Classification and Identification of Lie Algebras 0821843559, 9780821843550
The purpose of this book is to serve as a tool for researchers and practitioners who apply Lie algebras and Lie groups t
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Table of contents :
Front Matter
Copyright
Contents
Preface
Acknowledgements
Part 1. General Theory
Chapter 1. Introduction and Motivation
Chapter 2. Basic Concepts
2.1. Definitions
2.2. Levi theorem
2.3. Classification of complex simple Lie algebras
2.4. Chevalley cohomology of Lie algebras
Chapter 3. Invariants of the Coadjoint Representation of a Lie Algebra
3.1. Casimir operators and generalized Casimir invariants
3.2. Calculation of generalized Casimir invariants using the infinitesimal method
3.2.1. Formulation of the problem and number of functionally independent solutions.
3.2.2. Method of characteristics for a single PDE.
3.2.3. Solution of the system of PDEs (3.2).
3.3. Calculation of generalized Casimir invariants by the method of moving frames
Part 2. Recognition of a Lie Algebra Given by Its Structure Constants
Chapter 4. Identification of Lie Algebras through the Use of Invariants
4.1. Elementary invariants
4.2. More sophisticated invariants
Chapter 5. Decomposition into a Direct Sum
5.1. General theory and criteria
5.2. Algorithm
5.3. Examples
Chapter 6. Levi Decomposition. Identification of the Radicaland Levi Factor
6.1. Original algorithm
6.2. Modified algorithm
6.3. Examples
Chapter 7. The Nilradical of a Lie Algebra
7.1. General theory
7.2. Algorithm
7.3. Examples
7.4. Identification of the nilradical using the Killing form
Part 3. Nilpotent, Solvable and Levi Decomposable Lie Algebras
Chapter 8. Nilpotent Lie Algebras
8.1. Maximal Abelian ideals and their extensions
8.2. Classification of low-dimensional nilpotent Lie algebras
Chapter 9. Solvable Lie Algebras and Their Nilradicals
9.1. General structure of a solvable Lie algebra
9.2. General procedure for classifying all solvable Lie algebras with a given nilradical
9.3. Upper bound on the dimension of a solvable extension of a given nilradical
9.4. Particular classes of nilradicals and their solvable extensions
9.5. Vector fields realizing bases of the coadjoint representation of a solvable Lie algebra
Chapter 10. Solvable Lie Algebras with Abelian Nilradicals
10.1. Basic structural theorems
10.2. Decomposability properties of the solvable Lie algebras
10.2.1. Nilradicals of minimal dimension.
10.3. Solvable Lie algebras with centers of maximal dimension
10.4. Solvable Lie algebras with one nonnilpotent element and an n-dimensional Abelian nilradical
10.5. Solvable Lie algebras with two nonnilpotent elements and n-dimensional Abelian nilradical
10.6. Generalized Casimir invariants of solvable Lie algebras with Abelian nilradicals
10.6.1. General form of the generalized Casimir invariants and their number.
10.6.2. Diagonal structure matrices.
10.6.3. Casimir invariants in the case f = 1.
10.6.4. Casimir invariants for low-dimensional nilradicals.
10.6.5. Summary.
Chapter 11. Solvable Lie Algebras with Heisenberg Nilradical
11.1. The Heisenberg relations and the Heisenberg algebra
11.2. Classification of solvable Lie algebras with nilradical h(m)
11.2.1. Basic classification theorem.
11.3. The lowest dimensional case m = 1
11.4. The case m = 2
11.5. Generalized Casimir invariants
Chapter 12. Solvable Lie Algebras with Borel Nilradicals
12.1. Outer derivations of nilradicals of Borel subalgebras
12.2. Solvable extensions of the Borel nilradicals NR(b(g))
12.2.1. Solvable extensions of the Borel nilradicals of maximal dimension.
12.2.2 Solvable extensions of the split real form of the nilradical NR(b(g))
12.2.3. Solvable extensions of the Borel nilradicals of less than maximal dimension.
12.2.4. Solvable extensions of dimension n_{NR}+1.
12.3. Solvable Lie algebras with triangular nilradicals
12.3.1. The structure of the algebras of strictly upper triangular matrices and their derivations.
12.3.2. Illustration of the procedure for low dimensions.
12.4. Casimir invariants of nilpotent and solvable triangular Lie algebras
12.4.1. Invariants of nilpotent triangular Lie algebras.
12.4.2. Invariants of the solvable triangular Lie algebras.
12.4.3. Casimir invariants of solvable extensions of t(4).
12.4.4. General results.
12.4.5. Conclusions.
Chapter 13. Solvable Lie Algebras with Filiform and Quasifiliform Nilradicals
13.1. Classification of solvable Lie algebras with the model filiform nilradical n_{n,1}
13.1.1. Nilpotent algebra n_{n,1}.
13.1.2. Construction of solvable Lie algebras with the nilradical n_{n,1}.
13.1.3. Standard forms of solvable Lie algebras with the nilradical n_{n,1}.
13.1.4. The generalized Casimir invariants of n_{n,1} and of its solvable extensions.
13.2. Classification of solvable Lie algebras with the nilradical n_{n,2}
13.2.1. Nilpotent algebra n_{n,2} and its structure.
13.2.2. Construction of solvable Lie algebras with the nilradical n_{n,2}.
13.2.3. Generalized Casimir invariants of n_{n,2} and of its solvable extension.
13.3. Solvable Lie algebras with other filiform nilradicals
13.4. Example of an almost filiform nilradical
13.4.1. Automorphisms and derivations of the nilradical n_{n,3}.
13.4.2. Construction of solvable Lie algebras with the nilradical n_{n,3}.
13.4.3. Dimension n = 6.
13.4.4. Dimension n = 5.
13.5. Generalized Casimir invariants of nn,3 and of its solvable extensions
Chapter 14. Levi Decomposable Algebras
14.1. Levi decomposable algebras with a nilpotent radical
14.2. Levi decomposable algebras with nonnilpotent radicals
14.3. Levi decomposable algebras of low dimensions
14.3.1. Levi extensions of Abelian radicals.
14.3.2. Levi extensions of non-Abelian radicals with Abelian nilradicals.
14.3.3. Levi decomposable algebras with non-Abelian nilradicals.
Part 4. Low-Dimensional Lie Algebras
Chapter 15. Structure of the Lists ofLow-Dimensional Lie Algebras
15.1. Ordering of the lists
15.2. Computer-assisted identification of a given Lie algebra
Chapter 16. Lie Algebras up to Dimension 3
16.1. One-dimensional Lie algebra
16.2. Solvable two-dimensional Lie algebra with the nilradical n_{1,1}
16.3. Nilpotent three-dimensional Lie algebra
16.4. Solvable three-dimensional Lie algebras with the nilradical 2n_{1,1}
16.5. Simple three-dimensional Lie algebras
Chapter 17. Four-Dimensional Lie Algebras
17.1. Nilpotent four-dimensional Lie algebra
17.2. Solvable four-dimensional algebras with the nilradical 3n_{1,1}
17.3. Solvable four-dimensional Lie algebras with the nilradical n_{3,1}
17.4. Solvable four-dimensional Lie algebras with the nilradical 2n_{1,1}
Chapter 18. Five-Dimensional Lie Algebras
18.1. Nilpotent five-dimensional Lie algebras
18.2. Solvable five-dimensional Lie algebras with the nilradical 4n_{1,1}
18.3. Solvable five-dimensional Lie algebras with the nilradical n_{3,1} ⊕ n_{1,1}
18.4. Solvable five-dimensional Lie algebras with the nilradical n_{4,1}
18.5. Solvable five dimensional Lie algebras with the nilradical 3n_{1,1}
18.6. Solvable five-dimensional Lie algebras with the nilradical n_{3,1}
Chapter 19. Six-Dimensional Lie Algebras
19.1. Nilpotent six-dimensional Lie algebras
19.2. Solvable six-dimensional Lie algebras with the nilradical 5n_{1,1}
19.3. Solvable six-dimensional Lie algebras with the nilradical n_{3,1} ⊕ 2n_{1,1}
19.4. Solvable six-dimensional Lie algebras with the nilradical n_{4,1} ⊕ n_{1,1}
19.5. Solvable six-dimensional Lie algebras with the nilradical n_{5,1}
19.6. Solvable six-dimensional Lie algebras with the nilradical n_{5,2}
19.7. Solvable six-dimensional Lie algebras with the nilradical n_{5,3}
19.8. Solvable six-dimensional Lie algebras with the nilradical n_{5,4}
19.9. Solvable six-dimensional Lie algebras with the nilradical n_{5,5}
19.10. Solvable six-dimensional Lie algebra with the nilradical n_{5,6}
19.11. Solvable six-dimensional Lie algebras with the nilradical 4n_{1,1}
19.12. Solvable six-dimensional Lie algebras with the nilradical n_{3,1} ⊕ n_{1,1}
19.13. Solvable six-dimensional Lie algebra with the nilradical n_{4,1}
Bibliography
Index
Book Presentation by Authors
The monograph: Classification and Identification of Lie Algebras
Why to be interested in identification and classification of Lie algebras?
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Classification and Identification of Lie Algebras
Volume 33
C R M
CRM Monograph Series Centre de Recherches Mathématiques Montréal
Classification and Identification of Lie Algebras ˇ nobl Libor S Pavel Winternitz The Centre de Recherches Mathématiques (CRM) was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM receives funding from the Natural Sciences and Engineering Research Council (Canada), the FRQNT (Québec), the NSF (USA), and its partner universities (Université de Montréal, McGill, UQAM, Concordia, Université Laval, Université de Sherbrooke and University of Ottawa). It is affiliated with the lnstitut des Sciences Mathématiques (ISM). For more information visit www.crm.math.ca.
American Mathematical Society Providence, Rhode Island USA
The production of this volume was supported in part by the Fonds de recherche du Qu´ebec–Nature et technologies (FRQNT) and the Natural Sciences and Engineering Research Council of Canada (NSERC). 2010 Mathematics Subject Classification. Primary 17Bxx, 17B05 81Rxx, 81R05, 70Hxx, 37J15; Secondary 17B20, 17B30, 17B40, 70Sxx, 37Jxx.
For additional information and updates on this book, visit www.ams.org/bookpages/crmm-33
Library of Congress Cataloging-in-Publication Data ˇ Snobl, Libor, 1976- author. ˇ Classification and identification of Lie algebras / Libor Snobl, Pavel Winternitz. pages cm. – (CRM monograph series ; volume 33) Includes bibliographical references and index. ISBN 978-0-8218-4355-0 (alk. paper) 1. Lie algebras. 2. Lie superalgebras. I. Winternitz, Pavel, author. II. Title. QA252.3.S66 2014 512.482–dc23
2013034225
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1
19 18 17 16 15 14
Contents Preface
ix
Acknowledgements
xi
Part 1.
General Theory
1
Chapter 1. Introduction and Motivation
3
Chapter 2. Basic Concepts 2.1. Definitions 2.2. Levi theorem 2.3. Classification of complex simple Lie algebras 2.4. Chevalley cohomology of Lie algebras
11 11 17 17 20
Chapter 3. Invariants of the Coadjoint Representation of a Lie Algebra 3.1. Casimir operators and generalized Casimir invariants 3.2. Calculation of generalized Casimir invariants using the infinitesimal method 3.3. Calculation of generalized Casimir invariants by the method of moving frames
23 23
32
Part 2. Recognition of a Lie Algebra Given by Its Structure Constants
37
Chapter 4. Identification of Lie Algebras through the Use of Invariants 4.1. Elementary invariants 4.2. More sophisticated invariants
39 39 42
Chapter 5. Decomposition into a Direct Sum 5.1. General theory and criteria 5.2. Algorithm 5.3. Examples
47 47 56 57
Chapter 6. Levi Decomposition. Identification of the Radical and Levi Factor 6.1. Original algorithm 6.2. Modified algorithm 6.3. Examples
63 63 65 66
Chapter 7. The Nilradical of a Lie Algebra 7.1. General theory 7.2. Algorithm
71 71 75
v
24
vi
CONTENTS
7.3. 7.4. Part 3.
Examples Identification of the nilradical using the Killing form
79 84
Nilpotent, Solvable and Levi Decomposable Lie Algebras
87
Chapter 8. Nilpotent Lie Algebras 8.1. Maximal Abelian ideals and their extensions 8.2. Classification of low-dimensional nilpotent Lie algebras
89 89 93
Chapter 9. Solvable Lie Algebras and Their Nilradicals 99 9.1. General structure of a solvable Lie algebra 99 9.2. General procedure for classifying all solvable Lie algebras with a given nilradical 99 9.3. Upper bound on the dimension of a solvable extension of a given nilradical 103 9.4. Particular classes of nilradicals and their solvable extensions 105 9.5. Vector fields realizing bases of the coadjoint representation of a solvable Lie algebra 106 Chapter 10. Solvable Lie Algebras with Abelian Nilradicals 107 10.1. Basic structural theorems 107 10.2. Decomposability properties of the solvable Lie algebras 114 10.3. Solvable Lie algebras with centers of maximal dimension 116 10.4. Solvable Lie algebras with one nonnilpotent element and an n-dimensional Abelian nilradical 121 10.5. Solvable Lie algebras with two nonnilpotent elements and n-dimensional Abelian nilradical 123 10.6. Generalized Casimir invariants of solvable Lie algebras with Abelian nilradicals 125 Chapter 11. Solvable Lie Algebras with Heisenberg Nilradical 11.1. The Heisenberg relations and the Heisenberg algebra 11.2. Classification of solvable Lie algebras with nilradical h(m) 11.3. The lowest dimensional case m = 1 11.4. The case m = 2 11.5. Generalized Casimir invariants
131 131 132 134 135 136
Chapter 12. Solvable Lie Algebras with Borel Nilradicals 12.1. Outer derivations of nilradicals of Borel subalgebras 12.2. Solvable extensions of the Borel nilradicals NR b(g) 12.3. Solvable Lie algebras with triangular nilradicals 12.4. Casimir invariants of nilpotent and solvable triangular Lie algebras
141 141 146 153 162
Chapter 13. Solvable Lie Algebras with Filiform and Quasifiliform Nilradicals175 13.1. Classification of solvable Lie algebras with the model filiform nilradical nn,1 176 13.2. Classification of solvable Lie algebras with the nilradical nn,2 182 13.3. Solvable Lie algebras with other filiform nilradicals 189 13.4. Example of an almost filiform nilradical 190 13.5. Generalized Casimir invariants of nn,3 and of its solvable extensions199
CONTENTS
vii
Chapter 14. Levi Decomposable Algebras 14.1. Levi decomposable algebras with a nilpotent radical 14.2. Levi decomposable algebras with nonnilpotent radicals 14.3. Levi decomposable algebras of low dimensions
203 204 207 208
Part 4.
215
Low-Dimensional Lie Algebras
Chapter 15. Structure of the Lists of Low-Dimensional Lie Algebras 15.1. Ordering of the lists 15.2. Computer-assisted identification of a given Lie algebra
217 217 218
Chapter 16. Lie Algebras up to Dimension 3 16.1. One-dimensional Lie algebra 16.2. Solvable two-dimensional Lie algebra with the nilradical n1,1 16.3. Nilpotent three-dimensional Lie algebra 16.4. Solvable three-dimensional Lie algebras with the nilradical 2n1,1 16.5. Simple three-dimensional Lie algebras
225 225 225 225 226 226
Chapter 17. Four-Dimensional Lie Algebras 17.1. Nilpotent four-dimensional Lie algebra 17.2. Solvable four-dimensional algebras with the nilradical 3n1,1 17.3. Solvable four-dimensional Lie algebras with the nilradical n3,1 17.4. Solvable four-dimensional Lie algebras with the nilradical 2n1,1
227 227 227 228 229
Chapter 18. Five-Dimensional Lie Algebras 18.1. Nilpotent five-dimensional Lie algebras 18.2. Solvable five-dimensional Lie algebras with the nilradical 18.3. Solvable five-dimensional Lie algebras with the nilradical 18.4. Solvable five-dimensional Lie algebras with the nilradical 18.5. Solvable five dimensional Lie algebras with the nilradical 18.6. Solvable five-dimensional Lie algebras with the nilradical 18.7. Five-dimensional Levi decomposable Lie algebra
231 231 4n1,1 232 n3,1 ⊕ n1,1 235 n4,1 239 3n1,1 240 n3,1 241 241
Chapter 19. Six-Dimensional Lie Algebras 243 19.1. Nilpotent six-dimensional Lie algebras 243 19.2. Solvable six-dimensional Lie algebras with the nilradical 5n1,1 248 19.3. Solvable six-dimensional Lie algebras with the nilradical n3,1 ⊕ 2n1,1 253 19.4. Solvable six-dimensional Lie algebras with the nilradical n4,1 ⊕ n1,1 266 19.5. Solvable six-dimensional Lie algebras with the nilradical n5,1 271 19.6. Solvable six-dimensional Lie algebras with the nilradical n5,2 277 19.7. Solvable six-dimensional Lie algebras with the nilradical n5,3 279 19.8. Solvable six-dimensional Lie algebras with the nilradical n5,4 283 19.9. Solvable six-dimensional Lie algebras with the nilradical n5,5 285 19.10. Solvable six-dimensional Lie algebra with the nilradical n5,6 286 19.11. Solvable six-dimensional Lie algebras with the nilradical 4n1,1 286 19.12. Solvable six-dimensional Lie algebras with the nilradical n3,1 ⊕ n1,1 293 19.13. Solvable six-dimensional Lie algebra with the nilradical n4,1 296 19.14. Simple six-dimensional Lie algebra 296 19.15. Six-dimensional Levi decomposable Lie algebras 296
viii
CONTENTS
Bibliography
299
Index
305
Preface The purpose of this book is to serve as a tool for practitioners of Lie algebra and Lie group theory, i.e., for those who apply Lie algebras and Lie groups to solve problems arising in science and engineering. It is not intended to be a textbook on Lie theory, nor is it oriented towards one specific application, for instance the analysis of symmetries of differential equations. We restrict our attention to finitedimensional Lie algebras over the fields of complex and real numbers. In any application Lie algebras typically arise as sets of linear operators that commute with a given operator, say the Hamiltonian of a physical system. Alternatively, Lie groups arise as groups of (local) transformations leaving some object invariant; the corresponding Lie algebra then consists of vector fields generating 1-parameter subgroups. The object may be for instance the set of all solutions of a system of equations. The equations can be differential, difference, algebraic or integral ones, or some combination of such equations. They may be linear or nonlinear. In any case, the Lie algebra is realized by some operators in a basis that is usually not the standard one and that depends crucially on the manner in which it was obtained. The structure constants of Lie algebras can be calculated in any basis, but they in turn are basis dependent and reveal very little about the actual structure of the given Lie algebra. After the Lie algebra g associated with a studied problem is found, the next task that faces the researcher is to identify the Lie algebra as an abstract Lie algebra. In some cases g may be isomorphic to a known algebra given in some accessible list. This is certainly the case for semisimple Lie algebras in view of Cartan’s classification of all simple Lie algebras over the complex numbers, and subsequent classification of their real forms. The fundamental Levi theorem, stating that every finite dimensional Lie algebra is isomorphic to a semidirect sum of a semisimple Lie algebra and the maximal solvable ideal (the radical) greatly simplifies the task of identifying a given Lie algebra. The weak link is that no complete classification of solvable Lie algebras exists, nor can one be expected to be produced in the future. The problem addressed in this book is that of transforming a randomly obtained basis of a Lie algebra into a “canonical basis” in which all basis independent features of the Lie algebra are directly visible. For low dimensional Lie algebras (of dimension less or equal six) this makes it possible to identify the Lie algebra completely. In this book we give a representative list of all such Lie algebras. As stated above, in any dimension a complete identification can be performed for semisimple Lie algebras. We also describe some classes of nilpotent and solvable Lie algebras of arbitrary finite dimensions for which a complete classification exists and hence an exact identification is possible.
ix
x
PREFACE
The book has four parts. The first presents some general results and concepts that are used in the subsequent chapters. In particular such invariant notions as the dimension of ideals in the characteristic series, and the invariants of the coadjoint representation are introduced. In Part 2 we present algorithms that accomplish the following tasks: (1) An algorithm for determining whether the algebra g can be decomposed into a direct sum. If g is decomposable the algorithm provides a basis in which g is explicitly decomposed into a direct sum of indecomposable Lie subalgebras. (2) A further algorithm is presented to find the radical R(g) and the Levi factor, i.e., the semisimple component of g. (3) If the Lie algebra is solvable, for instance if it is the radical of a larger algebra, then it is necessary to identify its nilradical, i.e., the maximal nilpotent ideal. A rational (i.e., avoiding calculation of eigenvalues) algorithm for performing this is presented. The text includes many examples illustrating various situations that may arise in such computations. All these algorithms have been implemented on computers. Part 3 is devoted to solvable and nilpotent Lie algebras. While a complete classification of such algebras seems not to be feasible, it is possible to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones. Finite-dimensional solvable Lie algebras with Abelian, Heisenberg, Borel, filiform and quasifiliform nilradicals are presented in Part 3. Part 4 of the book consists of tables of all indecomposable Lie algebras of dimension n where 1 ≤ n ≤ 6. They are ordered in such a way as to make the identification of any given low-dimensional Lie algebra written in an arbitrary basis as simple as possible. Any Lie algebra up to dimension 6 is isomorphic to precisely one entry in the tables. Essential characteristics of each algebra including its Casimir invariants are also provided. The book is based on material that was previously dispersed in journal articles, many of them written by one or both of the authors of this book together with collaborators. The tables in Part 4 are based on older results and have been independently verified, in some cases corrected, unified and ordered by structural properties of the algebras (rather than by the way they were originally obtained). Libor Šnobl and Pavel Winternitz
Acknowledgements Our research was funded by multiple sources during the years it took us to write this book. Research of L. Šnobl was supported by the postdoctoral fellowship of the Centre de recherches mathématiques, Université de Montréal in 2004 – 2006. Next, his research at Czech Technical University in Prague was funded mainly by the research plans MSM210000018 and MSM6840770039 of the Ministry of Education of the Czech Republic. The research of P. Winternitz was partly supported by research grants from NSERC of Canada. LŠ thanks the Centre de recherches mathématiques, Université de Montréal for hospitality during numerous visits there while working on the manuscript. PW thanks the Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague for hospitality during his visits there. We thank Professors I. Anderson, A. G. Elashvili, J. Patera, Dr. A. Bihlo and D. Karásek for interesting and helpful discussions. We also thank all our colleagues and students with whom we collaborated on the results that are presented in this book. We are particularly indebted to A. Montpetit for his efficient help in transforming our manuscript into a publishable book. We also thank Ms. I. Mette of AMS Book Acquisitions for keeping us on track. L. Šnobl dedicates this book to his parents Libuše and Zdeněk. P. Winternitz dedicates this book to his wife Milada and his sons Peter and Michael. We both thank them for their support and encouragement.
xi
Part 1
General Theory
CHAPTER 1
Introduction and Motivation Lie groups and Lie algebras appear in science in many different guises. They may be a priori parts of the theory, like Lorentz or Galilei invariance of most physical theories, or the (semi)simple Lie groups of the Standard model in particle theory. Alternatively, specific Lie groups may appear as consequences of specific dynamics. Consider any physical system with dynamics described by a system of ordinary or partial differential equations. This system of equations will be invariant under some local Lie group of local point transformations, taking solutions into solutions. This symmetry group G and its Lie algebra g can be determined in an algorithmic manner [86]. The Lie algebra g is obtained as an algebra of vector fields, usually in some nonstandard basis, depending on the way in which the algorithm is applied. An immediate task is to identify the algebra found as being isomorphic to some known abstract Lie algebra. To do this we must transform it to a canonical basis in which all basis independent properties are manifest. Thus, if g is decomposable into a direct sum, it should be explicitly decomposed into components that are further indecomposable g = g1 ⊕ g 2 ⊕ · · · ⊕ g k .
