ad-invariant element (4.7.92). Then, making use of Proposition 4.7.7, we construct central elements ˜k = Xa · · · Xa ga1 ...ak , C 1 k

(4.7.99)

where (see (4.7.30))

  ga1 ...ak = ga1 b1 · · · gak bk Tr Tb1 · · · Tbk ,

Tb ≡ T (Xb ).

(4.7.100)

Central elements (4.7.99) differ from operators Ck constructed in (4.7.26), (4.7.29) in that the former are defined without complete symmetrization of indices in coefficients ga1 ...ak . In other words, tensor ga1 ...ak is cyclic in its indices (because of cyclic property of trace), but it is not totally symmetric, unlike da1 ...ak given by (4.7.29). However, we can consider also Ad-invariant element (4.7.98) and make use of Proposition 4.7.7 to find other central elements ˜k,σ = gaσ(1) aσ(2) ...aσ(k) Xa · Xa · · · Xa , C 1 2 k

(4.7.101)

where σ is an arbitrary permutation of k elements. Clearly, elements Ck given by ˜k,σ in ˜k,σ . Conversely, elements C (4.7.26), (4.7.29) are linear combinations of elements C (4.7.101) are linear combinations of elements Ck (where symmetric tensors da1 ...ak are Ad-invariant, but otherwise arbitrary). The latter property follows from the fact that any polynomial (4.7.101) can be expanded in basis of symmetric polynomials (4.7.17). Thus, the set of independent generators Ck given by (4.7.26) enables one to construct all elements (4.7.101) of center Z of algebra U (A). Moreover, Proposition 4.7.2 ensures that the set of elements Ck in (4.7.26) contains all r Casimir operators (r is rank of Lie algebra A). In the rest of this section we discuss algebraic structures that emerge in quantum integrable models of field theory and statistical mechanics. Let T be faithful n-dimensional representation of Lie algebra A. In the case of matrix algebra A, convenient choice is defining representation. We define (n × n) matrix L = gab Ta ⊗ Xb , whose elements are Lij = (Ta )ij gab Xb (here i, j = 1, . . . , n), i.e., Lij ∈ A. Matrix L does not depend on choice of basis is A, carry information on Lie algebra A and is obtained from split Casimir operator (4.7.82) by taking the first factor there in representation T . Defining commutation relations (4.7.6) for generators Xa are written in terms of matrix L as follows: 1 i1 i2 (r k k [Lkj11 δjk22 − δjk11 Lkj22 ] − [Lik1 δik2 − δki1 Lik2 ]rk1j1kj22 ) [Lij11 , Lij22 ] = 1 2 1 2 1 2 2 1 [r12 , L1 − L2 ], 2 where we use shorthand notations ⇒ [L1 , L2 ] =

L1 = gab Ta ⊗ In ⊗ Xb ,

L2 = gab In ⊗ Ta ⊗ Xb ,

(4.7.102)

r12 = gab Ta ⊗ Tb ⊗ I. (4.7.103)

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˜k given by (4.7.99) can be written in compact form Central elements C ˜k = Tr(Lk ). C

(4.7.104)

ˆ 13 , C ˆ 23 and C ˆ 12 defined in (4.7.85), with the first Operators (4.7.103) are elements C two factors taken in representation T . Therefore, Eq. (4.7.102) is a consequence of Eq. (4.7.84), where, again, the first two factors have to be taken in representation T . d T yield Problem 4.7.20. Show that the relations (4.7.102) together with [Ta , Tb ] = Cab d defining commutation relations (4.7.6) for generators Xa .

It is worth noting that the right-hand side of (4.7.102) can be written in several different ways due to identity (4.7.105) [r12 , L1 + L2 ] = 0, which is obtained from (4.7.88) by taking the first two factors in representation T . In particular, one writes (4.7.102) in a simpler form [L1 , L2 ] = [ r12 , L1 ]. Then the fact that elements (4.7.104) are central, is proven straightforwardly: [Tr1 (Lk1 ), L2 ] =

k 

Tr1 (Lk−i [L1 , L2 ]Li−1 1 1 )

i=1

=

k 

k Tr1 (Lk−i [r12 , L1 ]Li−1 1 1 ) = Tr1 ([ r12 , L1 ]) = 0.

(4.7.106)

i=1

Here Tr1 is trace of the first factor, in notation (4.7.103). This proof demonstrates the advantage of notations, and, indeed, the entire technique briefly described here, which turns out to be very convenient in more involved calculations. 3. Split Casimir operators r = gab Ta ⊗ Tb of Lie algebras s(n), so(n) and sp(2r) Formulas (4.7.102) and (4.7.106), as well as other applications of matrix Lie algebras, involve an important matrix r = gab Ta ⊗ Tb . It is obtained from split Casimir operator ˆ = gab Xa ⊗ Xb by replacing generators Xa by operators in defining representation, C Ta = T (Xa ). We are going to find explicit expressions for matrices r = gab Ta ⊗ Tb for Lie algebras s(n), so(n) and sp(2r). These expressions (and other quantities related to solutions to Yang–Baxter equation, see below) are instrumental in defining an important class of infinite-dimensional algebras, called Yangians, of s-, so- and sp-types. Yangians [26] have a number of remarkable properties. In particular, they play central role in describing numerous infinite-dimensional integrable quantum systems. Furthermore, they are Hopf algebras endowed with coproduct, co-unity and antipode. This points towards rich representation theory, which is of interest for both physics and mathematics [12]. a. Matrix r = gαβ Tα ⊗ Tβ for algebra g(n, C) The set of matrix units eij (i, j = 1, . . . , n) makes basis {Tα } in defining representation of Lie algebra g(n, C) (here α = (ij) is composite index). We define invariant metric in g(n, C) by (4.2.9): (4.7.107) gij,k = Tr(eij ek ) = δjk δi (its normalization is irrelevant for our purposes). We recall that for Lie algebras like g(n, C), which are neither simple nor semisimple, invariant metric need not coincide with Killing metric (the latter is degenerate and is inconvenient for us). Making use

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of invariant metric (4.7.107), one writes split Casimir operator for g(n, C) in defining representation (summation over i, j, k,  is assumed) r = gij,k eij ⊗ ek = (eij ⊗ eji ) ≡ P, gij,k

δjk δi

(4.7.108) δri δsj .

gij,k gk,rs

where = is inverse matrix to gij,k : = Operator P on the right-hand side is transposition operator (4.4.12). Indeed, it obeys (4.4.11). As an example, consider the “last ” set of basis vectors ekn (k = 1, . . . , n). We have P (ekn ⊗ en ) = (eij ekn ⊗ eji en ) = δjk δi (ein ⊗ ejn ) = (en ⊗ ekn ), where we use the relation eij ek = δjk ei . Similarly, one can show that for any two matrices A = Aij eij and B = Bij eij , the following operator relations, characteristic of transposition operator, hold (A ⊗ B) P = P (B ⊗ A) , ⇒ A1 P = P A2 ,

∀A, B ∈ g(n, C)

A2 P = P A1 ,

(4.7.109)

∀A ∈ g(n, C),

(4.7.110)

where A1 := (A ⊗ In ) and A2 := (In ⊗ A). In particular, it follows from (4.7.110) that the operator r ≡ P in g(n, C) found in (4.7.108) has Ad-invariance property: (A1 + A2 ) P = P (A1 + A2 )

∀A ∈ g(n, C).

(4.7.111)

b. Matrix r for algebra s(n, C) Algebra s(n, C) is subalgebra of Lie algebra g(n, C). We choose basis Tα (α = 0, 1, . . . , n2 − 1) in g(n, C) such that one of its elements is unit matrix T0 = In , while others make basis Ta (a = 1, . . . , n2 − 1) of traceless generators of s(n, C) with normalization (4.7.44): Tr(Ta Tb ) = c2 (T ) gab (here ||gab || is Killing metric in s(n, C)). Metric of g(n, C) is defined in the new basis in the same way as in (4.2.9), (4.7.107), namely, gαβ = Tr(Tα Tβ ). Since Tr(T0 Tb ) = 0 (∀b), this metric has block structure  n 0T gαβ = (4.7.112) , (α, β = 0, 1, . . . , n2 − 1), 0 c2 ||gab || where c2 ≡ c2 (T ), and 0 and 0T are zero column and row. It follows from the form of metric (4.7.112) that matrix r = gab Ta ⊗ Tb for s(n, C) is related to similar matrix, calculated in (4.7.108) for g(n, C) and independent of the choice of basis in g(n, C) : P = gαβ Tα ⊗ Tβ =

1 1 ab In ⊗ In + g Ta ⊗ Tb . n c2

So, matrix r for s(n, C) is r = gab Ta ⊗ Tb = c2

 P−

1 In ⊗ In n

 ,

(4.7.113)

where (In ⊗ In ) and P are unit operator and operator of transposition (4.7.108). Let us now make use of matrix formulas (4.4.12), (In ⊗ In )ij11ij22 = δji11 δji22 , and write (4.7.113) in component form, gab (Ta )ij11 (Tb )ij22 = c2



(P )ij11ij22 = δji12 δji21 , δji12 δji21 −

1 i1 i2 δ δ n j1 j2

(4.7.114)

 .

(4.7.115)

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Note that by setting i1 = j1 in (4.7.115) and summing over j1 we get zero on the righthand side, in agreement with the fact that Tr(Ta ) = 0. Now, we choose i2 = j1 , i1 = j2 in (4.7.115) and sum over j1 and j2 . We get gab Tr(Ta Tb ) = c2 (n2 − 1). We make use of (4.7.44) and gab gab = dimC (s(n, C)) = (n2 − 1) to obtain c2 = c2 (T ), which is the expected result. Recall that the coefficient c2 (T ) defined in (4.7.44) relates metric Tr(Ta Tb ) and killing metric gab = Tr(ad(Xa )ad(Xb )) and is a characteristic of defining representation of s(n, C). By definition, matrix r does not depend on the choice of basis in s(n, C). Therefore, the explicit formula (4.7.113) is valid in any basis, including basis of any of the real forms of complex algebra s(n, C). In other words, the formula (4.7.113) gives explicit expression for split Casimir operator in defining representation of any of real forms of s(n, C), including su(n). Note that operator (4.7.113) has Ad-invariance property (4.7.111), since both (In ⊗ In ) and P obey (4.7.111) for any (n × n) matrices A. Example. Matrices τα = − 2i σα , where σα (α = 1, 2, 3) are Pauli matrices, make basis in Lie algebras su(2) and s(2, C) in defining representation. It follows from the normalization property Tr(τα τβ ) = − 12 δαβ = 14 gαβ , where gαβ is Killing metric, that   c2 = 14 . Then, making use of (4.7.113), we obtain r = 18 σα ⊗ σα = 14 P − 12 I2 ⊗ I2 , which gives the famous identity I2 ⊗ I2 + σα ⊗ σα = 2P, or δab δcd + (σα )ab (σα )cd = 2δad δcb . To end up this item, we note that the relations (4.7.102) for Lie algebras g(n, C) and s(n, C), with matrices r given by (4.7.108) and (4.7.113), are written in one and the same form: c2 (4.7.116) [P, (L1 − L2 )], [L1 , L2 ] = 2 (where c2 = 1 for g(n)). The convenient form of defining commutation relations (4.7.116) for Lie algebras s(n) and g(n) is obtained by renormalizing the generators, Lij → c2 Lij . Recalling (4.7.110), one writes [L1 , L2 ] = P (L1 − L2 ).

(4.7.117)

Problem 4.7.21. Show that the relation (4.7.117) is equivalent to defining commutation relations (3.3.4) for g(n, C). c. Matrices r for algebras so(n, C) and sp(2r, C) (2r = n) and their real forms Let us write definitions (3.1.42) and (3.1.25) for elements of algebras so(n, C) and sp(2r, C) in a unified way in terms of metric matrix c = ||cjk || : AT c + c A = 0 ⇔ (Ta )T c + c Ta = 0,

(4.7.118)

where A is an element of so(n, C) or sp(2r, C) (2r = n), and matrix c = ||cjk || is symmetric for so(n, C) and antisymmetric for sp(2r, C) : cT =  c,

(4.7.119)

( = +1 for so(n, C) and  = −1 for sp(2r, C)). Concrete choice of matrix c is c = η = Ip,q , see (3.3.37), for so(n, C) (and its real form so(p, q)) and c = J, as given by (1.2.66), for sp(2r, C) (and its real forms sp(2r, R), sp(p, q)). Let metric matrix with upper indices ¯mj cjk = δkm . Metrics cjk and c ¯mj are used to raise ||¯ cmj || be inverse of ||cjk ||, i.e., c and lower indices of any tensor, zj

k2 ...

= cjk1 z k1 k2 ... ,

¯mk1 zk1 k2 ... . z mk2 ... = c

(4.7.120)

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Note that indices are raised and lowered on the left. This is important for sp(2r, C) cmj are antisymmetric. because matrices cjk and ¯ The component form of relations (4.7.118) is ¯k1 i2 + Ai2k c ¯i1 k2 = 0. ck1 i2 Ak1i1 + ci1 k2 Ak2i2 = 0 ⇔ Ai1k c 1

2

(4.7.121)

¯i1 i2 , one can construct another Ad-invariant Making use of invariant metrics ci1 i2 and c matrix (this is special for so(n, C) and sp(2r, C) as compared to g(n, C) and s(n, C)) ¯i1 i2 cj1 j2 . K i1ji12j2 = c

(4.7.122)

Ad-invariance follows from the formulas (4.7.121), which give (cf. (4.7.111)) (A1 + A2 ) K = 0 = K (A1 + A2 ), where we again use shorthand notations A1 = A ⊗ In and A2 = In ⊗ A. Matrix K i1ji12j2 defines Ad-invariant operator K for so(n, C) and sp(2r, C) in defining representation, over ei ⊗ej ) = and beyond operators In2 ≡ In ⊗In and P, which acts is the standard way, K · ( em ) K km . One can show that the space of Ad-invariant operators for so(n, C) and ( ek ⊗  ij sp(2r, C) is 3-dimensional, with basis made of operators In2 , P and K. Thus, the most general Ad-invariant operator in the cases of Lie algebras so(n, C) and sp(2r, C) (2r = n) is a linear combination (4.7.123) αIn2 + βP + γK, where α, β and γ are arbitrary parameters. Let us make use of definitions (4.7.118) and (4.7.121) of Lie algebras so(n, C) and sp(2r, C) to construct bases in these algebras. We note that general solution to Eqs. (4.7.118), (4.7.121) reads cji , Ai k = E i k − ck E  j ¯

(4.7.124)

where E i k is an arbitrary matrix. To see this, we note that any matrix E i k can be decomposed as follows: Ei k =

1 (E i k 2

− ck E  j cji ) + 12 (E i k + ck E  j cji ) ≡ (E− )i k + (E+ )i k . (4.7.125)

We use this decomposition in Eq. (4.7.121) and find that E− is a solution of Eq. (4.7.121) and cki (E+ )k j = 0, or (E+ )k j = 0. Thus, the general solution to Eqs. (4.7.118), (4.7.121) is indeed given by (4.7.124). In what follows we need matrix units with one upper and one lower index. Their definition requires qualification. Let es r be matrix units such that (cf. (3.3.2) and (3.3.3)) (es r )i k = δkr δsi ,

es r ek i = δkr es i .

(4.7.126)

We recall the rules for raising and lowering indices, Eq. (4.7.120), and relation (4.7.119) to obtain (er s )k i = δkr δsi ,

¯ri csk , (er s )i k = c

¯ri csk , (es r )k i = c

(er s )i k (e j )k p =  δs (er j )i p ⇒ er s e j =  δs er j ,

(er s ej  )k p = δp ¯ crk csj = csj (er )k p ,

(4.7.127) (4.7.128)

¯r (esj )k p . ¯r cjp = c (es r e j )k p = δsk c (4.7.129)

Problem 4.7.22. Derive the latter relations from the definition (4.7.126).

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The distinction between products of matrix units es r and er s (second formulas in (4.7.126) and (4.7.128)) is due to the fact that our original definition of matrix multiplication is (A · B)i j = Ai k B kj (rather than (A · B)i j = Ai k Bk j ). This distinction is relevant for algebra sp(2r, C) only. We now replace an arbitrary matrix E in (4.7.124) by matrix unit es r . In this way we obtain basis element Ts r for so(n, C) or sp(2r, C) in defining representation (cf. (3.3.39) and (3.3.47)): ¯ji = (es r )i k − (es r )k i = δkr δsi −  csk c ¯ri . (Ts r )i k = (es r )i k − ck (es r ) j c (4.7.130) In virtue of (4.7.120) and (4.7.127), it can be written in equivalent form: (Ts r )i k = (es r )i k − (er s )i k ⇒ Ts r = es r −  er s . We define invariant metric gsr,j = standard way,

  crr  c gsjr 

(4.7.131)

in space so(n, C) (or sp(2r, C)) in a

cr ) c2 gsjr ≡ Tr(Ts r Tj  ) = (es r −  er s )i k (ej  −  e j )k i = 2(δjr δs −  csj ¯ ⇒ gsr,j =

1 2 Tr(Tsr Tj ) = (crj cs −  csj cr ) = − gsr,j = − grs,j c2 c2 (4.7.132)

Since metric ||gsr,j || acts in space of tensors φj with symmetry property φj = −φj , the inverse metric of (4.7.132) can be defined by the relation gsr,j gj,nm =

1 n m (δ δ − δsm δrn ). 2 s r

This gives explicitly

c2 ¯jn c ¯m ). ( ¯ cjm ¯ cn − c 8 Finally, we obtain the expression for the split Casimir operator in defining representation c2 ¯jm c ¯n − ¯ gj,nm Tj ⊗ Tnm = cjn ¯ ( c cm ) [ck (ej k −  ek j ) ⊗ cmp (enp −  ep n )] 8 c2 ¯jm c ¯n − ¯ cjn ¯ = ( c cm )ck ej k ⊗ cmp enp 2 c2 (e n ⊗ enk −  enk ⊗ enk ), = 2 k or, with and without indices, c2 i1 i2 rir11ir22 ≡ g j,nm (Tj )i1r1 (Tnm )i2r2 = ci1 i2 cr1 r2 ) (4.7.133) (δ δ −  ¯ 2 r2 r1 c2 ⇒ r ≡ gab (Ta ⊗ Tb ) = (P −  K), (4.7.134) 2 gj,nm =

¯i1 i2 cj1 j2 , where, in accordance with (4.7.122), the matrix of operator K reads K i1ji12j2 = c indices a and b denote pairs (j) and (nm), and coefficient c2 = c2 (T ) is given in (4.7.44) and characterizes defining representations T of algebras so(n, C) and sp(n, C) (or their real forms). Problem 4.7.23. Show that the identity (4.7.134) is consistent with Tr(Ta ) = 0 and symmetry property (cTa )T = − (cTa ), see (4.7.118). Check the normalization coefficient c2 /2 in (4.7.134). Hint: set i1 = r2 , i2 = r1 in (4.7.133), sum over r1 and r2 , and make use of equality gab Tr(Ta Tb ) = c2 dimC (A), where dimC (A) = n(n−) is complex dimension 2 of algebra A = so(n, C) (for  = +1) or algebra A = sp(n, C) (for  = −1).

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By inserting expression (4.7.134) for r-matrix into (4.7.102), we obtain defining commutation relations for generators Li k of algebras so(n, C) and sp(n, C) : c2 [L1 , L2 ] = (4.7.135) [ (P −  K), L1 − L2 ], 4 where  = +1 refers to so(n, C), and  = −1 to sp(n, C). Problem 4.7.24. Making use of the first of (4.7.102), write (4.7.135) in component form. Choose matrix c in definition (4.7.122) of operator K as symmetric metric η = Ip,q for so(p, q) and antisymmetric metric J = −J −1 given by (1.2.66) for sp(2r, K). Show that commutation relations (4.7.135) coincide with defining relations (3.3.41) and (3.3.48) for algebras so(p, q) and sp(n, K), provided that generators Ljs in (3.3.41) and Mjs in (3.3.48) are expressed through generators Lij in (4.7.135) via Ljs = c1 Lij cis and Mjs =

1 i Lj c2

2

cis .

4. Yangians Y (s(n)), Y (so(n)) and Y (sp(2r)) Let us now define new associative algebras called Yangians. As we already pointed out, these algebras are related to infinite-dimensional symmetries of integrable systems of quantum field theory and statistical mechanics. Proposition 4.7.8. Defining commutation relations (4.7.116) for Lie algebras g(n, C) and s(n, C) can be cast in the form of equation R12 (u − v) L1 (u) L2 (v) = L2 (v) L1 (u) R12 (u − v),

(4.7.136)

where u, v are arbitrary complex spectral parameters, L1 (u) = c2 u 1 + L1 ,

L2 (u) = c2 u 1 + L2 ,

(4.7.137)

1 = In ⊗ In ⊗ I is unit operator, operators L1 and L2 are definedh in (4.7.103) and R12 (u) := R(u) ⊗ I,

R(u) := uIn ⊗ In + P.

(4.7.138)

Matrix R(u) in (4.7.138) is called Yang’s R-matrix and obeys Yang–Baxter equation R12 (u) R13 (u + v) R23 (v) = R23 (v) R13 (u + v) R12 (u),

(4.7.139)

R12 (v) := (R(v) ⊗ In ), R23 (u) := (In ⊗ R(u)), R13 (u) := (P ⊗ In )(In ⊗ R(u))(P ⊗ In ). Proof. Let us again emphasize that Eq. (4.7.136) is understood as equality between two operators belonging to Matn (C) ⊗ Matn (C) ⊗ U (A). We move the right-hand side of Eq. (4.7.136) to the left and then make use of (4.7.116) to obtain for the left-hand side (we do not write unit operators 1 to simplify formulas) ((u − v) + P ) L1 (u) L2 (v) − L2 (v) L1 (u) ((u − v) + P ) = (u − v)(L1 (u) L2 (v) − L2 (v) L1 (u)) + (L2 (u) L1 (v) − L2 (v) L1 (u)) P = (u − v)[L1 , L2 ] + c2 (u − v)(L1 − L2 ) P 1 = (u − v)([L1 , L2 ] − c2 [P, (L1 − L2 )]) = 0. 2 c2 can be removed from (4.7.137) by renormalizing generators, Li k → c2 Li k in (4.7.116), or by dilating the spectral parameter, u → u/c2 , in (4.7.137), (4.7.138). h Coefficient

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Here the first and third equalities are obtained by employing properties (4.7.109), (4.7.110) of transposition matrix P, and the second equality uses the definition (4.7.137) of operator L(u). We leave it to the reader to check that matrices (4.7.138) obey the Yang–Baxter equation (4.7.139).

Problem 4.7.25. Show that RLL-relation (4.7.136) and Yang–Baxter equation (4.7.139) have the following equivalent component form: Ri1j1i2j2 (u − v) Ljk11 (u) Ljk22 (v) = Lij22 (v) Lij11 (u) Rj1kj12k2 (u − v),

(4.7.140)

Ri1j1i2j2 (u) Rjk11i3j3 (u + v) Rj2kj23k3 (v) = Ri2j2i3j3 (v) Rij11jk33 (u + v) Rj1kj12k2 (u), (4.7.141) where ir , jr , kr = 1, . . . , n, and R-matrix (4.7.138) and L-operator (4.7.137) are, explicitly, Ri1j1i2j2 (u) = uδji11 δji22 + δji12 δji21 ,

Lij11 (u) = c2 uδji11 + Lij11 .

(4.7.142)

Let us now consider formulas (4.7.136) and (4.7.140) with Yang matrix (4.7.138) as equations for operator L(u), and search for their solution in the form of series Lij (u) = δji +

∞  1 (1) i 1 1 u−α (E (α) )ij , (E )j + 2 (E (2) )ij + 3 (E (3) )ij + · · · = u u u α=0

(4.7.143) where (E (0) )ij = δji . We insert (4.7.143) into Eqs. (4.7.136) and (4.7.140) and get ⎞ ⎛ ∞     ∞  1 1 P (α) (β) ⎝ u−α E1 v−β E2 ⎠ − + v u vu α=0 β=0 ⎛ =⎝

∞ 

⎞ (β) v−β E2 ⎠

Equating terms proportional to (β−1) E2 ] (β−1)

= E2



+

u−α v−β ,

(β) [E2 ,

(α−1)

E1



 P . vu

(α)

1 1 − v u

(β−1)

(α, β = 1, 2, . . . ).

u−α E1

α=0

β=0

(α) [E1 ,

∞ 

+

we find

(α−1) E1 ] (α−1)

P − P E1

E2

(4.7.144)

Explicit component form of these equations, with transposition matrix P defined in (4.4.12), is [(E (α) )ij11 , (E (β−1) )ij22 ] + [(E (β) )ij22 , (E (α−1) )ij11 ] = (E (β−1) )ij21 (E (α−1) )ij12 − (E (α−1) )ij21 (E (β−1) )ij12 .

(4.7.145)

This gives a chain of quadratic commutation relations for operators {(E (α) )ij }. One defines infinite-dimensional associative algebra as space of all polynomials made of products of operators {(E (α) )ij } with structure relations (4.7.144) (or, in compact

form, (4.7.136)). Algebra with generators {(E (α) )ij }, defined in this way, is Yangian of g(n) type, denoted by Y (g(n)). To extract its subalgebra Y (s(n)) (Yangian of s(n)

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type), one imposes extra constraint [26] on generating function (4.7.143) for generators Y (g(n)): qdet(L(u)) = 1, (4.7.146) where qdet(L(u)) is quantum determinant of matrix ||Lij (u)|| defined as follows (cf. (1.2.4) and (1.2.12)): 1 εi ...i Li1 (u) Lij22 (u + 1) · · · Lijnn (u + n − 1)εj1 ...jn qdet(L(u)) = n! 1 n j1 =

1 i (u + 1) Lijnn (u) εj1 ...jn . εi ...i Li1 (u + n − 1) · · · L jn−1 n−1 n! 1 n j1 (4.7.147)

Here εi1 ...in = εi1 ...in is totally antisymmetric tensor of rank n defined in (1.2.5). Problem 4.7.26.  Let elements Lij (u) be generating functions (4.7.143) for generators

(E (α) )ij of Yangian Y (g(n)), i.e., elements Lij (u) obey Eqs. (4.7.136), (4.7.140) with R-matrix given by (4.7.138). Show that the operator qdet(L(u)) defined by (4.7.147) commutes with all generators of Yangian Y (g(n)): [qdet(L(u)), Lij (v)] = 0, ∀u, v, i, j, and hence belongs to the center of algebra Y (g(n)). Remark 2. Let us set α = 1 in (4.7.144). Then, recalling that E (0) = In , one finds the following relations for operators (E (β−1) )ij : (1)

(β−1)

[E1 , E2

(β−1)

] = E2

(β−1)

P − P E2

.

(4.7.148)

For β = 2 these coincide with defining relations of Lie algebra g(n, C), see (4.7.117). So, elements (E (1) )ij generate universal enveloping algebra U (g(n, C)), which is embedded in Y (g(n)) as subalgebra. Moreover, Proposition 4.7.8 implies (compare expansions (4.7.142) and (4.7.143) of L-operator), that one can choose (E (1) )ij = c1 Lij and 2

(E (α) )ij = 0 for all α > 1 in the solution (4.7.143). This gives the simplest nontrivial representation of Yangian Y (g(n)) with defining relations (4.7.144). Let us set α = 2 in (4.7.144). Then Eq. (4.7.148) gives (β)

E2

(β)

P − P E2

(2)

(β−1)

= [E1 , E2

(β−1)

] − E2

(1)

E1

(1)

P + P E1

(β−1)

E2

. (4.7.149)

This enables one to express all Yangian generators

(E (β) )ij

for β > 2 through generators

(E (1) )ij and (E (2) )ij . In other words, the whole Yangian Y (g(n)) is generated by elements of two matrices E (1) and E (2) . Remark 3. Yang–Baxter equation (4.7.141) admits remarkable graphical representation (see Fig. 4.7.1) due to A.B. Zamolodchikov. Directed lines in Fig. 4.7.1 are interpreted as world lines of particles in 2-dimensional space-time, and parameters u, v and (u + v) turn out to be angles between the lines. Every intersection of lines is associated with matrix i Rijk (u)

= j

>  Z  u Z Z  Z ~

k



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268

k3

v i1 i2

j2

u j1 i3

Fig. 4.7.1.

k3 k2

i1

=

j3

u+v k1

u+v

j3

v

j1

k2

u

j2 k1

i2 i3

Graphic representation of the Yang–Baxter equation (4.7.141).

This matrix describes two-particle scattering in which the particles change their initial states i and j to final states  and k. The right part of Fig. 4.7.1 is obtained from left part by parallel translation of the world line of one of the particles over the opposite vertex of the triangle. Both left and right parts of Fig. 4.7.1 describe threeparticle scattering that occurs as a sequence of 2-particle collisions. As an example, the first event in the left part is described by scattering matrix Rij11ij22 (u), the second one by

Rjk1 ij3 (u + v) and the last one by Rjk2 jk3 (v). The product of these matrices taken in this 1 3 2 3 order, together with sum over internal lines, gives the left-hand side of the Yang–Baxter equation (4.7.141). Its right-hand side is obtained in a similar way, as shown in the right part of Fig. 4.7.1. Analogous graphic representation exists for RLL-relation (4.7.140). Remark 4. Yang–Baxter equation (4.7.139) is the associativity condition for Yangian Y (g(n)). Indeed, relations (4.7.140) enable one to interchange the operators L(u) and L(v). Third-order monomials Ljk11 (u) Ljk22 (v) Lij33 (w) can be transformed into a combination of monomials Ljk1 (w) Ljk2 (v) Lij33 (u), with another ordering of spectral 1 2 parameters (u, v, w) → (w, v, u), in two ways: either one first interchanges L1 (u), L2 (v), then L1 (u), L3 (w) and finally L2 (v), L3 (w), or one makes the same permutation beginning with interchange of L2 (v) and L3 (w) (here 1, 2, 3 label vector spaces where matrices ||Ljk || act). Thus, making use of (4.7.136), we have L1 (u) L2 (v) L3 (w) = (R23 (v − w) R13 (u − w) R12 (u − v))−1 · L3 (w) L2 (v) L1 (u) R23 (v − w) R13 (u − w) R12 (u − v), L1 (u) L2 (v) L3 (w) = (R12 (u − v) R13 (u − w) R23 (v − w))−1 · L3 (w) L2 (v) L1 (u) R12 (u − v) R13 (u − w) R23 (v − w). (4.7.150) Since, for associative algebra, the result must be independent of the order of permutations (compare with the derivation of the Jacobi identity for Lie algebras in the proof of Proposition 3.2.1), the comparison of the right-hand sides of equations (4.7.150) gives third-order relation for R-matrix (an analog of the Jacobi identity (3.2.32)) R12 (u − v) R13 (u − w) R23 (v − w) = R23 (v − w) R13 (u − w) R12 (u − v), which, modulo notations for spectral parameters, coincides with the Yang–Baxter equation (4.7.139). ˆ (g(n)) via Remark 5. Let us define mapping Δ: Y (g(n)) → Y (g(n))⊗Y ˆ kj (u). Δ(Lij (u)) = Lik (u)⊗L

(4.7.151)

˜ i (u) = Δ(Li (u)) obey the same commutation It is straightforward to see that matrices L j j relations (4.7.136), (4.7.140), so mapping Δ is homomorphism. Mapping (4.7.151)

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obeys coassociativity axiom (4.7.77) and hence defines coproduct in Yangian Y (g(n)). Moreover, one defines co-unity in Y (g(n)) by setting (Lij (u)) = δji . Problem 4.7.27. Show that all axioms (4.7.77) and (4.7.78) are satisfied for coproduct (4.7.151) and co-unity  of Yangian Y (g(n)). Thus, in accordance with Definition 4.7.5, Yangian Y (g(n)) has all properties of bialgebra. Mapping Δ defines the Yangian generators in direct product of its representations. Note that coproduct (4.7.151) for Yangian is not co-commutative, i.e., ˆ k (u) = Lk (u)⊗L ˆ i (u), unlike coproduct (4.7.69) in algebra Lie case. Making Li (u)⊗L k

j

j

k

use of quantum determinant (4.7.147), one can construct matrix ||(L−1 )kj (u)||, inverse of ||Lkj (u)|| (cf. (1.2.14)): 1 εi ...i Li2 (u + 1) · · · Lijnn (u + n − 1)εj1 ...jn . qdet(L(u)) (n − 1)! 1 n j2 (4.7.152) Consider now mapping S of algebra Y (g(n)) to itself, defined by (L−1 )j1i1 (u) =

S(Lij (u)) = (L−1 )ij (u)

(4.7.153)

(extension to the whole algebra Y (g(n)) is made by using the first of Eqs. (4.7.80)). This mapping obeys all axioms (4.7.79), (4.7.80) for antipode in bialgebra Y (g(n)). Thus, Yangian Y (g(n)) is actually Hopf algebra. Problem 4.7.28. Check that antipode (4.7.153) obeys all axioms listed in Definition 4.7.6. Problem 4.7.29. Show that the Yang–Baxter equation (4.7.139) is satisfied by the following R-matrix [24,25], which is Ad-invariant (see (4.7.123)) under so(n) and sp(2r) (2r = n):     n n (4.7.154) R(u) = u u + −  In2 + u + −  P −  u K. 2 2 Here operator K is defined in (4.7.122);  = +1 for so(n), and  = −1 for sp(2r). Define Yangians Y (so(n)) and Y (sp(2r)) through R-matrix (4.7.154) and relations (4.7.136), (4.7.140). By using expansion (4.7.143) in these relations, derive analogs of structure relations (4.7.144) for generators of Yangiansi Y (so(n)) and Y (sp(2r)): (α)

(β−2)

(α−1) (β−1) (α−2) (β) ] − 2[E1 , E2 ] + [E1 , E2 ] (α−1) (β−2) (α−2) (β−1) , E2 ] − [E1 , E2 ]) + b ([E1 (α−1) (β−2) (α−2) (β−1) (α−2) (β−2) + P (E1 E2 − E1 E2 + b E1 E2 ) (β−2) (α−1) (β−1) (α−2) (β−2) (α−2) E1 − E2 E1 + b E2 E1 )P − (E2 (α−2) (β−1) (α−1) (β−2) + (K (E1 E2 − E1 E2 ) (β−1) (α−2) (β−2) (α−1) E1 − E2 E1 ) K) = 0, − (E2

[E1 , E2

(4.7.155)

−  and n = 2r for Y (sp(2r)). Show that enveloping algebras U (so(n)) and where b = n 2 U (sp(2r)) are subalgebras of Y (so(n)) and Y (sp(2r)), respectively. Hint: write relations (4.7.155) for α = β = 2 and compare the result with (4.7.135).

i Yangians

of so- and sp-types are called extended, since they contain infinite-dimensional central subalgebra, see Problem 4.7.30. To remove this center, one sets equal to 1 the generating function of central elements z(u), see (4.7.156).

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Problem 4.7.30.  Let elements Lij (u) be generating functions (4.7.143) for generators

(E (α) )ij of Yangians Y (so(n)) and Y (sp(2r)), i.e., let Lij (u) obey Eqs. (4.7.136), (4.7.140) with R-matrix (4.7.154). Consider operator z(u) defined by z(u) K i1ji12j2 =  n Lik1 (u) Lik2 (u − b) K kj11kj22 =  n K i1ki2k Lkj11 (u − b) Lkj22 (u), 1

2

1 2

(4.7.156) n 2

where b = − (n = 2r for algebra Y (sp(2r))). Show that z(u) is generating function for central elements of Yangians Y (so(n)) and Y (sp(2r)). Detailed discussion (see [12]) of infinite-dimensional algebras Y (g(n)), Y (s(n)), Y (so(n)) and Y (sp(2r)) and their representations is beyond the scope of this book. We note only that Yangians were introduced in mid-1980’s by V.G. Drinfeld [26] who studied algebraic structures that emerged in the framework of quantum inverse problem method. This method was proposed and developed by L.D. Faddeev and collaborators for solving nontrivial models of quantum field theory and statistical physics, see, e.g., [27, 28] and references therein. R-matrices (4.7.138), (4.7.154) and relations (4.7.139), (4.7.136) and (4.7.140) are cornerstones of many of these models.

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Chapter 5

Compact Lie Algebras

5.1.

Definition and Main Properties of Compact Lie Algebras

We turn to the discussion of an important class of compact Lie algebras. One might think that it would be more appropriate to study them in Section 3 dedicated to general Lie algebras. However, the presentation of some facts concerning compact Lie algebras requires knowledge of basic notions of representation theory. This is the reason for postponing the discussion of compact Lie algebras to this Chapter. Compact Lie groups (see Definition 2.1.9) have been studied in Section 2.1.4. Compact Lie algebras are defined as Lie algebras of compact Lie groups. We recall that Lie algebra, constructed as tangent space of a Lie group manifold, is always real. Therefore, compact Lie algebras are real by definition. In what follows we use another definition of compact Lie algebra, which does not make explicit reference to compact Lie group, but, as we show below, is equivalent to that given above. Definition 5.1.1. Lie algebra A is compact, if it is real and admits positive-definite, nondegenerate scalar product which is invariant under adjoint action in A (adjoint action in A is defined by Eq. (3.2.66)). In other words, real Lie algebra A is compact, if there exists real symmetric bilinear form (A, B), such that for any A ∈ A one has (A, A) ≥ 0,

(5.1.1)

and equality here occurs for zero element A = 0 only. Furthermore, the following equality must hold for any three elements A, X, Y ∈ A (compare 271

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with (3.2.67) and (4.2.7)): ([A, X], Y ) + (X, [A, Y ]) = 0,

(5.1.2)

(ad(A) · X, Y ) + (X, ad(A) · Y ) = 0.

(5.1.3)

or

The invariance of scalar product under adjoint action in Lie algebra A is precisely the validity of relations (5.1.2), (5.1.3). Problem 5.1.1. Show that the relations (5.1.2), (5.1.3), written for basis elements Xa of Lie algebra A, are equivalent to invariance condition (4.7.34) of tensor dab under co-adjoint action in A. Let G be compact Lie group, and A(G) its (real) Lie algebra. Since A(G) is the space of adjoint representation of group G, it admits positive-definite nondegenerate scalar product invariant under the action of G: this is true for any real representation of compact group G, see remark to Proposition 4.5.2 in Section 4.5. This scalar product obeys (ad(gA (t)) · X, ad(gA (t)) · Y ) = (X, Y ),

(5.1.4)

where gA (t) is a curve in the group with tangent vector A ∈ A(G) at unit element. Adjoint representations of Lie group and its Lie algebra are related by (4.2.16), i.e., one has at small t ad(gA (t)) · X = X + t · ad(A) · X + O(t2 ), ad(gA (t)) · Y = Y + t · ad(A) · Y + O(t2 ).

(5.1.5)

Inserting (5.1.5) in (5.1.4) gives (5.1.3) at small t, so the property (5.1.3) is indeed inherent in Lie algebras of compact Lie groups. Thus, Lie algebras of compact Lie groups are compact in the sense of Definition 5.1.1. Remarkably, the opposite statement is true as well: for every compact, in the sense of Definition 5.1.1, Lie algebra A there exists compact Lie group whose algebra is A. We prove the latter statement in Section 5.3. The conclusion is that Definition 5.1.1 is equivalent to the definition of compact Lie algebra as an algebra of compact Lie group. The existence of invariant, positive-definite scalar product in Lie algebra is of great importance for gauge theories, the major building block of the Standard Model of elementary particle physics and its extensions. This is the reason for using compact Lie groups and algebras when constructing gauge theories.

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Since compact Lie algebra can always be thought of as algebra of compact Lie group, the results of Section 4.5 have their obvious analogs in representation theory of compact Lie algebras. Namely, the following propositions hold. Proposition 5.1.1. Reducible complex representation of compact Lie algebra is completely reducible. Proposition 5.1.2. Any complex representation of compact Lie algebra is equivalent to anti-Hermitian. The first of these propositions tells that any representation of compact Lie algebra is partitioned into direct sum of irreducible representations. The second proposition says that the space V of representation T of compact Lie algebra A can always be endowed with positive-definite Hermitian form such that T (A) · x , y  = −x , T (A) · y , for all x, y ∈ V and A ∈ A. We comment on this property in Section 5.3. Examples 1. According to Cartan’s criterion, every real semisimple Lie algebra A has invariant, nondegenerate scalar product (3.2.64), determined by the Killing metric gab given in (3.2.62). One recalls (4.2.18) and writes for this scalar producta (X, Y ) = −Tr(ad(X) · ad(Y )),

∀X, Y ∈ A.

(5.1.6)

Thus, if the Killing metric, and hence scalar product (5.1.6), is nondegenerate and positive-definite, then the Lie algebra is semisimple and compact. Conversely, the Killing form is positive-definite for any semisimple compact Lie algebra (see Proposition 5.1.3 in this regard). More general statement that the scalar product (5.1.6) is nondegenerate and positive-definite for any compact Lie algebra, is incorrect. As an example, compact Abelian Lie group U (1)×U (1)×· · ·×U (1) has compact Lie algebra (by definition), but its Killing metric is zero, so that the scalar product (5.1.6) is degenerate. a It

is convenient for what follows to change sign here as compared to (3.2.64).

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2. For any real matrix Lie algebra A, the invariant scalar product is the trace: (A, B) ≡ −2 Tr(A · B),

∀A, B ∈ A.

(5.1.7)

This agrees with the definition of metric (2.2.29) of matrix Lie group at unit element. Scalar product (5.1.7) is the special case of the scalar product (4.2.6), where one sets α = −2 for convenience, and chooses defining representation T of matrix Lie algebra A. The sign in (5.1.7) ensures that the scalar product is positive-definite for compact Lie algebra (see below). The invariance condition (5.1.3) is satisfied for the scalar product (5.1.7); this follows from the general result (4.2.8). Nontrivial part of Definition 5.1.1 in matrix Lie algebra case is positivedefiniteness (5.1.1) of the scalar product (5.1.7). The latter property is inherent in compact and only compact matrix Lie algebras. For simple compact matrix Lie algebra, scalar products (5.1.6) and (5.1.7) coincide modulo an overall numerical factor. Problem 5.1.2. Show that (A, A) = −2Tr(A2 ) is positive for all nonzero elements A of algebras so(n, R), su(n) and usp(2k). Show that, depending on A, this scalar product can be both positive and negative for algebras s(2, R) and A6R = A(SL(2, C)). Set of basis elements {Xa } in any compact (not necessarily matrix) Lie algebra A can be chosen orthonormal, (Xa , Xb ) = δab .

(5.1.8)

This is consistent with the standard normalization of generators in matrix algebras (cf. (3.3.21)) Tr(Xa · Xb ) = − 21 δab .

(5.1.9)

Problem 5.1.3. Making use of (5.1.2), show that with orthonormal basis obeying (5.1.8), structure constants of compact Lie algebra A are antisymmetric in upper and lower indicesb d b Cab = −Cad .

(5.1.10)

c , where g constants Cabd = gdc Cab dc is the Killing metric, are always antisymmetric in all three indices, see (3.2.71). However, the latter property by itself, does not imply (5.1.10).

b Structure

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Thus, there is basis in compact Lie algebra, in which its structure constants are antisymmetric in the sense (5.1.10). On the other hand, the following proposition holds. Proposition 5.1.3. If structure constants of real semisimple Lie d b = −Cad , then the scalar algebra A can be chosen antisymmetric, Cab product (5.1.6) is positive-definite, and algebra A is compact. d = Proof. Let structure constants of Lie algebra A be antisymmetric, Cab b −Cad . Then matrices of operators in adjoint representation ad(A), (A = Aa Xa ∈ A) are also antisymmetric, see (3.2.68): d b = −Aa Cad = −ad(A)bd . ad(A)db = Aa Cab

(5.1.11)

So, the scalar product (5.1.6) defined by Killing form (3.2.64) is invariant, nondegenerate (since A is semisimple) and positive-definite: (A, A) = −Tr(ad(A) · ad(A)) = −ad(A)db ad(A)bd = ad(A)db ad(A)db > 0,

∀A = 0.

(5.1.12)

In accordance with Definition 5.1.1, this means that Lie algebra A is compact. 5.2.

Structure of Compact Lie Algebras

Structure of compact Lie algebras is given by the following proposition. Proposition 5.2.1. Any compact Lie algebra A is a direct sum of a certain number of Abelian subalgebras u(1) and simple compact Lie algebras Ai (i = 1, 2, . . . , n), A = u(1) + · · · + u(1) + A1 + · · · + An .

(5.2.1)

This partition is unique. Proof. We begin with proving that any Abelian ideal I in compact Lie algebra A commutes with the whole algebra, i.e., I belongs to center Z of algebra A. Let us take X, Y ∈ I and A ∈ A. Then [X, Y ] = 0, and it follows from (5.1.2) that ([X, A], Y ) = 0.

(5.2.2)

Element [X, A] belongs to I, since X ∈ I and I is ideal. Let us choose Y = [X, A], then, in view of (5.2.2) we have (Y, Y ) = 0. Since the scalar

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product is not degenerate for compact Lie algebra, we have Y = [X, A] = 0, ∀X ∈ I, and hence I ⊂ Z, as desired. This shows that center Z in A is maximal Abelian ideal in A. Now, algebra A is decomposed into direct sum A = Z + Z ⊥,

(5.2.3)

where Z ⊥ is orthogonal complement in A to center Z. Vector space Z ⊥ is Lie subalgebra of A, since for A1 , A2 ∈ Z ⊥ and Z ∈ Z, one has from (5.1.2) that ([A1 , A2 ], Z) = −(A2 , [A1 , Z]) = 0

⇒

[A1 , A2 ] ∈ Z ⊥ .

Algebra Z ⊥ does not contain Abelian ideals by construction (otherwise Z could be extended), so Z ⊥ is semisimple or simple. Obviously, both subalgebras Z and Z ⊥ are compact. Semisimple compact Lie algebra Z ⊥ can in turn be decomposed into direct sum of simple compact Lie algebras. Indeed, let A¯ be ideal in Z ⊥ . Its orthogonal complement A¯⊥ in Z ⊥ is also ideal in Z ⊥ , since if A ∈ Z ⊥ , X ∈ A¯ and Y ∈ A¯⊥ , then one has from (5.1.2) that (X, [A, Y ]) = −([A, X], Y ) = 0

⇒

[A, Y ] ∈ A¯⊥ .

¯ A¯⊥ ] ⊂ A¯⊥ Ideals A¯ and A¯⊥ commute with each other, since from [A, ⊥ ⊥ ¯ ¯ ¯ ¯ ¯ and [A, A ] ⊂ A it follows that [A, A ] = 0. Therefore, we obtain the decomposition Z ⊥ = A¯ + A¯⊥ , where A¯ and A¯⊥ are semisimple subalgebras of Z ⊥ . This process continues until Z ⊥ is partitioned into direct sum of simple subalgebras A¯i . Uniqueness of the partition  A¯i (5.2.4) Z⊥ = i

is shown by the following argument. Let there be another decomposition into direct sum of simple subalgebras,  Aˆα . Z⊥ = α

Subspace A¯i ∩ Aˆα is subalgebra of Z ⊥ and, furthermore, it is ideal in Z ⊥ (since if A ∈ A¯i and A ∈ Aˆα , then for all B ∈ Z ⊥ one has [A, B] ∈ A¯i and

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[A, B] ∈ Aˆα ). Since A¯i is simple, one has either A¯i ∩ Aˆα = A¯i and hence Aˆα = A¯i , or A¯i ∩ Aˆα = ∅. This proves uniqueness of decomposition (5.2.4). Finally, any compact Abelian subalgebra Z is direct sum of a certain number of u(1) subalgebras (the number of these equals dimension of algebra Z). This proves that (5.2.3) yields the decomposition (5.2.1). Remark 1. Proposition 5.2.1 is generalized in the following way. Any complex Lie algebra A(C) which has compact Lie algebra A as one of its real forms, is represented in a unique way as direct sum of maximal Abelian ideal Z(C) and semisimple complex Lie subalgebra Z ⊥ (C). The latter is complexification of compact semisimple Lie algebra Z ⊥ (complex Lie algebra Z ⊥ (C) is semisimple, since it inherits nondegenerate Killing metric of its real form Z ⊥ ). Complex semisimple Lie algebra Z ⊥ (C) is, in its turn, direct sum of its complex subalgebras Ai (C), which are complexifications of simple compact Lie algebras Ai . The latter property is due to the fact that Z ⊥ (C) has invariant, nondegenerate and positive-definite Hermitian form inherited from nondegenerate positive-definite scalar product in Z ⊥ . Remark 2. Lie algebra A is reductive if its adjoint representation is completely reducible. This means that the space of adjoint representation, i.e., algebra A itself, is direct sum of invariant subspaces Xi : A = X1 + X2 + X3 + · · · ,

(5.2.5)

and each space Xi is a space of irreducible representation of algebra A. Invariance of subspace Xi under adjoint action of A implies that [A, Xi ] ⊂ Xi , which gives, in particular, [Xi , Xi ] ⊂ Xi . Thus, subspaces Xi ⊂ A are Lie subalgebras of A. On the other hand, it follows from invariance of two spaces Xj and Xk that [Xk , Xj ] ⊂ Xj and [Xj , Xk ] ⊂ Xk , so [Xj , Xk ] = 0, and formula (5.2.5) gives decomposition of reductive Lie algebra A into direct sum of its Lie subalgebras Xi . 1-dimensional subalgebras Xi in (5.2.5) make center Z in A, while other subalgebras in (5.2.5), which we denote by Ai , are simple, otherwise they would not be subspaces of irreducible subrepresentations of adjoint representation of A. Thus, the decomposition (5.2.5) actually is A = Z + A1 + A2 + · · ·

(5.2.6)  where center Z is direct sum of 1-dimensional Abelian subalgebras of A, and i Ai is direct sum of simple subalgebras of A. We compare this decomposition with (5.2.1) and conclude that compact Lie algebras are an important subclass of reductive Lie algebras.

Before coming to the next proposition, we note that complexification of simple real Lie algebra may not be simple. An example is simple real Lie algebra so(1, 3) (isomorphic to simple algebra A6R = A(SL(2, C)) of Lie group SL(2, C)), whose complexification so(4, C) = s(2, C) + s(2, C) is not simple. On the other hand, if complex Lie algebra is simple, then all its real forms are necessarily simple.

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Proposition 5.2.2. Every real form A of simple complex Lie algebra A(C) has one and only one invariant scalar product (modulo numerical prefactor) which is the scalar product (5.1.6) defined by Killing form (3.2.64) of algebra A. Proof. Let A be real form of simple complex Lie algebra A(C), and (X, Y ) be an invariant scalar product in A. Invariance condition (5.1.3) can be cast in the following form: ad(A)ab (Xa , Xd ) = −(Xb , Xc ) ad(A)cd ⇒ ad(A)T · η = −η · ad(A),

(5.2.7)

where {Xa } is basis in Lie algebra A and η = ||(Xa , Xd )||. Therefore, matrix η obeys the same invariance condition (4.7.35) as Killing metric g = ||gab ||. According to Proposition 4.7.3 (it is important here that complexification of A is simple algebra), matrix η is proportional to ||gab ||, which completes the proof. If compact Lie algebra is semisimple, then all invariant scalar products are listed as follows. Let, as an example, the algebra be direct sum A = A1 + A2 , where A1 and A2 are simple Lie algebras. Any element X ∈ A can be written as X = X1 + X2 , where X1 ∈ A1 , X2 ∈ A2 . Let ( , )1 and ( , )2 be invariant scalar products in A1 and A2 , respectively. Then the most general invariant scalar product in A is (X, Y ) = α1 (X1 , Y1 )1 + α2 (X2 , Y2 )2 ,

(5.2.8)

where α1 , α2 are arbitrary positive numbers. In other words, quadratic invariants (under adjoint action) in A are linear combinations of quadratic invariants in each of the simple subalgebras. 5.3.

Relation of Compact Lie Algebras to Compact Lie Groups

In this section we finally establish the relation of compact Lie algebras to compact Lie groups. Namely, we prove Proposition 5.3.2 below. Before doing that we define a convenient notion of differentiation in Lie algebra. Consider set X (A) of linear operators X acting in Lie algebra A and obeying X · ([A, B]) = [X · A, B] + [A, X · B],

∀A, B ∈ A.

(5.3.1)

These operators are called differentiations in Lie algebra A: the property (5.3.1) is similar to Leibnitz rule, if [A, B] is viewed as product. Problem 5.3.1. Show that the set of differentiations X (A) is Lie algebra.

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By comparing (4.2.15) and (5.3.1) one observes that operators ad(Y ), where Y ∈ A, are differentiations in Lie algebra A. We recall (see (3.3.53)) that the set of operators ad(Y ), where Y runs over Lie algebra A, is adjoint Lie algebra ad(A). Proposition 5.3.1. For any compact semisimple Lie algebra A, its Lie algebra of differentiations X (A) is isomorphic to adjoint Lie algebra ad(A). Proof. It is obvious that adjoint algebra ad(A) is embedded in X (A). Let us define quadratic form in X (A): (X1 , X2 ) = −Tr(X1 X2 ),

∀X1 , X2 ∈ X (A).

(5.3.2)

This form is symmetric, but it can be degenerate and/or not positive definite. However, its restriction to subalgebra ad(A) coincides with nondegenerate and positive definite Killing form of compact semisimple algebra A, so algebra X (A), as vector space, is decomposed into direct sum X (A) = ad(A) + X ⊥ , where X ⊥ is orthogonal complement to ad(A) in X (A), constructed with form (5.3.2). Indeed, every element Y ∈ X (A) can be written as Y = (Y, ei ) ei + Y ⊥ ,

Y ⊥ ≡ Y − (Y, ei ) ei ,

where ei is orthonormal basis in ad(A). Since for all j one has (Y ⊥ , ej ) = (Y − (Y, ei ) ei , ej ) = 0, then Y ⊥ ∈ X ⊥ . Now, if X ∈ X ⊥ , then for any A ∈ A, element [X, ad A] also belongs to X ⊥ , since for all A, B ∈ A we have Tr([X, ad A] · ad B) = Tr(X · [ad A, ad B]) = 0. Here we use the fact that [ad(A), ad(B)] ∈ ad(A). On the other hand, we have [X, adA] = ad(X · A). This follows from the chain of equalities (Xad A − (ad A)X)B = X([A, B]) − [A, XB] = [XA, B] + [A, XB] − [A, XB] = [XA, B]. Hence, the operator Z = ad(X · A) belongs to both X ⊥ and ad(A), and so the norm (Z, Z) (understood in the sense of Killing form in ad(A)) vanishes. This implies that Z = ad(X · A) also vanishes, i.e., (X · A) ∈ Ker(ad). According to Proposition 3.3.3, mapping A → ad(A) is one-to-one (Ker(ad) is trivial), so element (X · A) vanishes. This is true for all A, so operator X equals zero. Orthogonal complement to ad(A) is absent, algebra X (A) coincides with ad(A), and any operator with property (5.3.1) is an operator from algebra ad(A). We now formulate and prove the main proposition of this section. Proposition 5.3.2. For any compact Lie algebra A, there exists compact Lie group G whose Lie algebra is A. Proof. Consider decomposition (5.2.3). Center Z is Lie algebra of a product of compact Lie groups U (1), so we have only to consider semisimple Lie algebra A⊥ and show that ¯ whose Lie algebra is A(G) ¯ = A⊥ . Then the desired there exists compact Lie group G ¯ compact Lie group is G = U (1) ⊗ · · · ⊗ U (1) ⊗ G.

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¯ we consider all linear operators gˆ acting in A⊥ To construct compact Lie group G, and obeying (ˆ g · A, gˆ · B) = (A, B), [ˆ g · A, gˆ · B] = gˆ · [A, B],

(5.3.3)

∀A, B ∈ A⊥ ,

(5.3.4) A⊥ .

where ( , ) is invariant, positive-definite, nondegenerate scalar product in The set ˜ where multiplication is, as usual, consecutive action. of operators gˆ is Lie group G, ˜ is a subgroup of O(n) where n is dimension of algebra A⊥ , Because of (5.3.3), group G ˜ is compact. Now, it follows from (5.3.4) that elements X of Lie algebra therefore G ˜ of group G ˜ (these are also operators acting in A⊥ ) obey (5.3.1). In accordance A(G) with Proposition 5.3.1, any of these operators has the form X = ad(C), where C ∈ A⊥ , ˜ is embedded in ad(A⊥ ). Conversely, an element C ∈ A⊥ defines an so Lie algebra A(G) operator exp(t · ad(C)), at least for small t (in a given basis in A⊥ , operator ad(C) is represented by its matrix, so exp(t · ad(C)) can be thought of simply as exponential of ˜ a matrix). Operator exp(t · ad(C)) has properties (5.3.3), (5.3.4), so ad(C) ∈ A(G), ˜ and hence ad(A⊥ ) = A(G). ˜ As we already pointed out (see i.e., ad(A⊥ ) ⊂ A(G)

Proposition 3.3.3), semisimple Lie algebra A⊥ and its adjoint algebra ad(A⊥ ) are ˜ of compact Lie group G, ˜ constructed isomorphic, so we conclude that Lie algebra A(G) ˜ is the desired compact via (5.3.3), (5.3.4), is isomorphic to A⊥ . Hence, Lie group G ¯ group G. Let A be compact Lie algebra. It can be decomposed into direct sum (5.2.1). Therefore, the compact Lie group whose algebra is A, is G = U (1) × · · · × U (1) × G1 × · · · × Gn , where Gi are simple compact Lie groups. Groups Gi can be chosen simply connected, i.e., universal covering (see proposition (3) in the beginning of Section 3.5). We know that any representation of algebra Ai = A(Gi ) descends from representation of universal covering Gi , see Proposition 4.2.2. This observation and propositions of Section 4.5 yield Propositions 5.1.1, 5.1.2, once one recalls that T (gA (t)) = 1 + t · T (A) + O(t2 ) for small t.

Remark. Consider noncompact Lie group G. If it had faithful finite-dimensional unitary representation T, then arguments similar to those used in proving Proposition 5.3.2 would tell that G is compact,c in contradiction to the original assumption. Therefore, faithful unitary representations of noncompact groups must be infinite-dimensional. We emphasize that noncompact groups may have finite-dimensional unitary representations, which necessarily are not faithful. A simple example is trivial representation.

finite-dimensional unitary representation T defines isomorphism of G to matrix group H whose matrices are unitary. The latter is a subgroup of unitary group U (n), which is compact for finite dimension of representation n. Thus, H = G is compact. c Faithful

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Root Systems and Classification of Simple Lie Algebras

6.1.

Cartan Subalgebra. Rank of Lie Algebra and Cartan–Weyl Basis

We continue to study the structure of compact Lie algebras. According to Proposition 5.2.1, any compact Lie algebra is direct sum of Abelian u(1) subalgebras and simple Lie algebras. Further analysis of compact Lie algebras goes along the study of simple compact Lie algebras. Before coming to this study, we state without proof an important proposition that enables one to reduce the study of compact semisimple Lie algebras to the analysis of complex semisimple Lie algebras. Proposition 6.1.1. Every complex semisimple Lie algebra AC has one and only one compact real form A. Here uniqueness means that any two compact real forms of AC are isomorphic and related to each other by an inner automorphism in AC . Proposition 6.1.1 enables one to formulate the second part of Remark 1 in Section 5.2 in a stronger way: any complex semisimple Lie algebra is decomposed into direct sum of its simple Lie subalgebras, and each of these complex simple subalgebras has unique simple compact real form. On the one hand, this means that complete classification of simple complex finite-dimensional Lie algebras (to be studied below in Section 6.2.5) is sufficient to describe all semisimple complex finite-dimensional Lie algebras. On the other hand, this is the classification of compact Lie algebras. We show in this section that compact simple Lie algebra has 281

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a special basis called Cartan–Weyl basis. This basis plays a key role in the study of simple complex Lie algebras, and hence simple compact Lie algebras. Another fact useful for physics applications and related to Proposition 6.1.1 is that compact real form of complex semisimple Lie algebra AC is maximal compact subalgebra of AC , i.e., any compact real subalgebra of AC is embedded, as subalgebra, in its compact real form. 6.1.1.

Regular elements. Cartan subalgebra and rank of Lie algebra

Let A be simple complex Lie algebra. Let us choose an element A ∈ A and consider operator ad(A) in adjoint representation, ad(A)(X) = [A, X]∀X ∈ A. Operator ad(A) has zero eigenvalue, e.g., for X = A. The zero eigenvalue may be degenerate. The set of zero eigenvectors AA , i.e., elements X ∈ A such that [A, X] = 0, is Lie subalgebra of A. Problem 6.1.1. Show that AA is Lie subalgebra of A. Dimension of subalgebra AA , or, in other words, multiplicity rA of zero eigenvalue of operator ad(A), depends on the choice of A. Minimum value of this multiplicity, i.e., the value of rA such that rA ≤ rB , ∀B ∈ A, is rank of Lie algebra A. It is denoted by rank(A), and the corresponding elements A ∈ A are called regular elements of Lie algebra A. For a regular element A, Lie subalgebra AA in A is called regular subalgebra. All regular subalgebras AA (with different regular elements) are isomorphic to each other and are related by inner automorphisms in A. We also state the following proposition without proof (see [3, §62]). Proposition 6.1.2. Let A be simple complex Lie algebra. Then its regular subalgebra is commutative. Definition 6.1.1. Commutative regular subalgebra H in A is Cartan subalgebra. In accordance with the above, dimension of H equals rank of algebra A. Example. Consider simple Lie algebra s(n, C), which is complexification of compact algebra su(n). Let us choose the matrix B = e11 − e22 ∈ s(n, C), where ekj are matrix units, as a candidate for regular element. To find zero eigenvectors of ad(B), consider an arbitrary traceless n × n matrix X = xkj ekj ∈ s(n, C). We require that [B, X] = 0 and find that all coefficients xkj are arbitrary, except for x1k = xk1 = 0

∀k = 1,

x2k = xk2 = 0

∀k = 2.

Set of elements X with this property is the subalgebra s(n, C)B ⊂ s(n, C) of zero eigenvectors of operator ad(B). Dimension of s(n, C)B (i.e., multiplicity rB of zero eigenvalue) is equal to the number of arbitrary elements xjk , i.e., rB = (n − 2)2 + 1. This dimension is minimal for n = 2, 3 and not minimal for n > 3. Problem 6.1.2. Show that subalgebras s(2, C)B and s(3, C)B , where B = e11 − e22 , are commutative.

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2 not minimal for n > 3, let us choose another element To check  that rB = (n − 2) + 1 is  n a e , where a A= n k kk k = 0, k=1 k=1 ak = 0 and ak = aj for all k = j. An element X with zero eigenvalue of ad(A) obeys [A, X] = 0, which gives (ak − aj )xkj = 0 (no summation over k and  j), so that xkj = 0 for k = j, and xkk are arbitrary modulo the trace constraint k xkk = 0. So, dimension of algebra s(n, C)A (multiplicity rA ) equals rA = (n − 1). We have rB > rA for n > 3, as promised. One can show that multiplicity rA = (n − 1) is minimal, so it is equal to the rank of algebra s(n, C), and regular subalgebra (Cartan subalgebra) H = s(n, C)A is commutative and is generated by traceless diagonal matrices. Cartan subalgebra of su(n) has the same properties, but now elements of the diagonal matrices are pure imaginary.

Remark. Somewhat different but equivalent definitions of regular elements and Cartan subalgebra are given in [13].

6.1.2.

Cartan–Weyl basis

In the first place, let us give another definition of Cartan subalgebra, which, however, is equivalent to Definition 6.1.1. Definition 6.1.2. Let A be semisimple complex Lie algebra. Let us single out commutative subalgebra H ⊂ A with generators {Hi } (i = 1, 2, . . . , r), such that 1. [Hi , Hj ] = 0 (∀i, j)), and H is maximal commutative subalgebra of A, i.e., it cannot be extended by adding other elements of A commuting with Hi ; 2. matrices of operators ad(Hi ) are simultaneously diagonalizable for all i by appropriate choice of basis in A. This maximal commutative subalgebra H in A is Cartan subalgebra whose dimension dim(H) = r is rank of Lie algebra A. Cartan subalgebra H in A is unique, in the sense that any two Cartan subalgebras in A defined according to Definition 6.1.2 are isomorphic and are related by an automorphism in A. Definition 6.1.3. Let A be semisimple compact Lie algebra (which is by definition real). Cartan subalgebra H ⊂ A is maximal commutative subalgebra of A. Note that this definition follows from Definition 6.1.2, since the requirement 2 in Definition 6.1.2 is unnecessary: it is a consequence of requirement 1 and Proposition 5.1.1. Cartan subalgebra of compact (semi)simple Lie algebra A can be constructed as follows. One chooses an arbitrary element H1 ∈ A, then adds a nonzero element H2 = const · H1 that commutes with H1 , then adds the third element H3 that commutes with both H1 ,

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and H2 and is not a linear combination of H1 and H2 , and continues this process for as long as possible. The elements H1 , H2 , . . . , Hr are linearly independent. Linear space with basis vectors H1 , H2 , . . . , Hr is precisely Cartan subalgebra of A. Since algebra A is compact, and any representation of A is (anti-)Hermitian, matrices of operators ad(Hi ) are simultaneously diagonalizable by appropriate choice of basis in complexification of algebra A. The precise way of choosing elements H1 , H2 is irrelevant (in particular, it is irrelevant what element H1 one starts with), as all subalgebras are related to each other by inner automorphisms of A. Problem 6.1.3. Show that for any element H1 ∈ su(3) there is one and only one element H2 ∈ su(3) such that H2 = αH1 (α ∈ R) and [H1 , H2 ] = 0. Proposition 6.1.3. Let AC be semisimple complex Lie algebra, A its compact real form. Let H be Cartan subalgebra of A. Then complexification of H is Cartan subalgebra of AC . Before proving this proposition we emphasize that it tells (recalling uniqueness of Cartan subalgebra of AC ) that Lie algebra AC and its compact real form A have Cartan subalgebras with the same generators. Therefore, constructing Cartan subalgebra of semisimple complex Lie algebra is equivalent to constructing Cartan subalgebra of its compact real form, and vice versa. Proof. We prove Proposition 6.1.3 ad absurdum. Let H be Cartan subalgebra of A, Hi be generators of H, and Ta be other generators (convenient choice of the latter is discussed below). Then for any real αa there exists a set of real parameters di = di (αa ) such that [di Hi , αa Ta ] = 0

(6.1.1)

(summation over repeated indices is assumed). Let the complex subalgebra of AC with generators Hi be not Cartan subalgebra. Since all Hi commute with each other, this means that this commutative subalgebra is not maximal in AC , i.e., there exist complex βa such that [Hi , βa Ta ] = 0

∀i.

This gives n Tn = 0 βa Cia

∀i,

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n where Tn = {Hi , Ta } is complete set of generators in A and AC , Cia are real structure constants. Since Tn is basis in AC , we have n = 0 ∀i, n βa Cia

⇒

n n Re(βa ) · Cia = Im(βa ) · Cia = 0 ∀i, n,

which contradicts (6.1.1), once one chooses αa = Re(βa ) (or αa = Im(βa ), if all Re(βa ) vanish). Complex semisimple Lie algebra is partitioned into direct sum of several complex simple Lie algebras. Therefore, the study of semisimple complex Lie algebras boils down to the study of simple Lie algebras. The latter always have simple compact real form (see Proposition 6.1.1 and discussion there). Simple complex Lie algebra AC has scalar product (Killing form) (X, Y ) = −Tr(ad(X) · ad(Y )),

X, Y ∈ A.

(6.1.2)

This scalar product is invariant and nondegenerate. Let us choose basis in AC coinciding with a basis in its compact real form A. Structure constants of algebra AC are real in this basis. One can choose this basis in such a way that all basis elements in AC are divided into two groups { Hi }

(i = 1, . . . , r),

{ Ta }

(a = 1, . . . , dim(A) − r),

(6.1.3)

where {Hi } are generators of Cartan subalgebra H in AC , and elements Ta make basis in the orthogonal complement H⊥ to H: (Hi , Ta ) = −Tr (ad(Hi ) · ad(Ta )) = 0.

(6.1.4)

Invariance of the scalar product (6.1.2) gives (Hi , [Hj , Ta ]) = −([Hj , Hi ], Ta ) = 0, hence [Hj , Ta ] ∈ H⊥ for all Hj and Ta , and the following expansion holds: [Hj , Ta ] = Tb hj,ba .

(6.1.5)

n = dimH⊥ = dim(A) − r.

(6.1.6)

Let us denote

It follows from (6.1.5) that real matrices hj = ||hj,ab || ≡ ρ(Hj ) make n-dimensional matrix representation ρ of Cartan subalgebra H. Indeed, relation (6.1.5) can be viewed as adjoint action of generators Hi in subspace

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H⊥ , while the Jacobi identity implies that ρ([Hi , Hj ]) = [ρ(Hi ), ρ(Hj )]. Since [Hi , Hj ] = 0, we have [hi , hj ] = 0

(∀i, j).

(6.1.7)

Note that in the case we consider, Cartan subalgebra of complex algebra AC coincides with Cartan subalgebra of its compact real form, and matrices hi are simultaneously diagonalizable (in n-dimensional complex space). It follows from the property (6.1.4) that Killing metric in basis (6.1.3) has block-diagonal form, where one of the blocks is made of elements gij = (Hi , Hj ) = −Tr(ad(Hi ) · ad(Hj )) = gji ,

(6.1.8)

and the second one contains Xab = (Ta , Tb ) = −Tr(ad(Ta ) · ad(Tb )) = Xba .

(6.1.9)

Since we have chosen the basis in simple complex Lie algebra AC which coincides with basis in its real form A, Killing metric in AC is not only nondegenerate, but also real. So, both matrices ||gij || and X = ||Xab || are nondegenerate and real. Furthermore, we have hj,ca Xcb = ([Hj , Ta ], Tb ) = −(Ta , [Hj , Tb ]) = −Xac hj,cb , so matrices ||hj,ab || (j = 1, . . . , r) are antisymmetric modulo equivalence transformation, (X · hj )T = −X · hj

⇒

−1 hT . j = −X · hj · X

(6.1.10)

Consider a diagonalizable n × n real matrix h which has symmetry property (6.1.10): hT = −X · h · X −1 .

(6.1.11)

Let α and v be its eigenvalues and eigenvectors (in complex space), hab vb = α va , i.e., h · v = α v. Then the following proposition holds. Proposition 6.1.4. If α is an eigenvalue of n × n matrix h obeying (6.1.11), then (−α) is also an eigenvalue of h. If matrix h is real, and matrix X is nondegenerate, symmetric and real (cf. (6.1.9)), then all these eigenvalues are pure imaginary.

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Proof. Eigenvalue α of matrix h obeys characteristic identity det(h − α In ) = 0.

(6.1.12)

This identity, together with (6.1.11), gives 0 = det((h − α In )T ) = det(−X · (h + αIn ) · X −1 ) = (−1)n det(h + αIn ) ⇒

det(h + αIn ) = 0.

Thus, if α is an eigenvalue, then (−α) is also an eigenvalue. Let us introduce, in complex space Vn , where matrices X and h act, scalar product (v, u) = va∗ Xab ub ,

v, u ∈ Vn .

(6.1.13)

This scalar product is Hermitian for real symmetric matrix X. Also, scalar product (6.1.13) is not degenerate, since matrix X is not degenerate (in Lie algebra context, this is the block (6.1.9) of nondegenerate Killing metric). The scalar product (6.1.13) is chosen in such a way that real matrix h obeying (6.1.11) satisfies (h · v, u) = −(v, h · u),

∀ v, u ∈ Vn .

(6.1.14)

Let v be an eigenvector of matrix h with eigenvalue α. Since matrix h is real, we have (h · v, v) = α∗ (v , v). On the other hand, it follows from (6.1.14) that (h · v , v) = −(v , h · v) = −α (v , v). Since the scalar product is nondegenerate, (v , v) = 0, these relations give α∗ = −α. Matrices hk , defined in (6.1.5), commute with each other, see (6.1.7). So, they are diagonalized simultaneously for all k: (α)

hk,ab vb

= i αk va(α)

⇔

hk v (α) = i αk v (α) .

(6.1.15)

Here v (α) are (generally, complex) eigenvectors, and (i αj ) are eigenvalues of real matrices ||hj,ab ||. In accordance with Proposition 6.1.4, αj are real. So, every eigenvector v (α) is characterized by r eigenvalues (iα1 , . . . , iαr ), i.e., it is associated with a vector α = (α1 , . . . , αr ) in an r-dimensional real

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vector space. We show later on (see Proposition 6.2.2) that each vector α is associated with only one eigenvector v (α) , i.e., there is no degeneracy. Equation (6.1.15) shows that complex conjugate vector v (α)∗ ∈ Vn is also eigenvector of real matrices ||hj,ab ||. It is associated with eigenvalue vector (−α1 , . . . , −αr ) = −α, v (α)∗ = v (−α) .

(6.1.16)

Eigenvectors v (α) can be normalized in such a way that they are orthonormal in scalar product (6.1.13): (v (α) , v (β) ) = δα,β ,

δα,β :=

r 

δαj ,βj .

(6.1.17)

j=1

Indeed, the fact that they are orthogonal follows from equalities −iαj (v (α) , v (β) ) = (hj · v (α) , v (β) ) = −(v (α) , hj · v (β) ) = −iβj (v (α) , v (β) ), which give (v (α) , v (β) ) = 0 in the case αj = βj for at least one j = 1, . . . , r. Then, by choosing normalization of v (α) one can always satisfy (6.1.17). Thus, eigenvectors v (α) make complete orthonormal system in Vn . Using this system, we perform convenient basis transformation in complexified (α) subspace HC⊥ ⊂ AC . To this end, we make convolution of (6.1.5) with va and obtain  [Hj , Eα ] = i αj Eα , Eα ≡ va(α) Ta . (6.1.18) a

We note here that matrices hj cannot have eigenvectors v (α) with all eigenvalues equal to zero, α = (0, . . . , 0), otherwise subalgebra H could be extended. So, instead of basis elements Ta ∈ H⊥ ⊂ A we introduce new basis elements Eα ∈ HC⊥ ⊂ AC associated with r-dimensional real nonzero vectors α = (α1 , . . . , αr ). The number of basis elements Eα ∈ AC equals the dimension of space H⊥ given by (6.1.6). We emphasize again that eigenvectors v (α) are generally complex, so elements {Hi , Eα } make basis in complex algebra AC , rather than in its real form A. Nevertheless, any element Y of real compact algebra A can be written as linear combination of generators {Hi } (i = 1, . . . , r) of Cartan subalgebra, with real coefficients, and root generators {Eα }, with complex coefficients. In view of this fact, and also Proposition 6.1.1, the new basis {Hi , Eα } is often associated not only with complex algebra AC , but also with its compact real form A.

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Definition 6.1.4. Vectors of r-dimensional real vector space Vr with coordinates (α1 , α2 , . . . αr ) introduced in (6.1.15), are root vectors (or roots) of Lie algebra AC , and space Vr is called root space. In accordance with (6.1.18), one often uses the term “root” for basis generator Eα as well. We prefer to call Eα root generator. Definition 6.1.5. Basis of Lie algebra AC made of elements {Hk , Eα }, where k = 1, . . . , r and α are root vectors in Vr , is called Cartan–Weyl basis. In adjoint representation we have [ad Hk , ad Eα · ad Eβ ] = i (αk + βk ) ad Eα · ad Eβ . We evaluate trace of this equality and obtain Tr(ad Eα · ad Eβ ) = 0, for all root vectors α, β ∈ Vr such that α + β = 0 (in the sense of vector sum of α = (α1 , . . . , αr ) and β = (β1 , . . . , βr )). Accordingly, since Killing metric is nondegenerate, we have Tr(ad E−α · ad Eα ) = 0 for all α. This also follows from (6.1.17): (E−α , Eβ ) = −Tr(ad E−α · ad Eβ ) (β)

= −va(α)∗ Tr(ad Ta · ad Tb ) vb

= (v (α) , v (β) ) = δα,β .

So, the normalization of generators Eα , determined by normalization condition (6.1.17), is such that Killing metric in Cartan–Weyl basis reads (Hk , Hj ) = −Tr(ad(Hk ) · ad(Hj )) = gkj , (Hk , Eα ) = −Tr(ad(Hk ) · ad(Eα )) = 0 ,

(6.1.19)

(Eα , Eβ ) = −Tr(ad(Eα ) · ad(Eβ )) = δα,−β . Now, the Jacobi identity gives [Hk , [Eα , Eβ ]] = i (α + β)k [Eα , Eβ ].

(6.1.20)

So, if (α + β) is root, then [Eα , Eβ ] = Nα,β Eα+β ,

(6.1.21)

where Nα,β are constants. We determine them explicitly later on, see Proposition 6.2.3 and Remark 2 after it; importantly, these constants are

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nonzero. If (α + β) is not root, and α + β = 0, then [Eα , Eβ ] = 0. If α + β = 0, then (6.1.20) gives [Hk , [Eα , E−α ]] = 0, i.e., [Eα , E−α ] = xj Hj ,

(6.1.22)

where xi are constants. These constants are found for Cartan–Weyl basis normalized according to (6.1.19) by the chain of equalities xj gjk = (Hk , [Eα , E−α ]) = ([Hk , Eα ] , E−α ) = i αk (Eα , E−α ) = i αk , i.e., xj = iαj , where αj = g jk αk , and g jk are elements of real matrix inverse of ||gjk ||. Note that matrix elements ||gjk || are expressed through roots as follows:   α α Ckα Cjα = αk αj , (6.1.23) gkj = −Tr(ad(Hk ) · ad(Hj )) = − α

α

β = iαj δαβ are obtained from (6.1.18), and where structure constants Cjα summation in (6.1.23) runs over all roots α. Let us introduce scalar product in real root space Vr :

(x , y) = xi gij y j = xi g ij yj ,

(6.1.24)

where x = (x1 , . . . , xr ) and y = (y 1 , . . . , y r ) are two vectors in Vr . It follows from representation (6.1.23) that this scalar product, and hence metric gkj , is positive definite. Indeed, Eq. (6.1.23) gives for any vector y ∈ Vr  y k gkj y j = (yα)2 ≥ 0. α

So, the root space Vr has Euclidean metric. This is not a surprise, since basis elements Hi of Cartan subalgebra of AC are inherited from Cartan subalgebra of compact real form A. Positive-definiteness of metric gkj is consistent with the fact that Killing metric for compact semisimple Lie algebra is always positive-definite. Thus, the defining commutation relations (3.2.30) for Lie algebras A and AC in Cartan–Weyl basis {Hk , Eα } are written as follows: [Hi , Hk ] = 0,

[Hk , Eα ] = i αk Eα ,

[Eα , Eβ ] = Nα,β Eα+β ,

if (α + β) is root,

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if (α + β) is not root and α + β = 0, [Eα , E−α ] = i αk Hk .

(6.1.25)

Note that scalar product (α, β) of two roots α and β, given in (6.1.24), is invariant under the change of basis in Cartan subalgebra Hi → Hi = Aij Hj ,

||Aij || ∈ GL(r, R).

(6.1.26)

Indeed, according to (6.1.18), this change induces transformation of roots αi → αi = Aij αj , while the formula (6.1.23) gives the transformation of metric g → A · g · AT ⇒ g −1 → (AT )−1 ·g −1 ·A−1 . (6.1.27) Scalar product (6.1.24) remains intact under this transformation:

gij → Aik gkm Ajm

⇒

(α, β) = αT g −1 β  = αT AT · (AT )−1 · g −1 · A−1 · A β = αT g −1 β = (α, β). Using the transformation (6.1.27), we can transform Euclidean metric to orthonormal form gij = δij . So, we have proven the following proposition. Proposition 6.1.5. By linear change of basis in Cartan subalgebra, Eq. (6.1.26), metric gij in root space Vr can be cast in orthonormal form gij = δij . Scalar product (α, β) of roots α and β is invariant under transformations (6.1.26). Remark 1. Let us emphasize that there still remains some freedom in defining the scalar product (6.1.2). Namely, invariant scalar product for simple Lie algebra is defined modulo overall scale factor, (X, Y ) → λ2 (X, Y ), where λ ∈ R (see Corollary 4.7.1). This enables one to dilate all root vectors simultaneously by one and the same scale factor λ. We occasionally make use of this freedom in what follows. Remark 2. In accordance with Proposition 5.1.2, any representation of compact Lie algebra A is equivalent to anti-Hermitian representation, Hj† = −Hj , Ta† = −Ta . In this representation, in view of (6.1.16), (6.1.18), Cartan–Weyl generators of algebra AC obey Eα† = −E−α ,

Hj† = −Hj .

(6.1.28)

Instead of anti-Hermitian generators Hk and Ta one often uses Hermitian ˜ k = −iHk , T˜a = iTa , then coefficients in algebra A are generators H pure imaginary (see remark in Example 5 in Section 3.1.3). The structure

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constants hj,ab in (6.1.5) are multiplied by (−i) and also become pure imaginary: h∗j,ab = −hj,ab ,

(6.1.29)

while positive-definite metric (6.1.8), (6.1.9) is given by ˜ k ) · ad(H ˜ j )), gkj = +Tr(ad(H

Xab = +Tr(ad(T˜a ) · ad(T˜b )).

Matrices hj obey (6.1.10), (6.1.29) and, according to Proposition 6.1.4, have real eigenvalues α∗ = α; this agrees with the fact that hj are representations ˜ k , Eα → ˜ j . In accordance with the change Hk → H of Hermitian operators H ˜ Eα , one substitutes Hk → iHk , Eα → −iEα in defining commutation relations (6.1.25) and replaces structure constants Nα,β → −iNα,β . As a result, new Cartan–Weyl generators have, instead of (6.1.28), the properties Hj† = Hj .

Eα† = E−α ,

(6.1.30)

The defining commutation relations take convenient form [Hi , Hk ] = 0,

[Hk , Eα ] = αk Eα ,

[Eα , Eβ ] = Nα,β Eα+β , [Eα , Eβ ] = 0,

if (α + β) is root,

(6.1.31)

if (α + β) is not root and α + β = 0,

[Eα , E−α ] = αk Hk ,

αk = g kj αj .

(6.1.32)

The relations (6.1.30) and (6.1.31), (6.1.32) are particularly convenient in the study of Cartan–Weyl basis, so we use this formulation in what follows. Example. An example is Cartan–Weyl bais {Hj , Eij , Fij } in Lie algebra s(n, C), given by (3.3.6), (3.3.5). We recall that compact real form of s(n, C) is Lie algebra su(n). Defining commutation relations for s(n, C) in basis {Hj , Eij , Fij } are given in (3.3.7). By comparing (3.3.7) and (6.1.31), we see that elements Hj ∈ s(n, C) (j = 1, . . . , n − 1) make Cartan subalgebra in s(n, C), so rank of s(n, C) is (n − 1). Elements Eij and Fij are root generators similar to Eα and E−α . Note that the basis {Hi } in Cartan subalgebra, chosen in (3.3.5), is inconvenient, since the metric (6.1.8) is not diagonal. Orthonormal basis in Cartan subalgebra of s(n, C) is constructed in (3.3.15). 6.2.

Root Systems of Simple Lie Algebras

Theory of root systems of simple Lie algebras is presented also in books and lecture notes [3, 14, 15, 20–23].

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Properties of roots of simple Lie algebras

Let us prove the following theorem. Theorem 6.2.1. Let AC be simple finite-dimensional complex Lie algebra. If α and β are roots of AC , then 2(α, β)/α2 is integer, and vector σα (β) ≡ β − 2

(α, β) α α2

(6.2.1)

is also a root. Here the scalar product (α, β) is defined in (6.1.24), and α2 = (α, α). Proof. It follows from (6.1.31) and (6.1.32) that for any root vector α, three algebra elements   2 2 1 i Eα , E−α = E−α , (6.2.2) Hα = 2 (α Hi ), E+α = α α2 α2 generate Lie subalgebra s(2, C): [Hα , E±α ] = ± E±α ,

[E+α , E−α ] = 2 Hα .

(6.2.3)

As we have shown in Section 4.7.3 (see (4.7.62)), eigenvalues of Cartan generator Hα in any finite-dimensional representation of s(2, C) are integers or half-integers. In our context, a representation of subalgebra s(2, C) ⊂ AC with basis (6.2.2) and commutation relations (6.2.3) is defined by adjoint action in the whole algebra AC . Cartan–Weyl basis is constructed in such a way that all eigenvectors of operator Hα in AC are root generators Eβ , and one has for all roots α and β [Hα , Eβ ] =

αi (α, β) [Hi , Eβ ] = Eβ . α2 α2

(6.2.4)

Thus, Eβ are eigenvectors of operator ad(Hα ) with eigenvalues (αβ)/α2 . These must be integer or half-integer. So, 2(αβ)/α2 are integer for all roots α and β. Consider now decomposition of AC , viewed as the space of the reducible representation of algebra s(2, C) ⊂ AC , into invariant subspaces of irreducible representations of the latter algebra. These are necessarily highest weight representations, as we discussed in Section 4.7.3. Since every highest weight vector (see Definition 4.7.3) is an eigenvector of Hα , highest weight vectors are searched for among the root generators. Let us take a root generator Eβ (β = ±α), and act on it by raising operator Eα ∼ Eα . If

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α + β is a root, then [Eα , Eβ ] = Nα,β Eα+β , where Nα,β = 0. By applying the operator Eα several times, we get at some kth step (since algebra AC is finite-dimensional) [Eα , Eγ ] = 0,

Eγ = 0,

(6.2.5)

where γ = β + kα is a root, such that α + γ is not a root. Root generator Eγ is an eigenvector of operator ad(Hα ) (see (6.2.4)): [Hα , Eγ ] =

(α, γ) Eγ . α2

(6.2.6)

Since it obeys (6.2.5), Eγ is the highest weight vector with the highest weight (αγ) α2 , in accordance with Definition 4.7.3. We then use generator Eγ for constructing irreducible highest weight representation of s(2, C). Namely, all basis vectors of the space of irreducible representation are obtained from Eγ by applying lowering operator E−α ∼ E−α :  , [E−α , Eγ ] = Eγ−α   ] = Eγ−2 [E−α , Eγ−α α, .. .  [E−α , Eγ−j α]

=

 Eγ−(j+1) α

(6.2.7) = 0,

where  Eγ−k α = N−α, γ · N−α, γ−α · · · N−α, γ−(k−1)α · Eγ−k α

(6.2.8)

is renormalized generator Eγ−k α , and we recall that the lowering procedure (6.2.7) must terminate at some (j + 1)th step. Note that the root generator Eβ we started with is, modulo normalization, one of the generators of the family (6.2.7). Clearly, we have for all integer m from interval 0 ≤ m ≤ j + 1   [Eα , Eγ−m α ] = μm Eγ−(m−1) α ,

(6.2.9)

where yet unknown coefficients μm obey μ0 = 0 = μj+1 .

(6.2.10)

Let us find recurrence relation for μm by writing a chain of equalities    μm+1 Eγ−m α = [Eα , Eγ−(m+1)α ] = [Eα , [E−α , Eγ−m α ]]   = −[Eγ−m α , [Eα , E−α ]] − [E−α , [Eγ−m α , Eα ]]

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  = −[Eγ−m α , (α, H)] + μm [E−α , Eγ−(m−1) α ]  = [(α, γ) − m (α, α) + μm ] Eγ−m α.

Hence, we have μm+1 = μm + (α, γ) − m (α, α).

(6.2.11)

We make use of the initial condition μ0 = 0, see (6.2.10), and obtain the solution to the recurrence relation (6.2.11): μm = m (α, γ) −

m(m − 1) (α, α). 2

(6.2.12)

The property μj+1 = 0, see (6.2.10), gives 0 = (α, γ) −

j (α, α) 2

⇒

j=2

(α, γ) . (α, α)

(6.2.13)

Thus, we have proven that if α and γ are roots, and α + γ is not a root, then for (α, γ) = 0 there exists sequence (string) of roots γ,

γ − α,...,

γ−jα =γ−2

(α, γ) α, (α, α)

(6.2.14)

where the last root is determined by (6.2.13). This analysis shows that any root generator belongs to one of the families (6.2.7), i.e., any root β belongs to one of the strings. Note that root string (6.2.14) is invariant under reflection with respect to hyperplane Pα normal to vector α and containing the origin of root space. Indeed, this string is directed along vector α, and the last root σα (γ) = γ − 2

(α, γ) α, (α, α)

(6.2.15)

is the reflection of the first root with respect to the hyperplane Pα (see Fig. 6.2.1).

α

... γ–α

γ

... ξ

γ – jα

η = α (α,γ) α2 Pα Fig. 6.2.1.

α

α-string of root γ.

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The latter fact is a consequence of the property that any root γ can be represented (see Fig. 6.2.1) as a sum of vector ξ ∈ Pα normal to α, and vector η = α (α,γ) α2 directed along α. Linear transformation of root space σα , defined in (6.2.1), does not move vector ξ, and changes the sign of vector η = (α,γ) α2 α: σα (ξ) = ξ − 2

(α, ξ) α = ξ, α2

σα (η) =

(α, γ) (α, γ) σα (α) = (−α) = −η. α2 α2

Therefore, the transformation (6.2.15) is indeed reflection of γ with respect to hyperplane Pα . The latter result means that all roots of the string (6.2.14) are placed symmetrically against the hyperplane Pα , and if root β belongs to string (αβ) (6.2.14), then this string contains also vector β − 2 (αα) α, which therefore is a root. Definition 6.2.1. Transformations (6.2.1) are called Weyl reflections. Proposition 6.2.2. All roots α of simple finite-dimensional Lie algebra AC have unit multiplicities. If α is a root, then vectors 2α, 3α, . . . are not roots. Proof. Given a root generator Eα , one constructs algebra sα (2, C) ⊂ AC with basis (6.2.2). Let the multiplicity of root α be greater than 1 and/or vectors 2 α, 3 α, . . . , k α be roots of AC . Consider subspace Z ⊂ AC given by Z = sα (2, C) + Vα + V2α + V3α + · · · + Vkα ,

(6.2.16)

where Vβ is subspace of spanned by root generators with root β, and Vα is subspace in Vα , which is orthogonal to generator E−α (recall that (Eα , E−α ) = 1). Basis vectors of subspace Vα are root generators which are different from Eα but corresponding to root α. This subspace is nontrivial, if the multiplicity of root α is greater than 1. In accordance with (6.1.31), Z is subalgebra of AC which is invariant under adjoint action of sα (2, C). Problem 6.2.1. Prove the last statement. Prove, in particular, that orthogonality relation (Vα , E−α ) = 0 implies [E−α , Vα ] = 0. Hint: check equality (Hγ , [E−α , Vα ]) = 0 for any root γ.

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Let us take basis vectors in each of subspaces in (6.2.16) and form basis in Z in this way. Consider matrix adZ (Hα ) of operator Hα ∈ sα (2, C) in space Z. Since all basis vectors in Z are eigenvectors of Hα , and due to Eq. (6.2.4), trace of this operator (sum of eigenvalues) in space Z is TrZ (adZ (Hα )) = (nα + 2 n2α + · · · + k nkα ).