(1.1)
Each indecomposable component must be further identified. Let g now denote such an indecomposable Lie algebra. A fundamental theorem due to E. E. Levi [59, 71] tells us that any finite-dimensional Lie algebra can be represented as the semidirect sum (1.2)
g = p r,
[p, p] = p,
[r, r] ⊂ r,
[p, r] ⊆ r,
where the subalgebra p is semisimple and r is the radical of g, i.e., its maximal solvable ideal. If g is simple, we have r = 0 (an indecomposable semisimple algebra is actually simple). If g is solvable, we have p = 0. Algorithms realizing decompositions (1.1), (1.2) exist [102] and are presented below. In view of the Levi theorem (1.2) the classification of all finite-dimensional Lie algebras can be reduced to three steps: (1) Classification of all simple Lie algebras. This also provides a classification of all semisimple ones. (2) Classification of all solvable Lie algebras. (3) Classification of all possible linear actions of the semisimple algebra p on the radical r. Semisimple Lie algebras over the field of complex numbers C have been completely classified by W. Killing and É. Cartan [22, 60], over the field of real numbers R by É. Cartan in [20, 21] (the analysis in the real case was later simplified by F. Gantmacher in [50]). 3
4
1. INTRODUCTION AND MOTIVATION
The third step is basically a task of the representation theory of semisimple Lie algebras. The main unsolved problem is the classification of all finite dimensional solvable Lie algebras, a task that does not have a realistic solution. Even all nilpotent Lie algebras are impossible to classify. (For instance in [127] the authors claim that they have found 24 168 nonisomorphic 9-dimensional nilpotent algebras Lie algebras with a maximal Abelian ideal of dimension 7 alone.) A realistic partial classification problem is to classify all solvable Lie algebras with a given nilradical of an arbitrary finite dimension n. So far this has been done for certain series of nilpotent Lie algebras, namely Abelian, Heisenberg and Borel nilradicals, as well as certain filiform and quasifiliform algebras. The results of the original articles [83, 84, 106, 115–118, 123, 124] are presented in a unified manner in Part 3 of this book. An interesting physical application of the classification of low-dimensional Lie algebras is in general relativity. Indeed, the classification of Einstein spaces according to their isometry groups [95] is based on the work of Bianchi and his successors [6, 65]. The Petrov classification concerns Einstein spaces of dimension 4 and hence involves isometry groups of relatively low dimensions [95, 120]. String theory, brane cosmology and some other elementary particle theories going beyond the standard model require the use of higher-dimensional spaces. Any attempt at a Lie group classification of such spaces will require knowledge of higher-dimensional Lie groups, including solvable ones. One of the very useful applications of Lie group analysis in science is the identification of seemingly different problems that are mathematically equivalent. Indeed, let us consider two systems, say (A) and (B), of differential equations 2 U ; ∂X Uα , ∂X EaA (X, Uα , . . . ) = 0, i i Xj
EaB (x, u; ∂xi uα , ∂x2i xj uα , . . . ) = 0, (1.3)
= (X1 , . . . , Xp ), X
= (U1 , . . . , Uq ), U
u = (u1 , . . . , uq ), x = (x1 , . . . , xp ), 1 ≤ a ≤ N, 1 ≤ α ≤ q, 1 ≤ i, j, . . . ≤ p of the same order describing different physical processes. From the mathematical viewpoint, we may consider these two systems equivalent if there exists a local invertible transformation of the independent and dependent variables (1.4)
Xi = Λi (x, u),
Uα = Ωα (x, u)
transforming the systems (A) and (B) into each other. The transformation (1.4) (X) induced together with the mapping between the spaces of functions u(x) and U by it is called a point transformation. A necessary condition for the equivalence of the systems (A) and (B) is that they have isomorphic Lie algebras of infinitesimal point symmetries gA and gB . The algebras gA and gB are by construction realized by vector fields in variables U ) and (x, u), respectively. (X, If gA and gB are isomorphic then a local point transformation (1.4) may exist such that it transforms the vector fields of gA into those of gB and vice versa. The transformation is unique up to point transformations leaving gA or gB invariant. In any case, the transformation taking gA into gB will also take the system (A)
1. INTRODUCTION AND MOTIVATION
5
into a class of systems with the symmetry algebra gB , which includes (B). If the symmetry algebra gB is large enough, the family of systems with the symmetry algebra gB is significantly restricted and often contains only the system (B). In this case the systems (A) and (B) are necessarily equivalent. Let us consider an example of this procedure which will also demonstrate the usefulness of the identification algorithms presented in this book. The system (A) in our example will be the shallow water equations UT + U UX + V UY + HX = 0, VT + U VX + V VY + HY = 0,
(1.5)
HT + (U H)X + (V H)Y = 0 describing the motion of an incompressible fluid in shallow water without boundaries in a gravitational field perpendicular to the flat bottom of the infinite basin [92]. The dependent variables are U, V , the two components of the fluid velocity and H, the depth. The independent variables are the time T and the Cartesian spatial coordinates X, Y . In our mathematical analysis we have set the gravitational constant g = 1 in (1.5). The system (B) consists of the rotating shallow water equations for a fluid in a finite curved basin subject to the force of gravity (perpendicular to the surface at rest), and to a Coriolis force due to the rotation of the fluid. The equations are [68] ut + uux + vuy + (Z + h)x = f v, vt + uvx + vvy + (Z + h)y = −f u,
(1.6)
ht + (uh)x + (vh)y = 0. The dependent and independent variables have the same interpretation as their uppercase counterparts in system (1.5), f corresponds to the Coriolis force, Z(x, y) = A(x2 + y 2 )
(1.7)
describes the form of the bottom of the basin containing the fluid. The basin is chosen as a circular paraboloid. Instead of the parameter A it is convenient to introduce an angular frequency ω such that ω2 − f 2 A= 8 and denote R2 = 12 (ω − f ).
R1 = 12 (ω + f ),
By an explicit algorithmic computation one arrives at the generators of the point symmetries of these two sets of equations (1.5) and (1.6). For the free liquid (1.5) one finds the generators [27] PT = ∂T , (1.8)
PX = ∂X ,
PY = ∂Y ,
D1 = T ∂T + X∂X + Y ∂Y ,
GX = T ∂X + ∂U ,
GY = T ∂Y + ∂V ,
D2 = −T ∂T + U ∂U + V ∂V + 2H∂H ,
L1 = −Y ∂X + X∂Y − V ∂U + U ∂V , Π = T ∂T + T X∂X + T Y ∂Y + (X − T U )∂U + (Y − T V )∂V − 2T H∂H . 2
6
1. INTRODUCTION AND MOTIVATION
Their Lie brackets define the following structure of a Lie algebra gA
PT PX PY GX GY D1 D2 L1 Π
PT 0 0 0 −PX −PY −PT PT 0 −D1 + D2
PX 0 0 0 0 0 −PX 0 −PY −GX
PY 0 0 0 0 0 −PY 0 PX −GY
GX PX 0 0 0 0 0 −GX −GY 0
GY PY 0 0 0 0 0 −GY GX 0
D1 PT PX PY 0 0 0 0 0 −Π
D2 −PT 0 0 GX GY 0 0 0 Π
L1 0 PY −PX GY −GX 0 0 0 0
Π D1 − D2 GX GY 0 0 Π −Π 0 0
The radical of the Lie algebra gA is spanned by L1 , D1 + D2 , GX , GY , PX , PY , the nilradical by PX , PY , GX , GY and the Levi factor by PT , D1 − D2 , Π. Thus the choice of basis of the vector space spanned by the generators (1.8) is not the most natural one from the point of view of the structure of the algebra gA . When we introduce a new basis
(1.9)
e1 = PT , e2 = D1 − D2 , e3 = −Π, e4 = −(D1 + D2 ), e5 = L1 , e6 = PY , e7 = PX , e8 = GY , e9 = GX ,
the structure of the Lie algebra gA , i.e., the Levi decomposition, the nilradical and the properties of the action of the radical and of the Levi factor on the nilradical, become immediately obvious. Namely, the Lie brackets read
e1 e2 e3 e4 e5 e6 e7 e8 e9
e1 0 −2e1 e2 0 0 0 0 −e6 −e7
e2 2e1 0 −2e3 0 0 e6 e7 −e8 −e9
e3 −e2 2e3 0 0 0 −e8 −e9 0 0
e4 0 0 0 0 0 −e6 −e7 −e8 −e9
e5 0 0 0 0 0 −e7 e6 −e9 e8
e6 0 −e6 e8 e6 e7 0 0 0 0
e7 0 −e7 e9 e7 −e6 0 0 0 0
e8 e6 e8 0 e8 e9 0 0 0 0
e9 e7 e9 0 e9 −e8 0 0 0 0
The Levi factor is spanned by e1 , e2 , e3 with the standard Lie brackets of the algebra sl(2, R). The radical is spanned by e4 , e5 , e6 , e7 , e8 , e9 , the nilradical by e6 , e7 , e8 , e9 , the basis elements e4 , e5 ∈ R(gA )\ NR(gA ) commute with the Levi factor and act on the nilradical as a dilation and a rotation, respectively.
1. INTRODUCTION AND MOTIVATION
7
Similarly, one finds the Lie algebra of generators of point symmetries also for the rotating fluid in a circular paraboloidal basin (1.6). The generators read [68] P0 = ∂t , D = x∂x + y∂y + u∂u + v∂v + 2h∂h , Y1 = cos(R1 t)∂x − sin(R1 t)∂y − R1 sin(R1 t)∂u − R1 cos(R1 t)∂v , Y2 = sin(R1 t)∂x + cos(R1 t)∂y + R1 cos(R1 t)∂u − R1 sin(R1 t)∂v , Y3 = cos(R2 t)∂x + sin(R2 t)∂y − R2 sin(R2 t)∂u + R2 cos(R2 t)∂v , Y4 = sin(R2 t)∂x − cos(R2 t)∂y + R2 cos(R2 t)∂u + R2 sin(R2 t)∂v , R = y∂x − x∂y + v∂u − u∂v ,
(1.10)
1 cos(ωt)(x∂x + y∂y − u∂u − v∂v + f (y∂u − x∂v ) − 2h∂h ) 2 1 sin(ωt)(f (y∂x − x∂y + v∂u − u∂v ) − ω 2 (x∂u + y∂v ) + 2∂t ), + 2ω 1 K2 = − sin(ωt)(x∂x + y∂y − u∂u − v∂v + f (y∂u − x∂v ) − 2h∂h ) 2 1 cos(ωt)(f (y∂x − x∂y + v∂u − u∂v ) − ω 2 (x∂u + y∂v ) + 2∂t ). + 2ω Their Lie brackets define the Lie algebra gB (shown in Table 1.1) with the radical K1 =
R(gB ) = span{D, R, Y1 , Y2 , Y3 , Y4 }, the nilradical NR(gB ) = span{Y1 , Y2 , Y3 , Y4 }, f p = span P0 + R, K1 , K2 . 2 From the indefinite signature of the Killing form of the Levi factor p one may conclude that p is isomorphic to the simple algebra sl(2, R). Thus the structure of the algebra gB is rather similar to that of the algebra gA . Their relation is more transparent when gB is written in a basis resembling the basis (e1 , . . . , e9 ) introduced above, i.e., with the standard structure of the Levi factor and nonnilpotent basis elements of the radical chosen commuting with the Levi factor. A (nonunique) change of basis fulfilling these requirements is 1 f 1 f e˜1 = − P0 + R + K2 , e˜2 = −2K1 , e˜3 = P0 + R + K2 , ω 2 ω 2 (1.11) e˜4 = −D, e˜5 = R, e˜6 = Y1 − Y3 , e˜7 = Y2 + Y4 , e˜8 = −Y2 + Y4 , e˜9 = Y1 + Y3 .
and the Levi factor
When the Lie brackets of the basis elements (˜ e1 , . . . , e˜9 ) are computed they turn out to be the same as the Lie brackets of the algebra gA expressed in the basis (e1 , . . . , e9 ). That means that the symmetry algebras gA and gB are isomorphic. The isomorphism between the symmetry algebras is a necessary condition for the existence of a point transformation mapping the system (1.5) to the system (1.6). However, it is far from sufficient. The next step is to look for a point transformation that maps the algebra gB realized as the algebra of vector fields on a manifold M ⊂ R6 with the coordinates t, x, y, u, v, h to the algebra gA of the vector fields on the same manifold with the coordinates T, X, Y, U, V, H. If such a transformation exists, then the two vector realizations gA , gB of the same abstract Lie algebra are equivalent. (Both the
1. INTRODUCTION AND MOTIVATION 8
P0 −ωK2
0
P0
0
ωK2
K1 −ωK1 −1 f P0 + R ω 2
K2
Y4
Y3
Y2
Y1
0 0
0
0
0
D
−Y3
Y4
Y1
−Y2
0 0
0
0
0
R
0
0
0
0
−Y1 Y2
1 Y4 2
Y1 f +ω − Y2 2 1 − Y3 2
0
0
0
0
−Y2 −Y1
1 Y3 2
Y2 f +ω Y1 2 1 Y4 2
0
0
0
0
−Y3 −Y4
1 Y2 2
Y3 f −ω Y4 2 1 − Y1 2
0
0
0
0
−Y4 Y3
1 Y1 2
Y4 ω−f Y3 2 1 Y2 2
Table 1.1
K1 ωK1
0 0 1 − Y4 2 1 − Y3 2 1 − Y2 2 1 − Y1 2
0
K2 0 0 1 Y3 2 1 − Y4 2 1 Y1 2 1 − Y2 2
1 f P0 + R ω 2
D R
Y3
Y2
Y1
0 0 f +ω Y2 2 f +ω Y1 − 2 ω−f Y4 2 f −ω Y3 2 Y4
1. INTRODUCTION AND MOTIVATION
9
transformation and the equivalence can be local, defined in a neighborhood of a generic point of M only). Using computer algebra we find a locally invertible map φ (1.12)
X = φX (t, x, y, u, v, h),
Y = φY (t, x, y, u, v, h),
T = φT (t, x, y, u, v, h),
U = φU (t, x, y, u, v, h),
V = φV (t, x, y, u, v, h),
H = φH (t, x, y, u, v, h)
ej ). Explicitly it reads such that ej = φ∗ (˜ ω t , T = cot 2 f f 1 cos t x − sin t y , X= 2 2 2 sin (ω/2)t f 1 f t x + cos t y , Y =− sin 2 2 2 sin (ω/2)t 1 ω f ω f U= −2 sin t cos t u + 2 sin t sin t v 2ω 2 2 2 2 ω f f ω + sin t sin t f + cos t cos t ω x 2 2 2 2 (1.13) f f ω ω t cos t f − sin t cos t ω y , + sin 2 2 2 2 ω f ω f 1 2 sin t sin t u + 2 sin t cos t v V = 2ω 2 2 2 2 f f ω ω t cos t f − sin t cos t ω x + sin 2 2 2 2 f f ω ω t sin t f + cos t cos t ω y , − sin 2 2 2 2 2 ω H = Ch sin t , 2 where C is an arbitrary nonvanishing integration constant. Since the symmetry algebras of the systems of equations (1.5) and (1.6) are equivalent, it is possible that the two models are related by a point transformation. In order to verify that we perform an explicit change of independent and dependent variables (1.13) in the PDE system (1.5) and we find that we indeed recover the system of PDEs (1.6) provided that we set the integration constant C in (1.13) to 1 C = 2. ω Thus, the two dynamical systems (1.5) and (1.6) can be (locally) transformed one into the other by a point transformation and any solution of (1.5) gives rise to a (local) solution of (1.6) and vice versa. The point transformation (1.13) is highly nontrivial and its existence and explicit form would be very difficult to obtain without first noticing the isomorphism between the symmetry algebras gA and gB . We mention that the physical interpretations of these two mathematically isomorphic algebras are quite different. For instance the nilradical of gA corresponds to translations and Galilei transformations in the x and y directions. The nilradical of gB corresponds to transformations related to the Coriolis force and the presence
10
1. INTRODUCTION AND MOTIVATION
of the boundary (1.7). The physical interpretations of the sl(2, R) Levi factors of gA and gB are also different.
CHAPTER 2
Basic Concepts The purpose of this chapter is to establish the notation and clarify the definitions used in the following text. Our aim here is not a comprehensive introduction to the theory of Lie algebras which can be found in numerous textbooks together with detailed proofs of theorems and propositions, e.g., [41, 46, 59, 62, 72, 108, 109]. 2.1. Definitions A Lie algebra g is a vector space over a field F equipped with a multiplication (also called a Lie bracket), i.e., a bilinear map [ , ] : g × g → g such that (2.1) (2.2)
[y, x] = −[x, y] x, [y, z] + y, [z, x] + z, [x, y] = 0
(antisymmetry) (Jacobi identity)
for all elements x, y, z ∈ g. We shall consider the fields F = R, C only. A subalgebra h of the Lie algebra g is a vector subspace of g which is closed under the bracket, [h, h] ⊆ h.
(2.3)
An ideal i of the Lie algebra g is a subalgebra such that [i, g] ⊆ i.
(2.4)
The Lie algebra g itself and {0} are trivial ideals. A Lie algebra which does not possess any nontrivial ideal is called simple. Lemma 2.1. If i, j ⊂ g are ideals, so is [i, j] = span{[a, b] | a ∈ i, b ∈ j}. Proof. Using
x, [a, b] = [x, a], b + a, [x, b] ∈ [i, j]
for all x ∈ g, a ∈ i, b ∈ j. Thus [x, y] ∈ [i, j] holds also for any y ∈ span{[a, b] | a ∈ i, b ∈ j}. Three different series of ideals can be associated with any given Lie algebra. The dimensions of the ideals in each of these series are important characteristic features of the given Lie algebra. The derived series g = g(0) ⊇ g(1) ⊇ · · · ⊇ g(k) ⊇ · · · is defined recursively (2.5)
g(k) = [g(k−1) , g(k−1) ],
g(0) = g.
The second term in the series, namely g(1) = [g, g], is called the derived algebra of g and may be also denoted D(g) or g2 . If the derived series terminates, i.e., there exists k ∈ N such that g(k) = 0, then g is called a solvable Lie algebra. If the derived series contains only the algebra g itself, i.e., g(1) = g, then the Lie algebra g is called perfect. 11
12
2. BASIC CONCEPTS
The lower central series g = g1 ⊇ · · · ⊇ gk ⊇ · · · is again defined recursively gk = [gk−1 , g],
(2.6)
g1 = g.
If the lower central series terminates, i.e., there exists k ∈ N such that gk = 0, then g is called a nilpotent Lie algebra. The highest value of k for which we have gk = 0 is the degree of nilpotency of the nilpotent Lie algebra g. Obviously, a nilpotent Lie algebra is also solvable. An Abelian Lie algebra is nilpotent of degree 1. The upper central series is z1 (g) ⊆ · · · ⊆ zk (g) ⊆ · · · ⊆ g. In this series z1 is the center of g (2.7)
z1 (g) = C(g) = {x ∈ g | [x, y] = 0, ∀y ∈ g}.
Now let us consider the factor algebra f1 g/z1 (g). Its center is C(f1 ) = C g/z1 (g) . We define the second center z2 (g) of g to be the unique ideal in g such that (2.8) z2 (g)/z1 (g) = C g/z1 (g) . Recursively we define kth-center zk as the unique ideal in g such that its image under factorization by zk−1 (g) is the center of g/zk−1 (g), i.e., (2.9) zk (g)/zk−1 (g) = C g/zk−1 (g) . The union of the upper central series is the hypercenter ∞
z∞ (g) =
(2.10)
zi (g).
i=1
Since the higher centers zi (g) are ordered by inclusion, the hypercenter can be also viewed as the largest ideal in the sequence and coincides with zk (g) for all k larger than certain k0 ∈ N. The upper central series terminates, i.e., the hypercenter is equal to the whole algebra g, if and only if g is nilpotent [59]. The hypercenter is always a nilpotent ideal in g because we have k+1 zk (g) = zk (g), [zk (g), . . . , zk (g)] ⊆ g, [g, . . . , [g, zk (g)] . . . = 0 k + 1 times
k times
for any k ∈ N by construction. We shall call these three series of ideals the characteristic series of the algebra g. We shall use the notations DS, CS and US for sets of integers denoting the dimensions of subalgebras in the derived, lower central and upper central series, respectively. Corollary 2.2 (of Lemma 2.1). Let i ⊂ g be an ideal, so are ik and i(k) , k ∈ N. Lemma 2.3. For any Lie algebra g, • the sum of any two solvable ideals of g is a solvable ideal of g, • the sum of any two nilpotent ideals of g is a nilpotent ideal of g. Proof. See [59, 62, 139].