(6.2.17)

Here nβ is dimension of Vβ , and nα = (nα − 1) is dimension of Vα . On the other hand, since Hα is represented (see (6.2.3)) as commutator Hα = 1 2 [Eα , E−α ], we have, due to cyclic property of trace, TrZ (adZ (Hα )) = 12 TrZ ([adZ (Eα ), adZ (E−α )]) = 0.

(6.2.18)

By comparing (6.2.17) and (6.2.18) we get nα = 0 and nm α = 0 (∀m = 2, . . . , k), which is the desired result. Remark 1. Proposition 6.2.2 is not, generally speaking, valid for infinitedimensional Lie algebras, since in that case the space Z can be infinitedimensional, and trace TrZ may not have cyclic property. As an example, consider Lie algebra with infinite number of generators Lm = z m+1 ∂z (m = −1, 0, 1, 2, . . . ), which act in space of functions of z and obey defining relations [Lm , Lk ] = (k − m) Lm+k . Note that operators L−1 , L0 , L1 make Lie algebra s(2, C), where L0 plays a role of Cartan generator Hα with the root α = 1. Generators Lk (k = 0), in view of [L0 , Lk ] = k Lk , correspond to 1-dimensional root vectors kα = k. This demonstrates that Proposition 6.2.2 is not valid for the infinitedimensional Lie algebra. Proposition 6.2.3. Without affecting normalization (6.1.19), generators Eα can be chosen in such a way that all coefficients Nα,β in (6.1.31) obey 2 Nα,β =

(m + 1)(j − m) (α, α), 2

(6.2.19)

where root β = γ −(m+1)α belongs to α-root string starting with the highest root γ (m = 0, . . . , j − 1), and j = 2(α, γ)/(α, α). Thus, structure constants Nα,β are real and are determined by (6.2.19) modulo sign.

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Proof. Let us insert (6.2.8) in (6.2.9) and make use of (6.2.12). We obtain that if (α + γ) is not a root, then N−α, β+α Nα, β = μm+1 = (m + 1)

(j − m) (αα), 2

(6.2.20)

where we use the notation β ≡ γ − (m + 1) α for a root on the string generated from root γ by root α, and j = 2(α, γ)/(α, α). Let three roots α ˜, β˜ and γ˜ form a triangle α ˜ + β˜ + γ˜ = 0.

(6.2.21)

Nα,˜ ˜α ˜ γ = Nγ ˜, ˜ ,β˜ = Nβ,

(6.2.22)

Then one has

which follows from antisymmetry of structure constants Nα, ˜α ˜ and ˜ β˜ = −Nβ, identities ([Eα˜ , Eβ˜ ], Eγ˜ ) + (Eβ˜ , [Eα˜ , Eγ˜ ]) = 0,

(˜ α ↔ γ˜ ).

We set α ˜ = −α, β˜ = −β, γ˜ = α + β in (6.2.22) and obtain the equality N−α,α+β = −N−α,−β . Symmetry of the root system enables one to choose, without affecting normalization in (6.1.19), generators of Lie algebra AC in such a way that relations (6.1.31), (6.1.32) are invariant under Cartan isomorphism σ: Eα → σ(Eα ) = −E−α ,

Hi → σ(Hi ) = −Hi ,

(6.2.23)

therefore Nα,β = −N−α,−β .

(6.2.24)

This relation in the case of compact Lie algebras follows also from antiautomorphism (6.1.30) (see also [23, p. 175]). Thus, with appropriate choice of generators one has N−α, β+α = −N−α,−β = Nα, β . Inserting this equality in (6.2.20), we obtain (6.2.19). Remark 2. Theorem 6.2.1 and Propositions 6.2.2, 6.2.3 lead to the conclusion that simple Lie algebra AC is determined by its rank r and the system of its root vectors embedded in r-dimensional Euclidean space Vr . Indeed, metric in Vr is found from (6.1.23), while commutation relations follow from (6.1.25) and (6.2.19). This remark will be made more precise in what follows, in particular, in Theorem 6.2.3. The sign issue of coefficients Nα,β is beyond the scope of this book.

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Remark 3. Let us set γ  = γ − jα. The last equality in (6.2.7) takes the form [E−α , Eγ  ] = 0, i.e., root γ  is such that (γ  − α) is not a root, and root generator Eγ  is the lowest weight vector of irreducible representation of algebra (6.2.3) embedded in adjoint representation. Then the set of equalities (6.2.9) is written as follows: [Eα , Eγ  +m α ] = Nα,γ  +m α Eγ  +(m+1) α , where j = −2 (α,γ α2



)

(m = 0, 1, . . . , j),

(6.2.25)

= 2 (α,γ) α2 ≥ 0. We recall (6.2.19) and write

2 2 Nα,γ  +m α = Nα,γ−(j−m) α =

(m + 1)(j − m) (α, α), 2

(m = 0, 1, . . . , j).

(6.2.26) The set of equalities (6.2.25) terminates at m = j, since, in view of (6.2.5) and (6.2.26), we have [Eα , Eγ  +j α ] = [Eα , Eγ ] = 0; in other words, (γ  +(j+ 1) α) = (γ + α) is not a root. Equalities (6.2.26) ensure that Nα,γ  +m α = 0 for all m = 0, . . . , j − 1. In this way we obtain string of roots γ  , γ  + α, . . . , γ  + j α = γ,

j = −2

(α, γ  ) ≥ 0. (α, α)

(6.2.27)

This string is constructed in (6.2.25) from the lowest weight root generator Eγ  by the action of raising operator Eα . The string (6.2.27) coincides with the root string (6.2.14), but roots in it are ordered in the opposite way. Let us now make use of Theorem 6.2.1 and consider inequality 0≤

2(α, β) 2(α, β) = m n = 4 cos2 θ ≤ 4, α2 β2

(6.2.28)

where m, n are integers, and θ is angle between vectors α and β (it follows from Proposition 6.1.5 that, keeping scalar products of roots untouched, one can always cast metric gij into orthonormal form gij = δij , and then the angle between vectors α and β in (6.2.28) is well defined). In accordance with (6.2.28), we have the following options:

(a) α ∦ β

⇒

0≤m·n≤3

2

(α, β) = α2

0

±1

±1

±1

2

(α, β) = β2

0

±1

±2

±3

⇒

(6.2.29)

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plus interchange α ↔ β, which is equivalent to interchange of upper and lower rows in table (6.2.29);

(b) α  β ⇒ m · n = 4

2

(α, β) = α2

±2

±1

±4

2

(α, β) = β2

±2

±4

±1

α = ±β

α = ±2β

β = ±2α

.

(6.2.30) Upper and lower signs here, both in the case (a), Eq. (6.2.29) and in the case (b), Eq. (6.2.30), are coordinated. The case α = ±β in (6.2.30) is trivial, in the sense that either α = β, or α is obtained from β by reflection. The cases α = ±2β, β = ±2α are forbidden by Proposition 6.2.2. Thus, the nontrivial situation is (a), i.e., set of options (6.2.29). Yet another consequence of inequality (6.2.28) is that the string (6.2.14) cannot contain more than four roots (j ≤ 3). Lemma 6.2.1. Let α and β be two different roots. If (α, β) > 0, then α − β is also a root, and if (α, β) < 0, then α + β is also a root. Proof. Consider the case (α, β) < 0 (the case (α, β) > 0 is obtained by replacing β → −β and hence is studied in the same way). It follows from (α,β) (6.2.29) that either 2 (α,β) α2 = −1, or 2 β 2 = −1. Let us choose, without loss of generality, 2 (α,β) = −1. Then we obtain, by making use of Weyl α2 reflection, that σα (β) = β − 2

(α, β) α = β + α, α2

is a root. Corollary 6.2.1. If α and β are roots, and (β + α) is not a root, then (α, β) ≥ 0 (see also formula (6.2.14) with j ≥ 0). Likewise (replacing α by −α), if (β−α) is not a root, then (α, β) ≤ 0. Combining the two statements, we find that if α and β are roots, while (β + α) and (β − α) are not roots, then (α, β) = 0. This corresponds to the first case in (6.2.29). Example: Root system of Lie algebra s(3, C) (or su(3)). Subalgebra H in s(3, C) with diagonal traceless matrices is commutative Cartan subalgebra.

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Let us choose, as basis in ⎛ 1 0 H1 = ⎝0 −1 0 0

H, two independent ⎛ ⎞ 1 0 1 0⎠, H2 = √ ⎝0 3 0 0

matrices (3.3.25), ⎞ 0 0 1 0 ⎠, 0 −2

301

(6.2.31)

such that Tr(Hi Hj ) = 2δij . Other six generators of s(3, C) (dimension of s(3, C) is eight), which we denote by E±α , E±β , E±γ , can be chosen as follows: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 0 0 0 1 Eα = √16 ⎝0 0 0⎠, Eβ = √16 ⎝0 0 1⎠, Eγ = √16 ⎝0 0 0⎠, 0 0 0 0 0 0 0 0 0 (6.2.32) E−α = EαT ,

E−β = EβT ,

E−γ = EγT .

(6.2.33)

Defining commutation relations of algebra s(3, C) in this basis are [H1 , Eβ ] = −Eβ , [H1 , Eγ ] = Eγ , √ √ [H2 , Eβ ] = 3Eβ , [H2 , Eγ ] = 3Eγ ,

[H1 , Eα ] = 2Eα , [H2 , Eα ] = 0, [Eα , Eβ ] = Nα,β Eγ ,

[Eγ , E−α ] = Nγ,−α Eβ ,

[Eα , E−β ] = 0,

(6.2.34)

[Eγ , E−β ] = Nγ,−β Eα ,

[Eα , Eγ ] = [Eβ , Eγ ] = 0, (6.2.35)

[Eα , E−α ] = 16 H1 ,

√

[Eβ , E−β ] =

[Eγ , E−γ ] =

√

3 12 H2

+

3 12 H2

1 12 H1 ,

−

1 12 H1 ,

(6.2.36)

where Nα,β = −Nγ,−α = Nγ,−β = √16 . In view of (6.2.33), remaining commutators are straightforwardly obtained from (6.2.34) and (6.2.35) by transposition, so we do not write them here. Note only that the structure constants Nα,β automatically satisfy (6.2.24). In accordance with (6.2.34), there are six root vectors √ √ α = (α1 , α2 ) = (2, 0), β = (β1 , β2 ) = (−1, 3), γ = α + β = (1, 3), √ √ −α = (−2, 0), −β = (1, − 3), −γ = (−1, − 3). (6.2.37) They can be drawn in root diagram as shown in Fig. 6.2.2, where vectors α(1) = α, α(2) = β define integer-valued basis for the whole root system of s(3, C).

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α(2) = β

γ =α+β α(1) = α

Fig. 6.2.2.

Root diagram of Lie algebra s(3, C).

Problem 6.2.2. Check that identities (6.2.22) are satisfied for structure constants Nα,β given by (6.2.35). Making use of (6.2.37) and (6.1.23), we calculate metric gij , which turns out to be orthogonal and Euclidean:

1 0 gij = 2(αi αj + βi βj + γi γj ) = 12 . 0 1 The root generators (6.2.32) and (6.2.33) are normalized as in (6.1.19): (Eα , E−α ) = (Eβ , E−β ) = (Eγ , E−γ ) = 1, while the relations (6.2.36) are [Eα , E−α ] = αi Hi ,

[Eβ , E−β ] = β i Hi ,

[Eγ , E−γ ] = γ i Hi ,

(6.2.38)

as given in (6.1.32). Problem 6.2.3. Check the last statement. Hint: make use of (6.1.32) and compare it with (6.2.36). Note that another choice of generators in Cartan subalgebra, unlike the choice (6.2.31), can give rise to nonorthogonal metric gij , and that in turn can distort the diagram in Fig. 6.2.2. Problem 6.2.4. Give an illustration of Proposition 6.2.3, Lemma 6.2.1 and formula (6.2.14) by using root system of algebra s(3, C). Problem 6.2.5. Find all root vectors α = (α1 , α2 , α3 ) for algebra s(4, C) with the orthogonal choice (3.3.15) of basis {H1 , H2 , H3 } in Cartan subalgebra H ⊂ s(4, C). Draw all root vectors in the root vector space R3 as vertices of polyhedron. Describe this polyhedron.

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Weyl group and simple roots

We do not need any longer to consider compact real forms of complex Lie algebras. So, to simplify notations, we denote complex algebras by A instead of AC . Let us denote the set of all roots of simple Lie algebra A by Φ(A). Consider Weyl reflections (6.2.15) in Φ(A). In accordance with Theorem 6.2.1, root system is invariant under these reflections, so reflection operators make a group W . Definition 6.2.2. Group W of all Weyl reflections (6.2.15) is called Weyl group. Since the root system is finite for finite-dimensional Lie algebra, group W is also finite. Consider hyperplane Pα containing the origin in root space and normal to root α. These hyperplanes, for various α, divide r-dimensional root space into nonintersecting regions called Weyl chambers. As an example, in the case A = s(3, C), there are three reflecting hyperplanes (straight lines) Pα , Pβ , Pγ and, accordingly, six Weyl chambers Pγ

β

γ

Pβ α

Pα

Since Weyl reflections are performed with respect to hyperplanes Pα , Weyl group transforms one Weyl chamber to another. So, Weyl group W acts in the set of Weyl chambers, and this action is transitive (see Definition 7.1.1). The order of Weyl group is equal to the number of Weyl chambers. Let us now introduce the basis in root space Vr . To this end, we first choose one of Weyl chambers and vector x inside this chamber. We have (x, α) = 0, ∀α ∈ Φ(A), otherwise vector x would belong to one of the reflection hyperplanes. As x moves in one and the same chamber, scalar product (x, α) does not change sign, otherwise at some point one would have (x, α) = 0, meaning that x belongs to boundary hyperplane Pα .

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Let us introduce the definition: (1) α is positive root, if (x, α) > 0 ⇒ α ∈ Φ+ (x); (2) α is negative root, if (x, α) < 0 ⇒ α ∈ Φ− (x). Note that if α ∈ Φ+ (x), then for the opposite sign root we have (−α) ∈ Φ− (x). Lemma 6.2.2. If roots α(i) , i = 1, . . . , k ≤ r, are positive, and (α(i) , α(j) ) ≤ 0 ∀i, j (i = j), then α(i) are linearly independent. Proof. We prove this lemma ad absurdum. Let α() be roots obeying conditions of lemma, and k 

r α() = 0,

(6.2.39)

=1

where some r are nonzero. We make convolution of the left-hand side with vector x. Since (x, α() ) > 0 for all , some coefficients r in (6.2.39) are positive, and some are negative. Let us move all terms with r ≤ 0 to the right-hand side of (6.2.39): ≡

  i=1

si α(i) =

k 

tj α(j) ,

j=+1

where si ≥ 0, tj ≥ 0. This gives ⎞ ⎛  k  k     2 (i) (j) ⎠ ⎝ =  = si α , tj α si tj (α(i) , α(j) ) ≤ 0, i=1

j=+1

i=1 j=+1

where we recall that (α(i) , α(j) ) ≤ 0. This inequality shows that  = 0 and   si (α(i) , x) = tj (α(j) , x). 0 = (, x) = i

j

Since si ≥ 0, tj ≥ 0 and (x, α() ) > 0, ∀, we have si = 0 and tj = 0, i.e., r = 0 for all . Definition 6.2.3. Positive root is called simple, if it cannot be written as a sum of two positive roots.

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Theorem 6.2.2. There exists r = rank (A) linearly independent simple roots α(1) , α(2) , . . . , α(r) . Any positive root α can be written as α=

r 

ni α(i) ,

(6.2.40)

i=1

where ni ≥ 0 are integers, while any negative root β has the form β=

r 

mi α(i) ,

(6.2.41)

i=1

where mi ≤ 0 are negative integers. Furthermore, (α(i) · α(j) ) < 0 for all simple roots α(i) and α(j) , such that (α(i) + α(j) ) is a root (in other words, angle between these roots is always greater than 90◦ ), and (α(i) · α(j) ) = 0 if (α(i) + α(j) ) is not a root. This set Δ(x) of simple roots {α(1) , α(2) , . . . , α(r) } ⊂ Φ+ (x) is called base. Proof. (1) We note, in the first place, that difference of two simple roots α(i) and α(j) is not a root. Indeed, β = α(i) − α(j) cannot be a positive root (otherwise α(i) would be a sum of two positive roots), and β cannot be a negative root (otherwise α(j) = α(i) + (−β) would be a sum of two positive roots). Thus, β is not a root. (2) Since (α(i) − α(j) ) is not a root, one finds from Corollary 6.2.1 that (α(i) , α(j) ) ≤ 0, and if (α(i) + α(j) ) is a root, then (α(i) · α(j) ) < 0, whereas if (α(i) + α(j) ) is not a root, then (α(i) · α(j) ) = 0. Hence, all simple roots are linearly independent due to Lemma 6.2.2. (3) We make use of (6.1.23) and recall that the set Φ(A) consists of pairs of roots ±α. For this reason, any vector y ∈ Vr can be decomposed in positive roots α: ⎞ ⎛     ⎠ αi (α, y) = 2 αi αj g jk yk = ⎝ + αi (α, y). yi = α

α∈Φ+ (x)

α∈Φ− (x)

α∈Φ+ (x)

By definition, all positive roots are either simple or sums of other positive roots, which, in turn, are either simple or sums of positive roots, etc. Continuing this procedure, one eventually ends up in a sum of simple roots. Thus, any positive root, and hence any vector y ∈ Vr can be decomposed into a sum of simple roots. We conclude that simple roots make basis in r-dimensional root space Vr , so the number of simple roots is not smaller

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than r. Since, according to point (2), simple roots are linearly independent, their number cannot exceed r, and hence it is exactly r. (4) Let β be a positive root, which is not simple. Then the system of vectors {α(1) , . . . , α(r) , β}, is linearly dependent, and, according to Lemma 6.2.2, we have (α(j) , β) > 0 for at least one j. Due to Lemma 6.2.1 vector γ1 = β − α(j) is a root, and it is a positive root (otherwise α(j) would not be simple). Then either γ1 is a simple root belonging to the set {α(1) , . . . , α(r) } and hence β is a sum of two simple roots, or γ1 is again a positive root which is not simple, so there exists α() such that (α() , γ1 ) > 0 and hence γ2 = γ1 − α() is a positive root, etc. We continue this procedure until at kth step we get a simple root γk . In this way we design a procedure for decomposing any positive root into a sum (6.2.40) of r simple roots with nonnegative integer coefficients. Likewise, any negative root can be decomposed into a linear combination (6.2.41) with nonpositive integer coefficients. This proves Theorem 6.2.2.

Corollary 6.2.2. If any positive root can be decomposed into a sum (6.2.40) of positive roots α(1) , . . . , α(r) with nonnegative integer coefficients, then the set {α(1) , . . . , α(r) } is base in Δ(x), i.e., it is complete system of simple roots. Corollary 6.2.3. Point (1) of the proof of Theorem 6.2.2, the third row in (6.1.31) and (6.1.32) imply the following commutation relations (i)

[Eα(i) , E−α(j) ] = δij αk g kl Hl ≡ δij (α(i) , H).

(6.2.42)

One more property of simple Lie algebras is that any root generator Eα ∈ A is a multiple commutator of root generators Eα(i) (or E−α(k) ), corresponding to simple roots. In view of (6.1.21), this fact is consistent with Theorem 6.2.2 and follow from the proof of the following stronger statement. Theorem 6.2.3. Any simple Lie algebra is determined by the system Δ(x) of its simple roots. Proof. Recall, in the first place, that the system Δ(x) makes basis in Euclidean root space Vr , which is endowed with positive-definite metric gij and scalar product (6.1.24). To prove the theorem, it is sufficient

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to construct all roots of Lie algebra in terms of its simple roots; then Corollary 6.2.3 ensures that commutation relations are unambiguously reconstructed. If α is a root, then (−α) is also a root, so it is sufficient to construct positive roots only. A positive root represented as a sum of p simple roots (with unit coefficients; a root can be present in the sum more than once) is called root of order p. We construct complete root system via induction procedure in p. Roots of order 1 are known, since these are simple roots. Let roots of order (p − 1) be available. We first show that any root α of order p > 1 is written as γ + α(j) , where γ is a root of order (p − 1), which is known, and α(j) is a simple root. Let us make use of the argument of point (4) in the proof of Theorem 6.2.2. Let α be a positive root of order p > 1, then there exists a simple root α(j) , such that (α, α(j) ) > 0 and hence γ = α − α(j) is a root, and it is a positive root, otherwise α(j) would not be simple. The root γ is of order (p−1), since both roots α and γ = α−α(j) are positive, and, therefore, these roots are sums of simple roots with positive integer coefficients, see (6.2.40). This completes the argument. Now, let γ be a root of order (p − 1) and α(k) be a simple root. We ask whether the sum γ + α(k) is a root. To begin with, we set p = 2 and ask this question for a sum of two simple roots α(i) + α(k) . We recall that the difference of two simple roots, (α(i) − α(k) ), is not a root, and (α(i) , α(k) ) ≤ 0. Consider root string (compare with (6.2.27)) α(i) , α(i) + α(k) , α(i) + 2 α(k) , . . . , α(i) + j α(k)

(j ≥ 0).

(6.2.43)

In accordance with (6.2.27), the number j in (6.2.43) equalsa −2(α(i) , α(k) )/(α(k) )2 . In virtue of Proposition 6.2.3 (see also (6.2.26)), coefficient Nα(k) ,α(i) +m α(k) for m = 0 obeys (Nα(k) ,α(i) )2 = 12 (α(k) , α(k) ) j = −(α(k) , α(i) ).

(6.2.44)

Thus, if (α(i) , α(k) ) = 0, then j = 0 and (α(i) + α(k) ) is not a root, and (i)

(k)

,α hence [Eα(i) , Eα(k) ] = 0. If, on the other hand, j = −2 (α(α(k) )2 (i)

)

> 0, then

(k)

(α + α ) is a root, and generator Eα(i) +α(k) is equal to commutator [Eα(i) , Eα(k) ]. In this way one figures out whether (α(i) + α(k) ) is a root or not. Since (α(i) − α(j) ) is not a root, we find that α(i) is the lowest root (lowest weight vector). Now, let us come back to general question: given a root γ of order (p − 1) > 1, and a simple root α(k) , is the sum γ + α(k) a root (of order p) a As

we have seen, the number j for finite-dimensional simple Lie algebra takes values 0, 1, 2, 3, and the number of root vectors in a string cannot exceed 4.

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or not? If (γ, α(k) ) < 0, then Lemma 6.2.1 applies, and the sum γ + α(k) is a root. Suppose now that (γ, α(k) ) ≥ 0. Consider root string γ, γ − α(k) , γ − 2α(k) , . . . , γ − mα(k) ≡ γ 

(m ≥ 0),

(6.2.45)

which terminates at some m < p − 1, i.e., [γ − (m + 1)α(k) ] is not a root. We actually know the value of m. Indeed, all roots in (6.2.45) are positive (they cannot be negative because of the result (6.2.41) of Theorem 6.2.2) and have orders smaller than p. Therefore, the place where the string (6.2.45) terminates is known by assumption of induction (we know all roots of order smaller than p). Thus, [γ − (m + 1)α(k) ] is not a root, so [γ − mα(k) ] = γ  is the lowest root in α(k) -string (6.2.45). The question of whether or not the sum γ + α(k) is a root is equivalent to the question of whether or not the root γ is the highest in α(k) -string (6.2.45). The answer to the latter question is given by Eqs. (6.2.14) (6.2.27). In other words, γ + α(k) is a root, if the length (m + 1) of the string (6.2.45) is smaller than the length (j + 1) of the whole α(k) -string that starts with the lowest root γ  . If the length (m + 1) equals (j + 1), then γ is the highest root in the string (6.2.45), and the sum γ + α(k) is not a root. The length (j + 1) of the whole α(k) -string that starts with the lowest root γ  , is (see (6.2.27)) j+1=1−2

(γ  , α(k) ) (γ, α(k) ) = 1 + 2m − 2 . (k) 2 (α ) (α(k) )2

By comparing this with (m + 1), which is known by the assumption of induction, we find that if m > 2(α(k) , γ)/(α(k) )2 , (k)

, γ) then the sum γ +α(k) is a root, and in the opposite case we have 2 (α = (α(k) )2

m = j < p − 1, i.e., root γ is the highest in α(k) -string (6.2.45), and the sum γ + α(j) is not a root. Thus, we can construct all roots of algebra A, once its simple roots are known.

Remark 1. Despite the general character of this theorem, the procedure developed in its proof for constructing all positive root vectors as sums of simple roots is not particularly convenient for applications. In what follows (see Sections 6.2.3 and 6.2.5) we sometimes employ a different method of constructing complete root systems of simple Lie algebras out of its simple roots, which heavily uses the symmetry of root system under Weyl reflections.

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To summarize, Theorem 6.2.3 tells that simple finite-dimensional Lie algebra A is completely defined by the subset of its generators E±α(i) ,

Hα(i) = 2

(α(i) , H) , (α(i) )2

(6.2.46)

which correspond to all simple roots α(i) . To simplify formulas, we changed the definition of Cartan elements by a factor of 2 as compared with (6.2.2). Generators (6.2.46) obey defining commutation relations (compare with (6.2.4), (6.1.31), (6.1.32), (6.2.25) and (6.2.42)) [Eα(i) , E−α(j) ] = δi,j (α(i) , H),

(6.2.47)

[Hα(i) , Eα(j) ] = Kji Eα(j) ,

(6.2.48)

[Eα(i) , Eα(j) ] = Nα(i) ,α(j) Eα(i) +α(j) ,

(6.2.49)

[E (j) , . . . [Eα(j) , Eα(i) ] = (ad(Eα(j) ))   α

1−Kij

· Eα(i) = 0,

i = j,

1−Kij

(6.2.50) where the matrix with integer-valued elements Kij = 2

(α(i) , α(j) ) , (α(j) )2

(6.2.51)

is Cartan matrix; in accordance with (6.2.44), we have Nα2(i) ,α(j) = −(α(i) , α(j) ). One should add here similar relations for generators with negative roots E−α(i) , which we do not write down. The commutation relations (6.2.50), which follow from (6.2.43), are called Serre relations. Generators {E±α(i) , Hα(i) } (their total number is 3r) define, in accordance with Theorem 6.2.3, all of simple Lie algebra; this fact that has been proven by J.-P. Serre in 1966. Generators {E±α(i) , Hα(i) } are called Chevalley generators or Chevalley basis. Example. Let us again consider algebra s(3, C) (see Example in the end of Section 6.2.1). Simple roots in (6.2.37) are vectors α = α(1) and β = α(2) . Then Eα(1) = Eα ,

Eα(2) = Eβ ,

Hα(1) = 2(α, H)/(α)2 ,

Hα(2) = 2(β, H)/(β)2 , and formulas (6.2.34), (6.2.35) and (6.2.38) lead to equalities (6.2.47)– (6.2.49) and (6.2.50), where Nα(1) ,α(2) = √16 . Note that Serre relations

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(6.2.50) take the form [Eα(1) , [Eα(1) , Eα(2) ] = 0 and [Eα(2) , [Eα(2) , Eα(1) ] = 0; they are consequences of the last equalities in (6.2.35). Coming back to general theory, let us discuss possible form of Cartan matrix. Clearly, Kii = 2 and Kij ≤ 0 for i = j. In accordance with (6.2.29), there are the following options: Kij = Kji = cos(θij ) = − Kij Kji /2 = θij =

0 0 0 90o

−1 −1 −1/2 120o

−1 −2 √ − 2/2 135o

−1 −3 √ , − 3/2 150o (6.2.52)

where θij is angle between two simple roots α(j) and α(i) (defined with metric cast in orthonormal form gij = δij ; see Proposition 6.1.5). Of course, along with the option Kij = −1, Kji = −2 there is the option Kij = −2, Kji = −1, and along with Kij = −1, Kji = −3 there is the option Kij = −3, Kji = −1. Graphical representation of root vectors is called vector diagram. In accordance with (6.2.52), it is straightforward to draw all two-dimensional vector diagrams symmetric under Weyl reflections. These are shown in Fig. 6.2.3. Relative lengths of root vectors is given by the ratio K12 /K21 , while the overall scale is irrelevant for us (see Remark 1 after Proposition 6.1.5). Roots which are not simple, shown in Fig. 6.2.3, are obtained from simple roots by Weyl reflections. Note that the diagram A2 in Fig. 6.2.3 corresponds to root system of algebra s(3, C), which is shown in Fig. 6.2.2. 6.2.3.

Dynkin diagrams. Root systems of classical Lie algebras s(n, C), so(n, C), sp(2n, C)

It is clear from the above discussion that diagonal elements of Cartan matrix are always equal to 2, while nondiagonal elements carry all information α(2)

α(2) α(2) α

A2 Fig. 6.2.3.

(1)

α

B2

(1)

α(1)

G2

Two-dimensional root diagrams of types A2 , B2 and G2 .

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α(1)

α(2)

α(1)

|α(2) |2 = |α(1) |2

α(2)

|α(2) |2 = 2|α(1) |2

A2 Fig. 6.2.4.

< B2

α(1)


or x for i < j. Then all roots e − e(j) with i < j, including (6.2.61), are positive and can be written as sums of simple roots (6.2.61): e(i) − e(j) = α(i) + α(i+1) + · · · + α(j−1)

(i < j),

(6.2.62)

while all negative roots have representation e(i) − e(j) = −α(j) − α(j+1) − · · · − α(i−1)

(i > j).

(6.2.63)

These relations show that α(i) make basis of simple roots, whose number is n − 1 = rank(s(n, C)), and coefficients in expansion of any positive (or negative) root in vectors α(i) are positive (negative) integers.

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α(1)

α(2)

Fig. 6.2.5.

α(3)

.......

α(n−2)

α(n−1)

Dynkin diagram of type An−1 .

4. With our choice of simple roots (which is determined by the choice of vector x) we have (i)

(i+1)

(α(i) , α(j) ) = (ek −ek

(j)

(j+1)

) g kl (el −el

)=

1 (2δij −δi+1,j −δi,j+1 ). 2n

Also, in view of (6.2.60), we have (α(i) )2 = 1/n, so elements of (n − 1)dimensional Cartan matrix are ⎧ K = 2; (i) (j) (α , α ) ⎨ ii = (1 ≤ i, j ≤ n − 1). Kij = 2 = −1 if |i − j| = 1; K ⎩ ij (α(j) )2 Kij = 0 if |i − j| = 0, 1; This yields Dynkin diagram shown in Fig. 6.2.5. Algebras so(n, C). B- and D-series Algebra so(n, C) is vector space of complex antisymmetric n × n matrices. Its dimension is n(n − 1)/2. Basis in so(n, C) is made of matrices (3.3.31): Mi j = eij − eji ,

i < j.

(6.2.64)

These obey commutation relations (3.3.32): [Mi j , Mm n ] = δjm Mi n − δim Mj n + δin Mj m − δjn Mi m .

(6.2.65)

Basis in Cartan subalgebra is made of commuting Hermitian matrices H1 ≡ −i M1 2 ,

H2 ≡ −i M3 4 ,

H3 ≡ −i M5 6 , . . . ,

Hr ≡ −i M2r−1 2r , (6.2.66)

where 2r = n for even n and 2r = n − 1 for odd n. Thus, so(2r, C) and so(2r + 1, C) have the same rank r. Metric in root space Vr is diagonal: gab =

1 2

Tr(Ha Hb ) = − 12 Tr[(ei i+1 − ei+1 i )(ej j+1 − ej+1 j )] = δab , (6.2.67)

where i = 2a − 1, j = 2b − 1. We pause here to explain the relation of metric (6.2.67), calculated in defining representation, to metric (6.1.8), defined as trace in adjoint representation. We recall that for simple Lie algebras, invariant metric is unique modulo overall numerical factor (see

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Corollary 4.7.1). In particular, invariant metric calculated in defining representation coincides, modulo numerical factor, with Killing metric calculated in adjoint representation. The same is true for metrics (6.2.67) and (6.1.8), which are blocks in invariant metric (6.1.19) in Cartan basis, calculated in defining and adjoint representations. The normalization of metric drops out from the formula (6.2.51) for Cartan matrix. The number of roots in algebras so(2r, C) and so(2r + 1, C) equals 2r (2r − 1) − r = 2 r (r − 1), 2

(2r + 1) 2r − r = 2 r2 , 2

(6.2.68)

respectively, where we again use (6.1.6). Let us construct generators, corresponding to root vectors. It follows from (6.2.65) that

i [i M1 2 , M1 3 + iM2 3 ] = − M1 3 + M2 3 , (6.2.69)  where parameter  can take two values  = ±1 in order that the vector M1 3 + iM2 3 be eigenvector of operator ad(H1 ). Similar formula holds with replacement 3 → 4, so for  = ±1 generators ()

M1 k + iM2 k ≡ M(12,k)

(k = 3, 4)

(6.2.70)

are eigenvectors of element H1 of Cartan subalgebra (6.2.66) in adjoint representation. However, combinations (6.2.70) are not root generators, since they are not eigenvectors of element H2 of Cartan subalgebra: [M3 4 , M1 4 + iM2 4 ] = M1 3 + iM2 3 () () () () ⇒ [M3 4 , M(1 2,4) ] = M(1 2,3) , [M3 4 , M(1 2,3) ] = −M(1 2,4) .

(6.2.71)

To construct root generators, consider combination ()

()

M = (M1 3 + iM2 3 ) + i η(M1 4 + iM2 4 ) = M(1 2,3) + iηM(1 2,4) (η)

(η)

(,η)

(η,)

= −(M(3 4,1) + iM(3 4,2) ) ≡ M(1 2;3 4) = −M(3 4;1 2) ,

(6.2.72)

where  and η can independently take values ±1. Then we have from (6.2.66), (6.2.69) and (6.2.71) that [H1 , M ] = M ,

[H2 , M ] = η M ,

[H3 , M ] = 0, . . . , [Hr , M ] = 0. (6.2.73) Thus, we have four root generators M with  = ±1, η = ±1, which correspond to four roots (e(1) + ηe(2) ): (+1, ±1, 0, . . . , 0) = e(1) ± e(2) ,

(−1, ∓1, 0, . . . , 0) = −(e(1) ± e(2) ).

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Here e(a) are unit basis vectors in r-dimensional root space with coordinates (a) eb = δba (a, b = 1, . . . , r). The construction (6.2.72) is generalized to the case in which two generators Ha and Hb (a < b) of Cartan subalgebra are diagonalized. To this end, one considers invariant (under adjoint action of Ha and Hb ) subspace generated by {Mi j , Mi j+1 , Mi+1 j , Mi+1 j+1 },

(i = 2a − 1, j = 2b − 1),

whose linear combination of the type (6.2.72) (,η)

M(i i+1;j j+1) = (Mi j + iMi+1 j ) + i η(Mi j+1 + iMi+1 j+1 ) determines the roots: (,η)

M(i i+1;j j+1) ←→ (e(a) + ηe(b) ).

(6.2.74)

In this way we find that so(n, C) (for both even and odd n) has the set of roots (e(a) ± e(b) ), −(e(a) ± e(b) ) (a < b; a, b = 1, . . . , r). By definition of metric (6.2.67), these roots have the same length, and their number is r(r−1) · 4 = 2r(r − 1). For so(2r, C), this number coincides with the total 2 number of roots, so we have found all roots in this case. We still have to find additional roots for so(2r + 1, C). Before proceeding further, we give commutation relations for root generators (6.2.72), which are used in what follows (in particular, in the study of algebra G2 ): (,η)

(ρ,κ)

(,κ)

[M(1 2;3 4) , M(3 4;5 6) ] = (1 − η ρ) M(1 2;5 6) .

(6.2.75)

A. so(2r, C): Dr series We choose vector x = (x1 , . . . , xr ) in root space, which defines positive roots, such that xi > xj > 0 for i < j. As an example, x = (r, . . . , 2, 1). If i < j, then (x , e(i) ± e(j) ) = xi ± xj > 0, and roots e(i) ± e(j) (i < j) are positive, while roots −(e(i) ± e(j) ) (i < j) are negative. We can now choose the following (r − 1) simple roots: α(1) = e(1) − e(2) ,

α(2) = e(2) − e(3) , . . . , α(r−1) = e(r−1) − e(r) , (6.2.76)

while rth simple root is defined as α(r) = e(r−1) + e(r) .

(6.2.77)

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r−1 These r roots make basis, since e(i) = k=i α(k) + 12 (α(r) − α(r−1) ). Also, these r roots are simple, since for all positive roots we have e(i) − e(j) = α(i) + · · · + α(j−1) ,

(i < j ≤ r),

e(i) + e(j) = (e(i) − e(j) ) + 2(e(j) − e(r−1) ) + (e(r−1) − e(r) ) + (e(r−1) + e(r) ) = (α(i) + · · · + α(j−1) ) + 2(α(j) + · · · + α(r−2) ) + α(r−1) + α(r) ,

(i < j < r),

e(i) + e(r) = (e(i) − e(r−1) ) + (e(r−1) + e(r) ) = (α(i) + · · · + α(r−2) ) + α(r) ,

(i < r),

i.e., expansion coefficients are positive integers. Flipping signs here, we observe that all negative roots are linear combinations of simple roots (6.2.76), (6.2.77) with negative integer coefficients. Thus, all roots for so(2r, C) are (i,j)

α±

= (e(i) ± e(j) ),

(i,j)

−α±

(i,j)

= −(e(i) ± e(j) )

(1 ≤ i < j ≤ r). (6.2.78)

√

They have equal lengths |α± | = 2. Dynkin diagram for so(2r, C) is constructed from Cartan matrix. The latter is calculated by making use of simple roots (6.2.76), (6.2.77) and metric (6.2.67): Kij = 2

(α(i) , α(j) ) = (2 δij − δi,j−1 − δi−1,j ) (∀ i, j < r − 1), (α(j) )2

Kri = Kir = 2 So,

⎛

(α(i) , α(r) ) = (2δi,r − δi,r−2 ). (α(r) )2

2 −1 0 ... 0 0 0 ⎜−1 2 −1 . . . 0 0 0 ⎜ ⎜ 0 −1 2 ... 0 0 0 ⎜ ⎜ 0 0 −1 . . . 0 0 0 ⎜ ⎜ K =⎜ . . . ... . . . ⎜ ⎜ 0 0 0 ... 2 −1 0 ⎜ ⎜ 0 0 0 . . . −1 2 −1 ⎜ ⎝ 0 0 0 ... 0 −1 2 0 0 0 ... 0 −1 0

(6.2.79) ⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ .⎟. ⎟ 0⎟ ⎟ −1⎟ ⎟ 0⎠ 2

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α(r−1) α

(1)

α

(2)

.......

α

(r−3)

α

(r−2)

α(r) Fig. 6.2.6.

Dynkin diagram of type Dr .

Hence, we arrive at Dynkin diagram for algebra so(2r, C) shown in Fig. 6.2.6. Note that if we remove the vertex corresponding to the root α(r) (or α(r−1) ) from Dynkin diagram of so(2r, C) shown in Fig. 6.2.6, we obtain Dynkin diagram of s(r, C) shown in Fig. 6.2.5 (rank(s(r, C)) = r − 1). This means that algebra s(r, C) with root system generated by simple roots {α(1) , . . . , α(r−1) }, is subalgebra of so(2r, C), and this subalgebra is generated by Cartan elements H1 , . . . , Hr−1 and root generators Eα(1) , . . . , Eα(r−1) . Accordingly, we have for compact real forms that su(r) is subalgebra of so(2r). It is of interest also to consider algebras so(2r, C) for small r. i. Algebra so(8, C) has Dynkin diagram

It follows from the shape of this diagram that so(8, C) has additional symmetry called triality. ii. Algebra so(6, C) has Dynkin diagram

This diagram corresponds also to algebra s(4, C). So these two algebras are isomorphic. Isomorphic are also their compact real forms so(6) and su(4)). This agrees with the result of Section 3.3.1.2. iii. Algebra so(4, C) has Dynkin diagram

Hence, there is isomorphism so(4, C) = s(2, C) + s(2, C), and, for real compact forms, so(4) = su(2) + su(2). We have seen this in Section 3.3.1.2.

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B. so(2r + 1, C): Br -series We have already obtained 2r(r −1) roots (6.2.78) of algebra so(2r +1) (here we take into account that so(2r) ⊂ so(2r + 1)). However, the number of roots of so(2r + 1) equals 2r2 , see (6.2.68), so we have to find another 2r roots and root generators. We have not used index 2r + 1 yet. Consider now the operator M1 2r+1 . The only Cartan element that acts on it nontrivially is H1 = −i M1 2 : one has [H1 , M1 2r+1 ] = i M2 2r+1 . The right-hand side here involves M2 2r+1 , which nontrivially commutes with Cartan element H1 only: [H1 , M2 2r+1 ] = −i M1 2r+1 . Thus, linear space generated by two vectors Mi 2r+1 (i = 1, 2) is invariant under Cartan subalgebra, so the natural Ansatz for a root generator is a linear combination Mα = M1 2r+1 + α M2 2r+1 . We have [H1 , Mα ] = i M2 2r+1 − i α M1 2r+1 = −i α

M1 2r+1 −

1 M2 2r+1 , α

[Hb , Mα ] = 0 (b = 2, 3, . . . , r), and, therefore, Mα is root generator for α2 = −1 or α = ±i, root vectors being (−iα, 0, . . . , 0) = (±1, 0, . . . , 0). Replacing H1 by H2 , then by H3 , etc., we find 2r additional generators of so(2r + 1, C) whose roots are (±1, 0, . . . , 0), (0, ±1, 0, . . . , 0), . . . , (0, 0, . . . , ±1). Thus, with (6.2.74), all roots of so(2r + 1, C) are (a,b)

α±

= (e(a) ± e(b) ),

(a,b)

−α±

= −(e(a) ± e(b) ),

±e(a) .

(6.2.80)

√ (a,b) Their lengths are |α± | = 2 and |e(a) | = 1 (here we again use metric j+1 i+1 g ab = δ ab , see (6.2.67)). These root vectors written as ( e( 2 ) + η e( 2 ) ) i+1 and ρ e( 2 ) , where i, j = 1, 3, 5, . . . , 2r − 1, correspond to root elements (,η)

M(i i+1;j j+1) = Mi j + iMi+1 j + iηMi j+1 − η Mi+1 j+1 , (6.2.81) (ρ)

M(i) = Mi 2r+1 + iρMi+1 2r+1 , (,η)

(, η, ρ = ±1). (−,−η)

(ρ)

Using (6.2.64), we have (M(i i+1;j j+1) )† = −M(i i+1;j j+1) and (M(i) )† = (−ρ)

−M(i) . Therefore, Hermitian conjugation transforms generator with root vector α to generator with root vector (−α), in agreement with general theory, see Section 6.1.2.

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Problem 6.2.6. Check that root elements (6.2.81) obey commutation relations ()

(η)

[M(i) , M(i) ] = ( − η)H(i+1)/2 , (,η)

(ρ,κ)

[M(i i+1;j j+1) , M(i i+1;j j+1) ] = (1 − ηκ)( − ρ) H(i+1)/2 + (1 − ρ)(η − κ) H(j+1)/2 , ()

(η)

(,η)

[M(i) , M(j) ] = −M(i i+1;j j+1) , (ρ)

(,η)

(η)

[M(i) , M(i i+1;j j+1) ] = (1 − ρ) M(j) ,

(6.2.82)

and relations similar to (6.2.75), (,η)

(ρ,κ)

(,κ)

[M(i i+1;j j+1) , M(j j+1;k k+1) ] = (1 − η ρ) M(i i+1;k k+1) ,

(6.2.83)

where i, j, k = 1, 3, 5, . . . and i < j < k. Simple roots of so(2r + 1, C) are chosen as follows: α(1) = e(1) − e(2) , . . . , α(r−1) = e(r−1) − e(r) ,

α(r) = e(r)

(6.2.84)

(compare these with simple roots (6.2.76), (6.2.77) of algebra so(2r, C)), so for all positive roots (a < b) we obtain expansions with real positive coefficients e(a) − e(b) = α(a) + α(a+1) + · · · + α(b−1) , e(a) + e(b) = α(a) + α(a+1) + · · · + α(b−1) + 2(α(b) + α(b+1) + · · · + α(r) ), e(a) = α(a) + α(a+1) + · · · + α(r) .

(6.2.85)

Problem 6.2.7. Find all positive roots (6.2.85) of algebra so(7, C) by considering linear combinations of simple roots α(1) = e(1) − e(2) , α(2) = e(2) −e(3) , α(3) = e(3) and using only rules given in Lemma 6.2.1 and Remark 1 after Theorem 6.2.3. Negative roots are expressed via simple roots in a similar manner, with flipped sign of all coefficients. Cartan matrix is calculated by using (6.2.84) and reads Kij = 2

(α(i) , α(j) ) = (2 δij − δi,j−1 − δi−1,j ) (i, j ≤ r − 1), (α(j) )2

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Kir = 2

321

(α(i) , α(r) ) = (−2 δi,r−1 + 2δi,r ), (α(r) )2

Kri = 2

(α(r) , α(i) ) = (−δi,r−1 + 2δi,r ). (α(i) )2

In other words, ⎛

2 −1 0 ... 0 0 0 ⎜−1 2 −1 . . . 0 0 0 ⎜ ⎜ 0 −1 2 ... 0 0 0 ⎜ ⎜ 0 −1 . . . 0 0 0 ⎜ 0 K =⎜ ⎜ . . . ... . . . ⎜ ⎜ 0 0 0 . . . −1 2 −1 ⎜ ⎝ 0 0 0 ... 0 −1 2 0 0 0 ... 0 0 −1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟. .⎟ ⎟ 0⎟ ⎟ −2⎠ 2

Thus, Dynkin diagram for algebra so(2r + 1, C) has the form shown in Fig. 6.2.7. Algebras sp(2r, C): Cr -series In accordance with item 8 of Section 3.3.1, elements of Lie algebra sp(2r, C) are complex 2r × 2r matrices A, obeying symplectic relation (3.3.46),

0 Ir T (J · A) = J A, J = , (6.2.86) −Ir 0 meaning that matrix M ≡ (J · A) is symmetric. Therefore, dimension of sp(2r, C) equals the dimension of space of complex symmetric 2r × 2r matrices 2r(2r + 1)/2 = r(2r + 1). Let us write matrix A in block form

X Y A= , (6.2.87) Z W where X, Y, Z, W are complex r × r blocks. Then the symplectic relation (6.2.86) gives X = −W T , α(1)

α(2)

Fig. 6.2.7.