Thus in any given Lie algebra we have two unique ideals, the radical and the nilradical. The radical is the maximal solvable ideal and we shall denote it by R(g).
2.1. DEFINITIONS
13
It is unique because it is a sum of all solvable ideals. Similarly, the nilradical NR(g) is the maximal nilpotent ideal. The following relation between them holds 2 (2.11) R(g) ⊆ NR(g) ⊆ R(g). Both of them can vanish, then the algebra is semisimple. When the radical 2 of g is nonvanishing, also the nilradical is nonvanishing — either R(g) = 0, i.e., 2 both NR(g) and R(g) are Abelian and therefore coincide, or 0 = R(g) ⊆ NR(g). Lemma 2.4. For any subalgebra h of the Lie algebra g we have (2.12)
NR(g) ∩ h ⊆ NR(h),
(2.13)
R(g) ∩ h ⊆ R(h).
Notice that the equality sign in (2.12), instead of the ⊆ sign would, in general, not be appropriate. Indeed, consider the example of the two-dimensional nonAbelian Lie algebra span{e1 , e2 } with commutation relation (2.14)
[e1 , e2 ] = e1 .
The nilradical is NR(g) = span{e1 }. Consider the subalgebra s = span{e2 }. We have NR(g) ∩ s = {0} NR(s) = s. In relation to the nilradical the hypercenter satisfies z∞ (g) ⊆ NR(g).
(2.15)
The radical of a Lie algebra g is characterized by means of repeated nilradical formation as follows [139]: NR(g) ≡ (NR)1 (g) ⊆ (NR)2 (g) ⊆ (NR)3 (g) ⊆ · · · ⊆ (NR)∞ (g), (NR)i+1 (g)/(NR)i (g) = NR g/(NR)i (g) (i ∈ N),
NRi (g) ≡ (NR)∞ (g) = R(g). i∈N
We use lower indices for this repeated nilradical formation to avoid confusion with the lower central series of equation (2.6). There is always a first index j = j(g) for which (2.16)
(NR)j (g) = R(g)
and either j = 1 or else j > 1 with (NR)j−1 (g) R(g). The centralizer centg (h) of a given subalgebra h ⊆ g in g is the set of all elements in g commuting with all elements in h, i.e., (2.17)
centg (h) = {x ∈ g | [x, y] = 0, ∀y ∈ h}.
The normalizer normg (h) of a given subspace h ⊆ g in g is the set of all elements x in g such that [x, h] is in the subspace h for any h ∈ h, i.e., (2.18)
normg (h) = {x ∈ g | [x, y] ∈ h, ∀y ∈ h}.
When h is a subalgebra then necessarily h ⊆ normg (h). The normalizer of an ideal in g is the whole algebra g. Both the centralizer and the normalizer can be also taken with respect to a subset f ⊂ g, centf (h) = centg (h) ∩ f,
normf (h) = normg (h) ∩ f.
14
2. BASIC CONCEPTS
A representation ρ of a given Lie algebra g on a vector space V is a linear map of g into the space L (V ) of linear operators acting on V ρ : g → L (V ) : x → ρ(x) such that for any pair x, y of elements of g (2.19)
ρ([x, y]) = ρ(x) ◦ ρ(y) − ρ(y) ◦ ρ(x)
holds. When the map ρ is injective, the representation is called faithful. A subspace W of V is called invariant if ρ(g)W = {ρ(x)w| x ∈ g, w ∈ W } ⊆ W. A representation ρ of g on V is • reducible if a proper nonvanishing invariant subspace W of V exists, • irreducible if no nontrivial invariant subspace of V exists, • fully reducible when every invariant subspace W of V has an invariant , i.e., complement W , ρ(g)W ⊆W . (2.20) V =W ⊕W In particular, any irreducible representation is also fully reducible. An important criterion for irreducibility of a given representation is Theorem 2.5 (Schur lemma). Let g be a complex Lie algebra and ρ its representation on a finite-dimensional vector space V . (1) Let ρ be irreducible. Then any operator A on V which commutes with all ρ(x), [A, ρ(x)] = 0, ∀x ∈ g, has the form A = λ1 for some complex number λ. (2) Let ρ be fully reducible and such that every operator A on V which commutes with all ρ(x) has the form A = λ1 for some complex number λ. Then ρ is irreducible. For future reference we recall that the Schur lemma holds also in a more general situation where the representation of a Lie algebra g is replaced by an arbitrary set of operators S acting on the complex vector space V and by (ir)reducibility we understand the (non)existence of a nontrivial subspace invariant under all operators B ∈ S, respectively. In such a context the following corollary to the Schur lemma is useful: Corollary 2.6. Let S be a set of operators acting on a complex vector space V such that every traceless operator A on V which commutes with all B ∈ S is nilpotent. Then the vector space V cannot be written as a direct sum of two nontrivial subspaces invariant under S. A particular representation is defined for any Lie algebra g, namely the adjoint representation of a given Lie algebra g is a linear map of g into the space of linear operators acting on g ad : g → L (g) : x → ad(x) defined for any pair x, y of elements of g via (2.21)
ad(x) y = [x, y].
The image of ad is denoted by ad g.
2.1. DEFINITIONS
15
A well-known theorem allows us to express nilpotency of a given algebra in terms of operators in the adjoint representation. Theorem 2.7 (Engel theorem). A Lie algebra g is nilpotent if and only if k k ∈ N exists such that ad(x) = 0 for all x ∈ g, i.e., if all ad(x) are nilpotent operators on g. Similarly, the solvability of a given complex algebra can be formulated in terms of operators representing it in any faithful representation. Theorem 2.8 (Lie theorem). Let g be a Lie algebra and ρ its faithful representation on a complex vector space V , n = dim V . The algebra g is solvable if and only if a filtration of V by codimension 1 subspaces (Vk )nk=1 invariant with respect to the representation ρ exists, i.e., (2.22)
0 V1 V2 · · · Vn ≡ V,
(2.23)
ρ(g)Vk ⊂ Vk .
Remark 2.9. When the representation ρ is chosen to be the adjoint representation, the statement of the Lie theorem holds even when it is not faithful. As a direct consequence of the Lie and Engel theorems 2.8 and 2.7 we find that we have (2.24) D R(g) = [R(g), R(g)] ⊆ NR(g). and moreover [R(g), g] ⊆ NR(g). (For proofs of Theorems 2.7, 2.8 and their consequences see e.g. [59, p. 36 – 37, 48 – 51; 62, p. 42 – 49].) An automorphism Φ of a given Lie algebra g is a linear map Φ: g → g such that for any pair x, y of elements of g (2.25)
Φ([x, y]) = [Φ(x), Φ(y)].
A derivation D of a given Lie algebra g is a linear map D: g → g such that for any pair x, y of elements of g (2.26)
D([x, y]) = [D(x), y] + [x, D(y)].
If an element z ∈ g exists, such that D = ad(z),
i.e., D(x) = [z, x], ∀x ∈ g
the derivation D is called an inner derivation, any other one is an outer derivation. The set of all automorphisms of g with composition as the group law forms a Lie group Aut(g) and its Lie algebra coincides with the algebra Der(g) of all derivations of g. An ideal i in g is called a characteristic ideal if it is invariant with respect to every derivation D ∈ Der(g). Theorem 2.10. The radical, nilradical as well as any ideal belonging to the characteristic series of a finite-dimensional Lie algebra g is invariant under any derivation of g.
16
2. BASIC CONCEPTS
Proof. We notice that any of the mentioned ideals is by construction invariant under every automorphism of g. E.g., if the solvable ideals R(g) and Φ R(g) differed for some automorphism Φ ∈ Aut(g), their sum would be a larger solvable ideal in g, thus contradicting the maximality of R(g). For any derivation D ∈ Der(g) and any ideal i invariant under every automorphism we consider the one-parameter family of automorphisms Φ(t) = exp(tD). We have exp(tD)(i) = i,
∀t ∈ F (= R, C),
i.e., also (2.27)
D(i) ⊆ i.
An even stronger result than Theorem 2.10 holds. Namely, any derivation of a solvable Lie algebra maps the algebra into its nilradical (see, e.g., [62, Proposition 1.40] for a proof). A symmetric bilinear form B on a given Lie algebra g such that (2.28)
B(ad(x)y, z) + B(y, ad(x)z) = 0
for any triple x, y, z ∈ g is called ad-invariant or invariant. A symmetric bilinear form B is invariant with respect to automorphisms if (2.29) B Φ(x), Φ(y) = B(x, y) for any automorphism Φ of g and any pair x, y ∈ g. We recall that any form B invariant with respect to automorphisms is also ad-invariant. This statement follows from the fact that (2.29) implies upon differentiation (2.30) B(D(x), y) + B x, D(y) = 0 for any derivation D, in particular for all inner derivations. The converse is not true in general because outer derivations of g may exist or there may be discrete transformations in Aut(g) which are not obtained by exponentiation of elements in Der(g). The Killing form K of a given Lie algebra g is a symmetric bilinear form on g defined by (2.31) K(x, y) = Tr ad(x) · ad(y) . The Killing form is invariant with respect to automorphisms. In the particular case of complex simple Lie algebras, any invariant symmetric bilinear form is a multiple of the Killing form. The Killing form provides the following important criteria for semisimplicity and solvability. Theorem 2.11 (Cartan’s criteria). A Lie algebra g is • semisimple if and only if its Killing form is nondegenerate; • solvable if and only if the restriction of its Killing form to the derived algebra vanishes.
2.3. CLASSIFICATION OF COMPLEX SIMPLE LIE ALGEBRAS
17
2.2. Levi theorem In any Lie algebra g there exists a unique maximal solvable ideal R(g) called the radical as was already mentioned. The radical satisfies R(g) = 0 if and only if g is semisimple. On the other hand the radical R(g) coincides with g if and only if g is solvable. Theorem 2.12 (Levi theorem). Any finite-dimensional Lie algebra g can be decomposed into a semidirect sum g = p R(g)
(2.32)
where the complement p of the radical R(g) in g is a semisimple Lie algebra, isomorphic to the factor algebra g/ R(g). The semisimple Lie algebra p is called the Levi factor or Levi subalgebra of g. A sequel to the Levi theorem is the result of Malcev Theorem 2.13 (Malcev theorem). Any two Levi factors p1 and p2 of the Lie algebra g are isomorphically mapped one into the other by some inner automorphism Φ of the form (2.33)
Φ = exp(ad z)
where z ∈ NR(g). (For proofs see, e.g., [59, 62, 71, 73, 139]). We notice that the derived algebra g2 is necessarily contained in p NR(g) where NR(g) is the nilradical of g. The reason is that for any x ∈ p its adjoint representation ad|R(g) (x) is a derivation of the radical R(g), i.e., maps R(g) to NR(g). 2.3. Classification of complex simple Lie algebras Let us briefly review classical results concerning the classification of simple and semisimple Lie algebras. This classification belongs to the greatest achievements in the theory of Lie algebras. These results were originally obtained by W. Killing [60] and É. Cartan [22]. Let g be a complex Lie algebra. Any nilpotent subalgebra g0 of g coinciding with its normalizer normg (g0 ) is called a Cartan subalgebra. It can be constructed in the following way. Let x ∈ g. Consider the linear operator ad(x) ∈ L (g) and find its generalized nullspace k g0 (x) = lim ker ad(x) . k→∞
When dim g0 (x) is minimal, i.e., dim g0 (x) = min dim g0 (y), y∈g
we call the element x ∈ g regular. Proposition 2.14. Let x ∈ g be a regular element of the complex Lie algebra g. Then g0 (x) is a Cartan subalgebra of g. Any other Cartan subalgebra of g is related to g0 (x) by an automorphism of g.
18
2. BASIC CONCEPTS
Consequently, the dimension of the Cartan subalgebra g0 (x) is independent of the choice of the regular element x and is called the rank of the Lie algebra g. We point out that the proposition holds whether or not g is semisimple, i.e., any complex Lie algebra has a Cartan subalgebra unique up to automorphisms. The uniqueness is lost for real algebras; e.g., a finite number of distinct Cartan subalgebras exists for each finite-dimensional semisimple Lie algebra over R, see [63, 121]. Cartan subalgebras of semisimple algebras have special properties. The Cartan subalgebra g0 of a semisimple Lie algebra is Abelian. In addition, all elements of the Cartan subalgebra are ad-diagonalizable or semisimple, meaning that ad(h) ∈ gl(g) is diagonalizable for every h ∈ g0 . Therefore, there exist common eigenspaces gλ ⊂ g of all operators ad(h), h ∈ g0 and nonvanishing functionals λ ∈ g∗0 such that ad(h)eλ = λ(h) · eλ ,
h ∈ g0 , eλ ∈ gλ
g∗0
(where is the dual space of the vector space g0 .) These functionals λ are called roots of the semisimple Lie algebra g. The collection of all roots is called the root system of the algebra g and denoted by Δ. The diagonalizability of ad(h) implies that g = g0 {gλ | λ ∈ Δ} where stands for a direct sum of vector spaces. It is always possible to introduce an ordering among the roots via a choice of h0 ∈ g0 such that λ(h0 ) = 0 and λ(h0 ) ∈ R for all roots λ. This ordering is not unique but different choices give results equivalent up to automorphism of g. For any pair of roots λ, κ one writes λ > κ if and only if λ(h0 ) > κ(h0 ). Similarly one defines positive roots λ > 0, i.e., λ(h0 ) > 0 and negative roots λ < 0, i.e., λ(h0 ) < 0. The set of all positive roots is denoted Δ+ , the set of negative roots Δ− . We have Δ = Δ+ ∪ Δ− . Simple roots are positive roots which cannot be written as a sum of two positive roots. We denote the set of all simple roots by ΔS . We list the most important properties of the root system Δ and root subspaces gλ of a semisimple complex Lie algebra g: (1) The Killing form K of g when restricted to g0 × g0 is nondegenerate. (2) To any functional λ ∈ g∗0 we can associate a unique element hλ ∈ g0 such that (2.34)
λ(h) = K(hλ , h),
∀h ∈ g0
and we can define a nondegenerate bilinear symmetric form , on g∗0 so that λ, κ = K(hλ , hκ ),
∀λ, κ ∈ g∗0 .
(3) If λ is a root then so is −λ and no other multiple of λ is a root. (4) All root subspaces gλ are 1-dimensional. (5) [gλ , gκ ] = gλ+κ whenever λ, κ and λ + κ are roots. (6) When λ + κ is neither 0 nor a root we have [gλ , gκ ] = 0. (7) [gλ , g−λ ] ⊂ g0 . (8) There is a basis of g consisting of elements of the Cartan subalgebra g0 and of the root subspaces gλ such that the structure constants of g in this basis are integers; such a basis is called the Weyl – Chevalley basis of g and the real form of the Lie algebra g corresponding to this choice of basis is called the split real form of g. (9) Simple roots are linearly independent.
2.3. CLASSIFICATION OF COMPLEX SIMPLE LIE ALGEBRAS
19
(10) Any positive root is a linear combination of simple roots with nonnegative integer coefficients; therefore, the root system Δ is contained in the real subspace of g∗0 spanned by the simple roots, we denote this subspace by h∗ . (11) , defines a real scalar product on h∗ . (12) The whole Lie algebra g is obtained by multiple Lie brackets of root vectors eα where α ∈ ΔS or −α ∈ ΔS . (13) The root system Δ is invariant under all reflections Sλ of the form Sλ (α) = α − 2
α, λ λ, λ, λ
λ, α ∈ Δ;
all such reflections generate a finite group called the Weyl group of the root system Δ. (14) Any root is an image of some simple root under the action of some element of the Weyl group; in particular, it has the same length. It turns out that the structure of a semisimple complex Lie algebra is fully determined up to isomorphism by angles and relative lengths of its simple roots in the Euclidean space h∗ . This information is usually encoded either in the Cartan matrix A = (aκλ ) κ, λ , κ, λ ∈ ΔS aκλ = 2 λ, λ or equivalently in Dynkin diagrams. The Cartan matrix has only integer entries: 2 on the diagonal, 0, −1, −2, −3 off the diagonal. The Dynkin diagram associated with the Cartan matrix is a graph with vertices corresponding to the simple roots where the number of edges connecting the vertices labelled by κ and λ is equal to aκλ aλκ ∈ {0, 1, 2, 3}. Further, one distinguishes graphically between shorter and longer roots either by different symbols for vertices or different types of arrows connecting vertices. We shall use a convention that the arrow goes from the longer root to the shorter one, e.g., a subdiagram of the form u u κ λ implies the following values of the elements of the Cartan matrix aκλ = 2
κ, λ = −2, λ, λ
aλκ = 2
λ, κ = −1. κ, κ
The structure of any root system can be shown to be such that: (1) Simple components gk of a semisimple Lie algebra g correspond to connected subdiagrams of the Dynkin diagram of g. (2) There are no closed loops in Dynkin diagrams. (3) A connected Dynkin diagram is either simply laced meaning that it contains only simple edges and consequently all roots are of the same length, or the corresponding root system contains roots of precisely two different lengths. The fundamental classification result is due to W. Killing [60] and É. Cartan [22] whose computations were later significantly simplified by E. Dynkin [43, 44]. It states that a finite-dimensional complex simple Lie algebra g either belongs to one of the classical series of simple Lie algebras, or is one of the exceptional simple Lie algebras. The classical Lie algebras are
20
2. BASIC CONCEPTS
Table 2.1. Dynkin diagrams of simple Lie algebras. Al :
u α1
u α2
u··· u α3 αl−1
u αl
Bl :
u α1
u··· u u u α2 αl−2 αl−1 αl
Cl :
u α1
u··· u u u α2 αl−2 αl−1 αl uαl−1
Dl :
u α1
u··· u u α2 αl−3 αl−2 uα6
u αl
E6 :
u α1
u α2
u u α3 α4 uα7
u α5
E7 :
u α1
u α2
u α5
u α6
E8 :
u α1
u α2
u u α3 α4 u α8 u u α3 α4
u α5
u α6
F4 :
u α1
u u α2 α3
G2 :
u u α2 α1
u α7
u α4
• sl(l + 1, C) of rank l ≥ 1, also denoted Al , the algebra of traceless (l + 1) × (l + 1) matrices, • so(2l + 1, C) of rank l ≥ 2, also denoted Bl , the algebra of skew-symmetric (2l + 1) × (2l + 1) matrices, • sp(2l, C) of rank l ≥ 3, also denoted Cl , the algebra of 2l×2l matrices skewsymmetric with respect to a nondegenerate antisymmetric form on C2l , • so(2l, C) of rank l ≥ 4, also denoted Dl , the algebra of skew-symmetric 2l × 2l matrices. The five exceptional algebras are denoted by E6 , E7 , E8 , F4 , G2 . Out of these algebras, the algebras Al , Dl , E6 , E7 , E8 are simply laced. The corresponding Dynkin diagrams are given in Table 2.1. 2.4. Chevalley cohomology of Lie algebras Certain properties of Lie algebras are best described in terms of Chevalley (or Chevalley – Eilenberg) cohomology [29]. Therefore, let us review its definition here. Let us assume that a Lie algebra g and its representation ρ on a vector space V are given. The k-cochain c is a totally antisymmetric linear map from g×k =
2.4. CHEVALLEY COHOMOLOGY OF LIE ALGEBRAS
21
g × g × · · · × g to V . The vector space of all k-cochains is denoted by C k (g, V ; ρ). The direct sum of all C k (g, V ; ρ), k = 0, . . . , dim g is called the cochain complex and denoted by C • (g, V ; ρ). On the cochain complex the cohomology operator d is defined by its action on an arbitrary k-cochain (2.35)
d c(x1 , . . . , xk+1 ) =
k+1
(−1)i+1 ρ(xi )c(x1 , . . . , x ˆi , . . . , xk+1 )
i=1
+
(−1)i+j c([xi , xj ], x1 , . . . , x ˆi , . . . , x ˆj , . . . , xk+1 )
1≤i 1, then there is the element n2 1E1 − n1 1E2 of Q0 which has two components with nonvanishing trace, a contradiction. It follows that ρ = 1, and g is indecomposable.
56
5. DECOMPOSITION INTO A DIRECT SUM
5.2. Algorithm The results of the previous section can be summarized as an algorithm for establishing whether a Lie algebra g of finite dimension n is decomposable or not. Moreover, if g is decomposable, the algorithm will perform the decomposition. Step 1. Remove the maximal central component of g, if one exists. This is done using Theorem 5.7 and more specifically equations (5.21) and (5.22). From now on assume C(g) ⊆ D(g). Step 2. Determine the n×n matrices of the adjoint representation of g and find the centralizer A = CR (ad g) of the adjoint representation in R = Fn×n . Choose a basis for A in the form {a1 = 1n , a2 , . . . , as } with Tr ai = 0, 2 ≤ i ≤ s. Denote by A0 the traceless subset of A : A0 = {a2 , . . . , as }. Step 3. Determine whether g is absolutely indecomposable by calculating the traces of the products ai ak , 2 ≤ i, k ≤ s. The algebra g is absolutely indecomposable if and only if (5.55)
Tr ai ak = 0,
2 ≤ i, k ≤ s.