Y T = Y,

.......

Z T = Z,

α(r−2)

α(r−1)

>

Dynkin diagram of type Br .

(6.2.88) α(r)

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and we can choose the following basis in algebra sp(2r, C): ei,i − er+i,r+i ≡ Hi ei,j − er+j,r+i ≡ Xij ei,r+j + ej,r+i ≡ Yij ,

(i = 1, . . . , r), (i = j = 1, . . . , r),

er+i,j + er+j,i ≡ Zij

(6.2.89)

(i, j = 1, . . . , r).

It is straightforward to see that we have listed all basis elements of sp(2r, C). Diagonal elements Hi (i = 1, . . . , r) generate Cartan subalgebra, so rank of sp(2r, C) is equal to r, and the number of roots, given by (6.1.6), is equal to r(2r + 1) − r = 2r2 . Metric in root space Vr of sp(2r, C) is chosen as follows: gij =

1 2

Tr(Hi Hj ) = δij .

(6.2.90)

This metric coincides with metric (6.1.8), modulo numerical factor. The reason for that is the same as given after Eq. (6.2.67). We make use of (6.2.89) and obtain [Hi , Xkl ] = (e(k) − e(l) )i Xkl ,

[Hi , Ykl ] = (e(k) + e(l) )i Ykl ,

[Hi , Zkl ] = −(e(k) + e(l) )i Zkl , (k)

where, as always, ei (e(k) − e(l) )

(k = l),

(6.2.91)

= δik . Thus, roots of algebra sp(2r, C) are given by ±(e(k) + e(l) ) (k = l),

±2 e(k) ,

k, l = 1, . . . , r. (6.2.92)

The number of these roots equals r(r − 1) + r(r + 1) = 2r2 , so there are no other roots in sp(2r, C). Roots of the following form are treated as positive: (e(k) − e(l) )

k < l,

(e(k) + e(l) ) ∀k, l.

This corresponds to vector x with coordinates xk > xl > 0 for k < l. The basis of simple roots is then (e(1) − e(2) , . . . , e(r−1) − e(r) , 2 e(r) ).

(6.2.93)

Making use of metric (6.2.90), we calculate Cartan matrix Kij = 2

(α(i) , α(j) ) = (2 δij − δi,j−1 − δi−1,j ) (i, j < r − 1), (α(j) )2

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Kir = 2 Kri = 2 or

323

(α(i) , α(r) ) = (−δi,r−1 + 2δi,r ), (α(r) )2

(α(r) , α(i) ) = (−2 δi,r−1 + 2δi,r ), (α(i) )2

⎛

2 −1 0 ... 0 0 0 ⎜−1 2 −1 . . . 0 0 0 ⎜ ⎜ 0 −1 2 ... 0 0 0 ⎜ ⎜ 0 0 −1 . . . 0 0 0 ⎜ K =⎜ ⎜ . . . ... . . . ⎜ ⎜ 0 0 0 . . . −1 2 −1 ⎜ ⎝ 0 0 0 ... 0 −1 2 0 0 0 ... 0 0 −2

⎞ 0 0⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟. .⎟ ⎟ 0⎟ ⎟ −1⎠ 2

Dynkin diagram for sp(2r, C) is determined by the latter expression and is shown in Fig. 6.2.8. Remark. Root system of C-series (Lie algebra sp(2r, C)) with simple roots (6.2.93) is dual to root system of B-series (Lie algebra so(2r + 1, C)) with simple roots (6.2.84). Duality means that these systems transform to each other (modulo overall normalization of their vectors) under duality transformation α → α∨ = α/(α)2 .

(6.2.94)

Cartan matrix can be written as Kij = (α(i) , α(j)∨ ), so the transformation (6.2.94) is equivalent to transposition of Cartan matrix, Kij → Kji . 6.2.4.

Dynkin diagrams and classification of finite-dimensional simple Lie algebras

In this section we figure out what are other simple finite-dimensional Lie algebras over and beyond those discussed in the previous section, see also books [3,10] and lecture notes [20]. In other words, we are going to obtain complete classification of all simple finite-dimensional complex Lie algebras. This is done by listing all admissible Dynkin diagrams. By admissible α(1)

α(2)

Fig. 6.2.8.

.......

α(r−2)

α(r−1)


k), each containing the origin. A hyperplane Vk ⊂ Rn can be thought of as vector space whose basis is made of k orthonormal vectors  xi ∈ Rn (i = 1, . . . , k). In other words, any system of k mutually orthogonal xk ) in Rn defines a point in Gn,k . The systems related by rotations unit vectors ( x1 , . . . ,  and reflections in hyperplane Vk define one and the same point in Gn,k ; the latter transformations make a group O(k). There is an action of O(n) in Gn,k induced by the action on basis vectors in Vk : xk ) → ( x1  , . . . ,  xk  ) = (O ·  x1 , . . . , O ·  xk ), ( x1 , . . . , 

∀O ∈ O(n).

(7.2.46)

x1 , . . . ,  xk ) This action is transitive, since any hyperplane Vk ∈ Gn,k with basis vectors ( can be obtained by O(n) transformation from a reference hyperplane Ek ∈ Gn,k , whose ek ), where  em = (0, . . . , 0, 1, 0 . . . , 0) are basis vectors in basis vector system is ( e1 , . . . ,     m−1

Rn (in other words, Ek is the coordinate hyperplane of the first k axes) : xk ) = (O ·  e1 , . . . , O · ek ). ( x1 , . . . ,  xm )i for m ≤ k, i.e., its first k Here matrix ||Oij || ∈ O(n) is such that Oim = ( xk ). Stationary subgroup on O(n) leaves columns are coordinates of vectors ( x1 , . . . ,  the hyperplane Ek ∈ Gn,k intact under the transformation (7.2.46), and its elements are 

Ω1

0n−k,k

0k,n−k Ω2

 ,

Ω1 ∈ O(k),

Ω2 ∈ O(n − k).

So, the stationary subgroup is isomorphic to O(k) × O(n − k) and, in accordance with Proposition 7.1.2, we have Gn,k = O(n)/(O(k) × O(n − k)). The space Gn,k is called Grassmann manifold (or Grassmannian). Clearly, Gn,k = Gn,n−k and Gn+1,1 = RPn . Problem 7.2.11. Calculate the dimension of the Grassmann manifold Gn,k .

8. Complex projective space CPn is the space of nonzero complex vectors z = (z1 , z2 , . . . , zn+1 ) ∈ Cn+1 with identification z ∼ λz for all complex λ = 0. In other words, CPn is the space of complex “straight lines” in Cn+1  passing through the origin. We can normalize vectors z, so that i zi∗ zi = 1, then all vectors of the form eiφ z (∀φ ∈ R and given z) define one and the same point in CPn . There is transitive action of group U (n+ 1) in the space  of complex vectors normalized by i zi∗ zi = 1. Let us choose a point in CPn determined by reference vector z = (0, . . . , 0, 1). Its stationary subgroup has

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matrices

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 .. .

⎞

⎟ ⎟ ⎟ ⎟, ⎟ 0⎠ 0 ... 0 eiφ U

361

U ∈ U (n) , φ ∈ R.

Therefore, Proposition 7.1.2 gives CPn = U (n + 1)/(U (n) × U (1)). Making use of (7.2.28), one rewrites this relation as CPn = S 2n+1 /U (1). In the simplest case n = 1 we have CP1 = S 2 and S 2 = SU (2)/U (1) = S 3 /S 1 .

(7.2.47)

This is famous Hopf fibration of sphere S 3 , which treats this sphere as fiber bundle with base S 2 and fiber S 1 . Problem 7.2.12. Show that CPn = DnC ∪ CPn−1 , where DnC is open 2ndimensional ball, z1 z1∗ +· · ·+zn zn∗ < 1, and CPn−1 is the boundary of this ball, z1 z1∗ +· · ·+zn zn∗ = 1, with the identification (z1 , . . . , zn ) ∼ (eiφ z1 , . . . , eiφ zn ) for all φ ∈ R. Show, in particular, that CP1 = S 2 . Remark. We have shown in Example 3 that sphere S 2 is coset space S 2 = SO(3)/SO(2).

(7.2.48)

We see from (7.2.47) and (7.2.48) that S 2 is a coset space obtained from different spaces (SU (2) and SO(3) in our case), with the same orbits (U (1) = SO(2) = S 1 here). In other words, there exist different fiber bundles (SU (2) and SO(3)) with the same fibers (circles S 1 ) and the same base (sphere S 2 ). 9. Real Grassmann manifold Gn,k of Example 7 can be generalized to complex case. Complex Grassmann manifold GC n,k is the space of complex k-dimensional hyperplanes in Cn containing the origin. Like in Example 8, we find that GC n,k is coset space C GC n,k = Gn,n−k = U (n)/(U (k) × U (n − k)).

In some physics applications, important are coset spaces obtained from symplectic groups, namely, HPn−1 = USp(2n)/(USp(2n−2)×USp(2)), GH n,k = USp(2n)/(USp(2n−2k)×USp(2k)). Space HPn−1 is quaternionic projective space, while GH n,k is quaternionic Grassmann space.

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10. Consider sphere S 4n+3 defined by equation n+1 

q¯α qα = I2 ,

(7.2.49)

α=1

where qα (α = 1, . . . , n + 1) are quaternions (3.3.58) combined in quaternionic vector q = (q1 , . . . , qn+1 ) ∈ Hn+1 , 

(7.2.50)

and overbar denotes Hermitian conjugation. There is transitive action on the sphere (7.2.49) of group SU (n+1, H) whose elements are quaternionic matrices U obeying unitarity (3.3.61) (see Example 5 in Section 3.3.1.2). This action is defined in a standard way qα → Uαβ qβ . Let us choose a reference point in S 4n+3 whose quaternionic vector is  q = (0, . . . , 0, I2 ). Stationary subgroup of this vector contains quaternionic matrices ⎛ ⎞ 0 ⎜ ⎟ ⎜ .. ⎟ ⎜ U .⎟ ⎜ ⎟ , U ∈ SU (n, H), ⎜ ⎟ ⎝ 0⎠ 0 ... 0 I2 which make subgroup SU (n, H) ⊂ SU (n + 1, H). Hence, Proposition 7.1.2 tells that S 4n+3 = SU (n + 1, H)/SU (n, H) = USp(2n + 2)/USp(2n), where we use isomorphism SU (n, H) = USp(2n), which is the result of Problem 3.3.11. 11. Consider complex projective space CP2n+1 defined in Example 8. A pair of complex numbers z1 = x1 + i y1 and z2 = x2 + i y2 can be considered as quaternion (3.3.58) : q(z1 , z2 ) = (x1 + i σ1 y1 ) + (x2 + i σ1 y2 ) i σ3 = x1 + i σ1 y1 + i σ2 y2 + i σ3 x2 , (7.2.51) while multiplication of z1 and z2 by one and the same complex number λ = λ1 + iλ2 can be viewed as left multiplication of quaternion (7.2.51) by special quaternion Λ = λ1 + i σ1 λ2 : (7.2.52) q(λ z1 , λ z2 ) = Λ · q(z1 , z2 ). Making use of (7.2.51) and (7.2.52), one can think of space CP2n+1 as the space of quaternionic vectors (7.2.50) with identification  q ∼ Λ· q for all Λ = λ1 + i σ1 λ2 = 0, or the space of quaternionic vectors  q ∈ Hn+1 on sphere S 4n+3 given by (7.2.49) with identification ¯ · Λ = I2 . (7.2.53)  q ∼ Λ· q, Λ Since SU (n + 1, H) acts transitively in S 4n+3 (see Example 10), group SU (n + 1, H) transitively acts also in CP2n+1 . We choose a reference point in CP2n+1 whose vector is q0 = (0, . . . , 0, Λ), where quaternion Λ is special, Λ = λ1 + i σ1 λ2 , and obeys  ¯ · Λ = I2 Λ

⇒

λ1 = cos φ,

λ2 = sin φ.

(7.2.54)

The subspace of these quaternions Λ ∈ H makes a group U (1). Stationary subgroup of point  q0 ∈ CP2n+1 has quaternionic matrices of the form ⎛ ⎞ 0 ⎜ ⎟ ⎜ .. ⎟ ⎜ U .⎟ ⎜ ⎟, U ∈ SU (n, H), Λ = cos φ + i σ1 sin φ ∈ U (1). ⎜ ⎟ ⎝ 0⎠ 0 ... 0 Λ

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These matrices form subgroup SU (n, H) × U (1) ⊂ SU (n + 1, H). So, in accordance with Proposition 7.1.2 we have CP2n+1 = SU (n + 1, H)/(SU (n, H) × U (1)) = USp(2n + 2)/(USp(2n) × U (1)), where we again use isomorphism SU (n, H) = USp(2n) of Problem 3.3.11.

12. Let G be Lie group. Consider Lie group G × G and denote the first and second groups here by GL and GR , respectively, to distinguish them. Elements of GL ×GR are pairs (gL , gR ), where gL ∈ GL and gR ∈ GR . There is a subgroup GV of GL × GR which contains elements (g, g), ∀g ∈ G; it is isomorphic to G and is called diagonal subgroup. Coset space (GL ×GR )/GV is the space of equivalence classes with respect to equivalence relation (gL , gR ) ∼ (gL · g, gR · g).

(7.2.55)

−1 here and obtain that there is one and only one One can choose g = gR element in each coset which has the form (g  , e), where g  ∈ G and e is unit element in G. Thus, there is one-to-one correspondence between cosets and elements of group G, i.e.,

(GL × GR )/GV = G.

(7.2.56)

Left action of group GL × GR in coset space (GL × GR )/GV is determined by the relation −1 , e), (gL , gR ) · (g, e) = (gL · g, gR ) ∼ (gL · g · gR

so that the action is −1 . g → gL · g · gR

The case particularly important for physics is G = SU (n). In that case, the coset space (7.2.56) is [SU (n)L × SU (n)R ]/SU (n)V = SU (n). Probably the most important application has to do with the fact that Quantum Chromodynamics with n massless quark flavors (chiral limit) has global symmetry SU (n)L × SU (n)R which is spontaneously broken down to diagonal symmetry SU (n)V . 7.3.

Action of Group G in Coset Space G/H. Induced Representations

Let G be Lie group and H its Lie subgroup. Consider coset space G/H. In each coset, one chooses a representative k ∈ G; different k correspond to different cosets from G/H, and vice versa. The set of all representatives

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is a subset K in G, which is in one-to-one correspondence with G/H. Any element g0 ∈ G is represented in a unique way as g0 = k · h,

(7.3.1)

where h ∈ H. We considered in Example 3 of Section 7.1 the left action of group G in coset space G/H. Due to uniqueness of the representation (7.3.1), this action is written as (cf. (4.1.25)) g · k = k(g, k) · h(g, k),

∀k ∈ K, ∀g ∈ G,

(7.3.2)

where k(g, k) ∈ K and h(g, k) ∈ H. So, the left action of element g ∈ G in G/H induces the left action of g in subspace K ⊂ G, namely, g

k −→ k(g, k) = g · k · h(g, k)−1 .

(7.3.3)

The group property of action of G in G/H translates to the properties of mapping (7.3.3): k(g2 , k(g1 , k)) = k(g2 · g1 , k),

(7.3.4)

h(g2 , k(g1 , k)) · h(g1 , k) = h(g2 · g1 , k).

(7.3.5)

These are obtained by comparing the two equivalent expressions g2 · g1 · k = g2 · k(g1 , k)·h(g1 , k) = k(g2 , k(g1 , k)) · h(g2 , k(g1 , k))·h(g1 , k), g2 · g1 · k = k(g2 · g1 , k) · h(g2 · g1 , k). Relation (7.3.4) coincides with the standard group property (1.4.2). On the other hand, the second relation (7.3.5) does not imply that mapping h(., k): G → H (with prescribed k) is homomorphism, so it cannot be employed for constructing representation of group G in a straightforward manner. It is remarkable, however, that formulas (7.3.5) and (7.3.4) enable one to make use of the function h(g, k): G × K → H for constructing a representation of group G out of representation of its subgroup H, which is called induced representation. Let us study this construction in Lie group context (induced representations in the case of finite groups are discussed in Section 4.1.2). Let there be a representation of subgroup H ⊂ G in space V, i.e., element h ∈ H transforms v ∈ V to v → h · v ∈ V (we do not introduce special notation for this operation to simplify formulas below). The induced representation T of group G, which we are going to define, acts in space of

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functions v(k) (k ∈ K) taking values in V. In other words, the functions we are talking about are mappings from K = G/H to V. The definition is −1  · v(k(g −1 , k)) ≡ vg (k), T (g) · v(k) = h(g −1 , k)

∀g ∈ G.

(7.3.6)

Let us show that this is homomorphism, i.e., T obeys (4.1.1). We write T (g1 ) · T (g2 ) · v(k)

 −1 (7.3.7) = T (g1 ) · vg2 (k) = h(g1−1 , k) · vg2 (k(g1−1 , k))     −1 −1  −1 −1 · h(g2 , k(g1−1 , k)) · v k(g2−1 , k(g1−1 , k)) . = h(g1 , k)

On the other hand, we have   −1 T (g1 · g2 ) · v(k) = h (g1 · g2 )−1 , k · v(k((g1 · g2 )−1 , k)).

(7.3.8)

The two expressions (7.3.7) and (7.3.8) coincide due to (7.3.4) and (7.3.5), so that we have T (g2 )·T (g1 ) = T (g2 ·g1 ). Other homomorphism properties are checked in a similar way. Thus, Eq. (7.3.6) indeed defines a representation of G, which is representation induced from representation of subgroup H ⊂ G. Example 1. Consider a trivial representation of subgroup H in V: v → h · v = v (∀h ∈ H, ∀v ∈ V). In that case the definition (7.3.6) gives for induced representation of group G: [T (g) · v](k) = v(k(g −1 , k)).

(7.3.9)

In this case v(k) is a numerical function of representatives of elements of G/H. Example 2. If subgroup H is trivial (contains only unit element e ∈ G), then any of its representations is trivial, while K = G. In that case formulas (7.3.6), (7.3.9) are equivalent to the definition (4.5.12) of left regular representation of G. Other examples of induced representations are studied in the accompanying book. Detailed presentation of the method of induced representations is given also in book [18]. Remark. Induced representation acts on functions in coset space G/H in such a way that there is the dependence on the choice of representatives in cosets (i.e., on parameterization of G/H). Yet this dependence boils down to simple property: induced representations constructed with different sets of representatives in G/H, are equivalent.

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˜ be two sets of representatives of cosets in G/H, so Indeed, let K and K that g · k = k(g, k) · h(g, k),

˜ k) ˜ · h(g, ˜ k), ˜ g · k˜ = k(g,

where h, ˜ h ∈ H. In what follows we use the notations k, k˜ for representatives of one and the same coset, then k = k˜ · qk , where qk ∈ H is some function of k. Obviously, g k˜ and gk = g k˜ · qk belong to the same coset, and we have ˜ k) ˜ · qk(g,k) . k(g, k) = k(g, ˜ respectively, taking Let V and V˜ be spaces of functions on K and K, values in the space of representation of subgroup H. Induced representation T (g) in V is defined in (7.3.6), while induced representation T˜ (g) in V˜ , constructed with another choice of representatives, is defined in the same way as (7.3.6): ˜ = [h(g ˜ −1 , k)] ˜ −1 v˜(k(g ˜ −1 , k)). ˜ (T˜(g)˜ v )(k)

(7.3.10)

Let us construct isomorphism Λ : V → V˜ , such that T˜ (g)Λ = ΛT (g),

(7.3.11)

meaning that representations T (g) and T˜(g) are equivalent. Note that if the induced representation acted on functions on G/H, in the sense that it were independent of the choice of representatives, then mapping Λ would be ˜ = trivial: for k and k˜ belonging to the same coset one would have (Λv)(k) v(k). It turns out that this is not the case. Let us write ˜ k) ˜ · qk(g,k) · h(g, k). g · k˜ · qk = g · k = k(g, k) · h(g, k) = k(g, On the other hand, we have ˜ · qk = k(g, ˜ k) ˜ · h(g, ˜ k) ˜ · qk , (g · k) which gives ˜ k) ˜ · qk , qk(g,k) · h(g, k) = h(g, or ˜ k)] ˜ −1 = qk [h(g, k)]−1 q −1 . [h(g, k(g,k)

(7.3.12)

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This implies that mapping Λ we are after, is ˜ = qk v(k). (Λv)(k)

(7.3.13)

This mapping is obviously invertible, and hence it is isomorphism. To see that mapping (7.3.13) obeys (7.3.11), we write ˜ = [h(g ˜ −1 , k)] ˜ −1 (Λv)(k(g ˜ −1 k)) ˜ T˜ (g)(Λv)(k) ˜ −1 , k)] ˜ −1 qk(g−1 ,k) v(k(g −1 , k)) = [h(g = qk [h(g

−1

−1

, k)]

v(k(g

−1

(7.3.14)

˜ , k)) = (ΛT (g)v)(k),

where we consecutively use formulas (7.3.10), (7.3.13) and (7.3.12). This is the desired result. 7.4.

Models of Lobachevskian Geometry and Geometry of Spaces AdS and dS

Let us remind the properties of straight lines on a plane. • There is one and only one straight line passing through two different points. • Two different straight lines either cross at a single point or do not cross at all, i.e., are parallel to each other (this can be viewed as a corollary of the first property). One more property is given by Euclid’s fifth postulate whose contemporary formulation is • in a plane, given a line and a point not on it, at most one line parallel to the given line (i.e., not intersecting the given line) can be drawn through the point. Lobachevsky’s geometry is not Euclidean, as it is based on denial of Euclid’s fifth postulate. At the same time, it preserves the two first properties listed above. In this section we use methods of the theory of homogeneous spaces to construct explicit models of Lobachevskian geometry. We begin with linear-fractional transformation (1.4.37) in complex plane, z → w(z) =

az +b , cz +d

z ∈ C,

(7.4.1)



 a b ∈ SL(2, R), i.e., ad − bc = 1. Matrices ±A define one and c d the same transformation, so Eq. (7.4.1) defines the action of a group with real matrix A =

SL(2, R)/{±I2 } = PSL(2, R) in complex plane C. This group action is not transitive in C. Indeed, there are three orbits of the group P SL(2, R), which are: (1) real axis; (2)–(3) upper and lower half-planes. This is clear from equality z − z∗ , (7.4.2) w − w∗ = |c z + d|2 showing that each of the three orbits transforms to itself under (7.4.1). Consider one of the orbits, namely, upper half-plane P = {z ∈ C | Im(z) > 0}, where group PSL(2, R) acts

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transitively. Let us choose a reference point i in P and find its stationary subgroup H. We find from (7.4.1) equation i = (a i + b)/(c i + d) which gives a=d,

b = −c

⇒

a2 + b2 = 1

⇒

a = d = cos φ,

b = −c = sin φ.

Thus, any element h of the stationary subgroup H is written as 

cos φ sin φ , h= − sin φ cos φ

(7.4.3)

and hence H = SO(2). In accordance with Proposition 7.1.2, there is a smooth and one-to-one mapping of the upper half-plane to coset space P ↔ P SL(2, R)/SO(2) = SO ↑ (1, 2)/SO(2),

(7.4.4)

where we use isomorphism P SL(2, R) = SO ↑ (1, 2), see Problem 3.5.5. Remark 1. Any compact 2-dimensional Riemann surface (except for sphere) is diffeomorphic to P/Γ, where Γ is a discrete subgroup of the group of linear-fractional transformations P SL(2, R). One makes use of the fact that P SL(2, R) transitively acts in upper half-plane P and constructs a model of Lobachevskian non-Euclidean geometry, called Poincar´ e model. To this end, one defines Lobachevskian straight lines (Λ-lines) in P as half-circles (see Fig. 7.4.1) whose centers (point A) belong to real axis. One also treats as straight Λ-lines half-circles of infinite radius, i.e., “true” straight lines in complex plane, normal to real axis. These straight Λ-lines have the first two properties we recalled in the beginning of this section. Indeed, Fig. 7.4.1 clearly shows that there is only one half-circle passing through two different points z1 and z2 . So, two different straight Λ-lines either cross at one point or do not cross at all. On the other hand, it is clear that there always exist two different Λ-lines that cross each other but do not cross a given third Λ-line (in fact, there is infinite number of these pairs of Λ-lines). Thus, the model does violate Euclid’s fifth postulate and thus gives an example of Lobachevskian geometry. To introduce metric in P with straight lines defined as Λ-lines, we note, in the first place, that linear-fractional transformations (7.4.1) take Λ-lines to Λ-lines. This follows from Proposition 1.4.1 which ensures that linear-fractional transformations take circles to circles; also, they leave real axis invariant, as we have seen, so centers of Λ-lines remain on real axis. Given four points z1 , z2 , z3 , z4 ∈ P, one defines double ratio (z1 , z2 , z3 , z4 ) =

(z2 − z3 )(z1 − z4 ) (z2 − z3 )/(z2 − z4 ) = , (z1 − z3 )/(z1 − z4 ) (z2 − z4 )(z1 − z3 )

(7.4.5)

which is invariant under linear-fractional transformations (7.4.1) with any matrix A ∈ GL(2, C).

Fig. 7.4.1.

Straight Λ-lines in Poincar´ e model of Lobachevskian geometry.

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Problem 7.4.1. Show that the double ratio (7.4.5) is invariant under linear-fractional transformations (7.4.1). Now, consider two points z1 , z2 ∈ P. There is unique Λ-line passing through them. We define a function, which is invariant under linear-fractional transformations (7.4.1): ρ(z1 , z2 ) = ln(z1 , z2 , α, β),

(7.4.6)

where α and β are end points of the Λ-line, passing through z1 and z2 , see Fig. 7.4.1. For three points z1 , z2 and z3 in one Λ-line, as shown in Fig. 7.4.1, the following identity holds: ⇒

(z1 , z2 , α, β)(z2 , z3 , α, β) = (z1 , z3 , α, β)

ρ(z1 , z2 ) + ρ(z2 , z3 ) = ρ(z1 , z3 ).

Moreover, we will see momentarily that (z1 , z2 , α, β) is real, and that (z1 , z2 , α, β) > 1, so ρ(z1 , z2 ) > 0 and invariant object ρ(z1 , z2 ) can be postulated as the distance between points z1 and z2 in the non-Euclidean model on P. For given Λ-line, there exists linear-fractional transformation (7.4.1) that maps this Λ-line to imaginary axis; upon choosing b = −αa and d = −βc in (7.4.1) one maps the points α, z1 , z2 , β to the points 0, iy1 , iy2 , ∞ in imaginary axis, see Fig. 7.4.1, with y2 > y1 > 0. Invariance of the double ratio ensures that (z1 , z2 , α, β) = (iy1 , iy2 , 0, ∞) =

y2 |y1 + y2 | + |y1 − y2 | , = y1 |y1 + y2 | − |y1 − y2 |

(7.4.7)

where we use y2 > y1 > 0. Formula (7.4.7) immediately gives that (z1 , z2 , α, β) is real, and that (z1 , z2 , α, β) > 1. Problem 7.4.2. Write a point z of half-circle of radius r (see Fig. 7.4.1) as z = (α+β) + 2 reiφ , where φ ∈ R, and show directly that (z1 , z2 , α, β) ∈ R and (z1 , z2 , α, β) > 1. We note that

|z1 −z2 | ∗| |z1 −z2

= (z1 , z1∗ , z2∗ , z2 )1/2 is invariant under transformations (7.4.1), so |y1 − y2 | |z1 − z2 | = . |z1 − z2∗ | |y1 + y2 |

Therefore, Eq. (7.4.7) can be written in a convenient form 1+ (z1 , z2 , α, β) =

1−

|z1 −z2 | ∗| |z1 −z2 |z1 −z2 | ∗| |z1 −z2

.

(7.4.8)

e metric in P: We use (7.4.8) in (7.4.6), and in the limit z1 → z2 obtain Poincar´ ρ(z1 , z2 ) → ds =

|dz| y

⇒ ds2 =

dz dz ∗ , y2

(7.4.9)

where dz = z2 − z1 and y = Im(z2 ) = Im(z1 ). Half-plane P with metric (7.4.9) and distance between points (7.4.6) is called Poincar´ e model of Lobachevskian geometry. Problem 7.4.3. Show directly that transformation (7.4.1) is isometry of Poincar´ e model, i.e., it does not modify the metric (7.4.9): dw dw ∗ dz dz ∗ = . 2 (Im z) (Im w)2

(7.4.10)

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In accordance with (7.4.4) and (7.2.26), space P is diffeomorphic to one of the sheets − , which is a surface (7.2.22) in R2,1 : of two-sheet hyperboloid, say, S2,1 x20 − x21 − x22 = 1,

x0 ≥ 1.

(7.4.11) − S2,1

by mapping Let us show that Poincar´ e metric (7.4.9) is obtained from metric of space (7.4.4). We do this in two steps. We first construct mapping of upper sheet of hyperboloid − to unit disc B: |z| < 1 in complex plane, and in this way define non-Euclidean metric S2,1 in B. Then we make use of well-known linear-fractional map of unit disc B to upper halfplane P ⊂ C. It is useful for what follows to begin with somewhat more general setting and consider − + + − (one could consider Sp,q , but this would give nothing new, since Sq,p = Sp,q ). space Sp,q It is (hyper)surface (7.2.22) whose metric is induced from embedding pseudo-Euclidean space Rp,q with metric ds2 = dxk ηkj dxj =

p 

p+q 

(dxj )2 −

j=1

(dxj )2 .

(7.4.12)

j=p+1

− The space Sp,q is defined by equation

xa ηab xb − x20 = −1, where

||ηab ||

a, b = 1, . . . , p + q − 1,

(7.4.13)

= Ip,q−1 .

Problem 7.4.4. Show that metric (7.4.12) of space (7.4.13) is written in coordinates xa (a = 1, . . . , p + q − 1) as follows:   xa xb dxb , (7.4.14) ds2 = dxa ηab − x20 where x20 = 1 + xa xa and xa = ηab xb . − makes use of stereographic coordinates ka One parameterization of space Sp,q (a = 1, . . . , p + q − 1), related to Cartesian coordinates in Rp,q by

xa =

2 ka , 1 − k 2

x0 =

1 + k 2 , 1 − k 2

(7.4.15)

where k 2 := ka ηab kb = ka k a . It is straightforward to see that Eq. (7.4.13) is automatically satisfied. We have for differentials dx0 = 4

ka dk a , (1 − k 2 )2

dxa =

2 dka 1 − k 2

+ ka dx0 .

− We substitute this into (7.4.12) and derive metric of space Sp,q in stereographic coordinates dk a dk b ηab ds2 = dxa ηab dxb − dx20 = 4 . (7.4.16) (1 − k 2 )2

Note that metric (7.4.16) is conformally flat, as it differs from flat metric by conformal factor 4/(1 − k 2 )2 . − , i.e., choose q = 1. For p > 2 it is a multiWe now specify to the space Sp,1 dimensional generalization of two-sheet hyperboloid. In this case we have ||ηab || = Ip , i.e., metric (7.4.16) has Euclidean signature. The first formula in (7.4.15) gives the projection

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Fig. 7.4.2.

371

Stereographic mapping from “south pole”.

of hyperboloid xa xa + 1 = x20 in Rp,1 onto hyperplane Rp with x0 = 0, see Fig. 7.4.2. Stereographic mapping (7.4.15) is done from “south pole” x0 = −1, xa = 0, so the point ka = 0 (k 2 = 0) is mapped to x0 = 1 and xa = 0, and for k 2 −1 → −0 we have x0 → +∞ and | x| → ∞. Thus, Eq. (7.4.15) gives invertible map of the interior of a unit ball with − . coordinates k a in Rp to upper sheet of hyperboloid Sp,1

− in We insert ||ηab || = Ip in (7.4.16) and obtain metric of hyperboloid Sp,1 stereographic coordinates: dk a dk a ds2 = 4 . (7.4.17) (1 − k 2 )2

Like metric (7.4.16), this metric is conformally flat. We note for future reference that sphere S p is obtained by setting ηab = −Ip in (7.4.13) and (7.4.15). Positive metric on S p is defined by changing sign in (7.4.16), which gives dk a dk a ds2 = −dxa ηab dxb + dx20 = 4 . (7.4.18) (1 + ka ka )2 2 In the case p = 2 we obtain metric on S : dz dz ∗ , (7.4.19) ds2 = 4 (1 + z z ∗ )2 where z = k1 + ik2 . To get closer to Poincar´ e model of Lobachevskian geometry, we specify to upper − . In this case we have ||ηab || = diag(1, 1) and, in sheet of 2-dimensional hyperboloid S2,1 accordance with (7.4.15), this space is mapped to disc B: k 2 = k 2 + k 2 < 1. The metric 1

is given by

2

dk12 + dk22 dz dz ∗ =4 , (7.4.20) 2 2  (1 − z z ∗ )2 (1 − k ) where we introduce z = k1 + ik2 . Linear-fractional transformations of particular form az + b , a, b ∈ C, (7.4.21) z→w= ∗ b z + a∗ where   a b ∈ SU (1, 1), a a∗ − b b∗ = 1, U= ∗ ∗ a b (±U give the same transformation) define transitive action of group P SU (1, 1) = SU (1, 1)/{I2 , −I2 } in unit disc B. ds2 = 4

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Problem 7.4.5. Show that transformation (7.4.21) is isometry in B, i.e., they do not change the metric (7.4.20): ds2 = 4

dz dz ∗ dw dw ∗ =4 . (1 − z z ∗ )2 (1 − w w ∗ )2

(7.4.22)

In accordance with (7.4.21), stationary subgroup H ⊂ P SU (1, 1) of point z = 0 has b = 0, a∗ a = 1, and thus H = U (1). So, we have diffeomorphism P SU (1, 1)/U (1) = B,

(7.4.23)

which induces non-Euclidean metric (7.4.20) in B. The disc B endowed with metric (7.4.20) is called conformal model of Lobachevskian geometry. We now make the second step and obtain metric (7.4.9) in upper half-plane P = {z ∈ C | Im(z) > 0} from metric (7.4.20) in B. To this end, we map unit circle B to P. It is known that all linear-fractional diffeomorphisms of upper half-plane Im z > 0 to unit disc |w| < 1 are given by w = eiφ

z−a , z − a∗

∀a ∈ P,

∀φ ∈ R.

(7.4.24)

This map takes point z = a ∈ P to center w = 0 of disc |w| < 1, and real axis Im z = 0 to circle |w| = 1. In our case, we recall the choice of reference points in B and P (see the derivations of (7.4.4) and (7.4.23)) and set a = i, w = eiφ

z−i . z+i

Making use of this formula in (7.4.22), we obtain metric in Poincar´e half-plane P : ds2 =

dz dz ∗ dx2 + dy 2 = , (Imz)2 y2

(7.4.25)

where z = x + iy and y > 0. This metric coincides with (7.4.9). Remark 2. Straight lines in conformal Lobachevskian model B are images of Λ-lines obtained by map (7.4.24) of P to B. Since linear-fractional transformation (7.4.24) takes circles to circles (Proposition 1.4.1) and preserves angles ((7.4.24) is conformal transformation), straight lines in conformal model are circular arcs whose ends hit the boundary circle |z| = 1 at right angle. Remark 3. When constructing conformal and Poincar´ e models of Lobachevskian geometry, we employed diffeomorphism of upper sheet of hyperboloid (7.4.11) to unit disc B: z ∗ z < 1 and then to upper complex half-plane P: Im(z) > 0. Let us temporarily − . Infinite “points” of M (at x0 → ∞) denote the upper sheet of hyperboloid by M ≡ S2,1 1 ∗ are mapped to points of unit circle S : z z = 1 and points of real axis Im(z) = 0 in complex plane C, respectively. This property suggests the notion of conformal boundary ∂c M of upper sheet of hyperboloid, which is identified with the inverse image of the boundary of the disc B (i.e., unit circle) in conformal model, and inverse image of real axis in Poincar´ e model. In this way we can formally define manifold with boundary M ∪ ∂M (upper sheet of hyperboloid with added infinite points), whose image is closed disc z ∗ z ≤ 1 or upper half-plane with added real axis, Im(z) ≥ 0. The price to pay for such a “compactification” of upper sheet of hyperboloid is the divergence of its metric on the boundary ∂M, which shows up as the divergence of metric (7.4.20) at the boundary B (unit circle z ∗ z = 1) and metric (7.4.25) at the boundary of P (real axis Im(z) = 0).

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Nevertheless, metrics (7.4.20) and (7.4.25) are conformally flat everywhere in internal regions of B and P, respectively, including vicinities of the boundaries z ∗ z = 1 and Im(z) = 0. If one thinks of the upper sheet of hyperboloid as 2-dimensional space-time manifold, then its causal structure is determined by null vectors la with gab la lb = 0 (light rays). Hence, the causal structure does not change under conformal transformation of s2 = Ω(z)ds2 . We conclude that the “manifold with boundary” M ∪ ∂M metric ds2 → dˆ has well-defined conformal structure and straightforwardly studied behavior of light rays. This technical trick (Penrose method of conformal compactification of space-time, see Chapter “Conformal infinity” in book [19]) is very convenient for the study of various manifolds encountered in gravity and cosmology. To end up this section, we briefly discuss geometry of anti-de Sitter space AdS n and de Sitter space dS n , introduced in Examples 5 and 6 in Section 2.1.1. We consider, in particular, the way to introduce conformal boundaries of these manifolds, which is based on the method given in Remark 3. This topic is important for physics applications, including the correspondence between gravitational theories in AdS n and conformal field theories living on its conformal boundary (the latter being conformal to Minkowski space), broadly dubbed AdS/CFT correspondence. − − and dS n = S1,n (of We note that a convenient parameterizations of AdS n = Sn−1,2 unit “radius” M ) are given by (7.4.15), where metric matrices are (the choice we make here is convenient for unifying formulas for AdS n and dS n ) AdS n : dS n :

η = In−1,1

⇒ ⇒

η = I1,n−1

x21 + · · · + x2n−1 − x2n − x20 = −1, x21 − x22 − · · · − x2n − x20 = −1.

(7.4.26)

AdS n and dS n metrics in stereographic coordinates (7.4.15) are given by (7.4.16) with the same ηab . To obtain multi-dimensional analog of Poincar´e metric (7.4.9), (7.4.25) we make, in analogy to 2-dimensional case (7.4.24), conformal transformation of coordinates k a → za : ya za ba , (7.4.27) ka = 2 + ba , ya ≡ 2 − y   z 2 where b is an arbitrary unit vector, b2 = ba bd ηad = 1. This transformation is indeed conformal, since it is a composition of inversions and translations. We have, in coordinates {za }, 2 za ba , 1 − k 2 = − 2 y   z2 ⇒ ds2 = 4

dka dk a =

dza dz a ( z 2 y 2 )2

(7.4.28)

dk a dk b ηab dz a dz b ηab = . 2 2  (za ba )2 (1 − k )

(7.4.29)

The second equality in (7.4.29) is straightforward to obtain by noticing that metric keeps its conformal form under conformal transformations. Since b2 = 1, we can choose in both AdS n and dS n cases b = (1, 0, . . . , 0). Then (za ba ) = z1 , and AdS n and dS n metrics (7.4.29) become AdS n :

ds2 =

dS n :

ds2 =

2 2 − dzn dz12 + · · · + dzn−1

z12 2 2 − · · · − dzn dz12 − dzn−1

z12

,

(7.4.30)

.

(7.4.31)

Metric coefficients depend only on coordinate z1 , which is spacelike for AdS n and timelike for dS n . Note that with our conventions, AdS n and dS n metrics have different signatures;

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if necessary, the signature convention may be reversed by flipping the overall sign in (7.4.30) or (7.4.31). To restore the parameter M (“radius” of AdS n or dS n , see (2.1.6) and (2.1.7)), we rescale the coordinates, xi → M xi , in (7.4.13). Then, instead of (7.4.13) and (7.4.15), we have 2 ka 1 + k 2 , x0 = M , (7.4.32) xa ηab xb − x20 = −M 2 , xa = M 2  1−k 1 − k 2 and metric in stereographic coordinates becomes ds2 = 4 M 2

dk a dk b ηab dz a dz b ηab = M2 (za ba )2 (1 − k 2 )2

⎧ 2 2 dz 2 + · · · + dzn−1 − dzn ⎪ ⎪M 2 1 ⎪ ⎨ 2 z1 = 2 2 ⎪ dz12 − dzn−1 − · · · − dzn ⎪ 2 ⎪ ⎩M 2 z1

(7.4.33a)

for AdS n , (7.4.33b) for dS n .

The transformation (7.4.15) can be considered as a map of AdS n (or dS n ) to some region in Rn−1,1 (or in R1,n−1 ). Then conformal transformation (7.4.27) Rn−1,1 → Rn−1,1 (or R1,n−1 → R1,n−1 ) takes singularity points of metric (7.4.16), belonging to the surface k 2 = 1, to singularity points of metric (7.4.29), i.e., points of hyperplane (za ba ) = z1 = 0 in Rn−1,1 (or in R1,n−1 ). Like in 2-dimensional case, surfaces k 2 = 1 and (za ba ) = z1 = 0 can be viewed as images of conformal boundary of “compactified” (anti-)de Sitter space: for all k with k 2 < 1 and all za with z1 > 0, metric (7.4.33) is conformally flat, while it tends to infinity as k 2 → 1 and z1 → 0. It is instructive to obtain metrics (7.4.33b) for AdS n and dS n directly from (7.4.12), without an intermediate passage (7.4.32) to stereographic coordinates {ka }. We write (7.4.26) (with M = 1) as follows (it is convenient to relabel the coordinates x0 ↔ xn−1 in dS n case): AdS n :

x20 − x21 − · · · − x2n−2 + (xn − xn−1 )(xn + xn−1 ) = xμ xν δ¯μν + (xn − xn−1 )(xn + xn−1 ) = M 2 ,

dS n :

x20 + x21 + · · · + x2n−2 + (xn + xn−1 )(xn − xn−1 )

(7.4.34)

= xμ xν δμν + (xn − xn−1 )(xn + xn−1 ) = M 2 , where δ¯ = I1,n−2 , δ = In−1 . These formulas show that topologically AdS n = S 1 × Rn−1 and dS n = S n−1 × R, as shown in Figs. 7.4.3 and 7.4.4.

Fig. 7.4.3. Space AdS n is topologically S 1 × Rn−1 . Circle S 1 in plane x1 = · · · = xn−1 = 0 is given by x20 + x2n = M 2 , i.e., its radius is M .

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Fig. 7.4.4. Space dS n is topologically S n−1 × R1 . Sphere S n−1 in n-dimensional hyperplane xn−1 = 0 has radius M . Now, we choose new coordinates (u, z0 , z1 , . . . , zn−2 ) in AdS n and dS n , such that xn − xn−1 = u M 2 , x μ = M u zμ , xn + xn−1

μ = 0, 1, . . . , n − 2,

(7.4.35)

1 = − u (zμ z μ ), u

where z ν = δ¯νμ zμ for AdS n , and z ν = δνμ zμ = zν for dS n . One checks that Eq. (7.4.34) is satisfied automatically:   1 − u zμ z μ = M 2 . xμ xμ + (xn − xn−1 )(xn + xn−1 ) = M 2 u2 (zμ z μ ) + u M 2 u The first equality in (7.4.16) gives metrics in the new coordinates:  2  du 2 μ . − u dz dz ds2 = −dxμ dxμ − d(xn − xn−1 )d(xn + xn−1 )= M 2 μ u2

(7.4.36)

Parameterization (7.4.35) is singular at the point u = 0, which, in terms of coordinates xi , corresponds to two different asymptotics xn → xn−1 , xμ → 0 and xn + xn−1 → ±∞ (as u → ±0). Therefore, regions with u > 0 and u < 0 have to be studied separately; we e patch). consider the region with u > 0, i.e., xn > xn−1 (Poincar´ It follows from (7.4.36) that the cross sections of AdS n and dS n by hyperplanes xn − xn−1 = uM 2 for any given u are Minkowski spaces Rn−2,1 with metrics ds2 = −M 2 u2 dzμ dzν δ¯μν and flat Euclidean spaces Rn−1 , respectively. Formally speaking, the cross section obtained in the limit u → +∞ has the same flat metric structure. This limiting cross section is precisely the conformal boundary of AdS n or dS n . It is worth emphasizing that u is timelike coordinate in de Sitter space-time, so its conformal boundary is at infinite future (in the patch with u > 0). Continuing with Poincar´e patch u > 0, we make the change of variables in (7.4.36) : u = 1/ζ, where ζ > 0. The resulting metric is (i = 1, 2, . . . , n − 2) ⎧ (dζ 2 + dzi dzi − dz0 dz0 ) ⎪ ⎪ M2 for AdS n , ⎪ ⎨ 2 μ ζ2 (dζ − dzμ dz ) = (7.4.37) ds2 = M 2 ⎪ ζ2 (dζ 2 − dzi dzi − dz0 dz0 ) ⎪ n ⎪ ⎩ M2 for dS . ζ2 Modulo notations, it coincides with metric (7.4.33b). Metric (7.4.37) is generalization of Poincar´ e metric (7.4.25), so it is sometimes called Poincar´ e metric on AdS n and dS n ,

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and coordinates {ζ, zμ }, are called Poincar´ e coordinates. Conformal boundaries of AdS n and dS n correspond to the limit ζ → +0. Problem 7.4.6. Show that parameterization (7.4.35) and metrics (7.4.36), (7.4.37) coincide, modulo notations and trivial changes of coordinates on AdS n and dS n , with parameterizations (3.4.15), (3.4.20) and metrics (3.4.16), (3.4.21) used in Section 3.4.2.

7.5.

Metrics and Laplace Operators in Homogeneous Spaces

In this section we study a general method for constructing invariant metrics in homogeneous spaces G/H, which makes use of local properties of Lie group G and its Lie subgroup H. Within this approach we construct invariant Laplace operators on spaces G/H and demonstrate their relationship to quadratic Casimir operators of Lie algebra A(G). This topic has numerous applications in theoretical and mathematical physics. Before coming to our main theme, we find it timely to introduce some notions of differential geometry on smooth manifolds. 7.5.1.

Elements of differential geometry on smooth manifolds

Let M be smooth n-dimensional manifold. At its every point x ∈ M we define basis in tangent space Tx (M ), ea (x) (a = 1, . . . , n). We refer to the system of basis vectors {ea (x)} at given point x as vielbein.c Any vector v (x) ∈ Tx (M ) can be decomposed as v (x) = v a (x) ea (x).

(7.5.1)

Let U ⊂ M be local chart containing x, and xμ = (x1 , . . . , xn ) be coordinates of the point x. Let us denote components of vielbein vectors ea (x) in a given coordinate frame by eμa (x), where μ is coordinate index. Then, in accordance with (7.5.1), components of vector v (x) ∈ Tx (M ) in coordinate basis are v μ (x) = v a (x) eμa (x).

(7.5.2)

As usual, we assume that all functions of xμ in any chart U are smooth. Once functions v a (x) (or, equivalently, vectors v (x)) are specified everywhere in U , one says that there is vector field v (x) defined in U ⊂ M . c For

n = 4, the system { ea (x)} is called vielbein or tetrad.

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Another choice of coordinate frame xμ → (x )μ = f μ (x) in the same chart U ⊂ M leads to transformation of components of vectors ea (x) and v (x) given by the rule (2.2.5): eμa (x) → (e )μa (x ) =

∂(x )μ ν e (x). ∂xν a

(7.5.3)

v μ (x) → (v  )μ (x ) =

∂(x )μ ν v (x), ∂xν

(7.5.4)

 μ

) ν (cf. transformation dxμ → d(x )μ = ∂(x ∂xν dx and Eq. (7.5.4)). Another operation is the local change of basis in tangent space

eμa (x) → eμa (x) = Λba (x) eμb (x),

(7.5.5)

where matrices ||Λba (x)|| generically depend on point x and belong to GL(n, R). It induces the change in coordinates of vectors v a (x) → va (x) = (Λ−1 )ab (x) v b (x).

(7.5.6)

We assume that every tangent space is endowed with nondegenerate scalar product and vielbein ea (x) ∈ Tx (M ) is such that (ea (x), eb (x)) = ηab ,

(7.5.7)

where ηab is independent of x. Then the scalar product of two vectors v , u ∈ Tx (M ) is (v , u) = (v a ea (x), ub eb (x)) = v a (x) ηab ub (x).

(7.5.8)

The constant tensor ηab plays a role of “flat” metric, which is the same in tangent spaces at all x, and (7.5.7) is (generalized) orthonormality condition for basis ea (x) ∈ Tx (M ) at every point x. Even though ea (x) are constrained by (7.5.7), the freedom in the choice of basis vectors still remains, but matrices ||Λba (x)|| in (7.5.5) have to obey ηab = Λca (x) ηcd Λdb (x).

(7.5.9)

The constant metric η = ||ηab || can be cast into the standard form η = Ip,q , see (1.2.63), where p + q = n, so the property (7.5.9) means that matrices ||Λba || belong to pseudo-orthogonal subgroup SO(p, q) of the group GL(n, R). This is local (gauge) subgroup, in the sense that its matrices ||Λba || depend on point x.

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Flat metric matrix ||ηab || and its inverse ||η ab || enable one to raise and lower indices a, b, . . . labeling the vielbein vectors in tangent spaces Tx (M ) and components of vectors v : v a (x) = η ab vb (x) , va (x) = ηab v b (x). The vielbein is used for constructing second rank tensor eμa (x) η ab eνb (x) = g μν (x),

(7.5.10)

which defines metric in manifold M . This metric and its inverse, ||gμν ||, is used, in turn, to raise and lower indices, this time for components v μ (x) of vectors v ∈ Tx (M ) in coordinate basis: v μ (x) = g μν (x) vν (x),

vμ (x) = gμν (x) v ν (x).

These formulas are generalized in an obvious way to tensors with multiple tangent space indices a, b, . . . and coordinate indices μ, ν, . . . . Yet another convenient object is dual vielbein eaμ (x) = η ab eνb (x) gμν ,

(7.5.11)

whose matrix is inverse of ||eμa (x)||, eμa (x) eaν (x) = δνμ ,

eμa (x) ebμ (x) = δab .

In accordance with (7.5.3) and (7.5.5), components eμa (x) are transformed as follows: eaμ (x) → (e )aμ (x ) = (∂μ xν ) eaν (x),

(7.5.12)

eaμ (x) → eaμ (x) = (Λ−1 )ab (x) ebμ (x),

(7.5.13)

where ∂μ xν =

∂xν . ∂(x )μ

Making use of eμa (x) and eaμ (x), one can replace coordinate indices μ, ν, . . . by tangent space ones, a, b, . . . , and vice versa: v a (x) = v μ (x) eaμ (x),

v μ (x) = v a (x) eμa (x).

(7.5.14)

The consistency of the whole construction is illustrated by the fact that, in accordance with (7.5.3), components gμν indeed transform as components of metric tensor (see (2.2.13)) under coordinate transformation. Making use

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of (7.5.11), one has the standard relation for the scalar product (7.5.8) (cf. (2.2.12)), (v , u) = v a (x) ηab ub (x) = v a (x) eμa (x) gμν (x) eνb (x) ub (x) = v μ (x) gμν (x) v ν (x). This reiterates that the matrix gμν (x) = eaμ (x) ηab ebν (x),

(7.5.15)

which is inverse of (7.5.10), indeed has interpretation of metric. We emphasize that this metric is invariant under local (gauge) tangent space transformations (7.5.13). To define covariant derivative of a vector field v (x), one needs to equip manifold M with additional structure called connection. In local coordinate frame, covariant derivative Dμ reads Dν v μ (x) = ∂ν v μ (x) + Γμνλ (x) v λ (x),

(7.5.16)

where Γμνλ (x) are connection coefficients at point x ∈ M and ∂ν = ∂/∂xν . One requires that covariant derivative obeys Leibnitz rule,   Dν v μ (x) · uλ (x) = (Dν v μ (x)) · uλ (x) + v μ (x) · (Dν uλ (x)), and that covariant derivative of a scalar coincides with the conventional derivative, i.e., Dμ (v ν uν ) = (Dμ v ν ) · uν + (Dμ uν ) · v ν = ∂μ (v ν uν ).

(7.5.17)

Problem 7.5.1. Making use of (7.5.17), derive the formula Dν vμ (x) = ∂ν vμ (x) − Γλνμ (x) vλ (x).

(7.5.18)

These properties define the covariant derivative of a tensor of any rank, say, Dν aμλ = ∂ν aμλ + Γμνρ aρλ + Γλνρ aμρ ; Dν aμλ = ∂ν aμλ + Γμνρ aρλ − Γρνλ aμρ . Covariance of the operator Dν in (7.5.16) means that the components Dν v μ (x) transform homogeneously, as components of the second rank tensor, under coordinate transformation xμ → (x )μ : Dν v μ (x) → ∂ν xλ ∂ρ (x )μ Dλ v ρ (x).

(7.5.19)

In view of (7.5.4), this gives for the transformation rule of connection: (Γ )μνγ = ∂ν xλ ∂γ xρ Γξλρ ∂ξ (x )μ − ∂ν xλ ∂γ xρ ∂λ ∂ρ (x )μ .

(7.5.20)

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This transformation, generally speaking, is inhomogeneous; connection is not a tensor. However, one can decompose connection Γμνλ into symmetric and antisymmetric parts, Γμνλ (x) = Γμ(νλ) (x) + Γμ[νλ] (x),

(7.5.21)

where Γμ(νλ) = 12 (Γμνλ + Γμλν ),

Γμ[νλ] = 12 (Γμνλ − Γμλν ).

The second, inhomogeneous term on the right-hand side of (7.5.20) is symmetric with respect to ν ↔ γ. Therefore, antisymmetric part Γμ[νλ] is transformed homogeneously, i.e., it is a tensor, while symmetric part Γμ(νλ) transforms inhomogeneously. Coming to tangent space vector components, we define the covariant derivative Dν for components v a (x) in a way similar to (7.5.16): a Dν v a (x) = ∂ν v a (x) + ωνb (x) v b (x),

(7.5.22)

a where ων (x) = ||ωνb (x)|| plays the role of gauge field of local (gauge) group O(p, q). Indeed, the main requirement is that the covariant derivative (7.5.22) is transformed under the gauge transformation (7.5.6) as tangent space vector, Dν v a → (Λ−1 )ab Dν v b . This gives the transformation law

ω μ = Λ−1 ωμ Λ + Λ−1 ∂μ Λ,

(7.5.23)

which is characteristic of gauge fields. Note that for given μ, ωμ takes values in Lie algebra so(p, q). The gauge field ωμ (x) is called spin connection. Components of vielbein vectors eaμ (x) carry one coordinate and one tangent space index. Accordingly, covariant derivative of eaν (x) reads a b Dμ eaν (x) = ∂μ eaν (x) − Γλμν eaλ + ωμb eν .

(7.5.24)

Making use of this covariant derivative and antisymmetric components Γλ[μν] (which make a tensor), one defines antisymmetric tensor a Tμν = 12 (Dμ eaν (x) − Dν eaμ (x)) + Γλ[μν] eaλ .

(7.5.25)

This tensor is called torsion. Its explicit form is a a a b Tμν = 12 (∂μ eaν (x) − ∂ν eaμ (x) + ωμb ebν − ωνb eμ ).

(7.5.26)

This shows that torsion depends on spin connection only, and does not depend on connection Γμνλ .

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Antisymmetric tensors are conveniently thought of as differential forms. As an example, the definition (7.5.26) is written in terms of differential forms in a compact form d ea = T a − ωba ∧ eb ,

(7.5.27)

where we use the notations a a dxμ ∧ dxν , ωba = ωμb dxμ , ea = eaν dxν , d ea = ∂μ eaν dxμ ∧ dxν , T a = Tμν

and dxμ1 ∧ dxμ2 ∧ · · · ∧ dxμr = (−1)p(σ) dxμσ(1) ∧ dxμσ(2) ∧ · · · ∧ dxμσ(r) (here σ is transposition of (1, 2, . . . , r) and p(σ) is its parity). The latter r-form plays the role of basis in the space of rank r antisymmetric tensors. The relation (7.5.27) is called the first Cartan’s structure equation. An important class of metric manifolds is obtained by requiring that the two definitions of covariant derivatives, Eqs. (7.5.16) and (7.5.22), are consistent with each other in the sense that Dμ v a (x) = eaν (x) Dμ v ν (x). Then the Leibnitz rule implies that the covariant derivatives of the vielbein vectors vanish, a b eν = 0. Dμ eaν (x) = ∂μ eaν − Γλμν eaλ + ωμb

(7.5.28)

This means, in particular, that torsion (7.5.25) coincides with the antisymmetric part of connection Γλ[μν] eaλ . Importantly, it follows from Eq. (7.5.10) that Dμ gνλ = 0. Connection with this property is called metric connection. Specifying further, one requires that torsion tensor identically vanishes, a = 0. In view of (7.5.28) this means that Γλ[μν] = 0, i.e., connection Tμν a coefficients are symmetric, Γλμν = Γλ(μν) . Symmetric connection with Tμν = 0 and Dμ gνλ = 0 is called Levi-Civita connection, and metric manifolds with Levi-Civita connection are (pseudo-)Riemannian manifolds. Manifold is Riemannian, if its metric is Euclidean, ηab = δab , and pseudo-Riemannian otherwise. Locally, at a given point, metric gμν can be set equal to ηab by coordinate transformation, so gμν inherits signature of ηab . PseudoRiemannian geometry with η = I1,3 = diag(1, −1, −1, −1) is a crucial ingredient of General Relativity.

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The coefficients Γλμν of Levi-Civita connection are called Christoffel’s symbols. Making use of (7.5.28), one expresses Christoffel symbols through spin connection (and vice versa), Γλμν = eλa (∂μ eaν + ωμa b ebν )

⇒

ωμa b = −(∂μ eaν − Γλμν eaλ )eνb .

(7.5.29)

In its turn, spin connection is expressed through the vielbein components eaν (x). To see the latter property, we define, in a standard way, ωμ,λν = ωμa b ηac ecλ ebν , and using the symmetry Γλμν = Γλνμ , obtain from (7.5.28) that ωμ,λν − ων,λμ = gλρ (eρa ∂ν eaμ − eρa ∂μ eaν ).

(7.5.30)

Spin connection ωμ,λν is antisymmetric in indices λ and ν. This follows a || belong to Lie algebra so(p, q), and from the fact that matrices ωμ = ||ωμb hence obey Eq. (3.3.37), i.e., ηad ωμd b = ωμ,ab = −ωμ,ba ⇒ ωμ,λν = −ωμ,νλ .

(7.5.31)

Making use of this property and considering (7.5.30) with two permutations of indices λ ↔ ν and λ ↔ μ, we obtain 2ων,μλ = (ωμ,λν − ων,λμ ) + (ωμ,νλ − ωλ,νμ ) − (ωλ,μν − ων,μλ ) = gλρ (eρa ∂ν eaμ − eρa ∂μ eaν ) + gνρ (eρa ∂λ eaμ − eρa ∂μ eaλ )

(7.5.32)

− gμρ (eρa ∂ν eaλ − eρa ∂λ eaν ), or ωνa b = g μρ eaμ eλb ων,ρλ =

1 2

eaμ eλb (Cλνμ + Cνλμ − C μλν ),

(7.5.33)

where Cμνλ = gμρ (eρa ∂ν eaλ − eρa ∂λ eaν ). Problem 7.5.2. Show that Eq. (7.5.32) can be written as 2 ων,μλ = eaλ ∂ν eaμ − eaμ ∂ν eaλ + ∂λ gνμ − ∂μ gνλ .

(7.5.34)

Making use of (7.5.29), show that Christoffel symbols are expressed through metric tensor as follows: Γλμν = g λρ Γρμν ,

Γρμν =

1 2

(∂ν gρμ + ∂μ gρν − ∂ρ gμν ) .

Prove the identity ∂ν gρμ = Γρμν + Γμρν .

(7.5.35)

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Spin connection ωμ (x) = ||ωμa b (x)||, being gauge field, defines the field strength tensor, which is covariant under gauge transformations (7.5.5), (7.5.23): a a a d a d − ∂ν ωμb + ωμd ωνb − ωνd ωμb , Rμν a b = [Dμ , Dν ]a b = ∂μ ωνb

(7.5.36)

a || for covariant where we use the notation Dμ = ||Dμa b || = ||δba ∂μ + ωμb derivative matrix (7.5.22). In geometrical context, tensor Rμν a b is called curvature tensor. Since under gauge transformations (7.5.23) this operator transforms in covariant way, Dμ → Λ−1 Dμ Λ, field strength tensor is covariant,

μν = Λ−1 Rμν Λ. Rμν → R

(7.5.37)

μν = ||(R μν )a || Here Rμν = ||Rμν a b ||, and the transformed tensor R b a is obtained from (7.5.36), where ωμb is replaced by transformed spin connection (7.5.23). In Riemannian geometry, Christoffel symbols and spin connection are related to each other by (7.5.29). So, tensor Rμν a b can be written in terms of Christoffel symbol. To this end, we notice that the first of Eq. (7.5.29), a ebν (x) + eλa (x) ∂μ eaν (x), Γλμν = eλa (x) ωμb

(7.5.38)

formally coincides with the gauge transformation of spin connection, Eq. (7.5.23), where matrix elements of Λ are replaced by vielbein components, Λaν (x) → eaν (x). In view of this observation, Eq. (7.5.37) immediately gives Rμν λ ρ = eλa (x) Rμν a b ebρ (x),

(7.5.39)

where Rμν λ ρ = ∂μ Γλνρ − ∂ν Γλμρ + Γλμξ Γξνρ − Γλνξ Γξμρ .

(7.5.40)

In Riemannian geometry, tensor Rμν λ ρ is called Riemann curvature tensor, or simply Riemann tensor. This tensor has the following symmetry properties: Rμνλρ + cycle(μ, ν, ρ) = 0, Rμνλρ = −Rνμλρ , Rμνλρ = −Rμνρλ , Rμνλρ = Rλρμν ,

(7.5.41)

where Rμνλρ = Rμν σ ρ gλσ . The first two identities in (7.5.41) follow from (7.5.40), whereas the third one (antisymmetry of Rμνλρ in λ, ρ) is a consequence of the fact that matrices Rμν = ||Rμν ab || in (7.5.36) belong

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to Lie algebra so(p, q) (see (7.5.31)). The fourth identity in (7.5.41) can be obtained by using the representation Rμνσρ = gλσ Rμν λ ρ = 12 (∂μ ∂ρ gσν − ∂μ ∂σ gρν − ∂ν ∂ρ gσμ + ∂ν ∂σ gρμ ) + Γξσν Γξμρ − Γξσμ Γξνρ .

(7.5.42)

Problem 7.5.3. Derive the formula (7.5.42). Hint: make use of (7.5.35) and the identity ∂ν gρμ = Γρμν + Γμρν . The fourth of identities in (7.5.41) shows that the contraction of the first and third indices of curvature tensor gives second rank symmetric tensor Rνρ = Rμν μ ρ = Rμνσρ g μσ .

(7.5.43)

It is called Ricci tensor. One more geometric tensor is Cμνλρ = Rμνλρ + α(gμλ Rνρ + gνρ Rμλ − (μ ↔ ν)) + β(gμλ gνρ − (μ ↔ ν))R,

(7.5.44)

where R = Rνν is scalar curvature, and (n is number of dimensions) α=

1 1 , β= . 2−n (n − 1)(n − 2)

(7.5.45)

Tensor (7.5.44) is called Weyl tensor; it is traceless, Cμνλρ g μλ = 0.

(7.5.46)

Problem 7.5.4. Show that for any values of parameters α and β, Weyl tensor Cμνλρ has the same symmetry properties (7.5.41) as Riemann tensor Rμνλρ . Show that it is traceless, i.e., obeys (7.5.46), only if the parameters are given by (7.5.45). Example. Consider pseudo-Riemannian space with conformally flat metric gμν (x) = Ω(x)−2 ημν ,

g μν (x) = Ω(x)2 η μν ,

(7.5.47)

where ||ημν || = Ip,q is flat space metric. Using (7.5.47) in (7.5.35), (7.5.42), we find the expression for Riemann tensor:   Rμνλρ (x) = ηρν Kμλ + ηλμ Kνρ − ηλν Kμρ − ημρ Kνλ   2  Kνρ ≡ Ω−3 ∂ν ∂ρ (Ω), + ηλν ηρμ − ηλμ ηρν Ω−4 ∂ Ω , (7.5.48)

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where (∂Ω)2 curvature are

=

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η ξρ (∂ξ Ω)(∂ρ Ω). Ricci tensor (7.5.43) and scalar

Rνρ (x) = (n − 2)Ω−1 ∂ν ∂ρ (Ω) + ηνρ (Ω−1 ∂ 2 (Ω) + (1 − n)Ω−2 (∂Ω)2 ), (7.5.49) R(x) = g νρ Rνρ = (n − 1)(2Ω ∂ 2 (Ω) − n (∂Ω)2 ),

(7.5.50)

where n = p + q and ∂ 2 = η ξρ ∂ξ ∂ρ . It is now straightforward to see that Weyl tensor (7.5.44) vanishes for conformally flat space with metric (7.5.47). In fact, this is a consequence of more general fact that Weyl tensor is invariant under conformal transformations gμν → Λ(x)gμν . We point out that there is inverse statement, which we do not prove: if Weyl tensor vanishes for a pseudo-Riemannian manifold, then this manifold is conformally flat, i.e., its metric can be cast in the form (7.5.47). Problem 7.5.5. Derive the formulas (7.5.48), (7.5.49) and (7.5.50) for conformally flat metric (7.5.47). Show that Weyl tensor vanishes in this case. Show that Weyl tensor is invariant under conformal transformations gμν → Λ(x)gμν .

Problem 7.5.6. Consider anti-de Sitter space AdS n and de Sitter space dS n with metrics of signature (+, −, . . . , −) (signature of AdS n is opposite here to that used in Section 7.4). In accordance with (7.4.33a), their metrics are: AdS n :

ds2 = dxμ dxν gμν (x) = 4M 2

dxμ dxν ημν , (1 + xμ xμ )2

(7.5.51)

dS n :

ds2 = dxμ dxν gμν (x) = 4M 2

dxμ dxν ημν , (1 − xμ xμ )2

(7.5.52)

where η = I1,n−1 = diag(1, −1, . . . , −1), xμ xμ = ημν xμ xν and M is a real parameter (“radius” of AdS or dS). Consider also sphere S n whose metric is given by (7.5.51), but with ημν = δμν , see (7.4.18). Show that Riemann tensor, Ricci tensor and scalar curvature of anti-de Sitter space AdS n , de Sitter space dS n and sphere S n are 1 (n − 1) n(n − 1) (gμλ gνρ −gμρ gνλ ), Rνρ = ∓ gνρ , R = ∓ , 2 2 M M M2 where minus sign refers to dS n , whereas plus sign to AdS n and sphere S n . Hint: use the results (7.5.48), (7.5.49) and (7.5.50). Rμνλρ = ∓

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In non-Riemannian geometry with nonzero torsion, curvature tensor of a manifold can still be defined as the field strength tensor for spin connection (gauge field) ωμ (x), and it is given by the expression (7.5.36). The latter expression can be written in terms of differential forms: d ω ab + ω ad ∧ ω db =

1 2

R ab ,

(7.5.53)

where R ab = Rμν a b dxμ ∧ dxν is curvature 2-form. Equation (7.5.53) is called the second Cartan’s structure equation. The two Cartan’s structure equations (7.5.27) and (7.5.53) relate vielbein eμa (x) and spin connection a a with torsion and curvature tensors Tμν and Rμν a b . ωμb Remark. Spin connection ωμ (x) of manifold M is SO(p, q) gauge field which takes values in Lie algebra so(p, q). It enables one to make parallel transport of a tangent vector  v ∈ Tx (M ) from point x ∈ M to point y ∈ M along a curve Cyx ⊂ M . This parallel transport is governed by matrix 

 U (Cyx ) = P exp

Cyx

ωμ (z)dz μ ,

(7.5.54)

where P denotes ordering along the curve Cyx , so that for an intermediate point z ∈ Cyx we have U (Cyx ) = U (Cyz ) · U (Czx ) (similar construction is introduced in (3.1.51)). The components of tangent vector  v  (y) obtained upon the parallel transport to point y ∈ M are (v  )a (y) = U (Cyx )ab vb (x). Parallel transport along closed contour Cxx results in rotation of vector v(x) in Tx (M ) : the transported tangent vector is  v  (x) = v (x) ∈ Tx (M ). Since ωμ (z)dz μ ∈ so(p, q), one has U (Cxx ) ∈ SO(p, q). If U1 and U (Cxx ) 1 U2 are two (pseudo-)orthogonal matrices of parallel transport along two contours Cxx 2 , then their product U ·U is again parallel transport matrix, which corresponds and Cxx 1 2 1 ·C 2 obtained by consecutive travel along C 2 and then C 1 . Unit element to contour Cxx xx xx xx U = 1 corresponds to trivial contour staying at point x, while inverse element U (Cxx )−1 corresponds to the same contour Cxx but traveled in opposite direction. Thus, the subset of (pseudo-)orthogonal matrices U (Cxx ) governing parallel transport along all possible contractible contours Cxx ⊂ M with one and the same initial and final point x makes a subgroup Hx ∈ SO(p, q). It is called holonomy group of manifold M . Problem 7.5.7. Show that holonomy group of connected manifold M is independent of the reference point x ∈ M (groups Hx and Hy are isomorphic for all x, y ∈ M ).

7.5.2.

Invariant metrics in homogeneous spaces

We now turn to constructing invariant metrics on homogeneous spaces. Recall that in accordance with Section 7.3, homogeneous space G/H can be parameterized by elements of subset K ⊂ G containing representatives of cosets. In cases of interest (see, e.g., Section 7.2), the subset K can be chosen as smooth manifold embedded in the Lie group manifold G. This enables one to introduce metric on K, i.e., define interval squared ds2 (k1 , k2 ) for neighboring elements k1 and k2 in K. In view of one-to-one mapping

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K ↔ G/H, we refer to metric on K as metric on coset space G/H. Metric ds2 (k1 , k2 ) is called invariant, if for all g ∈ G and all neighboring k1 , k2 ∈ K one has (cf. (2.2.16)) ds2 (k1 , k2 ) = ds2 (k(g, k1 ), k(g, k2 )),

(7.5.55)

where elements k(g, ki ) are defined in (7.3.3): if k ∈ K is representative of coset kH ∈ G/H, then k(g, k) ∈ K is representative of coset g · kH. In other words, the interval does not change under the left action of group G on homogeneous space G/H. Our purpose here is to construct invariant metric on G/H = K. Without loss of generality we assume that unit element e ∈ G belongs to the set K. Indeed, this element belongs to coset e H = H, and if this coset is labeled by another element k0 ∈ K, then we can shift all elements K → k0−1 K, and obtain another subset K such that e ∈ K. Due to the property (7.5.55), metric is invariant under this shift. Let us choose g = k1−1 in (7.5.55), then we have for invariant metric (7.5.55) that ds2 (k1 , k2 ) = ds2 (e, k(k1−1 , k2 )),

(7.5.56)

where the right-hand side is metric on K near unit element. Therefore, once metric on K is defined near unit element (this requires only the knowledge of local properties of group G and its subgroup H), it is extended in unique way to the whole homogeneous space K = G/H. Let the dimensions be dim(G) = n, dim(H) = n − d, dim(G/H) = d. We choose basis in Lie algebra A(G) of group G, {Y1 , . . . , Yn−d , X1 , . . . , Xd },

(7.5.57)

such that its first elements Ya (a = 1, . . . , n − d) are generators of Lie algebra A(H) of subgroup H, and Xα (α = 1, . . . , d) are remaining generators of A(G). Neighborhood of unit element e ∈ G has coordinates {a1 , ..., an−d , b1 , ..., bd }, and elements g ∈ G in this neighborhood have the following form (cf. (7.3.1)): g = exp(aα Xα ) · exp(ba Ya ) = k(a) · h(b),

(7.5.58)

where b = (b1 , . . . , bn−d ) are parameters of group H, whereas parameters a = (a1 , . . . , ad ) are interpreted as local coordinates in homogeneous space G/H. This suggests the choice of subset K ∈ G parameterizing coset space

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G/H in a neighborhood of unity: k(a) = exp(aα Xα ) ∈ K.

(7.5.59)

Definition 7.5.1. Let A be Lie algebra with basis (7.5.57), H its subalgebra with generators {Y1 , . . . , Yn−d } and X be a subspace in A with basis {X1 , . . . , Xd }. Subalgebra H is normalizer of X in A, if defining commutation relations have the following form: d Yd , [Ya , Yb ] = Cab

(7.5.60)

β [Yb , Xα ] = Cbα Xβ ,

(7.5.61)

γ d Yd + Cαβ Xγ , [Xα , Xβ ] = Cαβ

(7.5.62)

β γ d d , Cbα , Cαβ and Cαβ are structure constants. where Cab

Equations (7.5.60) and (7.5.61) show that A is partitioned into two invariant spaces under adjoint action of subalgebra H, i.e., the representation of H in A is decomposable. Proposition 7.5.1. Let A be Lie algebra, H its Lie subalgebra, and let there exist non-degenerate invariant scalar product in A. Let A⊥ be orthogonal complement to H in A with respect to this scalar product. If A, as vector space, is direct sum of its vector spaces H and A⊥ , i.e., A = H+A⊥ , the subalgebra H is normalizer of A⊥ in A. Proof. We have to construct basis in A obeying defining relations (7.5.60)– (7.5.62). Equation (7.5.60) is satisfied automatically, as it merely reflects the fact that Ya are generators of subalgebra H, whereas Eq. (7.5.62) has the most general form. So, we are left with Eq. (7.5.61). The scalar product in A obeys ([A, X], Y ) + (X, [A, Y ]) = 0 for all A, X, Y ∈ A, see (5.1.2). We choose elements Xα in (7.5.57) in such a way that they belong to A⊥ and make basis in A⊥ . Then (Ya , Xβ ) = 0, ∀a, β, and for all Ya ∈ H and Xβ ∈ A⊥ we have (Ya , [Yb , Xβ ]) = −([Yb , Ya ], Xβ ) = 0. Therefore, [Yb , Xβ ] ∈ A⊥ , which is the desired result. Remark 1. Any subalgebra H is a normalizer of its orthogonal complement in compact Lie algebra A. This follows from the fact that there is invariant and positive definite scalar product in compact Lie algebra A. This implies

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that A is partitioned into direct sum A = H + A⊥ , where A⊥ is orthogonal complement to H in A with respect to this scalar product. Remark 2. If Lie algebra A is noncompact, not every its subalgebra H is, generally speaking, normalizer of the relevant subspace X . Consider, as an example, simple algebra A = s(2, R) with basis elements {h, e+ , e− } and structure constants (3.2.51). There exists 1-dimensional Lie subalgebra H in s(2, R), generated by element Y = e+ . To make contact with the above, we denote other elements by X1 = h, X2 = e− . Then, in accordance with (3.2.51), we have [Y, X1 ] = −Y , which contradicts (7.5.61). Let A = A(G) and H = A(H) be Lie algebras of Lie group G and its subgroup H ⊂ G. Let H be normalizer in A; in this case subgroup H is called normalizer in G. Equation (7.5.61) defines linear representation of subalgebra A(H), and hence, locally, of group H, in vector space K whose basis is made of generators Xa ∈ A(G). Let us choose the subset K parameterizing the coset space G/H by making use of these generators, as written in (7.5.58) and (7.5.59). Then Eq. (7.5.61) has important consequence h · K · h−1 ⊂ K ⇔ h · K · h−1 ⊂ K,

∀h ∈ H.

(7.5.63)

We use it heavily when constructing invariant metric in G/H. In what follows we limit ourselves to matrix Lie groups G (this is a technical assumption which can be dropped) and homogeneous spaces K = G/H which have the property (7.5.63). The latter implies that subgroup H is normalizer in G. Let us consider elements k(a) given by (7.5.59), which belong to space K = G/H. We construct matrix k −1 d k, where d k denotes differential of matrix k: ∂ d k(a) = d(exp(aα Xα )) = (d aα ) α exp(aα Xα ). ∂a It follows from (2.2.31) that k −1 ·d k is tangent vectord to G at unit element, i.e., k −1 · d k belongs to Lie algebra A(G). Hence, k −1 · d k can be expanded in the basis of Lie algebra A(G): k −1 · d k = α Xα + ω b Yb =  + ω,

(7.5.64)

α = Eμα daμ , ω b = ωμb daμ ,

(7.5.65)

where 1-forms k −1 · d k,  and ω are Maurer–Cartan forms. Their matrix structure is k −1 · d k ∈ A(G),  ∈ K, ω ∈ H ≡ A(H). d To

be precise, one speaks of co-tangent space.

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Proposition 7.5.2. Left action of group G in space K = G/H, which results in transformation g ·k = k(g, k)·h(g, k), Eq. (7.3.3), acts on Maurer– Cartan forms  ∈ K and ω ∈ H as follows: g

 = h(g, k) ·  · h(g, k)−1 ∈ K,  → 

(7.5.66)

g

ω → ω  = h(g, k) · ω · h(g, k)−1 − (d h(g, k)) · h(g, k)−1 ∈ A(H). (7.5.67) That is, the form  ∈ K is transformed homogeneously, whereas the form ω ∈ H ≡ A(H) is transformed like connection (7.5.23). Proof. The transformation of the forms  and ω is dictated by the transformation of tangent vector k −1 · d k: g

k −1 · d k → k(g, k)−1 · dk(g, k) =   α Xα + ω  a Ya =  +ω .

(7.5.68)

We insert here the expression (7.3.3) for k(g, k) and obtain k(g, k)−1 · d k(g, k) = h(g, k) · k −1 · g −1 · d(g · k · h(g, k)−1 ) = h(g, k) · k −1 · d(k · h(g, k)−1 ) = h(g, k) · (k −1 · dk) · h(g, k)−1 − dh(g, k) · h(g, k)−1 , where we use h · dh−1 = −dh · h−1 . We now recall that  ∈ K, and ω ∈ H. Then, in accordance with (7.5.63), we get h(g, k) ·  · h(g, k)−1 ∈ K, whereas h(g, k) · ω · h(g, k)−1 ∈ H, dh(g, k) · h(g, k)−1 ∈ H. This immediately gives (7.5.66) and (7.5.2). In accordance with Eq. (7.5.64), vector k −1 · dk tangential to G is a sum of two tangent vectors  and ω, where ω is directed along Lie algebra of subgroup H, while vector  is tangential to manifold K = G/H. Therefore, the vectors Eμα in (7.5.65) are naturally interpreted as vielbein (more precisely, dual vielbein similar to eaμ in (7.5.11)) in tangent space of manifold G/H. The adjoint transformation (7.5.66) is then the gauge transformation of vielbein, in line with (7.5.13). These observations suggest that the invariant metric in homogeneous space K = G/H is the trace Tr( · ), which is invariant under (7.5.66). So, we have (compare with the general definition of metric in (7.5.15)) ds2 = −Tr( · ) = daμ gμν (a) daν ,

(7.5.69)

where gμν (a) = Eμα (a) Eνβ (a) ηαβ ,

ηαβ = −Tr(Xα · Xβ ).

(7.5.70)

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The signature of “flat” metric ηαβ depends on group G and its subgroup H, and, in general, is pseudo-Euclidean. If group G is compact then metric ηαβ is obviously Euclidean. It is worth noting that if subgroup H is trivial, i.e., consists of a single unit element, then G/H = G, and metric (7.5.69), (7.5.70) coincides with left-invariant metric in group G given by (2.2.26), (2.2.32) and (2.2.33). In that case matrices ||Eμα (a)|| are analogs of matrices ||[L−1 ]ij || in (2.2.26), see Problem 2.2.1. Example 1. Consider homogeneous space S 2 = SU (2)/U (1), which we already encountered, see (7.2.47). The generators of SU (2) are τα = −iσα /2, where σα are Pauli matrices. 1-dimensional subgroup H = U (1) ⊂ SU (2) is embedded as follows:

e−ib 0 , (7.5.71) exp(2 b τ3 ) = 0 eib where b ∈ R. Homogeneous space SU (2)/U (1) is parameterized by matrices (see (7.5.59)) k(a1 , a2 ) = exp(A), where a1 , a2 ∈ R and A = −2(a1 τ1 + a2 τ2 ) =



0

a2 + ia1

−a2 + ia1

0

(7.5.72)

=

0

a

−a∗

0

.

(7.5.73) Matrix A obeys A2 = −|a|2 I2 , so elements (7.5.72) are written as (cf. (2.1.17)) k(a) = exp(A) = I2 cos |a| + A

sin |a| = I2 α0 + i(α1 σ1 + α2 σ2 ), |a|

(7.5.74)

where α0 = cos |a|,

αm = am sin |a|/|a|,

m = 1, 2,

so that (α0 )2 + (α1 )2 + (α2 )2 = 1,

α0 dα0 + αm dαm = 0.

(7.5.75)

One way to proceed is to choose parameters α1 and α2 as independent coordinates on SU (2)/U (1) = S 2 . Then we use (7.5.75) to obtain the

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Maurer–Cartan form (7.5.64):   k(α)−1 d k(α) =  + ω = (I2 α0 − iαm σm ) · d I2 α0 + iαk σk     = iσm α0 dαm − αm dα0 + iσ3 α1 dα2 − α2 dα1 . (7.5.76) This gives for the two Maurer–Cartan forms  = iσm

    αm αk α0 δmk + dαk , ω = iσ3 α1 dα2 − α2 dα1 . 0 α

(7.5.77)

The general formula (7.5.69) for invariant metric yields   (1 + (α0 )2 ) m k 0 2 α α dαm dαk ds = −Tr( · ) = (α ) δmk + (α0 )2 2

= gmk dαm dαk ,

(7.5.78)

where the function α0 (α1 , α2 ) is given by (7.5.75). Note that the explicit form (7.5.78) of metric on sphere S 2 is not standard. This is due to the unusual coordinate choice we have made so far. To construct, within the approach we study, more conventional metric on sphere (say, metric (7.4.19)), we employ other coordinates to parameterize the elements k(a) given by (7.5.74). Namely, we introduce new matrix Z and new complex parameter z instead of matrix A and parameter a: k(a) = exp(A) = I2 cos |a| + A

1 sin |a| = (I2 + Z), |a| 1 + |z|2

(7.5.79)

where Z=

0

z

−z ∗

0

=A

tan |a| |a|

⇒

1 cos |a| =  . 1 + |z|2

In terms of the new coordinates, Maurer–Cartan form reads

1  k(a) (I2 + Z) . 1 + |z|2 (7.5.80) Here ω ∈ u(1) is diagonal matrix, while matrix  has the form (7.5.73). Therefore, connection 1-form ω ∈ u(1) is series in even powers of Z in −1

1 d k(a) =  + ω =  (I2 − Z) · d 1 + |z|2



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(7.5.80), and vielbein 1-form  is series in odd powers. So, Eq. (7.5.80) gives ω=−

(dz ∗ z − dz z ∗ ) (dz z ∗ + dz ∗ z)I2 + 2 Z · d Z dZ = σ3 ,  = . 2 2 2(1 + |z| ) 2(1 + |z| ) (1 + |z|2 ) (7.5.81)

Finally, invariant metric (7.5.69) in the space SU (2)/U (1) = S 2 takes the form dz dz ∗ . (7.5.82) ds2 = −Tr( · ) = 2 (1 + |z|2 )2 Modulo an overall factor, this metric coincides with the metric (7.4.19) obtained in stereographic coordinates.  Problem 7.5.8. Consider homogeneous space CPn = U (n + 1)/(U (n) × U (1)), studied in Example 8 of Section 7.2. Making use of (7.5.64) and (7.5.69), obtain Fubini–Study metric for this space, ∂ ∂ dzi dzi∗ (dzi zi∗ ) (zk dzk∗ ) − = dzk dzi∗ ln(1 + |z|2 ), 1 + |z|2 (1 + |z|2 )2 ∂zk ∂zi∗ (7.5.83)  where i, k = 1, . . . , n and |z|2 = k |zk2 |. ds2 =

Problem 7.5.9. Making use of definitions (7.5.64) and (7.5.69), construct invariant metrics for spheres S k treated as homogeneous spaces (7.2.9) and (7.2.28). Hint: apply the technique developed for solving Problem 7.5.8. Problem 7.5.10. Construct invariant metric for homogeneous space (G × G)/GV , where G is matrix Lie group and GV is the diagonal subgroup, see Example 11 in Section 7.2. Problem 7.5.11. Construct invariant metrics for de Sitter and anti-de Sitter spaces, treated as homogeneous spaces in accordance with (7.2.26). Hint: apply the technique developed for solving Problem 7.5.8. Thus, metric in homogeneous space G/H is completely determined by Maurer–Cartan form  in decomposition (7.5.64). When studying concrete problems, the calculation of forms  and ω is sometimes simplified by making use of Campbell–Poincar´e identity, see Eq. (8.5.4) in the solution of Problem 3.1.2 in Section 8.5.

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To clarify the geometric meaning of the forms  and ω, we write Maurer– Cartan identity   (7.5.84) ∂μ (k −1 ∂ν k) − ∂ν (k −1 ∂μ k) = − k −1 ∂μ k , k −1 ∂ν k , whose compact form is d (k −1 · d k) = −(k −1 · d k) ∧ (k −1 · d k). Problem 7.5.12. Prove Maurer–Cartan identity (7.5.84). Making use of this identity in the expansion (7.5.64), we get d (α Xα + ω b Yb ) = −(α Xα + ω a Ya ) ∧ (β Xβ + ω b Yb ) = − 12 α ∧ β [Xα , Xβ ] − 12 ω a ∧ ω b [Ya , Yb ] − ω a ∧ β [Ya , Xβ ] d α = − 12 α ∧ β (Cαγ β Xγ + Cαb β Yb ) − 12 ω a ∧ ω b Cab Yd − ω b ∧ β Cbβ Xα .

This is equivalent to two relations α d α = − 12 γ ∧ β Cγαβ − ω b ∧ β Cbβ ,

(7.5.85)

b . d ω b = − 12 γ ∧ δ Cγb δ − 12 ω a ∧ ω d Cad

(7.5.86)

Recall that we interpret α as vielbein in G/H. We compare (7.5.85) with α is the first Cartan’s structure equation (7.5.27) and see that ω αβ = ω b Cbβ naturally interpreted as the spin connection form. Equation (7.5.86) can be written as α − ω αγ ∧ ω γβ , d ω αβ = − 21 γ ∧ δ Cγb δ Cbβ

(7.5.87)

and it is precisely the second Cartan’s structure equation (7.5.53). Thus, Eqs. (7.5.85) and (7.5.86) enable one to evaluate torsion and curvature tensors in G/H. One compares (7.5.27) with (7.5.85) and finds the torsion: α = − 21 Eμγ Eνβ Cγαβ . Tμν

(7.5.88)

By comparing (7.5.87) with the second Cartan’s structure equation (7.5.53) one finds curvature tensor in G/H: Rα β = Rμν α β daμ ∧ daν = −γ ∧ δ Cγb δ Cbαβ ⇒ Rμν α β = −Eμγ Eνδ Cγb δ Cbαβ .

(7.5.89)

Remark 1. In accordance with (7.5.61) and (7.5.63), we have representation of group H in tangent space Te (K) to manifold K = G/H at unit element e ∈ K. Action of H

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in this representation preserves flat metric ηαβ given by (7.5.70). This follows from the relation β β Tr([Yb , Xα ], Xγ ) + Tr(Xα , [Yb , Xγ ]) = 0 ⇒ Cbα ηβγ + ηαβ Cbγ = 0.

Therefore, group H in this representation is a subgroup of SO(p, d − p). Since connection ω in G/H takes values in Lie algebra A(H), the holonomy group of the manifold G/H (see definition of holonomy group in the end of Section 7.5.1) is either H itself or a subgroup of H.

Remark 2. Let Lie algebra A(G) and its subalgebra A(H) be such that the structure constants Cγαβ vanish, and defining relations of A(G), Eqs. (7.5.60)–(7.5.62), read β d Yd , [Yb , Xα ] = Cbα Xβ , [Ya , Yb ] = Cab

d [Xα , Xβ ] = Cαβ Yd .

(7.5.90) Then, in accordance with (7.5.88), torsion in G/H vanishes, and hence covariant derivatives of vielbein vanish too, Dλ Eμα = 0. In that case Eq. (7.5.89) implies that curvature tensor in G/H is covariantly constant, Dλ Rμν α β = 0. Manifold with covariantly constant curvature tensor is called symmetric space. So, if group G and its subgroup H admit basis in Lie algebra A(G) such that the defining relations have the form (7.5.90), then G/H is automatically symmetric space. We note that algebras with defining relations of the form (7.5.90) are called Z2 -graded Lie algebras, since A(G) = A(0) + A(1) where the subspaces A(0) and A(1) have basis elements Ya and Xα , respectively, and the relations (7.5.90) give [A(i) , A(j) ] = A(i+j)mod(2) . Example 2. To describe fairly broad class of symmetric spaces, consider Lie group G whose elements are (n + k) × (n + k) matrices. Let its Lie algebra be partitioned into two subspaces A(0) and A(1) whose matrices have block forms

Ann 0nk 0nn Bnk (0) ∈A , ∈ A(1) , (7.5.91) 0kn Dkk Ckn 0kk where Ann , Bnk , . . . are blocks of sizes (n × n), (n × k), etc., and 0nk are zero matrices. Then the subspace A(0) is normalizer subalgebra of A(1) in A(G), and the commutation relations in A(G) have the form (7.5.90). In this case the homogeneous space G/H, where Lie algebra of subgroup H

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is A(H) = A(0) , is symmetric space. Examples of torsionless symmetric spaces of this sort are SU (n + k)/(SU (n) × SU (k)),

SO(n + k)/(SO(n) × SO(k)),

(7.5.92)

where subgroups SU (n) × SU (k) and SO(n) × SO(k) are embedded in SU (n + k) and SO(n + k) in block-diagonal way (the two factors have Lie algebras with matrices A and D in (7.5.91)). All homogeneous spaces considered in Section 7.2 are actually symmetric spaces; this applies, in particular, to coset space SU (2)/U (1) = S 2 . Problem 7.5.13. Show that (GL × GR )/GV (Example 11 in Section 7.2) (L) (R) is symmetric space. Hint: Let Xa and Xa be generators of Lie algebras (L) (R) A(GL ) and A(GR ), respectively. Then Ya = Xa + Xa are generators of (L) (R) diagonal subalgebra A(GV ), and Xa = Xa − Xa . Check that generators {Ya , Xb } obey (7.5.90). Problem 7.5.14. Show that manifolds SO(2n)/U (n), SU (n)/SO(n), SU (2n)/Usp(2n), Usp(2n)/U (n) are symmetric spaces. Hint: check that the relevant Lie algebras have the property (7.5.90). Fairly simple homogeneous spaces with torsion are obtained in the following way. Let G be matrix Lie group, and let its Lie algebra be partitioned into two subspaces A(0) and A(1) , whose elements are

Ann 0nk 0nn Bnk (0) ∈A , ∈ A(1) . (7.5.93) 0kn 0kk Ckn Dkk Subspace A(0) is normalizer subalgebra of A(1) in A(G); one chooses subgroup H ⊂ G in such a way that A(0) is its Lie algebra, A(0) = A(H). The commutation relations in A(G) have the general form (7.5.60)–(7.5.62). γ in (7.5.62), the homogeneous Due to nonvanishing structure constants Cαβ space G/H generically has nonzero torsion (7.5.88). Problem 7.5.15. Consider homogeneous spaces SU (n + k)/SU (n), SO(n + k)/SO(n), where k ≥ 2, and calculate torsion components (7.5.88).