If F = C and (5.55) holds, then g is indecomposable. If (5.55) does not hold, or if F = R, proceed further. Step 4. Determine the Jacobson radical J(A) using the definition (5.7) (for S = A). Choose a basis x1 , . . . , xν for J(A) and a complementary basis b1 = 1n , . . . , bμ as in (5.33). If F = R and μ = 2, then verify whether the relation (5.56)
b22 = k1n mod J(A),
k < 0,
holds. If (5.56) does hold, then g is indecomposable but not absolutely indecomposable. In all other cases the algebra g is decomposable and we proceed to decompose it. Step 5. Run through the basis elements b2 , . . . , bμ until one is found with a reducible minimal polynomial as in (5.48). Call this element br . We now have a nonnilpotent traceless matrix br in CR (A). Using the invariant factors of br , or the rational roots theorem, and if necessary a more powerful factorization procedure, factor the minimal polynomial mb into two mutually prime monic nonconstant polynomials f1 and f2 , as in (5.48). We define the polynomials P1 , P2 as in equation (5.49). The matrix (5.57)
M = P1 (br )f1 (br )
is a nontrivial idempotent in A/J(A), i.e., (5.58)
M 2 = M mod J(A).
Step 6. Perform a change of basis that diagonalizes M and realizes the decomposition of g. This is done using a matrix G obtained by placing the row space of M in its first r rows and the row space of M − 1 in the last n − r rows. Thus columns 1, . . . , r and r + 1, . . . , n of G−1 are the bases of the eigenspaces of M corresponding to the eigenvalues 1 and 0, respectively. The new basis of g is then given as (5.59)
ei = Gji ej .
5.3. EXAMPLES
57
Step 7. We have decomposed g into the direct sum of two algebras, g1 and g2 . Repeat the algorithm, starting at Step 2, for each component and continue until we arrive at a decomposition into indecomposable components. The algorithm described above is efficient if the dimension ν of J(A) is not very large. If ν is large then a faster algorithm exists. We refer the reader to [102] for its formulation. 5.3. Examples Let us demonstrate the whole procedure on several examples. Example 5.11. First, we consider the algebras so(4) and so(1, 3). It is well known that so(4) is decomposable and so(1, 3) is indecomposable but not absolutely indecomposable. Let us derive these conclusions using the algorithm described above and also find an explicit decomposition. The algebras so(4) and so(1, 3) have the Lie brackets [x1 , x3 ] = −x2 , [x2 , x3 ] = x1 ,
[x1 , x5 ] = x6 ,
[x1 , x6 ] = −x5 , [x2 , x4 ] = −x6 , [x2 , x6 ] = x4 ,
[x3 , x4 ] = x5 ,
[x1 , x2 ] = x3 , (5.60)
[x3 , x5 ] = −x4 , [x4 , x5 ] = x3 ,
[x4 , x6 ] = x2 , [x5 , x6 ] = x1
where = 1 for so(4) and = −1 for so(1, 3). Step 1. The center of so(4) and so(1, 3) vanishes, i.e., there is no central component. Step 2. The centralizer CR (ad g) of the adjoint representation is u13 v13 (5.61) CR (ad g) = u, v ∈ F . v13 u13 We set (5.62)
a1 = 16 ,
a2 =
0 13
13 . 0
We have A0 = span{a2 }. Step 3. We have Tr a22 = 6. Therefore neither so(4) nor so(1, 3) is absolutely indecomposable. Step 4. The Jacobson radical J CR (ad g) vanishes and we can set b1 = a1 , b2 = a2 . We have (5.63)
b22 = b1 .
Consequently, so(4) is decomposable both as a real and as a complex Lie algebra, whereas the real algebra so(1, 3) is indecomposable but not absolutely indecomposable. We proceed to decompose so(4) and the complexification soC (1, 3) of so(1, 3). Step 5. The matrix b2 has the reducible minimal polynomial (t − 1)(t + 1) when = 1, 2 (5.64) mb2 (t) = t − = (t − i)(t + i) when = −1. We consider the two cases separately.
58
5. DECOMPOSITION INTO A DIRECT SUM
When = 1 we set f1 (t) = t−1 and f2 (t) = t+1. Solving (5.49) for polynomials P1 and P2 we find a particular solution P1 (t) = 12 t and P2 (t) = 1 − 12 t. Thus we have ⎞ ⎛ 1 0 0 −1 0 0 ⎜0 1 0 0 −1 0 ⎟ ⎟ ⎜ ⎜ 1⎜ 0 0 1 0 0 −1⎟ ⎟. (5.65) M=1 = P1 (b2 )f1 (b2 ) = ⎜ 0 1 0 0⎟ 2 ⎜−1 0 ⎟ ⎝ 0 −1 0 0 1 0⎠ 0 0 −1 0 0 1 When = −1 we set f1 (t) = t − i and f2 (t) P1 (t) = 12 i and P2 (t) = − 12 i. Thus we have ⎛ 1 0 0 −i ⎜0 1 0 0 ⎜ 1⎜ 0 0 1 0 (5.66) M=−1 = ⎜ 2⎜ ⎜i 0 0 1 ⎝0 i 0 0 0 0 i 0
= t + i. Solving (5.49) we find ⎞ 0 0 −i 0 ⎟ ⎟ 0 −i⎟ ⎟ 0 0⎟ ⎟ 1 0⎠ 0 1
Step 6. We find changes of bases diagonalizing the matrices (5.65), (5.66), respectively. We can choose the following new bases =1: (5.67)
x1 = 12 (x6 − x3 ),
x2 = 12 (x5 − x2 ),
x3 = 12 (x4 − x1 ),
x4 = 12 (x2 + x5 ),
x5 = 12 (x1 + x4 ),
x6 = 12 (x3 + x6 ),
= −1 : x1 = 12 (x3 + ix6 ), x2 = 12 (x2 + ix5 ), x3 = 12 (x1 + ix4 ), x4 = 12 (ix6 − x3 ), x5 = 12 (ix5 − x2 ), x6 = 12 (ix4 − x1 ).
In these new bases the algebras so(4) and soC (1, 3), respectively, split explicitly into direct sums of simple subalgebras spanned by x1 , x2 , x3 and x4 , x5 , x6 . We have (5.68)
[x1 , x2 ] = x3 ,
[x1 , x3 ] = −x2 ,
[x2 , x3 ] = x1 ,
[x4 , x5 ] = −x6 ,
[x4 , x6 ] = x5 ,
[x5 , x6 ] = −x4
[x1 , x2 ] = −x3 ,
[x1 , x3 ] = x2 ,
[x2 , x3 ] = −x1 ,
[x4 , x5 ] = x6 ,
[x4 , x6 ] = −x5 ,
[x5 , x6 ] = x4
when = 1, and (5.69) for = −1. Notice that these new bases are not unique, any bases of the row spaces of the matrices M and M − 1 can be used. Consequently, the Lie brackets of the summands can be obtained in another form when the same procedure is repeated. Step 7. The subalgebras arising in the decompositions (5.68) and (5.69) are simple, i.e., further indecomposable. Thus, we have decomposed the algebra so(4) into a direct sum of two subalgebras, both of which turn out to be isomorphic to so(3) upon further inspection (e.g., calculating the signatures of their Killing forms). We have (5.70)
so(4) = so(3) ⊕ so(3).
5.3. EXAMPLES
59
Similarly, we have found that the algebra so(1, 3) is indecomposable but not absolutely indecomposable. Its complexification soC (1, 3) decomposes into a direct sum of two three-dimensional simple Lie algebras. Any such algebra is isomorphic to sl(2, C), i.e., we have soC (1, 3) = sl(2, C) ⊕ sl(2, C).
(5.71)
Example 5.12. As a second example let us consider a solvable 6-dimensional Lie algebra g with the Lie brackets (5.72)
[x2 , x3 ] = x2 − x1 + x6 ,
[x2 , x4 ] = x2 − x1 + x6 ,
[x4 , x5 ] = x1 − x6 .
We have the following dimensions of the ideals in the characteristic series (5.73)
US = [2, 4],
CS = [6, 2, 1],
DS = [6, 2, 0].
The derived algebra is spanned by x1 − x6 and x2 − x1 + x6 whereas the two-dimensional center is C(g) = span{x1 , x6 }.
(5.74)
Thus the algebra g has a one-dimensional central component, which can be chosen as span{x6 }. We have a decomposition (5.75)
g = g1 ⊕ g2 ,
g1 = span{x6 },
g2 = span{˜ x1 , x ˜2 , x ˜3 , x ˜4 , x ˜5 }
where we have defined (5.76)
x ˜1 = x1 − x6 ,
x ˜ 2 = x2 ,
x ˜ 3 = x3 ,
x ˜ 4 = x4 ,
x ˜ 5 = x5 .
Notice that the characteristic series of the subalgebra g2 have dimensions (5.77)
US = [1, 3],
CS = [5, 2, 1],
DS = [5, 2, 0].
No such algebra appears in the list of indecomposable 5-dimensional Lie algebras in Chapter 18; thus it must be decomposable. We proceed to demonstrate this using our algorithm of Section 5.2 and to find the actual decomposition. The algebra g2 has no central component. The centralizer CR (ad g2 ) of the adjoint representation is ⎫ ⎧⎛ ⎞ u u−v w x y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟ ⎪ ⎜0 v 0 0 0 ⎬ ⎨ ⎟ ⎜ ⎟ u, v, w, x, y ∈ F . 0 0 v v − u 0 (5.78) A = CR (ad g2 ) = ⎜ ⎟ ⎜ ⎪ ⎪ ⎪ ⎪ ⎝0 0 0 u 0 ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 0 0 0 u The traceless part A0 of A consists of matrices as in (5.78) with 3u + 2v = 0. The condition (5.55) does not hold; thus, g2 is not indecomposable over the field C. The Jacobson radical is ⎫ ⎧⎛ ⎞ 0 0 w x y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟ ⎪ ⎬ ⎨⎜ ⎜0 0 0 0 0 ⎟ ⎟ w, x, y ∈ F . 0 0 0 0 0 (5.79) J(A) = ⎜ ⎟ ⎜ ⎪ ⎪ ⎪ ⎪ ⎝0 0 0 0 0 ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 0 0 0 0
60
5. DECOMPOSITION INTO A DIRECT SUM
We choose the basis of the complement of J(a) in A in the form ⎞ ⎛ 2 5 0 0 1 ⎜0 −3 0 0 0⎟ ⎟ ⎜ ⎜ (5.80) b1 = 15 , b2 = ⎜0 0 −3 −5 0⎟ ⎟. ⎝0 0 0 2 0⎠ 0 0 0 0 2 We have Tr b22 = 30, i.e., the traceless part A0 is not an (associative) subalgebra of A and the algebra g2 is decomposable, as we already concluded using results of Chapter 18. The minimal polynomial mb2 (t) = t3 − t2 − 8t + 12
(5.81)
of the matrix b2 is reducible, mb2 (t) = f1 (t)f2 (t) with (5.82)
f2 (t) = (t − 2)2 .
f1 (t) = t + 3,
We find the polynomials (5.83)
P1 (t) =
1 (7 − t), 25
P2 (t) =
1 25
such that (5.49) holds. The corresponding idempotent in ⎛ 1 1 0 0 ⎜0 0 0 0 ⎜ (5.84) M = P1 (b2 )f1 (b2 ) = ⎜ ⎜0 0 0 −1 ⎝0 0 0 1 0 0 0 0
CR (ad g2 ) is ⎞ 0 0⎟ ⎟ 0⎟ ⎟. 0⎠ 1
A change of basis diagonalizing the matrix M is (5.85)
x1 = x ˜1 ,
x2 = x ˜4 − x ˜3 ,
x3 = x ˜5 ,
x4 = x ˜2 − x ˜1 ,
x5 = x ˜3 .
The vector spaces span{x1 , x2 , x3 } and span{x4 , x5 } are ideals in g2 . The Lie brackets are (5.86)
[x2 , x3 ] = x1 ,
[x4 , x5 ] = x4 .
Thus the algebra g2 decomposes into a direct sum of the Heisenberg algebra h(1) = n3,1 and the solvable two-dimensional algebra s2,1 . Altogether we have found that a change of basis (5.87)
x1 = x1 − x6 ,
x2 = x4 − x3 ,
x3 = x5 ,
x4 = x2 − x1 + x6 ,
x5 = x3 ,
x6 = x6
performs an explicit decomposition of the six-dimensional solvable algebra g into a direct sum of three algebras (5.88)
g = h(1) ⊕ s2,1 ⊕ a1 ,
where h(1) = span{x1 − x6 , x4 − x3 , x5 },
s2,1 = span{x3 , x2 − x1 + x6 },
a1 = span{x6 }.
5.3. EXAMPLES
61
Notice that the polynomial mb2 (t) depends on the choice of the matrix b2 . Had we chosen instead of b2 a matrix b2 with vanishing (1, 5)-entry, the minimal polynomial would be mb2 (t) = t2 + t − 6 = (t + 3)(t − 2) and the rest of the computation, e.g., the factors f1 , f2 , would be different. Nevertheless, the decomposition into subalgebras span{˜ x1 , x ˜4 − x ˜3 , x ˜5 } and span{˜ x2 − x ˜1 , x ˜3 } would be the same.
CHAPTER 6
Levi Decomposition. Identification of the Radical and Levi Factor The Levi theorem 2.12 in its full generality states that for an arbitrary finitedimensional Lie algebra g over a field F of characteristic zero it is possible to construct a semidirect sum decomposition g = p r.
(6.1)
Here r ≡ R(g) is the radical of g, i.e., its maximal solvable ideal, and p is a semisimple Lie algebra. The radical is unique. The complement p, i.e., the Levi factor, is isomorphic to the factor algebra g/r and is unique up to automorphisms of g. In other words, given a basis of g, say (x1 , . . . , xn ), it is always possible to find a new basis (6.2)
{s1 , s2 , . . . , sσ , r1 , r2 , . . . , rρ },
σ + ρ = n,
such that r = span{r1 , . . . , rρ }, p = span{s1 , . . . , sσ }, and the commutation relations are such that (6.3)
[p, p] = p,
[r, r] r,
[p, r] ⊆ r.
The question that we address here is: how does one find a convenient change of basis that realizes the Levi decomposition? Notice that a Levi decomposition can be performed for both decomposable and indecomposable Lie algebras. From the point of view of identifying a Lie algebra g, it is usually preferable to first perform a direct decomposition into indecomposable components and then construct a Levi decomposition for each component. Since the factor algebra g/r ∼ = p is semisimple and hence perfect, i.e., its derived algebra D(p) satisfies D(p) = p, we have (6.4)
D(g) + r = g.
From (6.4) we obtain the isomorphism (6.5)
D(g)/[D(g) ∩ r] ∼ = g/r
and hence dim(D(g)/[D(g) ∩ r]) = n − ρ = σ. The problem of obtaining a Levi decomposition is one of linear algebra. We shall address it in two ways. 6.1. Original algorithm First, we present an algorithm as given in [102]. Its essence is a reduction of the general case to the case with an Abelian radical. We proceed in four steps. 63
64 6. LEVI DECOMPOSITION. IDENTIFICATION OF THE RADICAL AND LEVI FACTOR
Step 1. Find the radical r = R(g). This is a simple task of linear algebra, since we can use the property [59, 139] (6.6)
R(g) = {x ∈ g | K(x, y) = 0, ∀y ∈ D(g)},
where K(x, y) is the Killing form (2.31)
K(x, y) = Tr ad(x) ad(y) .
If g = R(g), then g is solvable and p = 0 in (6.1). If R(g) = 0, then g = p is semisimple. In both cases the Levi decomposition is trivial. Step 2. If 0 = p = g, we calculate the derived series (2.5) of g: g(1) = [g, g], . . . , g(k+1) = g(k) , g(k) , . . . till we arrive, after a finite number of steps, at a perfect Lie algebra (6.7)
g(k+1) = g(k) ,
g(k) = g(k−1) .
If we know the Levi decomposition of g(k) , i.e., (6.8) g(k) = p R g(k) then obtain the Levi decomposition of g by extending the basis (r1 , . . . , rτ ) of we R g(k) to a basis of R(g) : (r1 , . . . , rτ , rτ +1 , . . . , rρ ). Step 3. From now on we assume that g is perfect which also implies that the radical of g is nilpotent (since D(g) ⊆ p N R(g)). We consider two cases separately, namely that of an Abelian and a non-Abelian radical, respectively. If R(g) is Abelian, the problem is a simple task of linear algebra, solved below in Step 4. Let us first reduce the case of a non-Abelian radical to that of an Abelian one. We have (6.9) g = D(g), D R(g) = 0. We note that D R(g) is an ideal in g. Let us choose a basis for g in the form (6.10)
(e1 , . . . , eμ , r1 , . . . , rν , x1 , . . . , xσ ), μ+ν =ρ where (e1 , . . . , eμ ) is a basis for D R(g) , (r1 , . . . , rν ) for a complement of D R(g) in R(g), and (x1 , . . . , xσ) for a complement of R(g) in g. Next, we construct the factor algebra ¯ g = g/D R(g) satisfying ¯g = D(¯g) and D R(¯g) = 0. We have dim g¯ = n − μ < n and the commutation relations for ¯g are obtained by setting e1 = · · · = eμ = 0 in the commutation relations for g. Using the method described below (for Abelian radicals), we obtain the Levi decomposition (6.11)
¯ g=¯ p R(¯g)
of g¯. From the residue classes in ¯ p we construct their representative elements in g and denote by p1 the vector space spanned by them. We obtain a proper subalgebra of g, namely (6.12) g1 = D R(g) + p1 , satisfying dim g1 = n − ν < n. If D R(g) is Abelian, we find a Levi decomposition of g1 ; if not, we repeat Steps 2 and 3 until, after a finite number of steps, we arrive at an algebra gk with an Abelian radical. For it we obtain (6.13)
gk = p R(gk ).
6.2. MODIFIED ALGORITHM
65
The Levi decomposition of g is then (6.1) with p as in (6.13). Step 4. Finally, let us consider the case of an Abelian radical. We have μ = 0 in (6.10), and the commutation relations for g are (6.14) [xi , xk ] = cik l xl + fik q rq ,
[xi , rp ] = hip q rq , 1 ≤ i, k, l ≤ σ, 1 ≤ p, q ≤ ρ.
[rp , rq ] = 0,
Summation over repeated indices over the appropriate ranges is understood throughout. We now replace the basis elements xi by (6.15)
si = xi + bpi rp
and require that the si form a Lie algebra: (6.16)
[si , sk ] = cik l sl .
Equations (6.14) – (6.16) imply that the unknown coefficients bpi must satisfy a system of inhomogeneous linear equations (6.17)
cik l bql − hip q bpk + hkp q bpi = fik q , bpi
1 ≤ p, q ≤ ρ,
1 ≤ i, k ≤ σ.
1 2 σ(σ − 1)ρ/2
and equations. It follows The system (6.17) involves ρσ unknowns from the Levi theorem that the system is compatible and that a solution bpi can always be found. It is actually possible to solve (6.17) explicitly and analytically, making use of the second order Casimir operator of the factor algebra p = g/r [139]. From the computational point of view it is usually preferable to solve (6.17) directly in each specific case. 6.2. Modified algorithm A streamlined alternative formulation of the algorithm combines Steps 3 and 4 into one. Its basic idea is a construction of a proper subalgebraof the Lie algebra g which contains p, without constructing the factor algebra g/D R(g) . Repeating the procedure finitely many times one obtains a subalgebra of g which has vanishing radical, i.e., coincides with p. From the computational point of view it is equivalent to the original algorithm. We observe that D(r) ≡ r2 is a characteristic ideal of g and consequently the subalgebra ˜g = p D(r) of g is a Levi decomposable algebra with the Levi factor p and radical r2 . We shall proceed to construct its basis. Let us suppose that we have a basis of g of the form (6.10). Thus we have the following particular Lie brackets (6.18)
[xi , xj ] = cij k xk + dij p rp + fij l el ,
(6.19)
[xi , rp ] = gip q rq + hip m em .
Summation over repeated indices k = 1, . . . , σ, l, m = 1, . . . , μ and p, q = 1, . . . , ν applies throughout. A basis of ˜ g = p D(r) can be without loss of generality chosen in the form (6.20)
{e1 , . . . , eμ , x ˆ1 , . . . , x ˆσ }
where x ˆk ∈ span{xk , r1 , . . . , rν }, i.e., (6.21)
x ˆj = xj + bpj rp .
66 6. LEVI DECOMPOSITION. IDENTIFICATION OF THE RADICAL AND LEVI FACTOR
The span of the set (6.20) is by construction complementary to span{r1 , . . . , rν }. Therefore it forms a basis of p D(r) if and only if it closes under the Lie bracket. That is always true for commutators of the type [ˆ xi , el ] since D(r) is an ideal in g. It remains to satisfy l
[ˆ xi , x ˆj ] = cij k x ˆk + f$ ij el
(6.22) l
k for some constants f$ ij . Notice that the structure constants cjk are the same in (6.18) and (6.22) because they are the structure constants of the semisimple factor algebra g/r in the same basis xj mod r = x ˆj mod r, j = 1, . . . , σ. Substituting (6.21) into (6.22) and dropping any term proportional to el we obtain the following set of equations, to be satisfied for all 1 ≤ i < j ≤ σ
(6.23)
dij q rq + bpj gip q rq − bpi gjp q rq = cij k bqk rq ,
i.e., a set of 12 σ(σ − 1)ν linear inhomogeneous equations gjp q bpi − gip q bpj + cij k bqk = dij q
(6.24)
for σν unknowns bpi . Due to the Levi theorem the set of equations (6.24) always has a solution. Thus, once we find any particular solution of (6.24) we have a basis of ˜ g = p D(r). We repeat the procedure until we arrive at k ∈ N such that r(k) = 0. Notice that it is advantageous to suppose that g is perfect but the streamlined version of the procedure does not depend on it. Remark 6.1. An analogous procedure was given in [35] where the lower central series of the nilpotent radical r of a perfect Lie algebra g was used instead of the derived series employed here. Consequently, the procedure has to be repeated more times but the arising systems of equations (6.24) are smaller and thus easier to solve. However, a comparable computational simplification can be achieved in our algorithm by simply using a basis of the nilpotent radical r which respects the lower central series, as in (9.19) in Section 9.3 below. 6.3. Examples In order to illuminate the algorithms outlined above we conclude this chapter by two examples. Example 6.2. Let us consider the finite dimensional part of the algebra of infinitesimal point symmetries of the heat equation [86]. It is spanned by the following vector fields in R3 with coordinates t, x, u Y1 = 4t2 ∂t + 4xt∂x − (2t + x2 )u∂u , (6.25)
Y2 = 4t∂t + 2x∂x ,
Y3 = ∂t ,
Y4 = −2t∂x + xu∂u ,
Y5 = u∂u ,
Y6 = ∂x .