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Problem 7.5.16. Calculate torsion components (7.5.88) for space SU (3)/[U (1) × U (1)], where Lie algebra of subgroup U (1) × U (1) is Cartan subalgebra of SU (3). 7.5.3.

Regular representations and invariant vector fields on Lie groups

Consider linear space of smooth functions α(g) on n-dimensional Lie group G. Operators T(R) (h) of right regular representation with h ∈ G act on α(g) as follows (see Eq. (4.5.13)): [T(R) (h)α](g) = α(g · h).

(7.5.94)

Let element h belong to vicinity of unit element, then it can be written in the exponential form h = exp(ta Xa ), where Xa are generators of Lie algebra A(G), and t = (t1 , . . . , tn ) parameterize elements h. Definition (7.5.94) gives for small {ta }   ∂  (R)   [T (Xa )α](g) = a α g · h(t )  . (7.5.95) ∂t  t=0 This defines the right regular representation of Lie algebra A(G). The leftand right-hand sides of (7.5.95) can be viewed as functions of coordinates x = (x1 , . . . , xn ) parameterizing elements g = g(x) ∈ G. We now recall that g(x) · h(t ) = g(F (x, t)), see (2.1.11), and obtain   ∂  (R)   [T (Xa ) α](g) = a α F (x, t)  = Lm x) ∂m α(x), (7.5.96) a ( ∂t  t=0  where Lm x) = ∂t∂a F m (x, t)t=0 , see (2.2.27), and we use shorthand a ( notation α(x) = α(g(x)). Thus, generator Xi ∈ A(G) in right regular representation is first-order differential operator (left-invariant vector field on G; we explain nomenclature below) T(R) (Xa ) = Lm x)∂m ≡ ρL (Xa ). a (

(7.5.97)

Likewise, the left regular representation (4.5.12) of Lie group G induces the representation of its Lie algebra of the following form:   ∂  [T (R) (Xa )α](g) = a α h−1 (t) · g(x)  = −Ram (x) ∂m α(x). (7.5.98) ∂t  t=0 So, differential operators (right-invariant vector fields on G) T (R) (Xa ) = −Ram (x) ∂m ≡ ρR (Xa )

(7.5.99)

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are generators in left regular representation of Lie algebra A(G). d Xd . Differential operators ρL (Xa ) and ρR (Xa ) (generators [Xa , Xb ] = Cab of Lie algebra A(G) in regular representations) obey commutation relations d d ρL (Xd ), [ρR (Xa ), ρR (Xb )] = Cab ρR (Xd ), [ρL (Xa ), ρL (Xb )] = Cab (7.5.100) d are structure constants of A(G). where Cab Let T be matrix representation of Lie group G. Matrix elements ||Tαβ (g)|| are functions on group G. Operators T (R) (h) of left regular representation (4.5.12) act on these functions as follows:

[T (R) (h) · Tαβ ](g) = Tαβ (h−1 · g) = Tαγ (h−1 ) Tγβ (g).

(7.5.101)

We consider element h(t) = exp(ta Xa ) for small ta and, in the same way as in the case (7.5.98), obtain [T (R) (Xa ) · Tαβ ](g(x)) = −Ram ∂m Tαβ (g(x)) = −Tαγ (Xa )Tγβ (g(x)). (7.5.102) Relations (7.5.101) and (7.5.102) show that subspace of functions whose basis elements are Tαβ (g) is invariant with respect to left regular representation. Therefore, left regular representation is reducible and contains all matrix representations T of group G as sub-representations. The same result holds for right regular representation. Problem 7.5.17. Prove the analogs of the relations (7.5.101) and (7.5.102) for right regular representation (7.5.94), (7.5.96): [T(R) (h) · Tαβ ](g) = Tαγ (g) Tγβ (h), x)) = Tαγ (g(x)) Tγβ (Xa ). Lm a ∂m Tαβ (g(

(7.5.103) (7.5.104)

Let G be matrix Lie group and A(G) its Lie algebra. We have equalities (2.2.30) and (2.2.31), which can be written as Xa · g(x) = Ram (x)∂m g(x) ⇔ g(x)−1 · Xa = −Ram (x)∂m g(x)−1 , (7.5.105) x)∂m g(x) = ρL (Xk ) g(x). g(x) · Xi = Lm i (

(7.5.106)

These are relations (7.5.102) and (7.5.104) specified to defining representation T . Making use of (7.5.102) and (7.5.104), one checks explicitly the validity of commutation relations (7.5.100). As an example, one applies Eq. (7.5.104)

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twice and gets (T (g(x)) · T (Xa )) · T (Xb ) = ρL (Xa ) [T (g(x)) · T (Xb )] = ρL (Xa ) ρL (Xb ) T (g(x)),

(7.5.107)

where matrix indices are omitted for brevity. One permutes indices a and b here and constructs the difference of the two relations. Upon recalling commutation relations for T (Xa ), one finds that   d [ρL (Xa ) , ρL (Xb )] − Cab ρL (Xd ) T (g(x)) = 0. Since T is arbitrary, this gives (7.5.100). Problem 7.5.18. Consider left and right actions of group G on itself, −1 · g(x) · gR , g(x) → gL

∀ gL , gR ∈ G.

Making use of (2.2.30), (2.2.31) (or (7.5.105), (7.5.106)), show that they induce the following transformations of right- and left-invariant fields Ram (x) x): and Lm a ( x) → ad(gR )ab Lm x), Ram (x) → ad(gL )ab Rbm (x), Lm a ( a (

(7.5.108)

where ||ad(g)ab || is matrix of adjoint representation in basis Xa ∈ A(G). This property justifies the terms “left-invariant and right-invariant vector fields” m used for ρL (Xa ) = Lm a ∂m and ρR (Xa ) = −Ra ∂m .

7.5.4.

Laplace operators on Lie groups and homogeneous spaces

Let lie group G be semisimple, and ηab = Tr(ad(Xa ) · ad(Xb )) be the standard non-degenerate metric on its Lie algebra A(G). In regular representations ρL and ρR , given by (7.5.97) and ρR (7.5.99), quadratic Casimir operator C2 , Eq. (4.7.12), is written in terms of second-order differential operators x)∂m Ljb (x)∂j , ΔL ≡ ρL (C2 ) = η ab Lm a ( ΔR ≡ ρR (C2 ) = η ab Ram (x)∂m Rbj (x)∂j .

(7.5.109)

These are called Laplace operators on Lie group G. Their explicit form (7.5.108) shows that operators ΔL and ΔR are simultaneously left- and right-invariant.

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Recall that convenient invariant metric in Lie group, defined in (2.2.24), (2.2.26) and (2.2.29) is gnm (x) = [L−1 (x)]an [L−1 (x)]bm ηab = [R−1 (x)]an [R−1 (x)]bm ηab ab ab ⇐⇒ g nm (x) = [L(x)]na [L(x)]m = [R(x)]na [R(x)]m b η b η . (7.5.110) m ( x ) (or R ( x )) play a role of vielbein With this choice, matrix elements Lm a a m −1 a −1 a ea (x); and [L (x)]m (or [R (x)]m ) is dual vielbein. Here a is tangent space index and m is coordinate index similar to index μ in Section 7.5.1). Using metric (7.5.110), one constructs the standard Laplace–Beltrami operator on group manifold

 1 ∂n | det(g)| g nm ∂m , Δ=  | det(g)|

(7.5.111)

where det(g) = det(||gmn ||). Proposition 7.5.3. Operators Δ, ΔL and ΔR on group manifold, defined by (7.5.111) and (7.5.109), coincide: Δ = ΔL = ΔR .

(7.5.112)

Proof. Operators ΔL and ΔR can be written in unified way, Δ = η ab enb ∂n em a ∂m , m m x) or em x). Then we have for metric (7.5.110) where em a = La ( a = −Ra ( a b that gnm = em en ηab and | det(g)| = e2 · | det(η)|, where e = det ||eam ||. In these notations, operator (7.5.111) reads

1 ∂n e enb η ab em a ∂m e   1 n ab ab n ab n m (∂n e) eb η + η (∂n eb ) em = a ∂m + η eb ∂n ea ∂m e    = erd (∂n edr ) enb η ab + η ab (∂n enb ) em a ∂m + Δ    = −edr (∂n erd ) enb η ab + η ab (∂n enb ) em (7.5.113) a ∂m + Δ ,

Δ=

where we use the identity erd (∂n edr ) = −(∂n erd ) edr . Vector fields ρ(Xa ) = em a ∂m are Lie algebra generators in regular representations. They obey

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commutation relations (7.5.100), so that n d r m n m n d n [em a ∂m , eb ∂n ] = Cab ed ∂r ⇒ ea (∂m eb ) − eb (∂m ea ) = Cab ed .

(7.5.114) Convolution of the second relation with ean and antisymmetry of structure constants Cabd give a n a ad (∂n enb ) − em = 0. b en (∂m ea ) = Cab = Cabd η

Thus, the expression in square brackets in (7.5.113) vanishes, and we arrive at the desired result. Remark 1. We have shown, in particular, that if vector fields ρ(Xa ) = em a ∂m on some manifold make semisimple Lie algebra with defining relations (7.5.114), then there is an identity between Laplace–Beltrami operator and quadratic Casimir operator: η ab ena ∂n em b ∂m =

1 ∂n (e enb η ab em a ) ∂m ≡ Δ. e

(7.5.115)

Remark 2. Laplace operator Δ = ρL (C2 ) = ρR (C2 ) has a remarkable property that it commutes with vector fields (7.5.97) and (7.5.99): [ρL (Xa ), Δ] = 0 = [ρR (Xa ), Δ]. Example 1. Let us construct Laplace operator Δ for group G = SU (2) (sphere S 3 ). To this end, we make use of parameterization (2.1.17) of group 3 3 ¯α xα , SU (2), namely, g = α=0 σ α=0 xα xα = 1. Let us treat three coordinates x = (x1 , x2 , x3 ) as independent. Then we have g(x)−1 ∂m g(x) = (x0 − iσk xk )(∂m x0 + iσm )   xr xm = iσr x0 δrm + xk εkmr + , x0

(7.5.116)

where k, r, m = 1, 2, 3, and we use the identity x0 ∂m x0 + xm = 0. Now, SU (2) generators are Xr = iσr . Then, by comparing (7.5.116) with (7.5.106), we obtain   xr xm −1 r ⇒ Lkm (x) = (x0 δkm + xr εkrm ) . (L )m = x0 δrm + xk εkmr + x0 (7.5.117)

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So, generators in right regular representation (7.5.97) are Xk = iσk → ρL (Xk ) = Lkm (x)∂m = x0 ∂k + xr εkrm ∂m ,

(7.5.118)

and we have for metric (7.5.110) that g mr = δ kj Lkm Ljr = (δmr − xr xm ). Finally, Eq. (7.5.118) translates the general expression (7.5.109) for Laplace operator to ΔL = ρL (Xm ) ρL (Xm ) = ∂k2 − (xk ∂k )2 − 2(xk ∂k ).

(7.5.119)

Problem 7.5.19. Show that the differential operators ρL (Xk ) given by (7.5.118) obey commutation relations of su(2): [ρL (Xk ), ρL (Xm )] = −2 εkmr ρL (Xr ).

(7.5.120)

Check that Laplace operator (7.5.119) can be written in the standard form (7.5.111), where g nm = (δmn − xn xm ), | det(g)|−1/2 = | det(Lkm )| and Lkm are given in (7.5.117). Rewrite the Laplace operator (7.5.119) in stereographic projection coordinates (7.4.15) and in spherical coordinates (2.2.36). Since group G acts not only on itself, but also on homogeneous spaces G/H, generators of its Lie algebra A(G) can also be represented as differential operators (vector fields) on G/H. The pertinent construction is quite similar to that given above for representation of A(G) in terms of vector fields on G (rather than G/H). An analog of left regular representation in the case of G/H is induced representation (7.3.6). Let us discuss this construction in some detail. Consider the set K of elements k = k(a) ∈ G, which parameterize homogeneous space G/H, a = (a1 , . . . , ad ) are coordinates on K = G/H in some local patch. Let us choose basis in A(G) in the same way as in (7.5.60)– (7.5.62). Then the elements k(a) have the form (7.5.59). Equation (7.3.2) for g close to unity reads g(τ α , ta ) · k(a) = k(˜ a1 , . . . , a ˜d ) · exp(˜ba (a, τ, t)Ya ),

(7.5.121)

where g(τ α , ta ) = exp(τ α Xα + ta Ya ), a ˜β = a ˜β (a, τ, t) and parameters ta , τ α are small. Consider the simplest induced representation (7.3.9), and make use of (7.5.121) to write Eq. (7.3.9) as follows: [T (g(−τ α , −ta )) · v](k(a)) = v(k(˜ a1 , . . . , a ˜d )),

(7.5.122)

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where v is a smooth function in homogeneous space G/H. Upon differentiating (7.5.122) over ta (and τ α ) and then setting ta = 0 = τ α we get [T (Yb ) · v](k(a)) = −Rbβ (a)

∂ v(k(a)) ≡ ρ(Yb ) v(k(a)), ∂aβ

∂ v(k(a)) ≡ ρ(Xα ) v(k(a)), ∂aβ where matrices of right-invariant vector fields are   ∂˜ aβ  ∂˜ aβ  β β Rb (a) = , Rα (a) = . ∂tb τ =t=0 ∂τ α τ =t=0 β (a) [T (Xα ) · v](k(a)) = −Rα

(7.5.123) (7.5.124)

(7.5.125)

Thus, we have constructed representation ρ of algebra A(G) in terms of vector fields in G/H: β ρ(Yb ) = −Rbβ (a) ∂β , ρ(Xα ) = −Rα (a) ∂β .

(7.5.126)

Problem 7.5.20. Let G be matrix group. Upon differentiating Eq. (7.5.121), obtain the relations d

Yb · k(a) = Rbβ (a) ∂β k(a) + k(a) Rb (a) Yd ,

(7.5.127)

b

β Xα · k(a) = Rα (a) ∂β k(a) + k(a) Rα (a) Yb ,

where matrices R are defined in (7.5.125), and   ∂˜bd  ∂˜bd  d d Rb (a) = , Rβ (a) =   ∂tb  ∂τ β  τ =t=0

(7.5.128)

. τ =t=0

Problem 7.5.21. Let us choose k(a) in the form (7.5.59). Show that Eq. (7.5.127) is then written as Yb · k(a) = aα Cbβα ∂β k(a) + k(a) · Yb , i.e., ρ(Yb ) = −aα Cbβα ∂β . Let algebra A(G) be semisimple. Then Laplace operator on homogeneous space G/H is quadratic Casimir operator C2 = η ab Ya Yb + η αξ Xα Xξ , in the representation (7.5.123), (7.5.124): β ΔG/H = ρ(C2 ) = η ab Raβ (a) ∂β Rbγ (a) ∂γ + η αξ Rα (a) ∂β Rξγ (a) ∂γ , (7.5.129)

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where ||η ab || and ||η αξ || are inverse of matrices ηab = Tr(ad(Ya )ad(Yb )),

ηαβ = Tr(ad(Xα )ad(Xβ )).

Recall that Killing metric in A(G) in the case we consider is

0 ||ηab || . g= 0 ||ηαβ || Problem 7.5.22. Show that ΔG/H = Δ, i.e., operator (7.5.129) coincides with the standard Laplace–Beltrami operator (7.5.111), where metric gmn on G/H is defined by (7.5.64) and (7.5.69). Hint: make use of Remark 1 to Proposition 7.5.3. Example 2. Let us calculate Laplace operator (7.5.129) in homogeneous space SU (2)/U (1) = S 2 . Space SU (2)/U (1) is parameterized by elements (7.5.74): k(α) = I2 α0 + iσ1 α1 + iσ2 α2 ,

(7.5.130)

where parameters α0,1,2 ∈ R obey (7.5.75), and we consider α1 , α2 as independent. An element of local vicinity of unity in SU (2) can be written in exponential form g(t ) = exp(itk σk ), and we choose element of U (1) in the form (7.5.71), i.e., h = exp(−ibσ3 ). Then Eqs. (7.3.2), (7.5.121) for g(t ) close to unity (small ti and b), read α) (I2 − ibσ3 ), (I2 + itk σk ) k(α) = k(˜

(7.5.131)

which is equivalent to α). k(α) + i(tk σk k(α) + b k(α) σ3 ) = k(˜

(7.5.132)

We insert here Eq. (7.5.130) and compare coefficients of matrices I2 , σk in (7.5.132) to calculate α ˜ k and b to linear order in tk :   α1 α2 (α2 )2 + t α ˜ 1 = α1 + t1 α0 − + t3 2 α2 , 2 α0 α0   α1 α2 (α1 )2 − t3 2 α1 , α ˜ 2 = α2 + t1 0 + t2 α0 − (7.5.133) α α0 b = −t3 +

 1  2 t1 α − t2 α1 . α0

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Upon differentiating (7.5.131) over tk and then setting tk = 0, we obtain analogs of relations (7.5.127), (7.5.128):   α2 α2 X1 · k(α) = α0 ∂1 + 0 (α1 ∂2 − α2 ∂1 ) k(α) − 0 k(α) · Y, α α   1 α α1 (7.5.134) X2 · k(α) = α0 ∂2 − 0 (α1 ∂2 − α2 ∂1 ) k(α) + 0 k(α) · Y, α α  1  2 Y · k(α) = −2 α ∂2 − α ∂1 k(α) + k(α) · Y, where ∂m = ∂α∂m with (m = 1, 2) and we denote SU (2) generators by Xm = iσm and Y = iσ3 . It follows from (7.5.134) that representation of these generators in terms of right-invariant vector fields (7.5.126) is   α2 iσ1 → ρ(X1 ) = − α0 ∂1 + 0 (α1 ∂2 − α2 ∂1 ) , α   α1 1 (7.5.135) 0 2 iσ2 → ρ(X2 ) = − α ∂2 − 0 (α ∂2 − α ∂1 ) , α   iσ3 → ρ(X3 ) ≡ ρ(Y ) = 2 α1 ∂2 − α2 ∂1 . Problem 7.5.23. Show that operators (7.5.135) obey defining commutation relations (7.5.120) of Lie algebra su(2). Hint: make use of the fact that [α0 , (α1 ∂2 − α2 ∂1 )] = 0. Laplace operator (7.5.129), which is Casimir operator for Lie algebra su(2) in representation (7.5.135) is given by ΔSU(2)/U(1) = ρ(Xk ) ρ(Xk ) =

1 1 + (α0 )2 m 2 m ∂ − 2 α ∂ − (α ∂m )2 . m (α0 )2 m (α0 )2

It coincides with Laplace–Beltrami operator (7.5.111) on 2-dimensional sphere (7.5.75), written in coordinates αm = (α1 , α2 ) and having metric (7.5.78). 7.5.5.

Spherical functions on homogeneous spaces

Let G be compact Lie group and H its non-Abelian subgroup. Let T (λ) be irreducible representation of G whose dimension is Nλ and which acts in space Vλ . Representation T (λ) of group G is at the same time representation of its subgroup H ⊂ G, and the latter may be reducible. This means that Vλ contains subspaces which are invariant under action of subgroup H in representation T (λ) . Let Vλ have 1-dimensional invariant subspace Vλ0 of H, where H acts trivially. In that case representation T (λ) of group G is called class 1 with respect to subgroup H; let us consider this case. Let us choose

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orthonormal basis in Vλ with vectors |λ, m, where m = 0, 1, . . . , Nλ − 1, in such a way that the first vector |λ, 0 be basis vector in Vλ0 . It is invariant under H : T (λ) (h) |λ, 0 = |λ, 0,

∀h ∈ H.

(7.5.136)

We use Dirac notations here. In this basis, the matrix of operator T (λ) (g) has matrix elements (λ) (7.5.137) Tmr (g) = λ, m| T (λ) (g) |λ, r. It follows from (7.5.136) that matrix elements (λ)

Tm0 (g) = λ, m|T (λ) (g)|λ, 0 ,

m = 0, 1, . . . , Nλ − 1

(7.5.138)

are functions on homogeneous space G/H, since λ, m|T (λ) (k · h)|λ, 0 = λ, m|T (λ) (k) · T (λ) (h)|λ, 0 = λ, m|T (λ) (k)|λ, 0, Functions

(λ) Tm0

∀h ∈ H.

(7.5.139)

on G/H given by (7.5.138) are called adjoint spherical functions of (λ)

T (λ) .

Among these functions, a particular function T00 is invariant under representation right and left shifts in G by elements h ∈ H; this function is called zonal spherical function of representation T (λ) . (λ) Adjoint spherical functions Tm0 on G/H form space Y (λ) of a certain representation of group G. Indeed, let us choose the set of elements k ∈ G which parameterize homogeneous space G/H, see (7.3.1). Using homomorphic property of representation T (λ) and Eqs. (7.3.2), (7.5.139), we get Tmr (g −1 ) Tr0 (k) = Tm0 (k(g −1 , k)). (λ)

(λ)

(λ)

(7.5.140)

The left-hand side here can be viewed as the action of a linear operator ρ(g) in space Y (λ) , so that we have (λ) (λ) ρ(g) · Tm0 (k) = Tm0 (k(g −1 , k)). (7.5.141) This formula defines the mapping ρ of group G to space of linear operators acting in Y (λ) . These operators are associated with shifts in the homogeneous space G/H : k → k(g −1 , k). Homomorphicity of mapping ρ is checked as follows: ρ(g1 ) · ρ(g2 ) · Tm0 (k) = ρ(g1 ) · Tm0 (k(g2−1 , k)) = Tm0 (k(g2−1 , k(g1−1 , k)) (λ)

(λ)

(λ)

= Tm0 (k(g2−1 · g1−1 , k)) = ρ(g1 · g2 ) · Tm0 (k). (λ)

(λ)

Thus, ρ is indeed representation of G. In fact, it is nothing but a finite-dimensional representation embedded in the simplest version of induced representations; this observation follows from comparing (7.5.141) with (7.3.9), (7.5.122)). If element g is close to unity, operator ρ(g) given by (7.5.141) can be written as exponential of vector field (7.5.126), i.e., the representation of Lie algebra A(G) is given in terms of vector field (differential operators). Problem 7.5.24. Making use of Eqs. (7.5.123), (7.5.124) and (7.5.140), construct representation ρ of A(G) and prove that a) ∂α Tm0 (k(a)) = −Tmp (Ya ) Tp0 (k(a)), ρ(Ya ) · Tm0 (k(a)) = −Rα a ( (λ)

(λ)

(λ)

(λ)

ρ(Xα ) · Tm0 (k(a)) = −Rβ a) ∂β Tm0 (k(a)) = −Tmp (Xα ) Tp0 (k(a)). α ( (λ)

(λ)

(λ)

(λ)

(7.5.142)

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Let subgroup H in G be such that all irreducible representations T (λ) of G are class 1 representations with respect to H. In that case one makes use of Peter–Weyl theorem (Theorem 4.6.1) to prove that any function f (k) on homogeneous space G/H can be (λ) expanded in spherical functions Tm0 (k): f (k) =

 m,λ

Cλm Tm0 (k) = (λ)

 m,λ

Cλm λ, m|T (λ) (k)|λ, 0,

(7.5.143)

where k ∈ G parameterizes space G/H, and Cλm are expansion coefficients. Note that all functions (7.5.143), as functions on group G, are constant in each coset in G/H and hence make basis in space L2 (G/H, dμ) of functions on G/H (which is a subspace of L2 (G, dμ)). (λ) One more property of basis spherical functions Tm0 is that they are orthogonal. This follows from the orthogonality relations (4.6.37), which are written in our case as follows:  1 1 λν (ν)∗ (λ) dμ(g) Tm 0 (g) Tm 0 (g) δ δm,m = Nν V (7.5.144)   1 (ν)∗ (λ) dμ(k) dμ(h) J(k, h) Tm 0 (k · h) Tm 0 (k · h), = V where we set g = k · h and replace integral over group G by integral over coset space G/H and integral over subgroup H. To this end, measure dμ(g) in group G is given in terms of measure dμ(h) in subgroup H and measure dμ(k) in coset space G/H: dμ(g) = dμ(h) dμ(k) J(k, h). Here J(k, h) is Jacobian of the transformation from g to h, k. Making use of (7.5.139), (λ) we write Eq. (7.5.144) as orthogonality of spherical functions Tm0 on G/H: 1 1 λν δ δm,m = Nν V



(ν)∗

(λ)

dμ(k) J(k) Tm 0 (k) Tm 0 (k),

(7.5.145)

 where J(k) is integral over subgroup H, namely, J(k) = dμ(h) J(k, h). (λ) of G, matrix elements of T (λ) (g) are We recall that given representation T functions on group G obeying (7.5.102): (λ)

(λ)

(λ)

ρR (Xa ) · Tmm (g) = −Tmp (Xa ) Tpm (g),

(7.5.146)

where ρR (Xa ) = −Rμ a ∂μ . This property reiterates that left regular representation is reducible and contains all irreducible representations T (λ) . We recall further that quadratic Casimir operator C2 = ηab Xa Xb takes fixed value in irreducible representation T (λ) (see (4.7.11)): T (λ) (C2 ) = c(λ) Iλ . Then, in accordance with (7.5.146), action of Laplace operator Δ, defined in (7.5.112), (λ) (λ) on functions Tm,m (g) = Tm,m (g( x)) is (λ)

(λ)

(λ)

(λ)

Δ Tm,m (g) = Tm,p (C2 ) Tp,m (g) = c(λ) Tm,m (g),

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(λ)

i.e., all Tm,m (g( x)) are eigenfunctions of Laplace operator. Proceeding along the same (λ)

lines and using (7.5.142), one shows that spherical functions Tm,0 (k(a)), as functions on homogeneous space G/H, are eigenfunctions of Laplace operator ΔG/H given by (7.5.129): (λ)

(λ)

ΔG/H Tm,0 (k(a)) = c(λ) Tm,0 (k(a)).

(7.5.147)

Making use of this result, one constructs complete orthonormal sets of eigenfunctions of Laplace operators on concrete homogeneous spaces. Particularly important for physics is sphere S 2 = SO(3)/SO(2) = SU (2)/U (1) and its multi-dimensional analogs.

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Chapter 8

Solutions to Selected Problems

8.1.

Problem 1.1.9, Section 1.1.1

Show that any permutation belonging to group Sn can be decomposed into product of cycles with nonoverlapping sets of symbols. As an example, find such a decomposition for the permutation   1 2 3 4 5 6 7 8 9 10 ∈ S10 . (8.1.1) 3 6 4 1 2 8 10 5 9 7 We consider for concreteness permutation (8.1.1) and decompose it into product of cycles. To this end, we take element 1 in the upper row of (8.1.1). This element moves to position 3, element 3 moves to 4, and 4 to 1. Thus, permutation (8.1.1) contains cycle (1, 3, 4). We then take any element different from 1, 3, 4 in the upper row, say, element 2, which moves to position 6. Repeating the above procedure, we find that permutation (8.1.1) contains another cycle (2, 6, 8, 5). We then take element 7, which is absent in previous cycles, and find its cycle (7, 10). Finally, we have   1 2 3 4 5 6 7 8 9 10 3

6

4 1

2 8

10 5 9

7

= (1, 3, 4) · (2, 6, 8, 5) · (7, 10) · (9). This procedure applies in an obvious way to any permutation from Sn , which demonstrates that any permutation can be decomposed into product of cycles containing different symbols.

409

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8.2.

Problem 1.1.27, Section 1.1.4

Show that any cycle can be decomposed into a product of transpositions, namely, the following identity holds: (a1 , a2 , . . . , ak ) = (a1 , ak )(a1 , ak−1 ) · · · (a1 , a2 ).

(8.2.1)

Show that any transposition can be decomposed into a product of neighboring transpositions σi = (i, i + 1) : (i, j) = σi · σi+1 · · · σj−2 · σj−1 · σj−2 · · · σi+1 · σi .

(8.2.2)

We begin with the following formula (∀k > 2) (a1 , a2 , . . . , ak ) = (a1 , ak )(a1 , a2 , a3 , . . . , ak−1 ),

(8.2.3)

which is checked by straightforward calculation: (a1 , ak )(a1 , a2 , a3 , . . . , ak−1 )    a1 a2 a1 a2 . . . ak−1 ak · = ak a2 . . . ak−1 a1 a2 a3   a1 a2 . . . ak−2 ak−1 ak . = a2 a3 . . . ak−1 ak a1

. . . ak−2

ak−1

ak

. . . ak−1

a1

ak



Recall that the first permutation is on the right. Making use of graphic representation (1.1.28), we draw this identity as follows: a

(a1 , a2 , . . . , ak−1 ) =

•1 •

(a1 , ak ) =

•

a1

a

a

•2 . . . k−2 • ... • ... • ... ... • • a2

ak−2

ak−1

a

•

•k

•

• =

•

ak−1

•

a1

•

•

a1

a2

a

•

... •

• . . . k−1 •

a2

ak−1

a

•k = (a1 , . . . , ak ) • ak

ak

Making use of (8.2.3) consecutively for k = n, n − 1, . . . , 3: (a1 , a2 , . . . , an ) = (a1 , an )(a1 , . . . , an−1 ) = (a1 , an )(a1 , an−1 )(a1 , . . . , an−2 ) = . . . , we obtain (8.2.1). The fact that any permutation is a product of neighboring transpositions is proven by the following argument. By consecutive neighboring transpositions, one can move any object ai out of n objects {a1 , a2 , . . . , an } to the

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place occupied by any other object aj , and then move aj to the original place of ai , with other objects staying in their places. Say, for j > i we first transpose ai and ai+1 (this is transposition σi ), so that object ai occupies (i + 1)th place while ai+1 moves to ith place. Then we apply the neighboring transposition σi+1 to put ai at (i + 2)th place, etc. The last neighboring transposition σj−1 moves object ai to jth place, and every object am (m = i + 1, . . . , j) occupies (m − 1)th place. After that by using neighboring transpositions in a similar way, one moves object aj from right to left, so that it moves from (j − 1)th place to ith place. This gives the identity(8.2.2), which can be proven also by using (1.1.32). 8.3.

Problem 1.2.22, Section 1.2.5

Establish isomorphism Sp (p, q) = Sp(p, q). Let us introduce 2r × 2r matrix S, which we write in block form   P+ P− , (8.3.1) S= P− P+ where r × r blocks P ± are  Ip Ir + Ip,q + = P = 2 0

0



0

,

P

−

Ir − Ip,q = = 2

 0

0

0

Iq

 ,

(8.3.2)

and matrix Ip,q is defined in (1.2.63). Blocks P ± are projection operators, P +P + = P +,

P −P − = P −,

P + P − = P − P + = 0,

P + + P − = Ir .

Matrix S transposes 2q components of 2r-dimensional column, which follows from the explicit form of projectors, Eq. (8.3.2). In view of (8.3.2), we have S 2 = I2r , and         Ip,q 0 0 Ip,q 0 Ip,q 0 Ir S S= , S S= . 0 Ip,q 0 Ip,q −Ip,q 0 −Ir 0 Thus, T  ↔ S T S is isomorphism of Sp (p, q) = Sp(p, q). 8.4.

Problem 2.1.8, Section 2.1.2

Prove that the manifold O(p, q) with p ≥ 1 and q ≥ 1 is a union of four manifolds which are not connected to each other, each of which is a coset of subgroup SO↑ (p, q) ⊂ O(p, q). Hint: prove, in the first place, that matrices obeying (2.1.35) have | det(X)| ≥ 1, | det(W )| ≥ 1.

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Consider Eq. (2.1.35), which is equivalent to pseudo-orthogonality condition (1.2.74). Matrices Z T · Z and Y T · Y are positive-definite, i.e., all their eigenvalues λa (Z) and λα (Y ) are real and positive. The latter property follows from inequality v T (Z T · Z)v = (Z v )T · Z v ≥ 0,

∀v ∈ Rp ,

and similar inequality for matrix (Y T · Y ). Then the first and second relations in (2.1.35) give  (1 + λa (Z)) ≥ 1, (det(X))2 = det(X T · X) = a  (8.4.1) (det(W ))2 = det(W T · W ) = (1 + λα (Y )) ≥ 1, α

so matrices X and W are invertible. It follows from (8.4.1) that group manifold O(p, q) with p ≥ 1 and q ≥ 1 has at least four disconnected components which are determined by inequalities (cf. (2.1.32)) 1. det(X) ≥ 1, det(W ) ≥ 1; 2. det(X) ≤ −1, det(W ) ≤ −1; 3. det(X) ≥ 1, det(W ) ≤ −1; 4. det(X) ≤ −1, det(W ) ≥ 1.

(8.4.2)

Let us show that the number of disconnected components of O(p, q) with p ≥ 1 and q ≥ 1 cannot exceed 4. Any element O ∈ O(p, q) of the form (2.1.34) can be continuously deformed to   X  0p×q  , (8.4.3) O = 0q×p W  where 0p×q and 0q×p are zero matrices. To do this, one makes a replacement Z → t Z in (2.1.34), where t ∈ [0, 1], and finds matrices X(t), Y (t) and W (t) by solving (2.1.35). The third of Eqs. (2.1.35) can be written as Y (t) = t [X(t)T ]−1 · Z T · W (t). This shows that there always exists solution obeying Y (t)|t=0 = 0. Then X  = X(0) and W  = W (0). Since (X  )T · X  = Ip , (W  )T · W  = Iq , i.e., X  , W  are orthogonal matrices, the manifold of matrices of the form (8.4.3), which is submanifold of O(p, q), consists of four disconnected

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components 1. det(X  ) = 1, det(W  ) = 1; 2. det(X  ) = −1, det(W  ) = −1; 3. det(X  ) = 1, det(W  ) = −1; 4. det(X  ) = −1, det(W  ) = 1, which are not connected to each other. Thus, the number of disconnected components does not exceed 4, and hence it is equal to 4. We also obtain that the number of disconnected components in pseudo-orthogonal group SO(p, q) with p ≥ 1 and q ≥ 1, equals 2: these are component 1 (subgroup SO↑ (p, q)) and component 2, listed in (8.4.2). Clearly, the group manifold O(p, q) is a union of four cosets (cf. (2.1.33)) O(p, q) = (SO↑ (p, q) · Ip+q ) ∪ (SO↑ (p, q) · P ) ∪(SO↑ (p, q) · T ) ∪ (SO↑ (p, q) · P T ), where P = diag (1, . . . , 1, −1) ,

T = (−1, 1, . . . , 1) .

Proof of the latter fact is left to the reader. 8.5.

Problem 3.1.2, Section 3.1.2

Prove Campbell–Hausdorff formula (3.1.9), (3.1.10). Check the result (3.1.13) by making use of (3.1.10); check (3.1.13) directly as well, by expanding exponential factors in (3.1.9). We begin with the following equality d  −tF t(F +δF )  = e−tF · δF · etF , e ·e (8.5.1) dt where δF denotes variation of operator F . We make use of (3.1.12), integrate (8.5.1) over t from 0 to 1 and obtain  1 ∞  tk −F F · δe = dt (−1)k [F, . . . , [F , δF ] . . .] e k!  0 k=0

=

∞  k=0

(−1)k

k

1 [F, . . . , [F , δF ] . . .]. (k + 1)! 

(8.5.2)

k

We define operator ad(F ) and simple functions thereof: ad(F ) · A = [F, A], e

ad(F )

(ad(F ))k · A = [F, [F, . . . , [F , A],

 ·A=e ·A·e F

−F

k

,

(8.5.3)

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where A is an arbitrary operator, and exponential factor ead(F ) is understood as Taylor series. In this way we have for (8.5.2) that e−F · δeF =

1 − e−ad(F ) δF. ad(F )

(8.5.4)

Formula (8.5.4) is Campbell–Poincar´e identity. Its equivalent form is obtained by starting from et(F +δF ) · e−tF in (8.5.1) and reads (δeF ) e−F =

ead(F ) − 1 δF. ad(F )

(8.5.5)

The same identity is found by multiplying (8.5.4) by eF on the left and by e−F on the right, and then replacing, on the right-hand side of (8.5.4), this transformation by action of operator ead(F ) in accordance with its definition (8.5.3). Identities (8.5.4), (8.5.5) can also be cast in the forms δF =

ad(F ) ad(F ) (δeF ) e−F . e−F (δeF ), δF = ad(F ) 1 − e−ad(F ) e −1

(8.5.6)

It is convenient to expand the operators on the right-hand sides of (8.5.6): ∞  ad(F ) ln(ead(F ) ) 1 (1 − ead(F ) )k = = − k ead(F ) − 1 ead(F ) − 1 ead(F ) − 1 k=1

=

∞  (1 − ead(F ) )k−1

k

k=1

,

∞  ad(F ) (1 − ead(F ) )k−1 ad(F ) . = e k 1 − e−ad(F )

(8.5.7)

(8.5.8)

k=1

Let us now specify to the operator F = F (t) given by eF (t) = etA1 · etA2 ,

(8.5.9)

where A1 and A2 do not commute. We apply differential operator δ ≡ ∂t to (8.5.9), multiply by e−F (t) on the right and make use of (8.5.5). We obtain (A1 + et ad(A1 ) · A2 ) =

ead(F (t)) − 1 · ∂t F (t). ad(F (t))

(8.5.10)

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We recall (8.5.7) and write (8.5.10) in the following form: ∂t F = =

ad(F ) (A1 + et ad(A1 ) · A2 ) −1

ead(F )

∞  (1 − ead(F (t)) )n (A1 + et ad(A1 ) · A2 ). n + 1 n=0

(8.5.11)

It follows from (8.5.9) that F (0) = 0. Then using the equality ead(F (t)) = et ad(A1 ) · et ad(A2 ) , which can be obtained from (8.5.3), in (8.5.11) and integrating (8.5.11) over t from 0 to 1, we deduce Campbell–Hausdorff identity in the form (3.1.10). Note that factor (1 − ead(F (t)) )n = (1 − et ad(A1 ) · et ad(A2 ) )n in the sum in (8.5.11) (or (3.1.10)) is at least of order tn , since

 1 − et ad(A1 ) · et ad(A2 ) ∞ m  tm  k Cm (ad(A1 ))m−k · (ad(A2 ))k =− m! m=1 k=0

 t2 = −t ad(A1 ) + ad(A2 ) − (ad(A1 ))2 + 2ad(A1 ) · ad(A2 ) 2  t3 (ad(A1 ))3 + 3(ad(A1 ))2 ad(A2 ) + (ad(A2 ))2 − 3!  + 3ad(A1 ) (ad(A2 ))2 + (ad(A2 ))3 + o(t3 ), (8.5.12) is proportional to t. The power k of parameter t on expansion on the righthand side of (8.5.11) in t corresponds to power k+1 in A1 and A2 . Therefore, if one wishes to calculate F up to terms of order k in A1 and A2 , one need to know F (t) up to terms of order tk−1 . We make use of (8.5.11) and (8.5.12) to find ∂t F (0) = (A1 + A2 ), ∂t2 F (0) = [A1 , A2 ], 1 ∂t3 F (0) = [A1 − A2 , [A1 , A2 ]], . . . , 2 which yields the expansion (3.1.13). 8.6.

Problem 3.1.4, Section 3.1.3

Prove that the following element of SL(2, C)   −1 φ , φ ∈ C, φ = 0, 0 −1

(8.6.1)

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cannot be written as a single exponential exp(A), where A ∈ s(2, C), but can be expressed as a product of exponential factors exp(A1 )·exp(A2 ), where     1 0 0 −φ , A2 = , A1 , A2 ∈ s(2, C). A1 = i π 0 −1 0 0 Is Campbell–Hausdorff series (3.1.10) convergent for these A1 and A2 ? Any element of algebra s(2, C) can be written as A = z i σi , where σi are Pauli matrices, zi are complex numbers. This matrix is either diagonalizable or, since Tr(A) = 0, has the form   0 u A=B B −1 , 0 0 where B ∈ SL(2, C). Therefore, matrix g = eA

(8.6.2)

is either diagonalizable or is given by   1 φ g=B B −1 . 0 1 In the latter case we have Tr(g) = 2. On the other hand, matrix (8.6.1) is not diagonalizable, and its trace is equal to (−2). This proves that matrix (8.6.1) cannot be equal to single exponential of element A ∈ s(2, C). This argument shows that for g ∈ SL(2, C), either g or −g has exponential form (8.6.2). Thus, any element SL(2, C) can be represented either as a single exponent exp(z i σi ), or product of exponents (−I2 ) · exp(z i σi ) = exp(i π σ3 ) · exp(z i σi ). As a side remark, this shows that the group manifold SL(2, C) is connected, since any element is connected to unity by either smooth curve exp(t z i σi ) or smooth curve exp(i π t σ3 ) · exp(t z i σi ), where t ∈ [0, 1]. Thus, exponential representation (8.6.2) covers group SL(2, C) almost completely, with the exception of a subset in SL(2, C) with elements   −1 φ (8.6.3) B B −1 , 0 −1

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where B is any element of SL(2, C) and φ is an arbitrary nonzero complex number. This subset makes 2-dimensional complex surface in SL(2, C) (see also [7]), since, in fact, it is parameterized by two complex numbers. Indeed, any element of SL(2, C) can be written as     φ−1/2 0 a c , ad − bc = 1. B= 0 φ1/2 b d We insert this into (8.6.3) and obtain the following representation for matrix (8.6.3):     2 −1 − ab a −1 φ B −1 = . B 0 −1 −b2 −1 + ab This shows that matrices (8.6.3) are parameterized by two complex parameters a and b. The issue of convergence of Campbell–Hausdorff series is left to the reader. 8.7.

Problem 3.4.3, Section 3.4.2

Show that (3.4.17) is indeed the most general solution to the Killing equation (3.4.12) in de Sitter metric (3.4.16). Show that transformations parameterized by cα , ω α and λ are spatial rotations and boosts in Minkowski space R(1,n) that involve coordinates xn and xn+1 , as well as xα . The only nonvanishing connection coefficients for metric (3.4.16) are (we set t ≡ y 0 )   2t α −1 δαβ , Γ0αβ = M −1 exp δαβ . Γα 0β = Γβ0 = M M These are used in the Killing equation (3.4.12), ∂k m + ∂m k − 2Γlkm l = 0.

(8.7.1)

This equation with k = m = 0 shows that 0 is independent of t, 0 = 0 ( y ). Then Eq. (8.7.1) with k = 0, m = α has the general solution   M 2t y) + (8.7.2) ξα ( ∂α 0 , α = exp M 2 where ξ α are arbitrary functions of y β . Finally, Eq. (8.7.1) with k = α, m = β reads   2 2t δαβ 0 + M exp − ∂α ∂β 0 = 0. ∂α ξβ + ∂β ξα − M M Since on the left-hand side the last term depends on time while others do not, we have ∂α ∂β 0 = 0 and the parameter 0 (y) is linear in y. Then the remaining equation is

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straightforwardly solved, and the general solution is 0 = M λ + 2ω α y α , ξα = −ω αβ y β − cα + λy α − M −1 (ωα  y 2 − 2ωβ y β y α ), where ω αβ = −ω βα , cα , ωα and λ are arbitrary real parameters. Together with (8.7.2) this gives (3.4.17). Parameterization (3.4.15) is explicitly invariant under SO(n − 1) rotations of coordinates xα (and y α ). Besides these SO(n − 1) rotations parameterized by ω αβ , there are remaining Lorentz transformations in (n + 1)-dimensional Minkowski space, under which the de Sitter hyperboloid (3.4.15) is also invariant. In other words, the following transformations must be (infinitesimal) isometries of de Sitter space: (1) rotations in planes (xα , xn ): xn → xn − aα xα , xα → xα + aα xn , xn+1 → xn+1 ; (2) boosts in planes (xα , xn+1 ): xn+1 → xn+1 + bα xα , xα → xα + bα xn+1 , xn → xn ; (3) boost in plane (xn , xn+1 ): xn+1 → xn+1 + d · xn , xn → xn + d · xn+1 , xα → xα . Here aα , bα and d are parameters of transformations. These transformations are conveniently organized by introducing, instead of xn and xn+1 , null coordinates x± = xn+1 ± xn . Linear combinations of transformations (1), (2) and (3) can be written as α

α

(1 ) = (1) + (2): x+ → x+ , x− → x− + 2 cM xα , xα → xα + cM x+ ; (2 ) = (2) − (1): x+ → x+ + 2ω α xα , x− → x− , xα → xα + ω α x− ; (3 ) = (3): x+ → (1 + λ)x+ , x− → (1 − λ)x− , xα → xα . In terms of the null coordinates, the parameterization (3.4.15) reads   t x+ = M exp , M     t y2  t x− = −M exp − + exp , M M M   t . xα = y α exp M It is now clear that under transformation (1 ), “time” t does not change, t → t, and that this is spatial translation y α → y α + cα . The transformation (3 ) is infinitesimal scaling t → t + M λ, y α → y α − λy α , which leaves xα intact. Finally, for the transformation (2 ) one finds from x+ → x+ + 2ω α xα that t → t + 2ω α y α , and then xα → xα + ω α x− yields    2t y 2 − 2( ω y )y α − M 2 ω α exp − y α → y α + M −1 ω α  . M This shows that isometries (3.4.17) are indeed Lorentz transformations in Minkowski space where de Sitter space is embedded.

8.8.

Problem 3.3.16, Section 3.3.1.2

Establish isomorphism of real algebras so(5) and usp(4).

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Let us construct convenient basis in usp(4). To this end, we introduce a special basis in space of complex 4 × 4 matrices. To describe the latter, we consider five 4 × 4 matrices Γ1 = σ1 ⊗ σ2 , Γ2 = σ2 ⊗ σ2 , Γ3 = σ3 ⊗ σ2 , Γ4 = I2 ⊗ σ1 , Γ5 = Γ1 Γ2 Γ3 Γ4 = I2 ⊗ σ3 ,

(8.8.1)

where σi are Pauli matrices given by (2.1.18), and ⊗ is direct product of matrices defined in (4.3.7), (4.3.8). Pauli matrices are Hermitian, so all Γk are Hermitian, Γ†k = Γk ,

k = 1, . . . , 5.

(8.8.2)

It follows from identities (3.3.10) that matrices {Γi } obey anticommutation relations Γk Γj + Γj Γk = 2 δkj I4 .

(8.8.3)

Operators obeying these anticommutation relations are called generators of Clifford algebra of 5-dimensional space, and their explicit matrix realization (8.8.1) is Hermitian representation of Clifford algebra. We now construct operators kj = 1 (Γk Γj − Γj Γk ), M 4

j, k = 1, . . . , 5.

(8.8.4)

General theory of Clifford algebra is studied in detail in accompanying book. It shows, in particular, that any complex 4 × 4 matrix A can be written as linear combinationa kj , A = z I4 + zi Γi + zkj M

(8.8.5)

where z, zk , zkj = −zjk (k, j = 1, . . . , 5) are 16 independent complex kj = −M jk makes basis parameters. So, the set of matrices I4 , Γi and M in complex space of 4 × 4 matrices. We note that the validity of expansion (8.8.5) can be verified directly, by making use of explicit forms (8.8.1), (8.8.4) and the fact that any complex 2 × 2 matrix a can be written fact, this property is well known in the theory of Dirac matrices γ ν in 4-dimensional Minkowski space-time, ν = 0, 1, 2, 3. The correspondence between matrices Γj and Dirac matrices is Γ4 = γ0 , Γα = iγ α , α = 1, 2, 3, Γ5 = γ 5 , where matrix γ 5 is proportional to the product of four γ ν ’s. Standard anticommutation relations for Dirac matrices are equivalent to (8.8.3). The standard theory of Dirac matrices ensures that any 4×4 matrix is linear combination of unit matrix, five Dirac matrices, six matrices σμν = [γ μ , γ ν ] and four matrices [γ μ , γ 5 ]. This is precisely the expansion (8.8.5). a In

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as a = wI2 + wi σi (w, wi ∈ C), while any complex 4 × 4 matrix has representation I2 ⊗ a + σi ⊗ ai , where a, ai ∈ Mat2 (C). In view of Eq. (8.8.3), matrices (8.8.4) obey commutation relations il + δjl M ki + δik M lj + δil M jk . ij , M kl ] = δjk M [M

(8.8.6)

These coincide with commutation relations (3.3.32) of generators of Lie jk are antialgebra so(5). Note that in accordance with (8.8.2), matrices M Hermitian, kj , † = −M M kj

j, k = 1, . . . , 5.

(8.8.7)

On the other hand, explicit forms of Pauli matrices (2.1.18) and matrices Γi given by (8.8.1) lead to the relations T T T T ΓT 1 = −Γ1 , Γ2 = Γ2 , Γ3 = −Γ3 , Γ4 = Γ4 , Γ5 = Γ5 .

(8.8.8)

These can be written as ΓT k C = C Γk , where

(8.8.9)

 C = Γ3 Γ1 = i σ2 ⊗ I2 =

0

I2

−I2

0

 (8.8.10)

is real antisymmetric matrix entering symplectic condition (3.1.26) with r = 2. Making use of (8.8.9), we find that matrices (8.8.4) are not only anti-Hermitian, Eq. (8.8.7), but also symplectic, Eq. (3.1.26): kj . T C = −C M M kj

(8.8.11)

Algebra usp(4) is the space of anti-Hermitian and symplectic matrices A ∈ Mat4 (C), A† = −A, AT C = −CA.

(8.8.12)

Let us insert expansion (8.8.5) into (8.8.12). It follows from (8.8.2), (8.8.7), ∗ (8.8.9) and (8.8.11) that z = 0, zk = 0 and zkj = zkj , i.e., zkj ∈ R. Thus, matrix A belongs to usp(4), iff kj , A = ωkj M

ωkj = −ωjk , ωjk ∈ R.

kj make basis in real Lie algebra usp(4). Finally, In other words, matrices M kj kj ↔ Mkj of the set of matrices M there is one-to-one correspondence M given by (8.8.4) and Mkj ∈ so(5) in (3.3.31): these matrices obey the same

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kj make basis defining commutation relations (8.8.6) and (3.3.32). Thus, M both in usp(4) and so(5), so that there is isomorphism usp(4) = so(5). Remark 1. This proof of isomorphism usp(4) = so(5) is valid for any other 4-dimensional representationb (different from (8.8.1)) of 5-dimensional Clifford algebra (8.8.3), provided that all Γi are Hermitian. In any of these representations, the relation between ΓT i and Γi is given by (8.8.9). The matrix C entering (8.8.9) and symplectic condition may, however, have nonstandard forms. Remark 2. Real Lie algebra usp(4) is sometimes denoted by spin(5) in physics literature, and the pertinent Lie group USp(4) by Spin(5). Group USp(4) = Spin(5) covers group SO(5) twice (groups Spin(n) are defined in accompanying book). 8.9.

Problem 3.3.17, Section 3.3.1.2

Establish isomorphisms so(4, 1) = sp(1, 1) and so(3, 2) = sp(4, R). Let us proceed in analogy to the previous section. Consider five 4 × 4 matrices γ1 = i Γ 1 ,

γ2 = i Γ 2 , γ3 = i Γ 3 , γ4 = i Γ 4 , γ5 = Γ 5 ,

(8.9.1)

where matrices Γi are defined in (8.8.1). Making use of (8.8.2), (8.8.3) and (8.8.9), we obtain γj† Γ5 = Γ5 γj , γj γk + γk γj = 2 gkj , γjT C = C γj ,

j, k = 1, . . . , 5, (8.9.2)

where matrix C is given by (8.8.10), and ||gij || = diag(−1, −1, −1, −1, 1) = −I4,1 ,     σ3 0 I1,1 0 = . Γ5 = 0 σ3 0 I1,1

(8.9.3)

We introduce matrices  kj = 1 (γk γj − γj γk ), L 4 b We

j, k = 1, . . . , 5,

(8.9.4)

mean here both representations obtained from (8.8.1) by equivalence transformations and representations inequivalent to (8.8.1). Note that there are only two inequivalent 4-dimensional representations of 5-dimensional Clifford algebra (8.8.3), which differ by the sign of matrix Γ5 .

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which obey relations similar to (8.8.6), (8.8.7) and (8.8.11):  il + gjl L  ki + gik L  lj + gil L  jk ,  ij , L  kl ] = gjk L [L † L kj

 kj , Γ5 = −Γ5 L

T L kj

 kj . C = −C L

(8.9.5) (8.9.6)

Commutation relations (8.9.5) coincide with the defining relations (3.3.41) for generators Ljk of algebra so(4, 1), given in (3.3.39).  kj , Any complex 4 × 4 matrix A can be written as = z I4 + zi γi + zkj L where z, zk and zkj = −zjk are 16 independent complex parameters, cf. (8.8.5). We use (8.9.6) and check that this matrix obeys the constraints (3.1.32) which single out sp(1, 1), only if z = zj = 0 and zkj ∈ R. Thus, any  kj , so L  kj are basis elements in element A ∈ sp(1, 1) has the form A = zkj L real Lie algebra sp(1, 1). The isomorphism so(4, 1) = sp(1, 1) is then merely  kj ↔ Lkj , where Lkj are basis elements in algebra so(4, 1). L Isomorphism so(3, 2) = sp(4, R) is established in a similar way. Consider five 4 × 4 matrices (cf. (8.9.1)) γ1 = i Γ 1 , γ2 = Γ 2 , γ3 = i Γ 3 , γ4 = Γ 4 , γ5 = Γ 5 .

(8.9.7)

This time all matrices γj are real, γj∗ = γj (see the definition (8.8.1) of matrices Γj ), and we have γj γk + γk γj = 2 gij , γjT C = C γj ,

(8.9.8)

where ||gij || = diag(−1, 1, −1, 1, 1) in this case. We again define matrices  kj by (8.9.4), but now with γ-matrices given by (8.9.7). It is straightforL ward to see that their commutation relations (8.9.5) coincide with defining relations (3.3.41) for generators Ljk of so(2, 3), which are given in (3.3.39). Any real 4 × 4 matrix A obeying symplectic condition AT C = −CA has the  kj , where ωkj ∈ R. Thus, real matrices L  kj defined in (8.9.4) form A = ωkj L in terms of γ-matrices (8.9.7), make basis in sp(4, R), and the isomorphism  kj ↔ Lkj of basis elements of Lie algebras sp(4, R) in question is mapping L and so(3, 2). 8.10.

Problem 3.3.18, Section 3.3.1.2

Establish isomorphism of complex Lie algebras so(6, C) = s(4, C) and isomorphisms of their real forms (1) so(6) = su(4), (2) so(3, 3) = s(4, R), (3) so(2, 4) = su(2, 2). Show that (4) so(5, 1) = s(2, H), where H is quaternion field. Show that (5) s(2, H) = su∗ (4).

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1. To simplify calculations, we introduce the following auxiliary construction. Consider six 8 × 8 matrices ˜ k = σ2 ⊗ γk (k = 1, 2, 3, 4, 5), Γ ˜ 6 = ε1/2 σ1 ⊗ I4 , Γ 6

(8.10.1)

where 4 × 4 matrices γk (k = 1, . . . , 5) obey γk γj + γj γk = 2 gkj I4 , ||gkj || = diag(ε1 , ε2 , . . . , ε5 ) ,

ε2k = 1,

(8.10.2)

parameter ε6 in (8.10.1) is such that ε26 = 1, and σ1 , σ2 are standard 2 × 2 Pauli matrices. Matrices γk (k = 1, . . . , 5) can be straightforwardly constructed in terms of 4 × 4 matrices (8.8.1); examples are given in Section 8.9. Note that matrices γk with the property (8.10.2) are traceless, Tr(γk ) = 0. Indeed, for k = j we have γk = −γj γk γj−1 and therefore ˜ A (A = 1, . . . , 6) obey Tr(γk ) = −Tr(γk ). Due to (8.10.2), matrices Γ ˜A Γ ˜B + Γ ˜ A = 2 gAB I8 , ˜B Γ Γ ||gAB || = diag(ε1 , . . . , ε5 , ε6 ),

ε2A = 1.

(8.10.3)

Making use of matrices (8.10.1), we now construct operators MAB =

1 ˜ ˜ ˜ A ), ˜B Γ (ΓA ΓB − Γ 4

A, B = 1, . . . , 6,

(8.10.4)

whose commutation relations, in view of (8.10.3), are [MAB , MCD ] = gBC MAD + gBD MCA + gAC MDB + gAD MBC . (8.10.5) They coincide with commutation relations (3.3.32) for generators of Lie algebra so(p, q), where p + q = 6; p and q are the numbers of parameters εA equal to +1 and −1, respectively. We recall the explicit forms of Pauli matrices σ1 and σ2 and cast matrices MAB in block-diagonal form,  +  0 MAB , (8.10.6) MAB = − 0 MAB ± are (k, m = 1, . . . , 5), where 4 × 4 blocks MAB ± = Mkm

1 1/2 i [γk , γm ] , Mk±6 = ∓ε6 γk . 4 2

(8.10.7)

+ − These matrices are traceless. Matrices MAB (and also MAB ) obey commutation relations (8.10.5) and, therefore, make basis in algebra so(p, q). The − + } is related to the set {MAB } by automorphism in so(p, q): second set {MAB + − + − Mkm → Mkm , Mk6 → −Mk6 , so we do not need to consider it.

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Note that the auxiliary construction with 8 × 8 matrices (8.10.1) is, strictly speaking, unnecessary. We could have started directly with the + . The advantage of the approach definition (8.10.7) for 4 × 4 matrices MAB with 8 × 8 matrices is that objects with indices A = 1, . . . , 6 are treated on equal footing, whereas index A = 6 is special in Eq. (8.10.7). Together with unity, operators (8.10.7) make basis in 5-dimensional + given by (8.10.7) Clifford algebra (8.10.2), and fifteen 4 × 4 matrices MAB make basis in the space of all traceless 4×4 matrices. So, the set of matrices of the form + , A = ω AB MAB

ω AB = −ω BA ,

ω AB ∈ R,

(8.10.8)

on the one hand, is algebra so(p, q), and, on the other, is a real form of complex algebra s(4, C). This fact can be used for establishing isomorphisms of algebras so(p, q) with p + q = 6 and real forms of s(4, C). Complexification of these isomorphisms boils down to choosing complex parameters ω AB in (8.10.8); it immediately establishes isomorphism of complex algebras so(6, C) = s(4, C). It remains to be understood which of real forms of s(4, C) is isomorphic to so(p, q) for concrete values p = 0, 1, 2, 3 and q = 6 − p. 2. Let us choose Hermitian γ-matrices (8.10.2), γ1 = Γ 1 , γ2 = Γ 2 , γ3 = Γ 3 ,

γ4 = Γ 4 , γ5 = Γ 5 ,

(8.10.9)

where matrices Γj are defined in (8.8.1). We set ε6 = 1 in (8.10.1), then all ˜† = Γ ˜ A , while matrices M + are ˜ A in (8.10.1) are Hermitian, Γ matrices Γ AB A anti-Hermitian: + † + ) = −MAB . (MAB

(8.10.10)

In accordance to this choice, metric gAB in (8.10.3) and (8.10.5) reads gAB = diag(1, 1, 1, 1, 1, 1). Thus, space of matrices (8.10.8) is Lie algebra su(4) and Lie algebra so(6) at the same time, and hence su(4) = so(6). 3. We now choose pure imaginary γ-matrices (8.10.2), cf. (8.9.7): γ1 = Γ 1 , γ2 = i Γ 2 , γ3 = Γ 3 ,

γ4 = i Γ 4 , γ5 = i Γ 5 ,

(8.10.11)

˜ j , and also Γ ˜ 6 , are real, and set ε6 = 1 in (8.10.1). Then all matrices Γ + ∗ ˜ = Γ ˜ A , so matrices M Γ A AB are real too, while metric gAB in (8.10.3), (8.10.5) reads gAB = diag(1, −1, 1, −1, −1, 1). So, space of matrices (8.10.8)

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is at the same time Lie algebra s(4, R) and Lie algebra so(3, 3), and hence these real algebras are isomorphic. 4. Let us choose the five γ-matrices as follows: γ1 = Γ 1 , γ2 = Γ 2 , γ3 = Γ 3 ,

(8.10.12)

γ4 = Γ 4 , γ5 = i Γ 5 ,

˜ A given by (8.10.1) obey and set (ε6 )1/2 = i in (8.10.1). Then matrices Γ Eq. (8.10.3) with metric gAB = diag(1, 1, 1, 1, −1, −1).

(8.10.13)

˜† = Γ ˜ A (A = 1, . . . , 4), Γ ˜ † = −Γ ˜ 5, Γ ˜† = They also satisfy the relations Γ 5 6 A ˜ 6 , which can be written as −Γ ˜† = C · Γ ˜ A · C −1 , Γ A

˜ 5 = σ3 ⊗ Γ5 = ˜6 Γ C = iΓ

 Γ5 0

0



−Γ5

.

(8.10.14)

This gives † + † + MAB = −C · MAB · C −1 ⇒ (Mkm ) · Γ5 + Γ5 · Mkm = 0.

(8.10.15)

In this case matrices (8.10.8) make, at the same time, algebra so(4, 2) with commutation relations (8.10.5) and algebra su(2, 2) (matrix Γ5 in (8.10.15) has the form (8.9.3), which is precisely what is needed for pseudo-unitary Lie algebra su(2, 2)). This establishes isomorphism su(2, 2) = so(4, 2). Note that so(4, 2) is conformal algebra of Minkowski space R1,3 , and hence su(2, 2) = conf(R1,3 ). 5. Finally, we make a choice of the five γ-matrices most often used in physics, γ1 = σ2 ⊗ σ1 , γ2 = σ2 ⊗ σ2 , γ3 = σ2 ⊗ σ3 , γ4 = σ1 ⊗ I2 , γ5 = γ1 γ2 γ3 γ4 = σ3 ⊗ I2 ,

(8.10.16)

˜ A obey (8.10.3) and choose (ε6 )1/2 = i in (8.10.1). Then the six matrices Γ with metric gAB = diag(1, 1, 1, 1, 1, −1),

(8.10.17)

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+ and, therefore, matrices MAB make Lie algebra so(5, 1). Let us write all + 4 × 4 matrices MAB defined in (8.10.7) in explicit 2 × 2 block form:     0 1 1 0 sμ 1 1 I2 + + Mμ6 = γμ = , , M56 = γ5 = 2 2 sμ 0 2 2 0 −I2     0 −s 1 1 1 μ s 0 μν + + Mμν = [γμ , γν ] = = [γμ , γ5 ] = , , Mμ5 0 sμν 4 4 2 sμ 0

(8.10.18) where μ, ν = 1, . . . , 4, and we use the notations sμ = (−iσ1 , −iσ2 , −iσ3 , I2 ), sμ = (iσ1 , iσ2 , iσ3 , I2 ), 1 (sμ sν − sν sμ ) = 4 1 = (sμ sν − sν sμ ) = 4

sμν = sμν

i j η σj , 2 μν i j η σj , 2 μν

j = 1, 2, 3.

i ’t Hooft symbols ημν and η iμν are defined in (3.3.70). Making use of (8.10.18), any matrix A of the form (8.10.8) can be written as     a b aμ sμ , bμ sμ + AB = , (8.10.19) A = ω MAB = c d cμ sμ , dμ sμ

where d4 = −a4 , and 15 real parameters aμ , bμ , cμ , dj (j = 1, 2, 3) can be expressed in terms of 15 parameters ω AB ∈ R. Matrices a, b, c and d are interpreted as elements of real associative 4-dimensional quaternion algebra H whose basis is e = I2 , i = iσ1 , j = iσ2 , k = iσ3 . With this interpretation, one observes that the space of 2 × 2 matrices with quaternionic elements is Lie algebra, which is naturally denoted by g(2, H). The space of special quaternionic matrices A of the form (8.10.19) with constraint d4 = −a4 , i.e., Tr(A) = 0, is also Lie algebra denoted by s(2, H). Thus, formula (8.10.19), together with our observation after Eq. (8.10.17), establishes isomorphism so(5, 1) = s(2, H). The derivation of isomorphism s(2, H) = su∗ (4) is left to the reader. 8.11.

Problem 4.7.8, Section 4.7.2

Derive the value of the Casimir operator C2 in adjoint representation of su(n) by employing the fact that tensor product of defining representation [n] and antifundamental representation [¯ n] is partitioned into

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direct sum of (complexified) adjoint representation [n2 − 1] and singlet representation [1]. One makes use of the following identity valid for representation T (p) = T (1) ⊗ T (2) of Lie algebra (see (4.3.12)): (p)

Ta(p) · Tb

(1)

= Ta(1) Tb (1)

+ Tb (1)

(2)

⊗ I(2) + Ta(1) ⊗ Tb

(2)

⊗ Ta(2) + I(1) ⊗ Ta(2) Tb ,

(8.11.1)

(2)

where Tb and Tb denote the generator Xb in representations T (1) and n], then it follows from T (2) , respectively. We choose T (1) = [n] and T (2) = [¯ (1) (2) the definition of algebra su(n) that Tr(1) (Ta ) = 0, Tr(2) (Ta ) = 0, where Tr(1) and Tr(2) are traces in the pertinent spaces of representations. We take trace Tr1⊗2 = Tr(1) Tr(2) of identity (8.11.1) and get (p)

(1)

Tr1⊗2 (Ta(p) · Tb ) = Tr(1) (Ta(1) Tb ) Tr(2) I(2) (2)

+ Tr(1) I(1) Tr(2) (Ta(2) Tb ) = n δab , (8.11.2) where we recall that Tr(2) I(2) = Tr(1) I(1) = n and that the normalization of generators is (1)

(2)

Tr(1) (Ta(1) Tb ) = Tr(2) (Ta(2) Tb ) =

1 δab . 2

(8.11.3)

Since the representation T (p) = [n] ⊗ [¯ n] is isomorphic to direct sum [1] of T (ad) = [n2 − 1] and T [1] = [1], and operators Ta vanish (trivial (ad) (ad) representation), the left-hand side of (8.11.2) equals Tr1⊗2 (Ta · Tb ). (ad) 2 We recall that dim(T ) = (n − 1), set b = a an sum over a in (8.11.2). (ad) (ad) The result is Tr1⊗2 (Ta Ta ) = n(n2 − 1). On the other hand, we take trace of (4.7.39) written in adjoint representation and use Tr I(ad) = (n2 −1). Comparing the two results, we get for adjoint representation of su(n) that (ad) C2 = 1. 8.12.

Problem 4.7.9, Section 4.7.2

Making use of defining relations (3.3.41) and (3.3.48), find structure rs constants Cij, k of Lie algebras so(p, q) and sp(2r, K) and calculate their Killing metrics. Show that quadratic Casimir operators (4.7.12) for so(p, q)

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and sp(2r, K) read (so)

C2

=−

1 Lab Lab , 4(p + q − 2) 

(sp)

C2



=−

1 Mab M ab , 4(2r + 2) (8.12.1)





where Lab = η aa η bb La b and M ab = J aa J bb Ma b . Show that their values in defining (fundamental) representations of so(p, q) and sp(2r, K) are [p+q]

C2

=

(p + q − 1) , 2(p + q − 2)

[2r]

C2

=

(2r + 1) . 4(r + 1)

Let us write defining commutation relations (3.3.41) and (3.3.48) in uniform way: [Fij , Fk ] = cjk Fi + cj Fki + cik Fj + ci Fjk ⇔

rs [Fij , Fk ] = Cij, k Frs ,

(8.12.2)

−1 where cik = ηik , Fij = Lij for so(p, q) and cik = Jik , Fij = Mij for sp(2r, K). Accordingly, the symmetry properties are cik =  cki and Fik = − Fki , where  = +1 for so(p, q) and  = −1 for sp(2r, K). By comparing the two relations in (8.12.2) we get rs Cij, k =

1 {r s} {r s} {r s} {r s} (cjk δi δ + cj δk δi + cik δ δj + ci δj δk ), 2

(8.12.3)

{r s}

where δi δ = (δir δs −δis δr ) are combinations of δ-symbols, -symmetrized in indices r and s. It is straightforward to see that structure constants (8.12.3) automatically have right symmetry under permutations of indices i ↔ j and k ↔ : rs rs rs Cij, k = − Cji, k = − Cij, k .

We insert the structure constants (8.12.3) into definition (3.2.62) of Killing metric. Tedious but straightforward calculation gives (cf. (4.7.132)) rs k gij,i j  = Cij, k Ci j  , rs = 2 (n − 2 ) (cji cj  i −  cjj  ci i ) ,

(8.12.4)

where n = p + q for so(p, q) and n = 2r for sp(2r, K). One takes a note of symmetry properties of the killing metric gij,i j  and defines its inverse by

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the relation gij,i j  gi j  

gi j

,k

,k

=

429

= 12 (δik δj −  δjk δi ). Then Eq. (8.12.4) gives     1 ( ¯ci  ¯cj k − ¯ci k ¯cj  ), 8 (n − 2 )

(8.12.5)

where matrix ||¯ cij || is inverse of ||cij ||. We insert metric (8.12.5) into the definition (4.7.12) and find the quadratic Casimir operators for so(p, q) and sp(2r, K):  

C2 = gi j

,k

Fi j  Fk = −

1 Fk F k , 4 (n − 2 )

(8.12.6)

which is equivalent to (8.12.1). Using defining representations (3.3.39) and (3.3.47) of algebras so(p, q) and sp(2r, K) in (8.12.1) and recalling (4.7.37), we obtain the following values of the quadratic Casimir operator [n]

C2 =

(n − ) , 2(n − 2)

where n = p + q,  = +1 for so(p, q), and n = 2r,  = −1 for sp(2r, K). 8.13.

Problem 4.7.14, Section 4.7.2

Making use of isomorphism conf(Rp,q ) = so(p + 1, q + 1) (see Proposition 3.4.2), show that quadratic Casimir operator of conformal algebra conf(Rp,q ) with defining relations (3.4.6) is proportional to operator C2 = Lij Lij + P i Ki + K i Pi − 2 D2 .

(8.13.1)

Show that in representation T defined in (3.4.7), operator (4.7.50) reads [T ] C2 = 2Δ(n − Δ) + Σij Σij , where n = p + q. In accordance with formula (8.12.1), quadratic Casimir operator C2so for algebra so(p + 1, q + 1) is proportional to C2 = Lab Lab . We take into account the signature of metric and explicit correspondence (3.4.42) between the generators of algebras conf(Rp,q ) and so(p + 1, q + 1) to obtain C2 = Lab Lab = −2(L0 n+1 )2 + Lij Lij + (L0 i − Ln+1 i )(L0 i + Ln+1 i ) + (L0 i + Ln+1 i )(L0 i − Ln+1 i ) = Lij Lij + P i Ki + K i Pi − 2 D2 , which is precisely (8.13.1). We then insert generators (3.4.7) of conformal algebra into (8.13.1), with the generator Km written as Km = −(x)2 ∂m − 2xm Δ − 2xk Lmk ,

(x)2 ≡ xk xk ,

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and use the identities Lmk Lmk = 2 (xm ∂ k Lmk + Lmk xm ∂ k ) − (xm ∂k − xk ∂m )(xm ∂ k − xk ∂ m ) + Σkm Σkm , × (xm ∂k − xk ∂m )(xm ∂ k − xk ∂ m ) = 2(x)2 (∂)2 − 2(x∂)2 + 2(2 − n)(x∂), where (∂)2 = ∂k ∂ k , (x∂) = xk ∂ k and n = (p + q). As a result we obtain C2 = 2 (xm ∂ k Lmk + Lmk xm ∂ k ) − (xm ∂k − xk ∂m )(xm ∂ k − xk ∂ m ) + Σkm Σkm + ∂ m ((x)2 ∂m + 2xm Δ + 2xk Lmk ) + ((x)2 ∂m + 2xm Δ + 2xk Lmk )∂ m − 2((x ∂) + Δ)2 = −(2(x)2 (∂)2 − 2(x∂)2 + 2(2 − n)(x∂)) + Σkm Σkm + 2(1 − n)(x∂) + (2(x∂) + 2nΔ) + 2 ((x)2 ∂m + 2xm Δ)∂ m − 2((x ∂) + Δ)2 = Σkm Σkm + 2Δ(n − Δ), as desired. 8.14.

Problem 4.7.26, Section 4.7.4

Let elements Lij (u) be generating functions (4.7.143) for generators (E (α) )ij of Yangian Y (g(n)), i.e., elements Lij (u) obey Eqs. (4.7.136), (4.7.140) with R-matrix given by (4.7.138). Show that the operator qdet(L(u)) defined by (4.7.147) commutes with all generators of Yangian Y (g(n)): [qdet(L(u)), Lij (v)] = 0, ∀u, v, i, j, and hence belongs to the center of algebra Y (g(n)). Before embarking on the solution to the problem itself, we make a few technical remarks. Besides the standard R-matrix R(u) given by (4.7.138), it is convenient to introduce R-matrix ˇ R(u) = P R(u) = I + u P

⇒

ˇ R(−u) ˇ R(u) = (1 − u2 ) I,

where P is transposition matrix, and I is unit matrix. Let us denote V = ˇ k in V ⊗M : operators R ⊗(k−1) ⊗(M −k−1) ˇ ˇ k (u) = In ⊗ R(u) ⊗ In = I + u Pk , R ⊗(k−1) is identity operator, and Pk = In ⊗P (k+1)th factors in the product V ⊗M . Operators

⊗M In

Here I = of kth and

ˇ i (v) = R ˇ i (v) R ˇ k (u) ˇ k (u) R R

∀u, v ,

k < M.

(8.14.1) Cn

and define (8.14.2)

⊗(M −k−1) ⊗ In

is transposition (8.14.2) obey locality relations ∀i = k ± 1,

and Yang–Baxter equation in the form of braid group relations (cf. (1.1.34), (4.7.139)) ˇ k+1 (u + v) R ˇ k (v) = R ˇ k+1 (v) R ˇ k (u + v) R ˇ k+1 (u). ˇ k (u) R R

(8.14.3)

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The second equality in (8.14.1) gives 1 ˇ k (−u) ⇔ ˇ −1 (u) = R R k (1 − u2 )

R−1 k,k+1 (u) =

431

1 Rk,k+1 (−u). (1 − u2 )

(8.14.4)

The second relation here is written for the standard R-matrix (4.7.138) (cf. (8.14.2)) ⊗(k−1)

Rk,k+1 (u) = In

⊗(M −k−1)

⊗ R(u) ⊗ In

= u I + Pk ,

k < M,

(8.14.5) V ⊗M .

which acts in a nontrivial way only on kth and (k + 1)th factors in the product We need in what follows also the standard R-matrices (acting nontrivially only on kth and th factors in V ⊗M ) given by Rk, (u) = u I + Pk, , where Pk, is transposition matrix of kth and th factors in V ⊗M . ˇ k (u) are not invertible at points u = +1 and In accordance with (8.14.4), operators R u = −1, where they are proportional to symmetrizer and antisymmetrizer, respectively, of vectors in the product of kth and (k + 1)th factors in V ⊗M : ˇ k (+1) = I + Pk , R ˇ k (−1) = I − Pk ⇒ R ˇ k (+1)R ˇk (−1) = 0. R We need also the following operators acting in V ⊗M : ⊗(k−1)

Lk (u) = In

⊗(M −k)

⊗ ||Lij (u)|| ⊗ In

,

(8.14.6)

||Lij (u)||

where is n×n matrix of generating functions for generators of Yangian Y (g(n)). In view of (4.7.140), operators (8.14.6) obey (cf. (4.7.136)) ˇ k (u − v). ˇ k (u − v) Lk (u) Lk+1 (v) = Lk (v) Lk+1 (u) R (8.14.7) R Let Ar be antisymmetrizer in the space of rank r tensors V ⊗r . It acts on components ψi1 ...ir of tensor ψ ∈ V ⊗r in the following way (compare with symmetrizer (4.7.17)): 1  (−1)P (σ) ψiσ(1) ...iσ(r) , (8.14.8) ψi1 ...ir → [Ar · · · ψ]i1 ...ir = r! σ∈S r

where P (σ) is parity of permutation σ ∈ Sr . It follows from the definition (8.14.8) that for all r > n we have Ar = 0, while for r = n the total antisymmetrizer An has rank 1, i.e., it projects the whole space V ⊗n onto 1-dimensional subspace. Indeed, we compare (8.14.8) and (1.2.6) and see that 1 i1 ...in n εj1 ...jn . (8.14.9) ε (An )i1 ...i j1 ...jn = n! Antisymmetrizer Ar is explicitly constructed in terms of matrices (8.14.2) by making use of induction procedure ˇ r−1 (−r + 1) · · · R ˇ 2 (−2)R ˇ1 (−1) 1 A1 = I, Ar = Ar−1 · R r! 1 = Ar−1 · ((−1)r−1 Pr−1 · · · P2 P1 + · · · + Pr−1 Pr−2 − Pr−1 + I) , r (8.14.10) ⊗r where Ar−1 is antisymmetrizer of the first (r−1) factors in V , and I is unit operator in V ⊗r (we write 1 instead of I in what follows). The fact that Ar is indeed antisymmetrizer, and also that the second line in (8.14.10) is correct, follows from the identities ˇ k−1 (+1) ˇ k−1 (+1) · Ar = 0 = Ar · R R ⇔ Pk−1 · Ar = Ar · Pk−1 = −Ar , Ar · Ak = Ak · Ar = Ar ,

∀k ≤ r,

(8.14.11) (8.14.12)

where Ak is antisymmetrizer of the first k factors in V ⊗r (operator Ak acts trivially on the last (r − k) factors in V ⊗r ). We leave the proof of the identities (8.14.11) and (8.14.12) to the reader.

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Let us introduce the notations ˇ r−1 (−r + 1) · · · R ˇ 2 (−2)R ˇ1 (−1), ˇ (r−1→1) = R R ˇ (1→r−1) = R ˇ 1 (−1)R ˇ2 (−2) · · · R ˇ r−1 (−r + 1). R Then explicit form of antisymmetrizers (8.14.10) is Ar = =

1 ˇ ˇ (1→2) · R ˇ 1 (−1) R(1→r−1) · · · R ar 1 ˇ ˇ (2→1) · · · R ˇ (r−1→1) , R1 (−1) · R ar

where ar = (r! · · · 2! · 1!). Relation (8.14.7) gives   ˇ (r−1→1) L1 (u) L2 (u + 1) · · · Lr (u + r − 1) R   ˇ (r−1→1) , = L1 (u + 1) · · · Lr−1 (u + r − 1) Lr (u) R

(8.14.13)

(8.14.14)

which, together with the second formula in (8.14.13), yields an identity An L1 (u) L2 (u + 1) · · · Ln (u + n − 1) = L1 (u + n − 1) · · · Ln−1 (u + 1) Ln (u) An .

(8.14.15)

We act on (8.14.15) by operator An on the left (or on the right) and make use of (8.14.9) and (8.14.12) to obtain An L1 (u) · · · Ln (u + n − 1) = An L1 (u) · · · Ln (u + n − 1) An = L1 (u + n − 1) · · · Ln (u) An = An L1 (u + n − 1) · · · Ln (u) An

(8.14.16)

= qdet(L(u)) An , where we denote qdet(L(u)) ≡

1 εi ...i Li1 (u) · · · Lijnn (u + n − 1) εj1 ...jn n! 1 n j1

1 εi ...i Li1 (u + n − 1) · · · Lijnn (u) εj1 ...jn n! 1 n j1   = Tr1,...,n An L1 (u) · · · Ln (u + n − 1)   = Tr1,...,n An L1 (u + n − 1) · · · Ln (u) , =

(8.14.17)

and Tr1,...,n = Tr1 · · · Trn is the total trace of an operator acting in V ⊗n (here Trk is trace of an operator acting in the last, kth factor in V ⊗k , i.e., Trk (A1 ⊗· · ·⊗Ak−1 ⊗Ak ) = (A1 ⊗ · · · ⊗ Ak−1 )Tr(Ak )). We need yet another identity −1 −1 An · R−1 n,n+1 (u + n − 1) · · · R2,n+1 (u + 1)R1,n+1 (u) −1 = R−1 n,n+1 (u) · · · R1,n+1 (u + n − 1) · An ,

(8.14.18)

which is understood as equality of operators acting in V ⊗(n+1) , and which follows from identity ˇ (r−1→1) · R−1 (u + r − 1) · · · R−1 (u + 1) R−1 (u) R r,r+1 2,r+1 1,r+1 −1 −1 ˇ = R−1 r,r+1 (u) Rr−1,r+1 (u + r − 1) · · · R1,r+1 (u + 1) · R(r−1→1) .

(8.14.19)

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The latter is derived by multiple application of Yang–Baxter relations (4.7.139), (8.14.3), written as ˇ k (v) = R ˇ k (v) Rk,k+2 (u + v) Rk+1,k+2 (u). Rk,k+2 (u) Rk+1,k+2 (u + v) R

(8.14.20)

We compare (8.14.20) with (8.14.7) and find that the standard Yang’s R-matrices 1 Rik (4.7.138) realize representation T of elements Lij (u): T k (Lij (u)) = u j (u). From this viewpoint, identities (8.14.19) and (8.14.18) are representations of equalities (8.14.14) and (8.14.15). We also need representation T of the relations (8.14.16), which can be cast in the following form: An · R1,n+1 (u) · · · Rn,n+1 (u + n − 1) = Mn+1 (u) · An −1 −1 ⇒ An · R−1 n,n+1 (u + n − 1) · · · R1,n+1 (u) = Mn+1 (u) · An ,

(8.14.21)

where M k (u) = u(u + 1) · · · (u + n − 1) · T k (qdet(L(u)),

(8.14.22)

and (see (8.14.17)) Mn+1 (u) =

1 ε1→n R1,n+1 (u) · · · Rn,n+1 (u + n − 1) ε1→n . n!

(8.14.23)

We use here shorthand matrix notations ε1→n and ε1→n for components εi1 i2 ...in and εj1 j2 ...jn of totally antisymmetric ε-tensor. Now we make use of above notations and formulas to show that the operator qdet(L(u)), given by (8.14.17), commutes with all elements Lij (v) for any value of parameter v. To this end, we consider the chain of equalities (we use (4.7.136) in its beginning)   qdet(L(u)) Ln+1 (v) = Tr1,...,n An L1 (u) · · · Ln (u + n − 1) Ln+1 (v)  −1 = Tr1,...,n An R−1 n,n+1 (u − v + n − 1) · · · R1,n+1 (u − v) × Ln+1 (v)L1 (u) · · · Ln (u + n − 1)

 · R1,n+1 (u − v) · · · Rn,n+1 (u − v + n − 1)  −1 = Tr1,...,n Mn+1 (u − v) Ln+1 (v) An L1 (u) · · · Ln (u + n − 1) ·  · R1,n+1 (u − v) · · · Rn,n+1 (u − v + n − 1) −1 (u − v) Ln+1 (v)Tr1,...,n = Mn+1

× An R1,n+1 (u − v) · · · Rn,n+1 (u − v + n − 1) qdet(L(u))

  = M −1 (u − v)L(v)M (u − v) n+1 · qdet(L(u)).

(8.14.24)

Here we use (8.14.21) for expressions underlined once, and (8.14.16) for expression underlined twice. Let us calculate the matrix Mn+1 (u). We make use of the identities

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Pk,n+1 P,n+1 = P,k Pk,n+1 , ε1→n P,k = −ε1→n ∀, k ≤ n and obtain ε1→n R1,n+1 (u) · · · Rn,n+1 (u + n − 1)      = ε1→n u + P1,n+1 (u + 1) + P2,n+1 · · · (u + n − 1) + Pn,n+1 = ε1→n P1,n+1 u(u + 1) · · · (u + n − 2) + u P2,n+1 (u + 1) · · · (u + n − 2)

+ · · · + u(u + 1) · · · (u + n − 2) Pn,n+1 + u(u + 1) · · · (u + n − 1) = u(u + 1) · · · (u + n − 1) · ε1→n 1 +

 

1 P1,n+1 + · · · + Pn,n+1 . u−n+1

We insert this in (8.14.23) and use the relation ε1→n Pk,n+1 ε1→n = (n − 1)! I(n+1) (here I(n+1) is unit matrix in the space number (n + 1)) to obtain 1 ε1→n R1,n+1 (u) · · · Rn,n+1 (u + n − 1) ε1→n n!  1 = n (n − 1)! u(u + 1) · · · (u + n − 2) + n!u(u + 1) · · · (u + n − 1) I(n+1) n!

Mn+1 (u) =

= u(u + 1) · · · (u + n − 2)(u + n) I(n+1) .

(8.14.25)

Finally, we compare left- and right-hand sides in the chain of equalities (8.14.24) and make use of the fact that matrix M (u) is proportional to unit matrix, to find that [qdet(L(u)), L(v)] = 0. Hence, quantum determinant qdet(L(u)) is the generating function for central elementsc of Yangian Y (g(n)).

8.15.

Problem 4.7.30, Section 4.7.4

Let elements Lij (u) be generating functions (4.7.143) for generators (E (α) )ij of Yangians Y (so(n)) and Y (sp(2r)) (2r = n), i.e., let Lij (u) obey Eqs. (4.7.136), (4.7.140) with R-matrix (4.7.154). Consider operator z(u) defined by z(u) K i1ji12j2 =  n Lik11 (u) Lik22 (u − b) K kj11kj22 =  n K i1ki12k2 Lkj11 (u − b) Lkj22 (u),

(8.15.1)

where b = n2 −(n = 2r for algebra Y (sp(2r))). Show that z(u) is generating function for central elements of Yangians Y (so(n)) and Y (sp(2r)).

accordance with (8.14.22), polynomial (8.14.25) divided by u(u + 1) · · · (u + n − 1), u−n , is the generating function for central elements of which is a rational function (u+n−1) Yangian Y (g(n)) in representation T . c In

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It follows from the explicit form of R-matrix (4.7.154) that R(−b) =  b K.

(8.15.2)

We recall that b = n/2 − ,  = +1 for Y (so(n)) and  = −1, n = 2r for Y (sp(2r)). We set u − v = −b in Eq. (4.7.140), then the latter equation is written as K i1ki2k Lkj11 (u − b)Lkj22 (u) = Lik2 (u) Lik1 (u − b) K kj11kj22 = 1 2

2

where

1

 z(u) K i1ji12j2 , n

cj1 j2 . z(u) = ck1 k2 Lkj11 (u − b)Lkj22 (u) ¯

(8.15.3)

(8.15.4)

The second equality in (8.15.3) is actually a consequence of the first equality there. Indeed, we convolve the first equality in (8.15.3) on the right with matrix K j11j22 = ¯j1 j2 c1 2 , use the identity K i1ji12j2 K j1 j2 =  n K i1i2 , and an obvious consequence of c 1 2 1 2 (8.15.4), (8.15.5) K i1ki2k Lkj11 (u − b)Lkj22 (u)K j1j2 = z(u) K i1i2 . 1 2

1 2

1 2

This gives precisely the second equality in (8.15.3). Formula (8.15.3) is equivalent to (8.15.1). Let us set u = −b in Yang–Baxter equation (4.7.139) for R-matrices (4.7.154). Then we recall (8.15.2) and obtain analogs of identities (8.15.3): K12 R13 (v − b) R23 (v) = R23 (v) R13 (v − b) K12 = x(v) K12 ,

(8.15.6)

(v2 − 1)(v2 − b2 ).

The last equality in (8.15.6) is obtained, e.g., by inserting where x(v) = explicit form of R-matrices (4.7.154) to the left-hand side of (8.15.6) and using simple identities K12 P12 = K12 , K12 P13 P23 = K12 P13 ,

K12 K13 = K12 P23 ,

K12 K23 = K12 P13 ,

K12 K13 K23 = K12 P13 ,

K12 P13 K23 =  n K12 P13 .

We now show that element z(u) is central, i.e., it commutes with all generators of Yangians Y (so(n)) and Y (sp(2r)). In other words, it commutes with generating functions Lij (v) for all v. To this end, we write L3 (v) z(u) K12 = L3 (v) K12 L1 (u − b) L2 (u) K12 = K12 L3 (v) L1 (u − b) L2 (u) K12 = K12 R13 (u − v − b) L1 (u − b) L3 (v)R−1 13 (u − v − b)L2 (u) K12 = K12 R13 (u − v − b) L1 (u − b) R23 (u − v) −1 × L2 (u) L3 (v) R−1 23 (u − v)R13 (u − v − b) K12 −1 = x(u − v) K12 L1 (u − b) L2 (u) L3 (v) R−1 23 (u − v)R13 (u − v − b) K12

 −1 x(u − v) z(u) K12 L3 (v) R−1 23 (u − v)R13 (u − v − b) K12 n  = z(u) L3 (v) (K12 )2 = z(u) L3 (v) K12 . n =

where we use the representation (8.15.5) and relations (4.7.136); we apply the identity (8.15.6) to expressions underlined once and identity (8.15.3) to expressions underlined twice. We see that the element z(u) is central for all values of the parameter u, and hence z(u) is generating function of central elements of Yangians Y (so(n)) and Y (sp(2r)). It is

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worth noting that z(u) does not determine the complete center of algebras Y (so(n)) and Y (sp(2r)): it has to be supplemented by other elements constructed by using quantum determinants (4.7.147).

8.16.