Evaluation of the commutators gives the following Lie brackets of an abstract Lie algebra (with yi replacing the vector fields Yi ) (6.26)
[y1 , y2 ] = −4y1 ,
[y1 , y3 ] = −2y2 + 2y5 ,
[y1 , y6 ] = 2y4 ,
[y2 , y3 ] = −4y3 ,
[y2 , y4 ] = 2y4 ,
[y2 , y6 ] = −2y6 ,
[y3 , y4 ] = −2y6 ,
[y4 , y6 ] = −y5 .
6.3. EXAMPLES
67
In Steps 1 and 2 we find that the Lie algebra (6.26) is perfect and its radical is spanned by y4 , y5 , y6 . The radical (equal to the nilradical) is the Heisenberg algebra h(1) ≡ n3,1 . The derived algebra of the radical is spanned by y5 . The basis of the form (6.10) can be chosen as (6.27)
e1 = y5 ,
r1 = y4 ,
r2 = y6 ,
x1 = y1 ,
x2 = y2 ,
x3 = y3
with the Lie brackets [r1 , r2 ] = −e1 , [r1 , x2 ] = −2r1 , [r1 , x3 ] = 2r2 , [r2 , x1 ] = −2r1 , (6.28) [r2 , x2 ] = 2r2 , [x1 , x2 ] = −4x1 , [x1 , x3 ] = 2e1 − 2x2 , [x2 , x3 ] = −4x3 . Step 3 of the procedure now requires the construction of a 5-dimensional Lie algebra g/ span{e1 } with an Abelian radical. Its Lie brackets are (6.29)
˜2 ] = −2˜ r1 , [˜ r1 , x
[˜ r1 , x ˜3 ] = 2˜ r2 ,
[˜ r2 , x ˜1 ] = −2˜ r1 , [˜ r2 , x ˜2 ] = 2˜ r2 ,
˜2 ] = −4˜ x1 , [˜ x1 , x ˜3 ] = −2˜ x2 , [˜ x2 , x ˜3 ] = −4˜ x3 [˜ x1 , x
where tildes were used to distinguish residue classes in g/ span{e1 } from the corresponding elements in g. Its (Abelian) radical is spanned by r˜1 and r˜2 . In Step 4 the Levi factor of g/ span{e1 } is obtained by an explicit solution of (6.17) or by inspection; it can be chosen to be the span of x ˜1 , x ˜2 , x ˜3 . Thus, we have p D R(g) = span{e1 , x1 , x2 , x3 }. The same conclusion is immediate in our modified algorithm since e1 , x1 , x2 , x3 span a subalgebra of g by an inspection of (6.28). Next, we construct the Levi decomposition of pD R(g) = span{e1 , x1 , x2 , x3 } with the nonvanishing Lie brackets (6.30)
[x1 , x2 ] = −4x1 ,
[x1 , x3 ] = 2e1 − 2x2 ,
[x2 , x3 ] = −4x3 .
This step coincides in both versions of the algorithm. The solution of the set of linear equations (6.17) or, equivalently, (6.24) is b11 = 0,
b12 = −1,
b13 = 0
and leads to the desired basis for the Levi factor in the form (6.31)
x1 = y1 ,
x2 − e1 = y2 − y5 ,
x3 = y3 .
Notice that in this case the Levi factor of the subalgebra span{e1 , x1 , x2 , x3 } coincides with its derived algebra and is therefore unique. However, as a Levi factor of the whole algebra g it is not unique since a different choice of the preimage p1 of ¯p was possible in Step 3. As stated in Theorem 2.13, the Levi factor p is determined only up to inner automorphisms of g. We conclude that the Levi decomposition of the “heat algebra” g (6.25) is (6.32)
g sl(2, R) h(1)
where the Levi factor sl(2, R) has the basis (6.33)
Y1 = 4t2 ∂t + 4xt∂x − (2t + x2 )u∂u , Y2 − Y5 = 4t∂t + 2x∂x − u∂u ,
Y3 = ∂t
and the radical is the Heisenberg algebra spanned by Y4 , Y5 , Y6 . As we have just seen explicitly, the difference between the two versions of the algorithm is conceptual rather than computational. Therefore, in our second example we shall employ only the second, streamlined version of it.
68 6. LEVI DECOMPOSITION. IDENTIFICATION OF THE RADICAL AND LEVI FACTOR
Example 6.3. Let us consider an 8-dimensional Lie algebra g with the Lie brackets [y1 , y2 ] = 2y2 , [y1 , y3 ] = −2y3 + 2y6 , [y1 , y4 ] = 2y4 , (6.34)
[y1 , y5 ] = 2y5 ,
[y1 , y7 ] = 2y7 ,
[y2 , y3 ] = y1 + y5 + y8 ,
[y2 , y6 ] = y5 ,
[y2 , y8 ] = 2y4 ,
[y3 , y5 ] = −y4 + y6 ,
[y3 , y8 ] = y6 ,
[y4 , y8 ] = 2y4 ,
[y5 , y6 ] = y4 ,
[y5 , y8 ] = y5 ,
[y6 , y8 ] = y6 ,
[y7 , y8 ] = 2y7 .
Its radical is spanned by y4 , y5 , y6 , y7 , y8 and the nilradical by y4 , y5 , y6 , y7 . In fact, the radical {y4 , . . . , y8 } is isomorphic to the algebra s5,22,a=1,b=2 of Section 18.3 and the nilradical is decomposable into n3,1 ⊕ n1,1 where n3,1 ≡ h(1) is the Heisenberg algebra. We start by Step 2. The derived series is g(1) = D(g) = span{y1 + y8 , y2 , . . . , y7 },
(6.35)
g(2) = g(3) = span{y1 + y8 , y2 , y3 , y4 , y5 , y6 }.
Thus, the Levi factor of g is found once the Levi factor of g(2) is constructed using the algorithm of Section 6.2. The nilpotent non-Abelian radical of g(2) is spanned by y4 , y5 , y6 with a single nonvanishing Lie bracket [y5 , y6 ] = y4 . We choose the basis as in (6.10) (6.36)
e1 = y4 ,
r1 = y5 ,
˜=g The Lie brackets of g [r1 , r2 ] = e1 , (6.37)
(2)
r2 = y6 ,
x1 = y1 + y8 ,
in this basis become [r1 , x1 ] = −r1 ,
x3 = y3 .
[r1 , x3 ] = e1 − r2 ,
[r2 , x2 ] = −r1 ,
[r2 , x1 ] = r2 ,
x2 = y2 ,
[x1 , x2 ] = −2e1 + 2x2 ,
[x2 , x3 ] = r1 + x1 . [x1 , x3 ] = r2 − 2x3 , In order to find p D R(˜ g) we have to perform the change of basis (6.21). The conditions (6.24) reduce to the equations b22 = 0,
b13 = 0,
b12 + b21 = 0,
b23 − b11 = 1.
Thus, a particular solution of (6.24) is b11 = 1,
b21 = 0,
bp2 = bp3 = 0,
p = 1, 2,
which corresponds to the change of basis (6.38)
x ˆ1 = x1 + r1 = y1 + y5 + y8 ,
x ˆ2 = x2 = y2 , x ˆ3 = x3 = y3 . The vectors e1 , x ˆ1 , x ˆ2 , x ˆ3 form a basis of p D R(˜g) . In this basis we have the Lie brackets [ˆ x1 , x ˆ2 ] = −2e1 + 2ˆ x2 , [ˆ x1 , x ˆ3 ] = e1 − 2ˆ x3 , [ˆ x2 , x ˆ3 ] = x ˆ1 . The radical g) is spanned by e1 which coincides with the center of of p D R(˜ p D R(˜ g) . Thus, the Lie algebra (6.39) is decomposable into a direct sum of a simple algebra sl(2, F) and a central component spanned by e1 , and the algorithm of the previous chapter can be used. Alternatively, we use the change of basis (6.21) (6.39)
x ¯1 = x ˆ1 + ˆb11 e1 ,
x ¯2 = x ˆ2 + ˆb12 e1 ,
x ¯3 = x ˆ3 + ˆb13 e1
6.3. EXAMPLES
69
once again, arriving at the conditions (6.24) expressed as ˆb1 = 0, ˆb1 = −1, 2ˆb1 = −1. 1 2 3 Thus, we have constructed a basis {¯ x1 , x ¯2 , x ¯3 } of the Levi factor p of g in the form x ¯1 = x ˆ1 = y1 + y5 + y8 ,
x ¯2 = x ˆ2 − e1 = y2 − y4 ,
x ¯3 = x ˆ3 − 12 e1 = y3 − 12 y4 .
To sum up, the Levi factor of the algebra (6.34) is % & (6.40) p = span y1 + y5 + y8 , y2 − y4 , y3 − 12 y4 and is isomorphic to sl(2, F). Let us recall that the Levi factor (6.40) is generically far from unique. In our example, another choice for it is spanned by (6.41)
x ˜1 = y1 + y8 ,
x ˜2 = y2 − y4 ,
x ˜3 = y3 − y6
and, in fact, the choice (6.41) is more convenient because the Lie brackets of the algebra (6.34) are more compact when written in terms of x ˜1 , x ˜2 , x ˜3 . Different choices of Levi factors arise through different choices of the particular solutions of the systems of linear equations involved.
CHAPTER 7
The Nilradical of a Lie Algebra The aim of this section is to present an algorithm by means of which it is possible to explicitly calculate the maximal nilpotent ideal of a finite-dimensional Lie algebra g. This algorithm was first presented in [102]. It is completely rational: it requires no irrational operations such as eigenvalue calculations. 7.1. General theory Let us first present several results that underlie the algorithm. Lemma 7.1. For any epimorphism (i.e., surjective homomorphism) ε : g → g we have (7.1) ε NR(g) ⊆ NR(g ). Proof. For any nilpotent ideal i of g we have [g, i] ⊆ i,
in = 0,
n ∈ N.
Upon application of ε we have [g , ε(i)] = [ε(g), ε(i)] ⊆ ε(i), ε(in ) = 0. Hence ε(i) is a nilpotent ideal of g and ε NR(g) ⊆ NR(g ).
For our purposes an important corollary of Lemma 7.1 is that for any ideal i of g we have NR(g)/i ⊆ NR(g/i).
(7.2)
Note that the equality sign alone in (7.1) and (7.2) would be inappropriate. Example 7.2. Let us consider the algebra g = span{e1 , e2 },
(7.3)
[e1 , e2 ] = e1
and take g = span{f } (with f = 0). We consider an epimorphism ε : g → g , ε(e1 ) = 0, ε(e2 ) = f . Thus ε NR(g) = ε(span{e1 }) = 0 NR(g ) = g . Our example also shows that a finite-dimensional Lie algebra g with a nilpotent ideal b and a nilpotent factor algebra g/b is not necessarily nilpotent — it suffices to consider g as in (7.3) and b = ker ε = span{e1 }. A composition series for a finite-dimensional solvable Lie algebra g [59] is a chain of ideals (7.4)
g = g 0 g1 g2 · · · gs = 0 71
72
7. THE NILRADICAL OF A LIE ALGEBRA
such that gi is maximal among the ideals of g properly contained in gi−1 (1 ≤ i ≤ s). A composition series of a solvable Lie algebra always exists as a refinement of the derived series. Lemma 7.3. The nilradical of a finite-dimensional solvable Lie algebra g is characterized as the set of all elements x ∈ g for which [x, gi−1 ] ⊆ gi ,
(7.5)
1≤i≤s
in any chosen composition series (gi ). Proof. We shall use the notation (7.6)
[x1 , x2 , . . . , xj ] = x1 , [x2 , . . . , xj ] ,
defining multiple commutators recursively. Let us denote by x the set of all elements x ∈ g satisfying (7.5). Equation (7.5) implies that x is an ideal in g. Let us first assume x ∈ NR(g). We have gi−1 ⊇ ai ≡ [NR(g), gi−1 ] + gi ⊇ gi . Let us suppose that ai = gi . Then by construction ai = gi−1 . Hence [NR(g), NR(g), gi−1 ] + gi = [NR(g), ai ] + gi = [NR(g), gi−1 ] + gi = ai = gi−1 . By iteration we have [NR(g), . . . , NR(g), gi−1 ] + gi = gi−1 . On the other hand, there exists a natural number n for which NR(g)n = 0. Consequently, NR(g), . . . , NR(g), gi−1 ⊆ [NR(g), . . . , NR(g)] = 0, n times
so that we obtain gi = gi−1 , a contradiction. Hence we find [NR(g), gi−1 ] ⊆ gi for 1 ≤ i ≤ s and (7.5) holds for x ∈ NR(g). Therefore we have NR(g) ⊆ x. Conversely, let x1 , . . . , xs belong to the ideal x (the xj are not necessarily distinct). From (7.5) we have ad(x1 )g ⊆ g1 , ad(x2 ) ad(x1 )g ⊆ g2 , . . . , ad(xs ) ad(xs−1 ) · · · ad(x1 )g = gs = 0. Consequently the sth element xs of the lower central series of the ideal x is 0. Thus, by definition, the ideal x is nilpotent, x ⊆ NR(g). Since NR(g) ⊆ x we obtain NR(g) = x. We already know (cf. Example 7.2) that a finite-dimensional Lie algebra g with a nilpotent ideal b and a nilpotent factor algebra g/b is not necessarily nilpotent. However, the following lemma provides a useful nilpotency criterion. Lemma 7.4. Let g be solvable, i ⊂ g its nilpotent ideal and NR(g/i2 ) be the nilradical of the factor algebra g/i2 . Then the nilradical of g can be found using (7.7)
NR(g)/i2 = NR(g/i2 ).
Proof. It is clear that r = {x ∈ g|x + i2 ∈ NR(g/i2 )} is an ideal since Lie brackets in g and g/i2 differ only by elements of i2 and i2 ⊂ r, and that if r is nilpotent then it is maximal in g by (7.2). Therefore we only have to prove that it is nilpotent. In order to do that we have to show that there exists a positive integer N ∈ N such that (ad(x)|r )N = 0, ∀x ∈ r.
7.1. GENERAL THEORY
73
By assumption we know that K ∈ N exists such that (ad(x)|r )K : r → i2 . We use the identity (7.8)
J l J−l J J ad(x) a, ad(x) b ad(x) [a, b] = l l=0
and proceed by induction, showing in each step that there always exists an integer K such that (ad(x)|r )K : ij → ij+1 , j ≤ n ∈ N. Let M ∈ N be such that (ad(x)|r )M : ij → ij+1 , j ≤ n − 1. The ideal in = [i, in−1 ] is spanned by elements of the form [a, b], a ∈ i, b ∈ in−1 and we shall consider only these. Then for any l we have (ad(x))l a ∈ i and for l ≥ M it lies in i2 . Similarly (ad(x))J−l b ∈ in−1 and lies in in for J − l ≥ M . Finally for J = 2M we have using the identity (7.8) (ad(x)|r )2M in ⊆ [i, in ] + [i2 , in−1 ] = in+1 . Since the lower central series of i terminates after say Q iterations and we have shown that there is an integer K such that (ad(x))K maps ij into ij+1 for any j ∈ N, taking N = QK we have (ad(x)|r )N : r → 0,
∀x ∈ r,
i.e., r is a nilpotent ideal in g and consequently r is the nilradical of g.
Lemma 7.4 reduces the problem of finding the nilradical of g to that of finding the nilradical of g/i2 , a lower-dimensional problem. Once NR(g/i2 ) is found, the nilradical of g is spanned by a basis of i2 together with elements representing a basis of NR(g/i2 ). By a slight abuse of notation we may write NR(g) = NR(g/i2 ) i2 .
(7.9)
Example 7.5. Consider a 4-dimensional algebra g spanned by e1 , e2 , e3 , e4 with the nonvanishing Lie brackets (7.10)
[e1 , e4 ] = e1 ,
[e2 , e3 ] = e1 ,
[e2 , e4 ] = e2 .
and the nilpotent ideal i = span{e1 , e2 , e3 }. We have i2 = span{e1 }. Upon factorization we obtain the Lie algebra g/i2 spanned by e¯2 ≡ e2 + i2 , e¯3 and e¯4 with the only nonvanishing bracket (7.11)
[¯ e2 , e¯4 ] = e¯2 .
The nilradical of g/i is found to be NR(g/i2 ) = span{¯ e2 , e¯3 }. Using Lemma 7.4 we find that 2
(7.12)
NR(g) = span{e2 , e3 } i2 = span{e1 , e2 , e3 } = i.
Thus by inspection of the nilradical and of the derived algebra g2 = span{e1 , e2 } the algebra (7.10) is isomorphic to s4,11 of Section 17.3. A similar factorization is obtained also with respect to the hypercenter. Lemma 7.6. Let g be a Lie algebra with the hypercenter z∞ (g). Then the following relation holds (7.13) NR g/z∞ (g) = NR(g)/z∞ (g).
74
7. THE NILRADICAL OF A LIE ALGEBRA
Proof. The inclusion ⊇ is obvious, cf. (7.2). On the other hand, let i be the ideal in g such that z∞ (g) ⊂ i, i/z∞ (g) = NR g/z∞ (g) . By definition j, k ∈ N exist such that g, . . . , g, z∞ (g) = 0 j times
and i ⊂ z∞ (g). Thus, k
ij+k ⊂ g, . . . , g, ik = 0, j times
i.e., i is a nilpotent ideal in g, i ⊆ NR(g).
Example 7.7. Consider a 4-dimensional algebra g spanned by e1 , e2 , e3 , e4 with the nonvanishing Lie brackets (7.14)
[e2 , e4 ] = e1 ,
[e3 , e4 ] = e3 .
It has the 1-dimensional center z1 = span{e1 } and the 2-dimensional second center z2 = span{e1 , e2 } which coincides with the hypercenter. We have g/z∞ = span{¯ e3 , e¯4 } with (7.15)
[¯ e3 , e¯4 ] = e¯3 .
Thus the nilradical of g/z∞ is spanned by e¯3 and we find the nilradical of g in the form (7.16)
NR(g) = span{e3 } z∞ = span{e1 , e2 , e3 }.
Consequently, the algebra (7.14) is isomorphic to s4,1 of Section 17.2. Theorem 7.8. Let g be a finite-dimensional Lie algebra, let b be an ideal of g, and let m be the ideal of g containing b for which (7.17)
NR(g/b) = m/b
holds. We then have (7.18)
NR(g) = NR(m).
Proof. According to (7.1) we have NR(g) ⊆ NR(m).
(7.19)
By Theorem 2.10 we know that NR(m) is an ideal of g. By construction NR(m) is nilpotent. Hence NR(m) ⊆ NR(g).
(7.20)
But (7.19) and (7.20) imply (7.18). Example 7.9. Consider the algebra s5,39 of Section 18.5, i.e., g = span{e1 , e2 , e3 , e4 , e5 } with (7.21)
[e4 , e2 ] = e2 ,
[e5 , e3 ] = e3 ,
[e5 , e4 ] = e1 ,
and the ideal b = span{e3 }. We have g/b = span{¯ e1 , e¯2 , e¯4 , e¯5 } with (7.22)
[¯ e4 , e¯2 ] = e¯2 ,
[¯ e4 , e¯5 ] = e¯1 .
7.2. ALGORITHM
75
The nilradical of g/b is NR(g/b) = span{¯ e1 , e¯2 , e¯5 }. Thus, the ideal m of Theorem 7.8 is (7.23)
m = span{e1 , e2 , e3 , e5 }
and we conclude that (7.24)
NR(g) = NR(m) = span{e1 , e2 , e3 }.
As a consequence of Lemma 7.4 we find that for solvable Lie algebras g we have (7.25)
NR(g)/(g2 )2 = NR(g/(g2 )2 ).
Using Lemma 7.6 we next arrive at the algebra g/(g2 )2 /z∞ (g/(g2 )2 ) with an Abelian derived algebra and vanishing (hyper)center. Such algebras have the following useful property. Lemma 7.10. Let g be a finite-dimensional Lie algebra satisfying the conditions (7.26)
(g2 )2 = 0,
(7.27)
C(g) = 0.
Then we have (7.28)
centg (g2 ) = {x ∈ g | [x, g2 ] = 0} = g2 .
Proof. The centralizer centg (g2 ) of g2 in g is an ideal in g, i.e., a representation space for the representation Γ of g Γ : g → End centg (g2 ) , Γ(x)(u) = [x, u], x ∈ g, u ∈ centg (g2 ) with g2 in its kernel; hence Γ(g) ∼ = g/ ker Γ is the homomorphic image of g/g2 , which means that Γ(g) is Abelian. It follows that 0, centg (g2 ) = m0 + m where m0 is the g-invariant subset of the elements of centg (g2 ) which are annihilated by some power of Γ(x) for each x of g, whereas the complementary subspace 0 is the intersection of the g-invariant subspaces Γ(g)i centg (g2 ) (i ∈ N). It m 0 ⊆ g2 . However, since C(g) = 0, it follows that C(g) ∩ m0 = 0 follows that m and consequently m0 = 0. However, (7.26) implies that g2 ⊆ centg (g2 ). Hence 0 = g2 . centg (g2 ) = m 7.2. Algorithm The results of the previous section give rise to an algorithm for calculating the nilradical of a finite-dimensional Lie algebra g. Step 1. Determine the radical R(g) using the formulas (6.6), (2.31). In view of Lemma 2.4 we have (7.29) NR(g) = NR R(g) . From now on we replace R(g) by g and assume that g is solvable. In view of Lemma 2.4 we can build up the nilradical gradually from smaller nilpotent ideals.
76
7. THE NILRADICAL OF A LIE ALGEBRA
Step 2. Calculate the ideals D(g) = g2 = [g, g] and D2 (g) = [D(g), D(g)]. In view of (7.25) we obtain NR(g) from the nilradical of the factor algebra g/D2 (g) using NR(g)/D2 (g) = NR g/D2 (g) . From now on we consider the algebra g/D2 (g), i.e., we assume that g is solvable and its derived algebra D(g) = [g, g] is Abelian: (7.30)
[D(g), D(g)] = 0.
Step 3. Calculate the hypercenter z∞ (g) of g (defined in (2.10)). In view of Lemma 7.6 we obtain NR(g) from the nilradical of g/z∞ (g) using NR(g)/z∞ (g) = NR g/z∞ (g) . We have thus reduced the problem of finding the nilradical of an arbitrary finitedimensional Lie algebra to that of finding NR(g) for a solvable Lie algebra g satisfying (7.30) and (7.31)
C(g) = 0
(since z∞ (g) = 0 is equivalent to C(g) = 0). Step 4. Introduce a basis for the derived algebra D(g) of the solvable algebra g, satisfying (7.30) and (7.31). Extend this basis to a basis of g: (7.32)
D(g) = span{u1 , . . . , um },
g = span{u1 , . . . , um , x1 , . . . , xn−m }.
Step 5. Choose a specific element of D(g), say u ≡ u1 and an element of the complement, say xj . Define a chain of elements ukj in D(g) as (7.33)
u0j = u,
u1j = [xj , u0j ], . . . , upj = [xj , up−1j ]
such that {u0j , . . . , up−1j } are linearly independent but upj is a linear combination of the preceding elements. Call the space spanned by these elements (7.34)
Xj = span{u0j , . . . , up−1,j }.
Its dimension is p and Xj is by construction ad xj invariant. The matrix representing ad xj restricted to the space Xj has the form ⎞ ⎛ 0 0 ... 0 apj ⎜1 0 . . . 0 ap−1j ⎟ ⎟ ⎜ ⎜ .. .. ⎟ .. ⎟ . 1 . . (7.35) ad xj |Xj = ⎜ ⎟ ⎜ ⎟ ⎜ . .. 0 ⎝ a2j ⎠ 1 a1j where akj , 1 ≤ k ≤ p are some numbers. The characteristic polynomial of ad xj |Xj is (7.36)
Pj (λ) = λp − a1j λp−1 − a2j λp−2 − · · · − ap−1j λ − apj .
It follows from (7.35) that P (λ) is also the minimal polynomial of ad xj |Xj . Proceed to Step 6 if λ = 0 is an eigenvalue (i.e., if apj = 0), to Step 7 otherwise.
7.2. ALGORITHM
77
Step 6. Form the ideal (7.37)
b1 = [xj , D(g)].
(The subspace b1 is an ideal because D(g) is Abelian.) The fact that we have apj = 0 guarantees that b1 is properly contained in D(g). Furthermore we have b1 = 0 since xj ∈ D(g) and by Lemma 7.10 the centralizer of D(g) is D(g). Having formed the ideal b1 we use Theorem 7.8 and we have to determine the nilradical of g/b1 , a lower-dimensional task. We then form the ideal m of g defined by m/b1 = NR(g/b1 ). Finally we must find NR(m) (dim m < dim g) and put (7.38)
NR(g) = NR(m).
Step 7. We are now considering the case apj = 0. If the matrix ad xj |Xj has p distinct eigenvalues, i.e., if the polynomial Pj (λ) is square-free, proceed to Step 8. If not, then find a nonconstant polynomial gj (λ) that divides Pj (λ) and is square-free. Form the ideal (7.39) b2 = gj ad(xj ) D(g). Since gj (λ) is not a minimal polynomial of the matrix (7.35) we have b2 = 0. Since gj (ad xj ) is by construction a singular matrix b2 is properly contained in D(g) b2 = D(g) . Since D(g) is Abelian, b2 is an ideal in g. Proceed further as with the ideal b1 , i.e. put m/b2 = NR(g/b2 ), b2 ⊆ m, and find NR(m); we have NR(g) = NR(m). Step 8. We finally arrive at the case when the polynomial Pj (λ) is square-free and apj = 0. The label j is associated with the chosen element xj . We run through all values of 1 ≤ j ≤ n − m and repeat Steps 5, 6 and 7. If NR(g) is still not determined, i.e., we have Pj (λ) square-free and apj = 0 for all j, we proceed as follows. Take the element u = u0j ∈ D(g) of Step 5 and find its centralizer in g (7.40)
m = centg (u) = {y ∈ g | [y, u] = 0}.
Since D(g) is Abelian we have (7.41)
D(g) ⊆ m.
The crucial fact, justified below is (7.42)
NR(g) = NR(m).
If D(g) = m, then NR(m) is found by returning to Step 2 with m used instead of g. If D(g) = m we have (7.43)
NR(g) = D(g)
and the algorithm terminates. Let us now justify (7.42). Lemma 7.11. Let g be a Lie algebra satisfying all conditions of Step 8, namely (7.30), (7.31) and let the characteristic polynomial Pj (λ) be square-free with apj = 0 for all elements xj in the basis (7.32). Then NR(g) ⊆ centg (u) = m and consequently we have NR(g) = NR(m).
78
7. THE NILRADICAL OF A LIE ALGEBRA
Proof. The adjoint representation ad g|D(g) of g restricted to D(g) is a commutative algebra (this follows from the Jacobi identity since D(g) is Abelian) of matrices X ∈ Cm×m .
(7.44)
Any Abelian subalgebra of gl(m, C) can be transformed into a normal form in which all matrices X simultaneously have the block diagonal form ⎛ ⎞ X1 ⎜ ⎟ X2 ⎜ ⎟ (7.45) X=⎜ dim Xq = mq , 1 ≤ q ≤ k ⎟, .. ⎝ ⎠ . Xk where each block Xq ∈ gl(mq , C) has the form (7.46)
Yqmq = 0.
Xq = aq I + Yq ,
The matrices Yq are nilpotent matrices which form an Abelian subalgebra of sl(mq , C) [122]. We recall that an algebra g is nilpotent if and only if a positive integer k exists such that (7.47)
(ad y)k = 0,
(7.48)
Pj (λ) =
∀y ∈ g
(cf. the Engel theorem 2.7). Let y be an element of g such that (ad y)k−1 D(g) = 0, then we have (ad y)k g = 0 (since ad y : g → D(g)). Consequently, ad x|D(g) is nilpotent if and only if ad x|N R(g) is nilpotent, i.e., NR(g) consists of elements x ∈ g such that ad x|D(g) is a nilpotent operator. Since Pj (λ) is a square-free minimal polynomial such that apj = 0 we have p '
(λ − Kjk )
k=1
with p '
(7.49)
Kjk = 0,
Kjk = Kjl
for k = l.
k=1
We have Pj (ad xj )u =
p '
(ad xj − Kjk )u = 0.
k=1
The matrix X in (7.45) acts on the vector space V ∼ D(g) ∼ Cm . We decompose (k V into subspaces invariant under the matrices X of (7.45): V = q=1 Vq . We have (7.50) uq , uq ∈ Vq , ad x(u) = ad x(uq ), u= q
(7.51)
0=
q
Pj (ad xj )(uq ) =
q
p '
q
k=1
(ad xj − Kjk )uq .
Since all eigenvalues Kjk for j given are different, at most one term in each of the products in (7.51) annihilates uq . We conclude that if uq = 0 we have (7.52)
ad xj uq = λjq uq
where λjq = Kjk for some 1 ≤ k ≤ p.
7.3. EXAMPLES
79
Suppose that x ∈ g is in NR(g) and expand x in the basis (7.32): x = y, y ∈ D(g). We have K K K j (7.53) (ad x) u = (ad x) uq = ξ λjq uq = 0 q
q
i ξ i xi +
j
for K large enough (since we assumed x ∈ NR(g)). This implies that for a given q we must have ξ k λjq = 0. (7.54) uq = 0 or j
In turn this means ∀q.
ad x(uq ) = 0, Consequently ad x(u) = 0 and finally
x ∈ centg (u), i.e., NR(g) ⊆ centg (u) = m. If centg (u) is not nilpotent we must take its nilradical and we obtain (7.42). 7.3. Examples Let us present several examples to illustrate the nilradical algorithm. We shall also demonstrate how the methods described in this and previous Chapters can be used to identify the given Lie algebra. For simplicity we shall denote various cosets arising in the procedure by the same letter used for the corresponding element in g, e.g., ek ≡ ek + b1 . Example 7.12. Let g be a 6-dimensional solvable Lie algebra with nonvanishing brackets [e1 , e6 ] = −e1 ,
(7.55)
[e2 , e6 ] = −e2 ,
[e4 , e5 ] = e2 ,
[e3 , e5 ] = e1 ,
[e5 , e6 ] = −e5 .
Its derived algebra D(g) = span{e1 , e2 , e5 } is Abelian and the center C(g) vanishes. Thus we proceed immediately to Step 4 of the algorithm. We choose the basis (7.56)
u1 = e1 ,
u2 = e2 ,
u3 = e5 ,
x1 = e3 ,
x2 = e4 ,
x3 = e6 .
In Step 5 we consider u = u1 and x1 . We find [x1 , u1 ] = [e3 , e1 ] = 0. Thus P1 (λ) = λ and we proceed to Step 6. We have b1 = [x1 , D(g)] = [e3 , D(g)] = span{e1 }. The quotient algebra g/b1 is spanned by e2 + b1 , . . . , e6 + b1 . The algebra g/b1 has nontrivial center C(g/b1 ) = z∞ (g/b1 ) = span{e3 }. Therefore, we construct another quotient ˜ g = (g/b1 )/ span{e3 } = g/ span{e1 , e3 } and proceed to Step 5 with the basis (7.57)
u ˜1 = e2 ,
u ˜2 = e5 ,
x ˜1 = e4 ,
x ˜2 = e6
(+ span{e1 , e3 }).
We have again ˜1 ] = [e4 , e2 ] = 0 (mod span{e1 , e3 }), [˜ x1 , u
80
7. THE NILRADICAL OF A LIE ALGEBRA
˜1 = [˜ i.e., we proceed to Step 6, factor out b x1 , span{˜ u1 , u ˜1 }] = span{e2 } and the arising center span{˜ x1 } span{e4 } of the quotient algebra, arriving at the algebra (7.58)
ˆ1 }, gˆ = span{ˆ u1 , x u ˆ1 = e5
[ˆ x1 , u ˆ1 ] = u ˆ1 ,
(+ span{e1 , . . . , e4 }),
(+ span{e1 , . . . , e4 }).
x ˆ1 = e6
In Step 5 we have [ˆ x1 , u ˆ1 ] = u ˆ1 which implies P1 (λ) = λ − 1. The polynomial P1 is square-free with a nonvanishing constant term. Thus we proceed to Step 8 and form the centralizer = centgˆ (ˆ m u1 ) = span{u1 } = D(ˆg). Equation (7.43) implies that we have found the nilradical NR(ˆg) = span{u1 }. Retracing back our quotients we find that the nilradical of g˜ coincides with the preim of NR(ˆ age m g) under the quotient NR(˜ g) = span{˜ u1 , u ˜2 , x ˜1 } span{e2 , e4 , e5 } and similarly we find the nilradical of the original algebra g (7.59)
NR(g) = span{u1 , u2 , u3 , x1 , x2 } = span{e1 , e2 , e3 , e4 , e5 }
with the Lie brackets (7.60)
[e3 , e5 ] = e1 ,
[e4 , e5 ] = e2 .
Thus the algebra (7.55) must be isomorphic to one of the algebras listed in Section 19.5. By comparison of the dimensions of the derived algebra we conclude that it is the algebra s6,149 . As a matter of fact, it is expressed in the same basis. In the example just presented we twice had to construct the ideal b1 of (7.37). In our second example we will demonstrate the construction of the ideal b2 of (7.39). Example 7.13. Let g be a 6-dimensional solvable Lie algebra with nonvanishing brackets (7.61)
[e1 , e6 ] = −2e1 ,
[e2 , e4 ] = e1 ,
[e2 , e6 ] = −2e2 ,
[e3 , e5 ] = e1 ,
[e3 , e6 ] = −e3 − e5 ,
[e5 , e6 ] = −e5 .
Its derived algebra D(g) = span{e1 , e2 , e3 , e5 } is non-Abelian and its derived algebra satisfies D2 (g) = span{e1 }. Thus in Step 2 we factor out D2 (g) = span{e1 }. The quotient algebra g/D2 (g) has a nonvanishing center C g/D2 (g) = z∞ g/D2 (g) = span{e4 }, i.e., Step 3 instructs us to further remove z∞ g/D2 (g) . Thus we obtain the algebra ˜ g = g/ span{e1 , e4 } span{e2 , e3 , e5 , e6 }
(7.62) with the brackets (7.63)
[e2 , e6 ] = −2e2 ,
[e3 , e6 ] = −e3 − e5 ,
[e5 , e6 ] = −e5 .
u2 = e5 ,
x1 = e6 .
In Step 4 we choose the basis (7.64)
u1 = e3 ,
u3 = e2 ,
In Step 5 we consider u = u1 = e3 and x1 = e6 . We find [x1 , u1 ] = [e6 , e3 ] = e3 + e5 = u1 + u2 ≡ u11 , [x1 , u11 ] = [e6 , e3 + e5 ] = e3 + 2e5 = 2u11 − u1 .
7.3. EXAMPLES
81
Thus we have ad x1 |X1 in (7.35) in the following form 0 −1 . (7.65) ad x1 |X1 = 1 2 Its characteristic polynomial P1 (λ) = λ2 − 2λ + 1 = (λ − 1)2
(7.66)
does not have a root λ = 0 and is not square-free. Thus in Step 7 we are instructed to form the ideal (7.67)
b2 = (ad x1 − 1)D(g) = (ad e6 − 1)D(g) = span{e2 , e5 }
and factor it out of the algebra ˜ g. We obtain the algebra ˆ g=˜ g/ span{e2 , e5 } span{e3 , e6 },
[e3 , e6 ] = −e3 .
The nilradical of the algebra ˆ g is found to be NR(ˆg) = span{e3 } in Step 8, as in the previous example. Consequently, the nilradical of ˜g is the same as the nilradical of of NR(ˆ the preimage m g) = span{e2 , e3 , e5 } ⊂ ˜g. m is nilpotent, i.e., NR(˜ Finally, we include the (hyper)center The ideal m g) = m. 2 C(g/D (g)) and the second element of the derived series g(2) = D2 (g) = (g2 )2 to obtain the nilradical of g in the form NR(g) = span{e1 , e2 , e3 , e4 , e5 }.
(7.68) with the Lie brackets (7.69)
[e2 , e4 ] = e1 ,
[e3 , e5 ] = e1 .
Thus NR(g) is the Heisenberg algebra in two spatial dimensions n5,4 and the algebra (7.55) must be isomorphic to one of the algebras listed in Section 19.8. By inspection of the list of algebras in Section 19.8 with the same structure of characteristic series we identify it as the algebra s6,189 . Notice that which branch of the algorithm is followed is rather sensitive to the choice of basis in the algebra. Let us now consider an example where both b1 and b2 arise. Example 7.14. Let g be a 6-dimensional solvable Lie algebra spanned by f1 , . . . , f6 with the following Lie brackets [f1 , f4 ] = −3f1 − f2 , [f1 , f5 ] = −f1 − f2 , (7.70)
[f1 , f6 ] = −f3 ,
[f2 , f4 ] = f1 − f2 ,
[f2 , f5 ] = f1 + f2 ,
[f2 , f6 ] = f3 ,
[f4 , f5 ] = 2f1 + 2f2 ,
[f4 , f6 ] = 2f1 + 2f3 , [f5 , f6 ] = f1 + f2 .
[f3 , f4 ] = −2f3 ,
The Abelian derived algebra is expressed as D(g) = span{f1 , f2 , f3 } and the center C(g) vanishes. Thus, we proceed to Step 4. The basis (f1 , . . . , f6 ) has the required form (7.32) with the identification (7.71)
u1 = f1 ,
u2 = f2 ,
u3 = f3 ,
x1 = f4 ,
x2 = f5 ,
x3 = f6 .
In Step 5 we consider u01 = f1 and x1 = f4 . We have (7.72) u01 = f1 ,
u11 = [f4 , f1 ] = 3f1 + f2 ,
u21 = [f4 , 3f1 + f2 ] = 4u11 − 4u01 ,
82
7. THE NILRADICAL OF A LIE ALGEBRA
i.e., ad x1 |X1 in (7.35) takes the form ad x1 |X1
(7.73)
0 = 1
−4 . 4
Its characteristic polynomial P1 (λ) = λ2 − 4λ + 4 = (λ − 2)2
(7.74)
does not have a root λ = 0 but is not square-free. We proceed to Step 7 and construct the ideal (7.75)
b2 = (ad x1 − 2 · 1)D(g) = (ad f4 − 2 · 1)D(g) = span{f1 + f2 }
and factor it out of the algebra g, i.e., we identify f2 with −f1 . The factor algebra g/b2 span{f1 , f3 , f4 , f5 , f6 } has the Lie brackets (7.76) [f1 , f4 ] = −2f1 ,
[f1 , f6 ] = −f3 ,
[f3 , f4 ] = −2f3 ,
[f4 , f6 ] = 2f1 + 2f3 .
It has the Abelian derived algebra D(g/b2 ) = span{f1 , f3 } and the 1-dimensional (hyper)center (7.77)
C(g/b2 ) = span{f5 }.
˜= In Step 3 we factor out C(g/b2 ), i.e., we consider the 4-dimensional algebra g span{f1 , f3 , f4 , f6 } with the Lie brackets (7.76). Again, the basis is already ordered as in Step 4. We proceed to Step 5 with u ˜ = f1 and x ˜1 = f4 . We find that (7.78)
[˜ x1 , u ˜] = [f4 , f1 ] = 2f1 = 2˜ u.
The characteristic polynomial is P1 (λ) = λ − 2,
(7.79)
i.e., square-free without vanishing root. We proceed to Step 8 and have to repeat the procedure for another choice of the vector x ˜. We take the vector x ˜2 = f6 and find (7.80)
u ˜12 = [˜ x2 , u ˜] = [f6 , f1 ] = f3 ,
u ˜22 = [˜ x2 , u ˜12 ] = [f6 , f3 ] = 0.
The characteristic polynomial is (7.81)
P2 (λ) = λ2 ,
with root λ = 0. We proceed to Step 6 and construct the ideal (7.82)
b1 = [˜ x2 , D(g)] = [f6 , span{f1 , f3 }] = span{f3 }.
We factor b1 out of the algebra ˜ g and obtain the algebra gˆ span{f1 , f4 , f6 } with the Lie brackets (7.83)
[f1 , f4 ] = −2f1 ,
[f4 , f6 ] = 2f1 .
and return to Step 2. The algebra ˆ g has an Abelian derived algebra and the 1-dimensional (hyper)center (7.84)
C(ˆ g) = span{f1 − f6 }.
We factor z(ˆ g) out of ˆ g, i.e., identify f6 with f1 , and we obtain the two-dimensional solvable Lie algebra (7.85)
[f1 , f4 ] = −2f1
which has the nilradical f1 , as can be found either by inspection or repeating Steps 4 to 8 once again.
7.3. EXAMPLES
83
Finally, we lift the ideal
span{f1 } = NR g/b2 /C(g/b2 )/b1 /C(ˆg)
(7.86) to an ideal in g, (7.87)
NR(g) = span{f1 }span{f1 − f6 }span{f3 }span{f5 }span{f1 + f2 } = span{f1 , f2 , f3 , f5 , f6 }
since both ideals m arising in Steps 6 and 7 are nilpotent. The nilradical NR(g) has the Lie brackets [f1 , f5 ] = −f1 − f2 ,
[f1 , f6 ] = −f3 ,
[f2 , f6 ] = f3 ,
[f2 , f5 ] = f1 + f2 ,
[f5 , f6 ] = f1 + f2 .
Computation of its characteristic series reveals that it is isomorphic to the nilpotent Lie algebra n5,4 . Thus, the algebra (7.70) must be isomorphic to one of the algebras listed in Section 19.5. The dimensions of its derived algebra and of its center imply that it is isomorphic to the algebra s6,149 , i.e., to the algebra (7.55) of Example 7.12. It is indeed so, the algebra (7.70) is obtained from the algebra s6,149 by the change of basis f1 = e1 + e5 ,
(7.88)
f2 = e1 − e5 ,
f4 = 2(e3 − e5 + e6 ),
f5 = 2e3 ,
f3 = e2 , f6 = e4 + e5 .
The nilradical (7.87) of the algebra (7.70) is of course just the nilradical (7.59) expressed in another basis (7.88). However, as we have seen, the branching of the algorithm and the complexity of computation were different. In our last example we will present an algebra for which neither ideals b1 nor b2 arise. Example 7.15. Let g be a 6-dimensional solvable Lie algebra with nonvanishing brackets (7.89)
[e1 , e5 ] = −e1 ,
[e1 , e6 ] = e1 ,
[e2 , e5 ] = −2e2 ,
[e3 , e6 ] = −e3 ,
[e4 , e5 ] = −4e4 ,
[e4 , e6 ] = −2e4 .
[e3 , e5 ] = −3e3 ,
Its derived algebra D(g) = span{e1 , e2 , e3 , e4 } is Abelian and g has a vanishing center. Thus we proceed to Step 4. We choose the basis (7.90)
u1 = e1 ,
u2 = e2 ,
u3 = e3 ,
u4 = e4 ,
x1 = e5 ,
x2 = e6 .
In Step 5 we consider u = u1 = e1 and x1 = e5 . We find [x1 , u] = u. Thus we have X1 = span{e1 } and ad x1 |X1 = 1 in (7.35). The characteristic polynomial P1 (λ) = λ − 1 is square-free and without vanishing root. We proceed directly to Step 8 and are instructed to repeat Step 5 with u = u1 = e1 and x2 = e6 . We find [x2 , u] = −u,
X2 = span{u} = span{e1 },
ad x2 |X2 = −1,
P2 (λ) = λ + 1.