Problem 6.2.14, Section 6.2.5

It is known (see, e.g., [10]) that 14-dimensional algebra G2 is subalgebra of 21-dimensional algebra so(7, C). Describe explicitly embedding of G2 into so(7, C). Making use of this embedding, construct explicit 7-dimensional matrix representation for generators of algebra G2 . Elements Hi (i = 1, 2, 3) make basis in Cartan subalgebra of rank-3 algebra so(7, C). Basis elements in Cartan subalgebra of rank-2 algebra G2 are any two elements of the set  1 = H2 − H 3 , H  2 = H3 − H 1 , H  3 = H1 − H 2 . H

(8.16.1)

Generators (6.2.81) of algebra so(7, C) obey ± ± [H1 , M(1) ] = ±M(1) ,

± [H2 , M(1) ] = 0,

(±±)

[H2 , M(34;56) ] = ±M(34;56) ,

(±±)

(±±)

[H1 , M(34;56) ] = 0,

(±±)

± [H3 , M(1) ] = 0, (±±)

[H3 , M(34;56) ] = ±M(34;56) .

(8.16.2)

We introduce linear combinations (±)

(∓∓)

(±)

(∓∓)

1 g∓ = κM(1) + M(34;56) ,

(±)

(∓∓)

2 g∓ = κM(3) + M(56;12) ,

3 = κM(5) + M(12;34) , g∓

(8.16.3)

where κ is yet undetermined constant. They obey the following commutation relations: 1 1 [H1 − H2 , g± ] = ∓g± , 1 [H2 − H3 , g± ] = 0,

1 1 [H3 − H1 , g± ] = ±g± ,

(cycle 1, 2, 3),

1 1 [g+ , g− ] = 2 κ2 H1 − 4 (H2 + H3 ), (cycle 1, 2, 3),

1 [g∓ ,

2 g∓ ]

=κ

2

(±±) M(12;34)

+

(∓) 4 κ M(5)

  4 (∓) (±±) =κ M(12;34) + M(5) . κ 2

(8.16.4) (8.16.5)

(8.16.6)

It follows from (8.16.5) and (8.16.6) that subalgebra of so(7, C), which we are going to identify with G2 and which contains, in particular, elements

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i g∓ , is closed, provided that κ2 = 4. Then 1 1 3 − H  2 ), (cycle 1, 2, 3), [g+ , g− ] = 4 (2 H1 − H2 − H3 ) = 4 (H

(8.16.7) 1 2 3 [g± , g± ] = 4 g∓ , (cycle 1, 2, 3).

Let us set (±∓)

G1± ≡ M(34;56) ,

(±∓)

G2± ≡ M(56;12) ,

(±∓)

G3± ≡ M(12;34) ,

(8.16.8)

Making use of (6.2.75) and (6.2.82) we get  2 , G1± ] = ∓G1± , [H  1 , G1± ] = ±2G1± , (cycle 1, 2, 3),  3 , G1± ] = ∓G1± , [H [H 1 − H 2 − H  3 ), (cycle 1, 2, 3), [G1+ , G1− ] = 4 (−H2 + H3 ) = −4/3 (2 H (8.16.9) [G3± , G1± ] = 2 G2∓ ,

[G1± , G2± ] = 2 G3∓ ,

1 2 [g± , g∓ ] = (κ2 + 2) G3∓ , 2 3 ] = 2 g∓ , [G1± , g±

[G2± , G3± ] = 2 G1∓ ,

3 2 [G1± , g∓ ] = −2 g∓ ,

(cycle 1, 2, 3),

(8.16.10)

We compare commutation relations (8.16.7)–(8.16.10) with (6.2.133) and see that embedding of G2 into Lie algebra so(7, C) is given by Eqs. (8.16.3) i with root and (8.16.8) with κ2 = 4. The relationship of generators Gi± , g∓ vectors of G2 is given in (6.2.132) and in Problem 6.2.13. We now insert definitions (6.2.66) and (6.2.81) in Eqs. (8.16.1), (8.16.3) and (8.16.8) to obtain the following explicit expressions for Cartan–Weyl generators of algebra G2 in terms of generators Mi j of algebra so(7, C):  1 = i(M5 6 − M3 4 ), H  2 = i(M1 2 − M5 6 ), H G1± = M3 5 + M4 6 ± i(M4 5 − M3 6 ), G2± = M5 1 + M6 2 ± i(M6 1 − M5 2 ), G3± = M1 3 + M2 4 ± i(M2 3 − M1 4 ), 1 g∓ 2 g∓ 3 g∓

(8.16.11)

= (κM1 7 + M3 5 − M4 6 ) ± i(κM2 7 − M4 5 − M3 6 ), = (κM3 7 + M5 1 − M6 2 ) ± i(κM4 7 − M6 1 − M5 2 ), = (κM5 7 + M1 3 − M2 4 ) ± i(κM6 7 − M2 3 − M1 4 ).

Different values κ = 2 and κ = −2 give two different embeddings of G2 in so(7, C). Cartan–Weyl basis (8.16.11) is straightforwardly converted into

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the basis of compact real form of algebra G2 (which is embedded in compact Lie algebra so(7)):  1 = M5 6 − M3 4 , H  2 = M1 2 − M5 6 , H G1 = M3 5 + M4 6 , G1 = M4 5 − M3 6 , G2 = M5 1 + M6 2 , G2 = M6 1 − M5 2 , G3 = M1 3 + M2 4 , G3 = M2 3 − M1 4 , g1 = κM1 7 + M3 5 − M4 6 , g1 = κM2 7 − M4 5 − M3 6 ,

(8.16.12)

g2 = κM3 7 + M5 1 − M6 2 , g2 = κM4 7 − M6 1 − M5 2 , g3 = κM5 7 + M1 3 − M2 4 , g3 = κM6 7 − M2 3 − M1 4 . Finally, we make use of matrix representation (6.2.64), namely, Mi j = eij − eji , where eij are matrix units and i, j = 1, . . . , 7, and insert it in (8.16.11) and (8.16.12). In this way we obtain explicit 7-dimensional matrix representation of the generators of complex algebra G2 and its compact real form. 8.17.

Problems 7.1.1, 7.1.3, Section 7.1

Let SO(2) be a subgroup of SO(3) embedded in the following way: ⎞ ⎛ 0 ⎟ ⎜ g ⎟ ⎜ (8.17.1) Og = ⎜ ⎟, g ∈ SO(2). ⎝ 0 ⎠ 0 0 1 Show that there is one-to-one correspondence between coset space SO(3)/SO(2) and 2-dimensional sphere, SO(3)/SO(2) = S 2 . Let O be matrix from SO(3), a be unit vector on sphere S 2 ∈ R3 with coordinates ai , i = 1, 2, 3. Action of element O ∈ SO(3) on vector a gives vector b with components bi = Oij aj . Show that this is transitive action of SO(3) in S 2 , and that stationary subgroup of any point on S 2 is isomorphic to SO(2). Orthogonality property OT O = I3 of matrix O ∈ SO(3) implies that its  rows (ei )j = Oij and columns (ej )i = Oij make two orthonormal systems of vectors in R3 . Let us denote Oi3 = xi , then x2i = 1 and, therefore, there is one-to-one correspondence between a point on S 2 and a subset of orthogonal matrices O whose third column is given by one and the same vector (x1 , x2 , x3 ). Let us show that this subset coincides with the left

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coset of subgroup SO(2) ⊂ SO(3) defined in (8.17.1). Let us take a matrix O ∈ SO(3) and multiply it on the right by a matrix (8.17.1) belonging to SO(2): ⎞⎛ ⎛ ⎞ x1 0 ⎟⎜ g ⎜ O ⎟ ⎟ ⎜ βγ ⎜ ⎟ αβ O  = O · Og = ⎜ ⎟⎜ ⎟ ⎝ x2 ⎠ ⎝ 0 ⎠ O3β ⎛ ⎜ (O · g) ⎜ αγ =⎜ ⎝ (O · g)3γ

x1 x2 x3

x3 ⎞

0

0

1

⎟ ⎟ ⎟ ∈ SO(3), ⎠

where α, β, γ = 1, 2. This product is an element of coset of SO(2) with respect to O ∈ SO(3). All these elements have one and the same third column, but we still have to show that vector (x1 , x2 , x3 ) unambiguously determines matrix O , modulo right multiplication by Og ∈ SO(2) ⊂ SO(3). Matrix g ∈ SO(2) can be chosen in such a way that the 2-dimensional vector (O · g)3γ in the third row of O  has the form (−b, 0), where b ≥ 0. We then use the fact that rows and columns  of matrix O  also make two orthonormal vector systems in R3 , set b = 1 − x23 and express the remaining elements of O  through elements (x1 , x2 , x3 ) of the third column of matrix O . To this end, we have to consider separately the cases b = 0 and b = 0. For b = 0, i.e., x3 = 1, we find that O  has the form (8.17.1), and hence O  ∈ SO(2); in that case the coset I3 · SO(2) 2 corresponds to the point x = (0, 0, 1) ∈ S . For b = 1 − x23 = 0 we have ⎞ ⎛x x 1 3 −x2 a x1 b 1 ⎟ ⎜ 2 2 O · Og = O(xi ) = ⎝ x2bx3 x1 a x2 ⎠, a = 2 , xi = 1, b −b 0 x3 (8.17.2) and since det O(xi ) = a b = 1 for O(xi ) ∈ SO(3), we have a = 1/b. Therefore, matrix O(xi ) is completely determined by the point x = (x1 , x2 , x3 ) on unit sphere S 2 . Any matrix O ∈ SO(3) whose third column is given by vector x with x3 = 1, can be obtained from (8.17.2) by multiplication on the right by some matrix Og ∈ SO(2). In other words, elements (8.17.2) are in one-to-one correspondence with points on a sphere x21 + x22 + x23 = 1 (x3 = 0), and, together with element I3 · SO(2) which

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corresponds to point x = (0, 0, 1), label cosets in SO(3)/SO(2). In this way we establish isomorphism SO(3)/SO(2) = S 2 . Any matrix O ∈ SO(3) whose third column is given by vector (x1 , x2 , x3 ) (say, matrix O(xi ) given by (8.17.2)) move the point ⎛ ⎞ 0 ⎜ ⎟ e3 = ⎝0⎠ ∈ S 2 1 to a point ⎛ ⎞ x1 ⎜ ⎟ x = ⎝x2 ⎠ ∈ S 2 . x3 Therefore, the action of SO(3) in S 2 is transitive. Clearly, the point e3 is stationary under the action of all elements (8.17.1) which form subgroup SO(2) ⊂ SO(3), hence SO(2) is the stationary subgroup of the point e3 ∈ S 2 . Finally, since the point e3 ∈ S 2 can be obtained from any other point x ∈ S 2 by the action of element O(xi )−1 ∈ SO(3), elements Og of the stationary subgroup Hx of the point x ∈ S 2 are obtained from elements (8.17.1) of the stationary subgroup He3 = SO(2) by automorphism Og = O(xi ) · Og · O(xi )−1 , and therefore Hx = SO(2) for all x ∈ S 2 . 8.18.

Problem 7.2.4, Section 7.2

Show that parameterization (7.2.13) covers unit sphere S n−1 once and only (n−1) are once, provided that the ranges of angles φm ≡ φm φ1 ∈ [0, 2π),

φi ∈ [0, π] (i = 2, . . . , n − 1).

(8.18.1)

We begin with calculating the volume Ω(S n−1 ) of unit sphere S n−1 , which is a surface in Rn defined by equation x 2 = 1, where x = (x , . . . , x ) ∈ Rn . To this end, we calculate an auxiliary integral A =  1n −x2n in two ways. On the one hand, we have d xe   ∞  2 2 n dx e−x = π n/2 . A = dn x e−x = (8.18.2) −∞

On the other hand, we write a vector x ∈ Rn of length r as x = rx, where x = (x1 , . . . , xn ) is vector on a unit sphere in Rn . We have for differentials

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dxi = dr xi + r dxi , so integration measure in Rn is written as follows: dn x = dx1 ∧ · · · ∧ dxn = rn−1 dr dx, dx :=

n 

(−)i−1 xi (dx1 ∧ · · · ∧ dxi−1 ∧ dxi+1 ∧ · · · ∧ dxn ),

(8.18.3) (8.18.4)

i=1

where ∧ is exterior (wedge) product, and dx denotes measure on unit sphere S n−1 ⊂ Rn . In view of (8.18.3), integral (8.18.2) is written as    ∞ 2 1 (8.18.5) rn−1 dr e−r = Ω(S n−1 ) Γ(n/2), A= dx 2 0  where ( dx) = Ω(S n−1 ) is the volume of unit sphere S n−1 . By comparing Eqs. (8.18.2) and (8.18.5) we obtain the general formula Ω(S n−1 ) =

2π n/2 . Γ(n/2)

(8.18.6)

Let us now discuss parameterization (7.2.13), (7.2.14) of sphere S n−1 . We note that when angles φi (i = 1, . . . , n−1) are varied in intervals [0, π/2], the parameterization (7.2.13) covers all points on the sphere S n−1 belonging to positive “quadrant” of Rn with xi ≥ 0 (∀i). In accordance with (7.2.13) and (7.2.14), replacement φk → π − φk , k = 2, . . . , n − 1, (each of these replacements leaves the sign of sin φk intact, the sign of cos φk gets flipped, while the ranges of angles (8.18.1) do not change), independently of other coordinates, changes sign of a given coordinate xk of unit vector x ∈ Rn , except for the first one, x1 . The sign of coordinate x1 can be flipped, without affecting signs of other coordinates, by replacement φ1 → 2π − φ1 . Thus, ranges of angles (8.18.1) enable one to have any set of signs of coordinates xi and hence get points of S n−1 in any “quadrant”. It remains to be seen that parameterization (7.2.13), (8.18.1) covers the sphere S n−1 only once (i.e., mapping (7.2.13) of the region (8.18.1) to unit sphere S n−1 is one-to-one except, possibly, for region of measure zero). To this end, we calculate the volume of the region of unit sphere covered by parameterization (7.2.13), (8.18.1). In the first place, we make use of the relations n 1  xi dxi , x1 i=2   n  1 xi dxi , . . . x1 dx1 + dx2 = − x2 i=3

xi dxi = 0 ⇒ dx1 = −

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and rewrite the measure (8.18.4) on unit sphere S n−1 : 1 1 dx2 ∧ · · · ∧ dxn = − dx1 ∧ dx3 · · · ∧ dxn = · · · x1 x2

dx =

(8.18.7)

With parameterization (7.2.13), we obtain the measure on unit sphere S n−1 in terms of angular variables, 1 dx2 ∧ · · · ∧ dxn = (−1)n−1 sφ2 s2φ3 · · · sn−2 φn−1 dφ1 dφ2 · · · dφn−1 . x1 (8.18.8) Then the integral over the range (7.2.14) gives for the volume of unit sphere S n−1 : dx =



 |dx| =

Ω(S n−1 ) =



2π

0

dφ1



π

0

dφ2 sφ2

π 0

dφ3 s2φ3 · · ·

 0

π

dφn−1 sn−2 φn−1

= 2π ·

1 Γ(1)Γ( 12 ) Γ( 32 )Γ( 12 ) Γ(2)Γ( 12 ) Γ( n−1 2 )Γ( 2 ) · · ··· n 3 5 Γ(2) Γ( 2 ) Γ( 2 ) Γ( 2 )

= 2π ·

(Γ( 12 ))n−2 2(π)n/2 = . n Γ( 2 ) Γ( n2 )

(8.18.9)

The fact that this result coincides with the formula (8.18.6) means that limits of integration in (8.18.9), i.e., ranges in (8.18.1), are correct. When calculating the multiple integral (8.18.9), we use the result 



π

(sφ )μ dφ = B

0

1 μ+1 , 2 2

 =

Γ( 12 )Γ( μ+1 2 ) Γ( μ+2 2 )

,

(8.18.10)

which follows from general formula  0

π/2

(c2φ )x−1/2 (s2φ )y−1/2 dφ =

1 Γ(x)Γ(y) . 2 Γ(x + y)

(8.18.11)

To conclude, parameterization (7.2.13) covers unit sphere S n−1 precisely once, provided that the angles vary in the ranges (8.18.1). 8.19.

Problem 7.2.6, Section 7.2

Show that unit sphere S 2n−1 is covered once and only once, if the ranges (n) (n−1) (i = 1, . . . , n − 1) in matrix of phases ψi (i = 1, . . . , n) and angles φi

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(7.2.30) are (n)

ψi

∈ [0, 2π),

(n−1)

φi

∈ [0, π/2].

(8.19.1)

Homogeneous space S 2n−1 = U (n)/U (n − 1) is parameterized on the basis of relations 

z1∗ z1 + · · · + zn∗ zn = 1 ⇒ z = eiψ1 x1 , . . . , eiψn xn ⇒ xi xi = 1,

(8.19.2)

where (x1 , . . . , xn ) are coordinates on unit sphere S n−1 . We choose the parameterization of coordinates xi given by (7.2.14), which depends on ˆk + iˆ xk , (n − 1) angles. With representation (8.19.2) for coordinates zk = x we obtain parameterization of the sphere S 2n−1 in terms of (n − 1) angles φk (k = 1, . . . , n − 1) and n phases ψk (k = 1, . . . , n): x ˆ1 = sψ1 x1 = sψ1 sφ1 sφ2 · · · sφn−1 , x ˆ1 = cψ1 x1 = cψ1 sφ1 sφ2 · · · sφn−1 , x ˆ2 = sψ2 x2 = sψ2 cφ1 sφ2 · · · sφn−1 , x ˆ2

= cψ2 x2 = cψ2 cφ1 sφ2 · · · sφn−1 , . . . , x ˆn−2 = sψn−2 cφn−3 sφn−2 sφn−1 ,

(8.19.3)

x ˆn−2 = cψn−2 cφn−3 sφn−2 sφn−1 , x ˆn−1 = sψn−1 cφn−2 sφn−1 , xˆn−1 = cψn−1 cφn−2 sφn−1 , ˆn = cψn xn = cψn cφn−1 . x ˆn = sψn xn = sψn cφn−1 , x Arguments which repeat those used in the solution of previous problem in Section 8.18 show that any point of unit sphere S 2n−1 is obtained within this parameterization, if the ranges of angles and phases are given by (8.19.1). It remains to be shown that this parameterization covers the sphere only once. To this end, we calculate the volume of the region of S 2n−1 covered by this parameterization with range (8.19.1). We note that xk = (cψk xk dψk + sψk dxk ) ∧ (−sψk xk dψk + cψk dxk ) dˆ xk ∧ dˆ = xk dψk dxk . Also, we have dx1 ∧ dx2 ∧ · · · ∧ dxk = 0,

x1 x2 · · · xn

= sφ1 cφ1 s2φ2 cφ2 · · · skφk cφk · · · sn−1 φn−1 cφn−1 .

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Then the measure dˆ x on S 2n−1 is written in terms of variables (8.19.3) as follows: x2 ∧ dˆ x2 ) ∧ · · · ∧ (dˆ xn dˆ xn ) dˆ x ∧ (dˆ dˆ x= 1 x ˆ1 =

(−sψ1 x1 dψ1 + cψ1 dx1 ) ∧ (x2 dψ2 dx2 ) ∧ · · · ∧ (xn dψn dxn ) sψ1 x1

= − dψ1 ∧ (x2 dψ2 dx2 ) ∧ · · · ∧ (xn dψn dxn ) = (−)n(n+1)/2 (x2 · · · xn ) (dx2 ∧ · · · ∧ dxn ) (dψ1 dψ2 · · · dψn ) = (−1)(n+1)(n+2)/2 (x1 x2 · · · xn ) sφ2 s2φ3 · · · × sn−2 φn−1 (dφ1 · · · dφn−1 )(dψ1 · · · dψn ) = (−1)(n+1)(n+2)/2 cφ1 sφ1 cφ2 s3φ2 cφ3 s5φ3 · · · ×cφn−1 s2n−3 φn−1 (dφ1 · · · dφn−1 )(dψ1 · · · dψn ), where we use (8.18.7). For the chosen range (8.19.1) of phases and angles, we have   π/2  π/2  π/2 3 |dˆ x| = sφ1 dsφ1 sφ2 dsφ2 s5φ3 dsφ3 · · · 0



× =

0

0

π/2

s2n−3 φn−1 dsφn−1

 0

0



2π

...

0

2π

(dψ1 · · · dψn )

1 1 1 1 2 πn · · ··· (2π)n = . 2 4 6 2n − 2 (n − 1)!

This coincides with the volume of unit sphere S 2n−1 , calculated with (8.18.6), and shows that the parameterization (8.19.3) covers unit sphere S 2n−1 precisely once (modulo, possibly, measure zero region), provided that the angles φk and phases ψk vary in the range (8.19.1). 8.20.

Problem 7.5.8, Section 7.5.2

Consider homogeneous space CPn = U (n + 1)/(U (n) × U (1)), studied in Example 8 of Section 7.2. Making use of (7.5.64) and (7.5.69), obtain Fubini–Study metric for this space, ∂ ∂ dzi dzi∗ (dzi zi∗ ) (zk dzk∗ ) − = dzk dzi∗ ln(1+|z|2), 1 + |z|2 (1 + |z|2 )2 ∂zk ∂zi∗  where i, k = 1, . . . , n and |z|2 = k |zk2 |. ds2 =

(8.20.1)

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In accordance with (7.5.59), we choose representative k(a) of an element of coset space CPn = U (n + 1)/(U (n) × U (1)) in exponential form: ⎛

⎞ a1 ⎜ ⎟ .. ⎟ ⎜ ⎜ .⎟ 0n k = exp(A) ∈ U (n + 1), A = ⎜ ⎟ ∈ u(n + 1), ⎜ ⎟ ⎝ an ⎠ −a∗1 ... −a∗n 0 (8.20.2)  ∗ 2 where 0n is zero n × n matrix, and ak ∈ C. Let us denote |a| = k ak ak . Matrix A obeys ⎛ 3

2

A = −|a| A,

⎜ ⎜ ⎜ A =⎜ ⎜ ⎝

0 .. .

−||ai a∗k ||

2

0

...

0

⎞

⎟ ⎟ ⎟ ⎟, ⎟ 0 ⎠ −|a|2

i, k = 1, . . . , n.

Making use of these relations, we get (1 − cos |a|) sin |a| +A |a|2 |a|   1 Z2  = In+1 +  +Z , 1 + |z|2 1 + 1 + |z|2

k = exp(A) = In+1 + A2

(8.20.3)

where we introduce matrix Z and new coordinates zi , ⎛

⎞ z1 ⎜ ⎟ .. ⎟ tan(|a|) ⎜ ⎜ . ⎟ 0n Z =A =⎜ ⎟, ⎜ ⎟ |a| ⎝ zn ⎠ −z1∗ . . . −zn∗ 0

z i = ai

tan |a| . |a|

Matrix k −1 is obtained from k by replacing ai → −ai (or zi → −zi ):

k

−1

1

= exp(−A) = In+1 +  1 + |z|2



 Z2  −Z , 1 + 1 + |z|2

(8.20.4)

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so Maurer–Cartan form (7.5.64) is k −1 · d k =  + ω  =

Z2

Z



 In+1 +  − 1 + |z|2 (1 + 1 + |z|2 ) 1 + |z|2   Z Z2  +d × d . 1 + |z|2 (1 + 1 + |z|2 ) 1 + |z|2 (8.20.5)

In accordance with (7.5.64), vielbein form  is obtained by collecting here odd powers of matrix Z,   Z2 Z  d  = In+1 +  2 2 1 + |z| (1 + 1 + |z| ) 1 + |z|2 Z Z2   − d 1 + |z|2 (1 + 1 + |z|2 ) 1 + |z|2 dZ Z (dZ) Z + Z (d |z|2 )  =  − 1 + |z|2 (1 + |z|2 )(1 + 1 + |z|2 ) ⎛ ⎞ v1 ⎜ ⎟ .. ⎟ ⎜ ⎜ 0n . ⎟ =⎜ ⎟, ⎜ ⎟ ⎝ vn ⎠ −v1∗ ... −vn∗ 0

(8.20.6)

where dzk zk (dzi zi∗ )  vk =  . − 1 + |z|2 (1 + |z|2 )(1 + 1 + |z|2 )

(8.20.7)

Accordingly, (U (n) × U (1))-connection form ω in CPn is a sum of terms in (8.20.5) with even powers of Z:   Z2 Z2   ω = In+1 +  d 1 + |z|2 (1 + 1 + |z|2 ) 1 + |z|2 (1 + 1 + |z|2 ) Z Z − d 2 1 + |z| 1 + |z|2

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2Z 2 (dZ) Z + Z 2 (d|z|2 ) (dZ) Z − Z (dZ)   + =  1 + |z|2 (1 + 1 + |z|2 ) 2 (1 + |z|2 )(1 + 1 + |z|2 )2 ⎞ ⎛ 0 ⎟ ⎜ . ⎟ ⎜ ⎜ ||ωik || .. ⎟ =⎜ ⎟, ⎟ ⎜ ⎝ 0 ⎠ 0 ... 0 ω0

447

(8.20.8)

where U (n)-connection is zi dzk∗ − dzi zk∗ zi zk∗ ((z, dz ∗ ) − (dz, z ∗ ))   ωik =  , − 1 + |z|2 (1 + 1 + |z|2 ) 2 (1 + |z|2 )(1 + 1 + |z|2 )2 whereas we have for U (1)-connection ω0 =

(dz, z ∗ ) − (z, dz ∗ ) = −Tr(||ωik ||). 2 (1 + |z|2 )

To calculate metric in CPn , we insert expressions (8.20.6) and (8.20.7) in (7.5.69). In this way we obtain Fubini–Study metric   1 (dzk∗ zk ) (dzi zi∗ ) 1 ∗ dz ) − − Tr( · ) = (vk∗ vk ) = (dz , k k 2 (1 + |z|2 ) (1 + |z|2 ) which can also be written as ds2 = dzk dzi∗ ·

∂2 ln(1 + |z|2 ). ∂zk ∂zi∗ 2

K Complex manifolds whose metric has special form ds2 = dzk dzi∗ · ∂z∂k ∂z ∗, i ∗ are K¨ ahler manifolds, and function K(zi , zk ) is called K¨ ahler potential.

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Selected Bibliography

Here we give far from being a complete list of monographs, which addresses the issues raised in our book. • A. Barut and R. Raczka, Theory of Group Representations and Applications, 2nd Revised Edition, World Scientific (1986). • H. Weyl, Symmetry, Princeton University Press (1952). • H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications (1950). • H. Weyl, The Classical Groups: Their Invariants and Representations, Princeton University Press (1997). • E. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic Press (1959). • N.Ja. Vilenkin, Special Functions and the Theory of Group Representations (Translations of Mathematical Monographs), American Mathematical Society (1968). • P.I. Golod and A.U. Klimyk, Mathematical Foundations of Symmetry Theory, Regular and Chaotic Dynamics, Moscow (2001), in Russian. • B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry — Methods and Applications, Springer (1992). • D.P. Zelobenko, Compact Lie Groups and Their Representations (Translations of Mathematical Monographs), American Mathematical Society (1978). • A.A. Kirillov, Elements of the Theory of Representations, Springer (1976). • A.A. Kirillov, Lectures on the Orbit Method (Graduate Studies in Mathematics, Volume 64) American Mathematical Society, Providence (2004). 449

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• F. Klein, Development of Mathematics in the 19th Century, Lie Groups: History, Frontiers and Applications, Math. Sci. Press, Brookline, MA (1979). • V.D. Lyakhovsky and A.A. Bolokhov, Symmetry Groups and Elementary Particles, Editorial URSS (2002), in Russian. • M.A. Naimark, A.I. Stern, Theory of Group Representations, Comprehensive Studies in Mathematics, Vol. 246, Springer (1982). • S.P. Novikov and I.A. Taimanov, Modern Geometric Structures and Fields (Graduate Studies in Mathematics, Vol. 71), American Mathematical Society, Providence (2006). • L.S. Pontryagin, Topological Groups, Gordon & Breach Science Pub; 3rd Edition (1986). • M.M. Postnikov, Lie Groups and Lie Algebras. Lectures in Geometry, Semester V, Editorial URSS (1994). • M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley Publishing Company (1962). • H. Georgi, Lie Algebras in Particle Physics, Westview press, Advanced Book Program (1999). • P. Ramond, Group Theory. A Physicist’s Survey, Cambridge University Press (2010). • B.G. Wybourne, Classical Groups for Physicists, A Wiley-Interscience Publication (1973). • W. Fulton and J. Harris, Representation Theory. A First Course, Springer-Verlag (1991).

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References

[Monographs] [1] H. Weyl, Symmetry, Princeton University Press (1952). [2] F. Klein, Development of Mathematics in the 19th Century, Lie Groups: History, Frontiers and Applications, Math. Sci. Press, Brookline, MA (1979). [3] L.S. Pontryagin, Topological Groups, Gordon & Breach Science Pub., 3rd Edition (1986). [4] P.I. Golod and A.U. Klimyk, Mathematical Foundations of Symmetry Theory, Regular and Chaotic Dynamics, Moscow (2001); in Russian. [5] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (A Course of Theoretical Physics, Vol. 2), Pergamon Press (1971). [6] B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry — Methods and Applications, Springer-Verlag (1992). [7] I.L. Buchbinder and S.M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity, Or a Walk Through Superspace, IOP, Bristol and Philadelphia, (1995). [8] D.P. Zelobenko, Compact Lie Groups and Their Representations (Translations of Mathematical Monographs), American Mathematical Society (1978). [9] J.-P. Serre, Linear Representations of Finite Groups (Graduate Texts in Mathematics, Vol. 42), Springer-Verlag (1977). [10] J.E. Hamphreys, Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics, Vol. 9), Springer-Verlag (1994). [11] C. Chevalley, Theory of Lie Groups, Vol. III of General Theory of Lie Algebras, Princeton University Press (1955). [12] A. Molev, Yangians and Classical Lie Algebras (Mathematical Surveys and Monographs, Vol. 143), American Mathematical Society (2007). [13] J.-P. Serre, Lie Algebras and Lie Groups. Lectures Given at Harvard University, 1964 (Lecture Notes in Mathematics), Springer-Verlag (1992). [14] N. Bourbaki, Lie Groups and Lie Algebras. Chapters 4–6, Springer-Verlag (2002). 451

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[15] H. Georgi, Lie Algebras in Particle Physics, Westview press, Advanced Book Program, 1999. [16] M.M. Postnikov, Lie Groups and Lie Algebras. Lectures in Geometry, Semester V, Editorial URSS (1994). [17] N.Ja. Vilenkin, Special Functions and the Theory of Group Representations (Translations of Mathematical Monographs), American Mathematical Society (1968). [18] M.B. Mensky, Induced Representations Method: Space-Time and Concept of Particles, Moscow, Nauka (1976), in Russian. [19] R. Penrose and W. Rindler, Spinors and Spacetime, Vol. 2, Cambridge University Press (1986). [Papers and Reviews] [20] D. Olive, Gauge Theories and Lie Algebras with some Applications to Spontaneous Symmetry Breaking and Integrable Dynamical Systems. Lectures given at the University of Virginia, 1982. Preprint (1983). [21] G. Racah, Group Theory and Spectroscopy. Lectures delivered at the Institute for Advanced Study, Princeton, 1951. Preprint CERN 61-8 (1961). [22] E.B. Dynkin, Classification of Simple Lie Groups, Selected Papers of E.B. Dynkin with Commentary 14 (2000) p. 23. [23] H. Weyl, Representation Theory of Continuous Semi-simple Groups Through Linear Transformations, Uspekhi Mat. Nauk 4 (1938) 201–257 (in Russian). [24] A.B. Zamolodchikov and Al.B. Zamolodchikov, Factorized S Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, Annals Phys. 120 (1979) 253. [25] N.Yu. Reshetikhin, Integrable Models of Quantum One-Dimensional Magnets with O (N ) and Sp (2K) Symmetry, Theor. Math. Phys. 63(3) (1985) 555–569. [26] V.G. Drinfeld, Hopf algebras and the quantum Yang–Baxter equation, Sov. Math. Dokl. 32 (1985) 254; V.G. Drinfeld, Quantum Groups, in Proceedings of the Int. Congress of Mathematics, Vol. 1 (Berkeley, 1986), p. 798. [27] E.K. Sklyanin, L.A. Takhtajan, and L.D. Faddeev, The Quantum Inverse Problem Method. I, Theor. Math. Phys. 40 (1980) 688. [28] L.D. Faddeev, How Algebraic Bethe Ansatz works for integrable model, in Symtries quantiques, Les Houches (1996), arXiv:hep-th/9605187; L.D. Faddeev, N.Y. Reshetikhin and L.A. Takhtajan, Quantization of Lie Groups and Lie Algebras, Algebraic Analysis: Papers Dedicated to Professor Mikio Sato on the Occasion of His Sixtieth Birthday, Academic Press (2014) 129.

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Index

Abelian group, 1 action of group, 54–56 adjoint action of group, 59 Ado’s theorem, 124 algebra, 28 group, 218, 219 Lorentz, 115 automorphism, 49

center of Lie algebra, 129 character of representation, 189, 199, 202, 220–224, 226–231 Chevalley basis, 309 Christoffel’s symbols, 382 complex structure, 130, 133 comultiplication, 253 conjugacy class, 17, 22–26, 230 connection, 164, 379 coefficients, 379 Levi-Civita, 381 metric, 381 spin, 380 coproduct, 253–270 coset, 12 coset space, 13, 346, 363 O(n)/(O(k) × O(n − k)) = Gn,k , 360 O(n + 1)/(O(n) × Z2 ) = RPn , 359 SO(3)/SO(2) = S 2 , 361 SU (n + 1, H)/(SU (n, H) × U (1)) = CP2n+1 , 363 SU (n + 1, H)/SU (n, H) = S 4n+3 , 362 S 3 /S 1 = S 2 , 361 S 2n+1 /U (1) = CPn , 361 U (n)/(U (k) × U (n − k)) = GC n,k , 361 U (n + 1)/(U (n) × U (1)) = CPn , 361 U Sp(2n)/(U Sp(2n − 2) × U Sp(2)) = HPn−1 , 361 U Sp(2n)/(U Sp(2n − 2k) × U Sp(2k)) = GH n,k , 361

bialgebra, 256 bilinear form, 38, 57 Burnside theorem, 206 Campbell–Hausdorff formula, 106, 415 Campbell–Poincar´ e identity, 393, 414 Cartan matrix, 309, 310 Cartan subalgebra, 140, 282–285, 292 Cartan’s structure equation, 381, 386, 394 Cartan–Weyl basis, 140, 282, 289–292 Casimir coefficient, 247 cubic c3 , 247 quadratic c2 , 247 Casimir operator, 237–248, 259 su(2), 243 su(3), 243 cubic, 243 quadratic, 243, 244, 401 s(2, C), 249 su(n), 245 split (polarized), 257 Cayley’s table, 10 Cayley’s theorem, 26, 51 center of group, 17, 26 453

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U Sp(2n + 2)/(U Sp(2n) × U (1)) = CP2n+1 , 363 U Sp(2n + 2)/U Sp(2n) = S 4n+3 , 362 covariant derivative, 379 Coxeter generators, 20 curvature tensor, 383 Riemann, 383 derivative of mapping, 94, 191 diffeomorphism, 70 direct product of matrices, 31, 198 of representations, 197–200 of vector spaces, 197 direct sum of Lie algebras, 127 of representations, 201–203 of vector spaces, 201 Dynkin diagram, 311, 324–331 s(n, C) = An−1 , 314 so(2r + 1, C) = Br , 321 so(2r, C) = Dr , 318 sp(2r, C) = Cr , 323 E6 , 331 E7 , 331 E8 , 331 F4 , 331 G2 , 331 epimorphism, 49, 52 factor algebra, 127 factor group, 14 finite group, 1 Gell-Mann matrices, 145 generators of group, 5 Sn , 20 Grassmann manifold complex, 361 quaternionic, 361 real, 360 Grassmannian, see Grassmann manifold group, 1 Bn , braid group, 20 Dn , dihedral, 4, 185 GL(n, K) (K = R, C), 9, 36, 75, 89 IO(p, q), 57, 82 O(1, 1), 78, 89

O(1, n), 46, 80 O(2), 8 O(n) = O(n, R), 9, 46, 77, 88 O(n, C), 41–42, 89 O(p, q), 45, 78, 82–86, 89 P SL(n, C), projective, 44 P SU (n), 44, 89 P SU (p, q), projective, 44 Q2n , di-cyclic, 5 SL(n, K) (K = R, C), 9, 36, 38, 76, 89 SO(2), 7, 75, 87, 232, 233 SO(3), 78, 87, 172–175 SO(n) = SO(n, R), 10, 46, 77 SO(n, C), 41–42 SO(p, q), 45 SU (2), 76, 87, 172–175 SU (n), 44, 76 SU (p, q), 43 Sn , symmetric, 5, 18–26, 185 Sp(2r, C), 43 Sp(2r, C), 42 Sp(2r, R), 44, 89 Sp(p, q), 45 U (1), 2, 75, 87, 232, 233 U (n), 44, 76, 88 U (p, q), 43, 89 U Sp(2r) = Sp(r), 44, 88 Zn = Cn , cyclic, 2, 185 Conf(R1,1 ), Conf(R2 ), conformal in 2 dimensions, 64 Conf(Rp,q ), conformal, 61–64, 82–86, 162–169 anti-de Sitter, 47 compact, 86–89 de Sitter, 47, 73 direct product, 16 holonomy, 386, 395 linear inhomogeneous, 55 Lorentz, 46, 80, 81 matrix, 36, 55 orthochronous, 81 proper orthochronous, 81 semisimple, 15, 129 simple, 15, 129 group action free, 348 transitive, 345 group algebra, 218

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group isomorphism, 3 group parameterization SO(3), Euler, 351 SO(n), 352 U (n), 356 Haar measure, 99–101, 219 Harish-Chandra isomorphism, 243 Hermitian form, 40 homeomorphism, 70 homogeneous space, 345–349, see coset space O(n)/O(n − 1) = S n−1 , 352 SO(3)/SO(2) = S 2 , 349 SO(n)/SO(n − 1) = S n−1 , 352 SO(n, 1)/SO(n − 1, 1) = dS n , 355 SO(n − 1, 2)/SO(n − 1, 1) = AdS n , 355 SU (n)/SU (n − 1) = S 2n−1 , 355 U (n)/U (n − 1) = S 2n−1 , 355 Rn /Zn = Tn , 349 homomorphism antilinear, 124 image, 50 kernel, 50, 151, 186 linear, 123, 151 of groups, 49 Hopf algebra, 256, 260, 269, 270 Hopf fiber bundle, 361 ideal, 126 index of subgroup, 13 invariant subspace, 205, 207 isometry, 165, 369 conformal, 168 isomorphism of groups, 49, 53 D2 = Z2 × Z2 , 16 D3 = S3 , 11, 223 D6 = Z2 × D3 , 16 isomorphism of Lie algebras, 124, 152–161 s(2, K) = sp(2, K), 153 s(2, R) = su(1, 1), 152 so(1, 5) = s(2, H) = su∗ (4), 161 so(2, 1) = s(2, R), 153 so(2, 2) = s(2, R) + s(2, R), 157 so(2, 4) = su(2, 2), 161 so(3) = su(2) = usp(2), 152 so(3, 2) = sp(4, R), 160 so(3, 3) = s(4, R), 161

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so(4) = su(2) + su(2), 154 so(4, 1) = sp(1, 1), 160 so(4, C) = s(2, C) + s(2, C), 157 so(5) = usp(4), 160 so(5, C) = sp(4, C), 160 so(6) = su(4), 161 so(6, C) = s(4, C), 161 so∗ (4) = su(1, 1) + su(2), 161 so∗ (6) = su(1, 3), 161 usp(2) = su(2), 153 usp(2r) = su(r, H), 153 conf(Rp,q ) = so(p + 1, q + 1), 170 so(1, 3) = s(2, C), 157 isomorphism of Lie groups SO(3) = SU (2)/Z2 , 172 U Sp(2r) = SU (r, H), 154 local, 172 Jacobi identity, 116, 119, 122, 123, 125, 137, 195, 289 Killing form, 136, 278 Killing metric, 136, 138, 196, 273, 278, 286, 289 Killing vector, 165 conformal, 168, 169 Lagrange theorem, 14, 26 Laplace operator on homogeneous space, 403 on Lie group, 399 Laplace–Beltrami operator, 400 Leibnitz rule, 278 Lie algebra, 122 Z2 -graded, 395 g(n, C), 109, 139 s(2, C), 108, 132, 138, 248 s(2, R), 138 s(n, C), 109, 139–141, 292 s(n, R), 139–141 so(1, 3), 147 so(1, n), 115 so(2), 108 so(3), 146 so(n) = so(n, R), 111, 146 so(n, C), 111, 146 so(p, q), 115, 147–148 so∗ (n), 150 sp(2r, K), 112, 148 sp(p, q), 113, 149

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su(2), 141, 248 su(3), 144–146 su(n), 111, 142–144 su(p, q), 115, 149 su∗ (n), 150 u(n), 110 u(p, q), 115, 149 usp(2r), 114 G2 , 342 E6 , 330, 337–338 E7 , 330, 336–337 E8 , 330, 333–335 F4 , 330, 338–340 G2 , 330, 340 Abelian, 125 adjoint, 152, 279 compact, 271–275 complex, 105, 122 complexification, 133 differentiation, 278 dimension E6 , 338 E7 , 337 E8 , 335 F4 , 340 G2 , 342 Kohno–Drinfeld, 258 matrix, 105 realification, 129–134 reductive, 277 semisimple, 127, 275, 281 simple, 127, 140, 208, 277, 281 solvable, 127 universal enveloping algebra, 235 Lie subalgebra, 125 normalizer, 388 regular, 282 Lobachevskian geometry, 65, 367–372 manifold compact, 86 connected, 71 K¨ ahler, 447 metric, 93 pseudo-Riemannian, 381 Riemannian, 381 simply connected, 71 smooth, 69 mapping, 47–48 codomain, 48

domain, 48 exponential, 117, 123 image, 47, 48 matrix, 28 Cabbibo–Kobayashi–Maskawa, 359 determinant, 28–30 Hermitian, 28 mixing, 358 orthogonal, 28 polar decomposition, 31 Pontecorvo–Maki–Nakagawa– Sakata, 359 special, 9 unitary, 28 matrix units es r , 139, 263 Maurer–Cartan form, 389, 392 Maurer–Cartan identity, 394 metric, 93, 378 Fubini–Study, 393, 444 invariant, 95, 387 left-invariant, 95–99 left-translation-invariant, 95 pseudo-Riemannian, 93 Riemannian, 93 right-invariant, 95–99 right-translation-invariant, 95 monomorphism, 49, 51 orbit of group, 345 order of group, 1, 15 partition, see Young tableau Pauli matrices, 77, 141 Peter–Weyl theorem, 232 Pfaffian, 32–34 Poincar´ e group, 58 Poincar´ e metric, 369, 376 Poincar´ e model, 368, 369 presentation of Sn , 20 of finite group, 5 quotient quotient quotient quotient

algebra, 127 group, 14, 53 set, 186 space, 205

rank of Lie algebra, 282 s(n, C) = An−1 , 312

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Index so(2r + 1, C) = Br , 314 so(2r, C) = Dr , 314 sp(2r, C) = Cr , 322 real form, 134–136, 139, 150–151 compact, 281, 282 regular element, 282 representation adjoint, 182, 195, 208 co-adjoint, 184, 196 completely reducible, 205, 207, 215 complex, 179, 190 conjugate, 182, 195 contragredient, 181, 194 decomposable, 203, 205, 207 defining, 181, 193 dimension, 179, 190 equivalent, 188, 190 faithful, 186 fundamental, 181 induced, 187, 364 irreducible, 204, 205, 207, 208 of group, 179 of Lie algebra, 190 su(2) and s(2, C), 248–253 real, 179, 190 reducible, 204, 205, 207 regular, 185, 219, 220, 231, 365 trivial, 193 unitary, 180, 191, 216 with highest weight, 250 Ricci tensor, 384 Riemann tensor, 383 root, 289, 293–302 negative, 304 positive, 304 simple, 304–309 root generator, 289 root space, 289 root system s(3, C), 300, 302 s(n, C) = An−1 , 312–314 so(2r + 1, C) = Br , 321 so(2r + 1, C) = Br , 319 so(2r, C) = Dr , 316–318 sp(2r, C) = Cr , 323 sp(2r, C) = Cr , 321 su(3), 300–302 G2 , 342 E6 , 337–338 E7 , 336–337

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E8 , 333–335 F4 , 338–340 G2 , 340 root vector, see Root Schur’s lemma, 209–212 semidirect product of groups, 58–61 Serre relations, 309 space anti-de Sitter AdS n , 73, 87, 373–376 de Sitter dS n , 73, 87, 373–376 homogeneous, see Homogeneous space of representation, 179 projective CPn , 360, 362 projective RPn , 71, 78 projective quaternionic HPn , 361 symmetric, 395–396 tangent, 73, 89–94, 117 vector, 27 complexification, 129 realification, 129 spherical function adjoint, 406 orthonormality, 407 zonal, 406 structure constants, 125 antisymmetry, 137, 275 subgroup, 11 An , alternating, 22 normal, 12, 53 stationary, 347 tensor, 197 Ad-invariant, 241–242, 259 anti-selfdual, 155, 157, 160 invariant, 199 rank, 197 selfdual, 155, 157, 160 tensor product, see Direct product ’t Hooft symbols, 155 torsion, 380, 396 universal covering group, 172, 192 Vandermonde determinant, 34 vector field left-invariant, 397 right-invariant, 397 Vielbein, 376

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weight of representation highest, 250 lowest, 250 Weyl chamber, 303, 337 Weyl group, 303, 332, 339 Weyl reflection, 296, 303, 339 Weyl tensor, 384

Yang–Baxter equation, 265, 267 Yangian, 260 Y (g(n)), 266 Y (so(n)), Y (sp(2r)), 269 Young tableau, 23

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