We proceed to Step 8. Since we have exhausted all xi and in both cases found square-free characteristic polynomial of ad xj |Xj without a root equal to 0, we construct (7.91)
m = centg (e1 ) = span{e1 , e2 , e3 , e4 }
84
7. THE NILRADICAL OF A LIE ALGEBRA
which equals the derived algebra D(g). Thus the algorithm terminates and we have found the nilradical of g, NR(g) = m = span{e1 , e2 , e3 , e4 }.
(7.92)
The nilradical is Abelian, thus the algebra (7.89) must be contained in the list presented in Section 19.11 among the algebras characterized by the values (7.93)
US = [0],
CS = [6, 4],
DS = [6, 4, 0].
We compute the Casimir invariants using the procedure described in Section 3.2 and find two functionally independent rational invariants (7.94)
I1 =
e1 e3 , e22
I2 =
e21 e4 . e32
The only algebra in Section 19.11 of the structure (7.93) which has two rational invariants is s6,213 for special values of the parameters a, b, c, d. Thus the algebra (7.89) must be contained in the class s6,213 and by an explicit calculation one finds that it is so for the values a = −1,
(7.95)
b = −2,
c = 2,
d = 3.
The change of basis taking us from s6,213 to the algebra (7.89) is (7.96)
e1 = e4 ,
e2 = e3 ,
e3 = e1 ,
e4 = e2 ,
e5 = e5 + 2e6 ,
e6 = −e5 .
Notice that also in this example the sequence of steps depends on the chosen basis (7.90). If we had chosen instead u1 = e2 and x1 = e6 , the ideal b1 of Step 6 would arise in our computation and more steps would be necessary to perform. 7.4. Identification of the nilradical using the Killing form Another method which is often used to find the nilradical of the given Lie algebra g is based on the properties of the Killing form. Its essential advantage is its computational simplicity. Its major disadvantage is that it does not always lead to a conclusive answer. Let g be a solvable Lie algebra. From the general properties of derivations of solvable Lie algebras we know that ad x : g → NR(g)
(7.97) for any x ∈ g and (7.98)
ad y : NRk (g) → NRk+1 (g)
for any k ≥ 1 and y ∈ NR(g). Thus, the composition ad y ◦ ad x of these two operators is nilpotent and consequently (7.99)
K(y, x) = Tr(ad y ◦ ad x) = 0.
To sum up, we have found that (7.100)
NR(g) ⊆ g⊥ ,
where g⊥ is the orthogonal complement of g with respect to the Killing form, (7.101)
g⊥ = {x ∈ g | K(x, g) = 0}.
Unfortunately, equality does not in general hold in (7.100), as the example of a complex 3-dimensional solvable algebra (7.102)
g = span{e1 , e2 , e3 },
[e1 , e3 ] = e1 ,
[e2 , e3 ] = ie2
7.4. IDENTIFICATION OF THE NILRADICAL USING THE KILLING FORM
85
with vanishing Killing form clearly demonstrates. On the other hand, given the relative computational complexity of the algorithm described in Section 7.2 it appears to be reasonable to first compute the ideal g⊥ and check whether it is nilpotent. If yes, we have identified the nilradical of g, (7.103)
NR(g) = g⊥ .
If not, one should compute the codimension of g2 + z∞ (g) in g⊥ . If it equals one, the nilradical is (7.104)
NR(g) = g2 + z∞ (g)
on dimensional grounds. Only if the codimension is larger than 1, the relations (7.105)
g2 + z∞ (g) ⊆ NR(g) g⊥
are insufficient to determine the nilradical of g immediately and we have to perform the algorithm as described in Section 7.2. We observe that in all examples discussed in Section 7.3 the nilradical coincides with g⊥ .
Part 3
Nilpotent, Solvable and Levi Decomposable Lie Algebras
CHAPTER 8
Nilpotent Lie Algebras The classification of low-dimensional nilpotent Lie algebras as performed in [77] is based on the fact that such algebras necessarily possess a nonvanishing Abelian ideal. Examples of such ideals are the last nonvanishing ideal in the lower central series or the center. Let us assume that we want to classify nilpotent Lie algebras n of given dimension n with a k-dimensional maximal Abelian ideal a. The procedure starts with the classification of nonequivalent representations of (n − k)-dimensional nilpotent algebras on the maximal Abelian ideal a. We are only interested in representations by nilpotent matrices. The knowledge of these defines the Lie brackets of the form [n, a]. The remaining Lie brackets, i.e., between pairs of vectors in the set n\a are now specified modulo a and different nonequivalent possibilities for them must be found. Contrary to the case of the radical or the nilradical of a Lie algebra, the maximal Abelian ideal of a nilpotent Lie algebra is not necessarily unique. The reason for this is that the sum of Abelian ideals is not necessarily an Abelian ideal. Consequently, the maximal one cannot be constructed simply by taking the sum of all Abelian ideals. It may even happen that different maximal Abelian ideals in a given nilpotent algebra have different dimensions. Proper care must be taken in order to avoid redundancies in the classification obtained in this manner. 8.1. Maximal Abelian ideals and their extensions Let us assume that n is a nilpotent Lie algebra and a is one of its maximal Abelian ideals (possibly unique). Let (e1 , . . . , ek ) be a basis of a and (ek+1 , . . . , en ) its extension to a basis of n. Let us construct the adjoint representation of n, restrict it to the ideal a and find ad|a (x), x ∈ n. For any x ∈ a we have ad|a (x) = 0. On the other hand, all ad|a (ej ), k < j ≤ n must be linearly independent nilpotent matrices; otherwise the ideal a would not be maximal or the algebra n would not be nilpotent. Therefore, the representation of n x ∈ n → ad|a (x) gives rise to a faithful representation of the nilpotent factor algebra n/a on a by nilpotent matrices. Such a representation can have dimension at most k(k − 1)/2 and we obtain the restriction n−k ≤
k(k − 1) 2
which leads to the lower bound on the dimension k of the maximal Abelian ideal √ 1 1 + 8n − 1 ≤ k ≤ n. (8.1) 2 89
90
8. NILPOTENT LIE ALGEBRAS
(The bound (8.1) is due to V. V. Morozov, [77].) This relation greatly facilitates the classification of nilpotent algebras in dimensions n = 1, 2, 3, 4. In these dimensions we have n k 1 1 2 2 3 2, 3 4 3, 4 and consequently either n is Abelian (n = a) or n/a is one-dimensional and the classification is completed by considering all possible canonical forms of nilpotent 2 × 2 or 3 × 3 matrices. In higher dimensions we may have n − k ≥ 2 and we proceed as follows. Assuming that n, k are given we consider step by step all nilpotent Lie algebras m of dimension n − k. Since n − k < n these are assumed to be already known. We find all their inequivalent representations on the k-dimensional vector space a expressed entirely in terms of nilpotent matrices. For each of these representations we presume that m = n/a and reconstruct from it the Lie brackets in n. As was noted above, some of these are defined only modulo a. Finally, we employ a convenient change of basis in n that preserves the already fixed brackets [ei , ej ], i ≤ k < j in order to bring the nilpotent algebra n to a chosen (and therefore unique) canonical form. Such a transformation allows a simplification of the brackets [ei , ej ], k < i, j. We note that nilpotent algebras arising in this way may be decomposable and that algebras constructed starting from different maximal Abelian ideals may turn out to be isomorphic. These questions must be addressed once the computation is finished. The decomposable algebras must be deleted from the list. Isomorphic algebras must be identified and precisely one representative of each isomorphism class chosen and declared “canonical.” As an example showing all the aspects of the classification procedure let us consider the case dim n = 5. The restriction (8.1) allows three possible values of k = 3, 4, 5. For the obtained nilpotent algebras we use the notation nn,j , to be used consistently in the tables presented below. The letter n stands for “nilpotent,” the first subscript gives the dimension of the algebra, i.e., n = 5 here. The second subscript j = 1, 2, 3, . . . labels nonisomorphic indecomposable Lie algebras of the dimension n. When k = 5 the nilpotent algebra itself is Abelian and consequently decomposable. For k = 4 we have one element e5 outside of the maximal Abelian ideal a = span{e1 , e2 , e3 , e4 }. Therefore, the only information needed for the full knowledge of n is the action of e5 on the Abelian algebra a. This is specified in terms of the nilpotent linear operator A (8.2)
A : a → a : [x, e5 ] = A(x).
Such an operator A can be chosen in its Jordan canonical form and will have zeros on the diagonal. When any of the irreducible blocks on the diagonal is 1 × 1 the corresponding algebra n is decomposable. Indeed, it has a nontrivial central
8.1. MAXIMAL ABELIAN IDEALS AND THEIR EXTENSIONS
component. Consequently, ⎛ 0 ⎜0 (8.3) A=⎜ ⎝0 0
91
there are only two nonequivalent possibilities ⎞ ⎛ ⎞ 0 1 0 0 1 0 0 ⎜ ⎟ 0 1 0⎟ ⎟ , A˜ = ⎜0 0 0 0⎟ . ⎠ ⎝ 0 0 1 0 0 0 1⎠ 0 0 0 0 0 0 0
The corresponding algebras are n5,5 (cf. Section 18.1) e1 e2 e3 e4
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 e1 e2 e3
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 e1 . 0 e3
˜5,1 and n e1 e2 e3 e4
˜5,1 is written in a form which naturally arose during the procedure The algebra n but is not the most convenient one from the point of view of its algebraic structure, namely the characteristic series. Therefore we perform a change of basis in order to arrive at its form n5,1 presented in Section 18.1 e1 e2 e3 e4
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 . e1 e2
Any 5-dimensional indecomposable nilpotent algebra with a 4-dimensional maximal Abelian nilradical is isomorphic to n5,1 or n5,5 . Finally, we consider the case k = 3. That means that we have two elements e4 , e5 acting on the maximal Abelian ideal a = span{e1 , e2 , e3 }. The two-dimensional factor algebra m = n/a is Abelian (since no two-dimensional indecomposable nilpotent algebra exists). We write its representation on a in terms of two operators (8.4)
A1,2 = − ad|a (e4,5 ),
i.e., [x, e4 ] = A1 (x), [x, e5 ] = A2 (x).
Thus we need to classify all two-dimensional Abelian subalgebras of sl(3, F) that are realized by nilpotent matrices. All subalgebras of sl(3, F) are known [136]. Three nonequivalent representations of n/a on a in terms of nilpotent matrices exist, namely ⎞ ⎞ ⎛ ⎛ 0 0 1 0 0 0 A1 = ⎝0 0 0⎠ , A2 = ⎝0 0 1⎠ , (8.5) 0 0 0 0 0 0 ⎞ ⎞ ⎛ ⎛ 0 1 0 0 0 1 (8.6) A¯1 = ⎝0 0 1⎠ , A¯2 = ⎝0 0 0⎠ , 0 0 0 0 0 0
92
(8.7)
8. NILPOTENT LIE ALGEBRAS
⎞ ⎞ ⎛ ⎛ 0 1 0 0 0 1 A˜1 = ⎝0 0 0⎠ , A˜2 = ⎝0 0 0⎠ , 0 0 0 0 0 0
distinguished by the maximal rank of a linear combination a1 A1 + a2 A2 and the dimension of the common nullspace of A1 and A2 . For each of these representations we have to investigate the distinct possibilities for the Lie bracket [e4 , e5 ]. In general we have (8.8)
[e4 , e5 ] = αe1 + βe2 + γe3 .
In the first case (8.5), we can perform a change of basis e4 = e4 − βe3 ,
e5 = e5 + αe3
to obtain [e4 , e5 ] = γe3 . When γ = 0, the vector space span{e1 , e2 , e4 , e5 } is a maximal Abelian ideal of dimension 4, the case already investigated above. When γ = 0 we rescale e1 , e2 , e3 by γ to get [e4 , e5 ] = e3 . Dropping the primes we find the algebra n5,2 e1 e2 e3 e4 e5 e1 0 0 0 0 0 e2 0 0 0 0 . e3 0 e2 e1 e4 0 e3 Similarly, considering the case (8.6) we can simplify [e4 , e5 ] by a suitable addition of e2 , e3 to e4 , e5 into the form [e4 , e5 ] = γe3 . The two distinct subcases are γ = 0 and γ = 0. The first case gives rise to the ˜5,4 algebra n e1 e2 e3 e4 e5 e1 0 0 0 0 0 e2 0 0 0 e1 e3 0 e1 e2 e4 0 0 which is again presented in Section 18.1 in a modified form n5,4 , better suited to the structure of the characteristic series and obtained by the change of basis e3 = −e4 , e4 = e3 e1 e2 e3 e4 e5 e1 0 0 0 0 0 e2 0 0 0 e1 . e3 0 e1 0 e4 0 e2 The case γ = 0 is brought to the form n5,6 e1 e2 e3 e4 by a rescaling of e1,2,3 .
e1 0
e2 0 0
e3 0 0 0
e4 0 0 e1 0
e5 0 e1 e2 e3
8.2. CLASSIFICATION OF LOW-DIMENSIONAL NILPOTENT LIE ALGEBRAS
93
Finally, for the case (8.7) we add a multiple of e3 to e4 in order to obtain [e4 , e5 ] = βe2 + γe3 . Now either β = γ = 0 and we obtain the algebra n5,3 e1 e2 e3 e4
e1 0
e2 0 0
e3 0 0 0
e4 0 e1 0 0
e5 0 0 e1 0
or at least one of β, γ is nonvanishing. In that case we may assume without loss of generality that β = 0 (otherwise we interchange e2 ↔ e3 and e4 ↔ e5 ) and perform a change of basis γ e1 = βe1 , e2 = βe2 + γe3 , e3 = βe3 , e4 = e4 , e5 = e5 − e4 . β This change of basis preserves the brackets defined by the matrices A˜1,2 , i.e., the Lie brackets between e1,2,3 and e4,5 , and (8.8) reduces to [e4 , e5 ] = e2 . The corresponding nilpotent algebra, however, turns out to be isomorphic to the algebra n5,4 constructed above upon the interchange of e4 and e5 and the change of sign of e2 . This completes the construction of all nonisomorphic indecomposable 5-dimensional nilpotent Lie algebras. 8.2. Classification of low-dimensional nilpotent Lie algebras Complete lists of isomorphism classes of nilpotent Lie algebras have been published for dimensions n ≤ 7 [51, 52, 77, 107, 110] and partial results exist for n = 8 and 9 [125–127]. The classification is performed along similar lines as that for n = 5 given in Section 8.1. The method due to V. V. Morozov [77] consists of several steps. (1) Choose a dimension k ≤ n − 1 respecting the bound (8.1) and denote the basis for the corresponding maximal Abelian ideal a by (e1 , . . . , ek ). (2) The remaining basis elements (ek+1 , . . . , en ) must form a nilpotent factor algebra n/a represented by k × k matrices acting on a as in (8.2) and (8.4). The factor algebra n/a has dimension less than n, i.e., we can assume that it is already known and has a prescribed fixed structure. For each nilpotent (n − k)-dimensional nilpotent algebra n/a we thus have to consider its representations, i.e., nonequivalent sets of linearly independent nilpotent matrices A1 , . . . , An−k with Aj ∈ Fk,k . If n/a is Abelian we have what is called an Abelian Nilpotent Subalgebra (ANS) of sl(k, F) (i.e., Abelian Lie algebra realized by nilpotent matrices) [64, 122, 137]. (3) The classification of Abelian nilpotent subalgebras of sl(k, F) is somewhat simplified by the existence of the so-called Kravchuk normal form of Maximal Abelian Nilpotent Subalgebras (MANS) of sl(k, F), i.e., ANS that are their own centralizers in sl(k, F). Each MANS is partly characterized by its Kravchuk signature (λ μ ν) which is a triplet of nonnegative integers satisfying (8.9)
λ ≥ 1,
ν ≥ 1,
λ+μ+ν =k
94
8. NILPOTENT LIE ALGEBRAS
where λ is the dimension of the common nullspace of the set of matrices Aj and ν is the codimension of their common image. Both λ and ν (and hence also μ) are basis independent (cf. Section 10, (10.11)). In the case n = 5, k = 3 considered in Section 8.1, the factor algebra is indeed Abelian. The three cases (8.5), (8.6) and (8.7) correspond to Kravchuk signatures (2 0 1), (1 1 1) and (1 0 2), respectively. If the factor algebra n/a is nilpotent but not Abelian, it is also partly characterized by its Kravchuk signature and can be presented in the normal form [122]. For chosen values of (λμν) we have to find all nonequivalent k-dimensional representations of n/a. (4) Once a representation A of the factor algebra n/a on the Abelian ideal a is given (8.10) A : n/a → gl(a),
A(ea )ei =
k
j
(Aa )i ej ,
a = k + 1, . . . , n, i = 1, . . . , k
j=1
the Lie brackets [ei , ea ] = −A(ea )ei = −
k
j
(Aa )i ej
j=1
are fixed. The Lie brackets between ea and eb , a, b = k + 1, . . . , n have in general the form (8.11)
[ea , eb ] =
n
Bab c ec +
k
Cab i ei .
i=1
c=k+1
Equivalently, we can express them in terms of skew symmetric bilinear maps B : n/a × n/a → n/a,
C : n/a × n/a → a
as (8.12)
[ea , eb ] = B(ea , eb ) + C(ea , eb ).
The map B represents the given Lie bracket on the factor algebra n/a, B( , ) = [ , ] mod a and therefore is known. The Jacobi identities (ea , eb , ec ) imply that the map C : n/a × n/a → a is a 2-cocycle for the representation A of the algebra n/a, (8.13) A(x)C(y, z) + A(y)C(z, x) + A(z)C(x, y) + C x, B(y, z) + C y, B(z, x) + C z, B(x, y) = 0 (cf. Section 2.4). Any change of basis of the form (8.14)
ea → ea = ea +
k
Mai ei
i=1
k
defines an operator M : n/a → a, M (ea ) = i=1 Mai ei . In the language of Chevalley cohomology the map M defines also the 2-coboundary dM : n/a × n/a → a dM (x, y) = A(x)M (y) − A(y)M (x) − M B(x, y) .
8.2. CLASSIFICATION OF LOW-DIMENSIONAL NILPOTENT LIE ALGEBRAS
95
The change of basis (8.14) modifies the Lie brackets (8.11). It leaves invariant the map B in (8.12), (i.e., its matrix Bab c in (8.11)) but induces a change of the map C which can be expressed as (8.15)
C → C = C + dM.
Therefore, nilpotent algebras with the same algebraic structure of n/a and representation A whose maps B differ by a coboundary are necessarily isomorphic. The classes of 2-cocycles modulo 2-coboundaries form the second cohomology group H 2 (n/a, a; A). Therefore, the cohomology theory can be employed in the identification of algebras which are not related by any transformation of the form (8.14), provided that the 2nd cohomology group H 2 (n/a, a; A) can be calculated. On the other hand, a change of basis of the form (8.14) is not the only transformation of ei , ea which is allowed. In addition we can take linear combinations of the elements ea induced by any automorphism of n/a and also change the basis of a in a way that commutes with the representation A. Last but not least, neither the representation A nor the maximal Abelian ideal of the given algebra n is in general unique (several nonequivalent maximal Abelian ideals often exist, sometimes of different dimension). This leads to further possible isomorphisms between the nilpotent algebras found by the above procedure. That means that cohomology arguments alone are not sufficient to eliminate all the isomorphisms between the arising algebras. Other classification methods have been devised, e.g., the Skjelbred – Sund construction of central extensions of nilpotent algebras based also on the cohomology theory of nilpotent Lie algebras [112]. It uses the center of n instead of a, i.e., constructs all n-dimensional nilpotent algebras inductively from the knowledge of all lower dimensional ones. This method was successfully employed by M.P. Gong in his classification of 7-dimensional nilpotent algebras [51] which provides an independent check and several minor corrections1 to the classification [110] performed by C. Seeley by a combination of other methods. We mention that the number of indecomposable nonisomorphic nilpotent Lie algebras grows dramatically with dimension. Namely, in dimension 2 there is no such algebra, in dimensions 3 and 4 there is only one each, in dimension 5 there are 6 algebras. In dimension 6 the numbers start to differ depending on the field F = C, R; there are 20 complex algebras and 24 real algebras. In dimension 7 continuous 1-parametric families of nonisomorphic algebras appear. The list of [51] which is probably the most reliable one presents the algebras organized into 119 classes over complex numbers and 24 additional classes over real numbers. A computerized classification of 9-dimensional nilpotent algebras with 7-dimensional maximal Abelian ideal in [127] claims to have found 24 168 classes of nonisomorphic nilpotent algebras. A complete list of nilpotent Lie algebras up to dimension 6 is given in the respective chapters devoted to Lie algebras of different dimensions in Part 4. The maximal Abelian ideals a, which were used in the classification of nilpotent algebras, are either 1 The list [110] almost coincides with [51] up to four corrections, only one substantial (the Lie brackets do not define an algebra), the others being restrictions on the parameter ranges or corrections of typographical errors.
96
8. NILPOTENT LIE ALGEBRAS
• uniquely specified as an ideal in the characteristic series or a centralizer of such, or • specified as a member of a class of equivalent ideals, described by their specific properties, which can be brought to the form presented in Part 4 by a suitable automorphism of n (in this case we use the symbol ∼). When there are several nonequivalent choices of the maximal Abelian ideal a, usually the one allowing its complement in n to be an Abelian subalgebra was preferred if one exists. “s.a.” is a shorthand notation for “subalgebra.” The ideals employed are as follows: • One-dimensional nilpotent Lie algebra n1,1 : a = n1,1 • Three-dimensional nilpotent Lie algebra n3,1 : a ∼ {any 2-dim. s.a. of n3,1 } • Four-dimensional nilpotent Lie algebra n4,1 : a = cent D(n4,1 ) = cent z2 (n4,1 ) • Five-dimensional nilpotent Lie algebras n5,1 : a = unique 4-dim. Abelian subalgebra of n5,1 n5,2 : a = D(n5,2 ) = z2 (n5,2 ) n5,3 : a ∼ {any 3-dim. Abelian s.a. of n5,3 } n5,4 : a = z2 (n5,4 ) n5,5 : a = cent D(n5,5 ) = cent z3 (n5,5 ) n5,6 : a = D(n5,6 ) = z3 (n5,6 ) • Six-dimensional nilpotent Lie algebras n6,1 : a ∼ {any 4-dim. Abelian s.a. of n6,1 } n6,2 : a ∼ {any 4-dim. Abelian s.a. of n6,2 such that [a, n6,2 ] = D(n6,2 )} n6,3 : a ∼ {any 4-dim. Abelian s.a. of n6,3 } n6,4 : a = cent D(n6,4 ) = z2 (n6,4 ) n6,5 : a = cent D(n6,5 ) = z2 (n6,5 ) n6,6 : a = cent D(n6,6 ) = z2 (n6,6 ) n6,7 : a = cent D(n6,7 ) = cent z2 (n6,7 ) n6,8 : a = z2 (n6,8 ) n6,9 : a = z2 (n6,9 ) n6,10 : a = D(n6,10 ) = z3 (n6,10 ) % & n6,11 : a ∼ any 4-dim. Abelian s.a. of cent D(n6,11 ) n6,12 : a = cent D(n6,12 ) = cent z2 (n6,12 ) 2 n6,13 : a = cent D(n6,13 ) = cent z (n6,13 ) n6,14 : a = D(n6,14 ) = z3 (n6,14 ) n6,15 : a = cent D(n6,15 ) = z3 (n6,15 ) n6,16 : a = z3 (n6,16 ) n6,17 : a = cent D(n6,17 ) n6,18 : a = (n6,18 )3 = z3 (n6,18 ) n6,19 : a = (n6,19 )3 = z3 (n6,19 )
8.2. CLASSIFICATION OF LOW-DIMENSIONAL NILPOTENT LIE ALGEBRAS
97
n6,20 : a = cent D(n6,20 ) = cent z4 (n6,20 ) n6,21 : a = D(n6,21 ) = z4 (n6,21 ) n6,22 : a = D(n6,22 ) = z4 (n6,22 ) The identification of different nilpotent Lie algebras in the lists in Part 4 can be accomplished easily over the field of complex numbers by determination of dimensions US, CS and DS of ideals in the characteristic series and the dimension of the algebra of derivations of n when the sets US, CS and DS coincide. Over the field of real numbers the identification is complicated by the existence of several different real forms of some of the complex algebras. Their differentiation is of course not possible by the above mentioned “dimensional” invariants. It turns out that these nonisomorphic real forms can be distinguished by the signatures of the Killing forms of their algebras of derivations. (The Killing form of a nilpotent Lie algebra itself automatically vanishes and consequently is useless for any identification.)
CHAPTER 9
Solvable Lie Algebras and Their Nilradicals 9.1. General structure of a solvable Lie algebra Let us recall that every solvable Lie algebra s contains a unique maximal nilpotent ideal called the nilradical n = NR(s). For a solvable Lie algebra s the dimension of the nilradical n satisfies [78] dim n ≥
(9.1)
1 2
dim s
(see Section 9.3 for an improved bound and its derivation). Any solvable Lie algebra s can be written as the algebraic sum of the nilradical n and a complementary linear space F s = F n.
(9.2) 2
The derived algebra s = [s, s] of a solvable Lie algebra s is contained in the nilradical (9.3)
s2 ⊆ n.
because all derivations map s into n, in particular any inner derivation. An element x of s is nilpotent if it satisfies k ad(x) y = x, . . . , x, [x, y] . . . = 0, ∀y ∈ s for some positive integer k. A set of elements {x1 , . . . , xk } of s is called linearly nilindependent if no nontrivial linear combination of them is nilpotent. Use will be made of the adjoint representation defined in (2.21) and of its restriction to the nilradical n of s denoted by ad|n . This restriction is realized by matrices A ∈ Fn×n where F is the ground field. If x is a nilpotent element of s, it will be represented by a nilpotent matrix in the adjoint representation and consequently also ad|n (x) will be nilpotent. A set of matrices in Fn×n will be called linearly nilindependent if no nontrivial combination of them is a nilpotent matrix. From now on we shall assume that the solvable algebra s is finite-dimensional with dim s = s and indecomposable, i.e., cannot be decomposed into a direct sum of two or more subalgebras. For the purpose of the classification we further assume that s is nonnilpotent. 9.2. General procedure for classifying all solvable Lie algebras with a given nilradical Let us consider a given nilpotent Lie algebra n of dimension n as a nilradical of a solvable Lie algebra s of dimension s. We wish to find all such (indecomposable, nonnilpotent) solvable Lie algebras. Any such solvable algebra will be called a 99
100
9. SOLVABLE LIE ALGEBRAS AND THEIR NILRADICALS
solvable extension of the given nilradical n. Let us choose a basis (e1 , . . . , en ) of n and extend it to a basis of s (9.4)
(e1 , . . . , en , f1 , . . . , ff ),
where n + f = s.
Since the derived algebra of a solvable Lie algebra is contained in the nilradical, the commutation relations for the solvable Lie algebra s in the basis (9.4) can be written as [ei , ej ] = Nij k ek ,
(9.5) (9.6)
[fα , ei ] =
(9.7)
[fα , fβ ] =
(Aα )ji ej , γαβ i ei ,
i, j ∈ {1, . . . , n}, α, β ∈ {1, . . . , f },
where Latin subscripts and superscripts refer to the basis of the nilradical n, Greek ones to the basis of the complementary space F . Summation over repeated indices is to be understood. The structure constants Nij k of the nilradical are assumed to be known (in the chosen basis (e1 , . . . , en )). The Jacobi identities for {ei , ej , ek } are already satisfied. The remaining Jacobi identities must be imposed. The triplets {fα , ei , ej }, {fα , fβ , ei }, {fα , fβ , fδ } imply (9.8)
Nij k (Aα )lk + Njk l (Aα )ki − Nik l (Aα )kj = 0,
(9.9)
([Aα , Aβ ])ji = γαβ k Nik j ,
(9.10)
γβδ i (Aα )ki + γαβ i (Aδ )ki + γδα i (Aβ )ki = 0,
respectively. Equation (9.8) is a system of n2 (n − 1)/2 linear homogeneous equations for n2 unknowns, i.e., the matrix elements (Aα )ji , for each value of α ∈ {1, . . . , f } separately (since Nij k are known). It follows from (9.8) that the matrix Aα represents a derivation Dα of the nilradical. In the adjoint representation of the Lie algebra s restricted to the nilradical n we have (9.11)
D1 = ad|n (f1 ), . . . , Df = ad|n (ff ) (A1 , . . . , Af )
where Dα (ei ) = (Aα )ji ej . Finding all sets of matrices {Aα } satisfying (9.8) is equivalent to finding all sets of nilindependent derivations of the nilradical. They must be nilindependent since otherwise the nilradical would be larger (it would contain one or more of the elements of F ). This also means that the derivations are outer ones: inner derivations of n are always represented by nilpotent matrices. For simplicity, we shall change the symbol Aα used for the matrices to Dα , so that the derivation and its matrix are denoted identically. As we have just seen, in order to have nontrivial solvable extensions our nilpotent algebra n must possess at least one nonnilpotent derivation. This is not automatic. A nilpotent algebra n which has only nilpotent derivations and consequently is not a nilradical of any solvable Lie algebra is called characteristically nilpotent. Characteristically nilpotent Lie algebras exist in any dimension n ≥ 7 and are in fact prevalent among nilpotent algebras of higher dimensions in the sense that they form an open subset of the variety of nilpotent algebras in the Zariski topology.
9.2. GENERAL CLASSIFICATION PROCEDURE
101
The simplest example of such an algebra is e1 e2 e3 e4 e5 e6
e1 0 0 0 0 0 0
e2 0 0 0 0 0 0
e3 0 0 0 0 0 −e1
e4 0 0 0 0 e1 −e1
e5 0 0 0 −e1 0 −e2
e6 0 0 e1 e1 e2 0
e7 0 e1 e2 e3 e4 e5
which was first presented in [47]. Relations (9.9) exist for f ≥ 2 and determine the properties of the set of matrices (D1 , . . . , Df ). In particular, if the nilradical is Abelian we have Nij k = 0 and hence the matrices commute (9.12)
[Dα , Dβ ] = 0.
In general the relation (9.12) does not hold. However, the space F is not uniquely defined by the solvable algebra s. In the next paragraph we will discuss “allowed transformations” of the basis of s that will be used to classify the Lie algebras s into equivalence classes. In all cases so far considered it turns out that F can be chosen so that matrices Dα commute, i.e., (9.12) holds in a suitably chosen basis. The relations (9.10) represent a set of linear homogeneous equations for the structure constants γαβ i once the matrices Dα are known. Nontrivial equations (9.10) exist for f ≥ 3. A classification of all solvable Lie algebras s with the same nilradical n amounts to a classification of all matrices Dα and constants γαβ i satisfying (9.8), (9.9) and (9.10) under the following “allowed transformations”: (1) Redefinition of the space F (9.13) f˜α = fα + r j ej , r j ∈ F. α
α
(2) Change of basis in the nilradical n (9.14)
ei = Sij ej ,
S ∈ Aut(n) ⊆ GL(n, F).
Thus the matrices S form the group of automorphisms of the nilradical, expressed in the chosen basis (e1 , . . . , en ). By definition, they leave the set of commutation relations (9.5) invariant and consequently respect all basis independent properties of the nilradical (in particular all ideals in the derived and lower and upper central series). (3) Change of basis in F (9.15) f˜i = Gj fj , G ∈ GL(f, F). i
The derivations Dα are by definition 1-cocycles i.e., in Z 1 (n, n; adn ). The inner derivations are 1-coboundaries, i.e., in B 1 (n, n; adn ). The transformation (9.13) transforms the corresponding matrices Dα as follows α )ki ek , [f˜α , ei ] = [fα , ei ] + rαj [ej , ei ] = (D i.e., (9.16)
α )i = (Dα )i − r k Njk i . (D j j α
The equivalence defined by (9.16) implies that Dα should be viewed as elements of the first cohomology group H 1 (n, n; adn ), i.e., equivalence classes, rather than
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9. SOLVABLE LIE ALGEBRAS AND THEIR NILRADICALS
outer derivations themselves. In other words, we combine inner derivations with the outer derivations to modify the matrices Dα . The matrices Dα defined in (9.6) must commute when the nilradical n is Abelian. Our experience leads us to the following conjecture Conjecture 9.1. The matrices Dα for non-Abelian nilradicals can be trans α that commute pairwise. In formed by the redefinition (9.16) into matrices D the cohomological formulation that means that a representative basis of the set % & [F ] = span [Dα ], α ∈ {1, . . . , f } can be chosen to consist of commuting operaα. tors D The classification of solvable extensions has been performed for the following classes of nilpotent Lie algebras: Heisenberg algebras h(N ) (where dim h(N ) = 2N + 1, N ≥ 1) [106], Abelian Lie algebras an , n ≥ 1 [83, 84], nilradicals of Borel subalgebras [118], in particular “triangular” Lie algebras t(N ), (dim t(N ) = N (N − 1)/2, N ≥ 2) [123, 124], naturally graded and Z-graded nilradicals of maximal degree of nilpotency [3, 19, 116, 117] and some other special types of nilradicals [115, 133]. Moreover, all solvable algebras in dimensions up to 6 have been found in this way [79, 80, 129]. In all these cases the Conjecture 9.1 was confirmed. Let us present several low dimensional examples in order to demonstrate the procedure. First, let us consider the case of 3-dimensional solvable Lie algebras. In this case the restriction (9.1) shows that the dimension of the nilradical dim NR(s) is 2 or 3. When dim NR(s) is 3, the algebra is equal to its nilradical, i.e., nilpotent, and was already found before. When dim NR(s) = 2 we have an Abelian nilradical and the solvable algebra s is determined once the action of one nonnilpotent element f1 on the nilradical n = NR(s) = span{e1 , e2 } is specified. Any change of basis in the nilradical is allowed because any regular linear map is an automorphism of n and consequently the task is reduced to the classification of 2 × 2 nonnilpotent matrices with respect to conjugation and overall rescaling. We find the following canonical forms for the matrix D1 . • Over the field of complex numbers the matrix D1 has one of the following forms 1 1 1 0 , 0 1 0 a where the parameter a satisfies 0 < |a| ≤ 1, • Over the field of real numbers the matrix forms 1 α 1 1 0 , , 0 −1 α 0 a
if |a| = 1 then arg(a) ≤ π. D1 has one of the following 1 1
where the parameters a, α satisfy −1 ≤ a ≤ 1, a = 0, α ≥ 0. The condition α 1 a = 0 arises from the restriction to indecomposable algebras. The matrix −1 α is present only over the field R because over the field C it is upon rescaling conjugated to ( 10 a0 ) with the choice a = (α + i)/(α − i). The corresponding solvable algebras as listed in Section 16.4 are s3,1 :
e1 e2
e1 0
e2 0 0
e3 −e1 , ae2
9.3. UPPER BOUND ON THE DIMENSION OF A SOLVABLE EXTENSION
s3,3
e1 0
e1 e2
e2 0 0
103
e3 −αe1 + e2 −e1 − αe2
which is isomorphic to s3,1 over the field C, and s3,2
e1 0
e1 e2
e2 0 0
e3 −e1 . −e1 − e2
A similar investigation can be performed in any dimension when the nilradical is Abelian and has codimension one in s. For more details see Section 10.4. When the codimension of the Abelian nilradical is greater than one the situation becomes more involved — one has to classify all Abelian subalgebras of the matrix algebra gl(n) and then use this classification in the construction of nonisomorphic solvable algebras with the Abelian nilradical n, see, e.g., Section 10.5. When the nilradical n is not Abelian its Lie brackets put restrictions on automorphisms of n, i.e., it is no longer an arbitrary regular linear map. Consequently, not all changes of basis in the nilradical are allowed. Similarly, the space of derivations of n is limited. As an example let us consider solvable Lie algebras with the Heisenberg nilradical, i.e., n = n3,1 e1 e2
e1 0
e2 0 0
e3 0 . e1
Any derivation D and any automorphism φ must preserve the ideal n2 = span{e1 } and in addition D(e1 ) or φ(e1 ) is determined by the action of D, φ, respectively, on e2 , e3 . We find that an arbitrary derivation of n has the matrix form Tr X b , X ∈ F2,2 , b ∈ F2 (9.17) D= 0 X and any automorphism takes the form det C d , C ∈ F2,2 , (9.18) φ= 0 C
det C = 0,
d ∈ F2 .
Solvable Lie algebras with Heisenberg nilradical h(m) will be considered below in Chapter 11 for an arbitrary dimension n = 2N + 1. The lowest dimension n = 3 will be treated as a special case in Section 11.3. 9.3. Upper bound on the dimension of a solvable extension of a given nilradical Let us discuss in some detail the upper bound on the maximal number f of nonnilpotent elements fa we can add to a given nilradical n. We recall that there is a simple bound f ≤n implied by (9.1) but as we shall see it is never saturated for non-Abelian nilradicals and can be easily improved as was shown in [113]. We start by choosing a convenient basis E of the nilpotent Lie algebra n. We first choose some complement m1 of n2 in n, n = n2 m1
104
9. SOLVABLE LIE ALGEBRAS AND THEIR NILRADICALS
and denote m1 = dim m1 . We construct a basis Em1 = (en−m1 +1 , . . . , en ) of m1 . In the next step, we recall that n2 = [n, n] = [m1 n2 , m1 n2 ] = [m1 , m1 ] + n3 , (the last sum is not necessarily direct). Consequently, we can choose a complement m2 of n3 in n2 such that m2 ⊆ [m1 , m1 ]. We write its basis Em2 in the form of some subset of Lie brackets of vectors in Em1 , i.e. Em2 = (en−m1 −m2 +1 , . . . , en−m1 ) where m2 = dim m2 and for any k ∈ {n − m1 − m2 + 1, n − m1 } a pair yk , zk ∈ Em1 exists such that ek = [yk , zk ] (yk , zk are not necessarily unique). Proceeding by induction we have nk = [mk−1 nk , m1 n2 ] = [mk−1 , m1 ] + nk+1 and we find a complement mk of nk+1 in nk , nk = nk+1 mk , * ) mk ⊆ [mk−1 , m1 ], mk = dim mk and a basis Emk = en+1−k m , . . . , en−k−1 m i i i=1 i=1 of mk such that (9.19)
∀ej ∈ Emk
∃yj ∈ Emk−1 , zj ∈ Em1 :
ej = [yj , zj ].
Together the elements of the bases Emk form a basis E = (e1 , . . . , en ) of the whole nilpotent algebra n. The main advantage of working in the basis E lies in the fact that any automorphism φ, or any derivation D, is fully specified once its action on the elements of the basis Em1 of m1 is known due to the definition of an automorphism, (2.25), or a derivation, (2.26), together with (9.19), respectively. In all our computations concerning solvable extensions of the given nilradicals and in the lists containing the classification of low dimensional solvable Lie algebras we shall use bases of indecomposable nilradicals satisfying (9.19). For decomposable nilradicals we use bases satisfying (9.19) up to permutation of basis elements (in order not to obscure the explicit decomposition). The matrix of any derivation D of n is by construction upper block triangular in the basis E ⎛ ⎞ DmK mK . . . DmK m2 DmK m1 ⎜ .. .. ⎟ .. ⎜ . . . ⎟ (9.20) D=⎜ ⎟. ⎝ Dm2 m2 Dm2 m1 ⎠ Dm1 m1 Due to the relation D([ek , ej ]) = [D(ek ), ej ]+[ek , D(ej )] the elements of the diagonal blocks Dmk mk , k = 2, . . . , K are linear functions of elements of Dm1 m1 . The elements in the general block Dmj mk , 2 ≤ k ≤ j ≤ K are linear functions of elements in the last column blocks Dm1 m1 , . . . , Dmj−k+1 m1 . Inner derivations by definition map nk → nk+1 , i.e., have strictly upper triangular block structure. Therefore, any set of outer derivations {D1 , . . . , Df } such that [Dj , Dk ] ∈ Inn(n) for all j, k = 1, . . . , f must have commuting m1 m1 -submatrices, (9.21)
[(Dj )m1 m1 , (Dk )m1 m1 ] = 0.
An iterative consequence of the definition of a derivation (2.26) n n n D [x, y] = [Dj x, Dn−j y] j j=0
9.4. PARTICULAR CLASSES OF NILRADICALS AND THEIR SOLVABLE EXTENSIONS 105
implies that the derivation D is nilpotent if and only if its submatrix Dm1 m1 is nilpotent. This equivalence implies that derivations D1 , . . . , Df are linearly nilindependent if and only if their submatrices (D1 )m1 m1 , . . . , (Df )m1 m1 are linearly nilindependent. Together with (9.21) this means that the number f of linearly nilindependent outer derivations of the given nilpotent algebra n commuting to inner derivations is bounded from above by the maximal number of linearly nilindependent commuting matrices of dimension m1 × m1 . This number is m1 = dim n − dim n2 . Thus we have Proposition 9.2. Let n be a nilpotent Lie algebra and s a solvable Lie algebra with the nilradical n. Let dim n = n, dim s = n + f . Then f satisfies (9.22)
f ≤ dim n − dim n2 .
The bound (9.22) is saturated for many classes of nilpotent Lie algebras whose solvable extensions were investigated — e.g., Abelian [84], (reproduced in Chapter 10), naturally graded filiform nn,1 , Qn [3, 116] (partly reproduced in Section 13.1), a decomposable central extension of nn,1 in [133] and triangular [123] (reproduced in Section 12.3). On the other hand, it is obvious that even the improved bound (9.22) cannot give a precise estimate of the maximal dimension of a solvable extension in all cases. In particular, we have always dim n − dim n2 ≥ 2, i.e., characteristically nilpotent Lie algebras cannot be easily detected using (9.22). Similarly, the bound (9.22) is not saturated in the case of Heisenberg nilradicals h [106] (reproduced in Chapter 11) where the maximal number of nonnilpotent elements is in fact equal to (dim h + 1)/2 < dim h − 1. We notice that a different improvement of the bound (9.1) was found in [81] (9.23)
f ≤ dim n − dim C(s).
It turns out that the estimate (9.22) is in most cases more restrictive than (9.23). In addition, (9.22) does not involve a priori knowledge of the algebra s whereas (9.23) requires it. 9.4. Particular classes of nilradicals and their solvable extensions In the following chapters we shall consider several classes of nilradicals and present some results concerning their solvable extensions. The nilpotent algebras whose solvable extensions have been classified with various degrees of detail, can be divided into three major classes: • Algebras with low degree of nilpotency. The algebras investigated so far in this class are the Abelian and Heisenberg algebras (in arbitrary finite dimensions). These algebras possess large algebras of derivations that have well-understood properties. E.g., for an Abelian nilradical, any linear transformation is a derivation and any regular linear map is an automorphism. Consequently, the construction of solvable extensions is reduced to the consideration of Abelian subalgebras in gl(n) and their equivalence. Similarly, for Heisenberg algebras h(n), the task is reduced to the study of Abelian subalgebras of sp(2n). • Nilradicals of Borel subalgebras of simple Lie algebras. Nilpotent algebras in this class have a very particular structure given by the corresponding root diagram. Consequently, all derivations of such algebras can be
106
9. SOLVABLE LIE ALGEBRAS AND THEIR NILRADICALS
found in explicit form using cohomological arguments. This was done by G.F. Leger and E.M. Luks in [67]. A prime example of a nilradical in this class is the algebra of strictly upper triangular matrices. • Algebras with a high degree of nilpotency. The structure of Lie brackets of such algebras usually significantly restricts the algebra of derivations. Therefore the algebras of derivations can often be written down explicitly in arbitrary dimension and similarly for the automorphisms. Consequently, many explicit lists of solvable algebras with nilradicals in this class are known. 9.5. Vector fields realizing bases of the coadjoint representation of a solvable Lie algebra When calculating invariants of the coadjoint representation of a solvable Lie algebra s we use the vector fields (3.1). When we use the basis (9.4) of s, i.e., (e1 , . . . , en , f1 , . . . , ff ), in which the Lie brackets are expressed as in (9.5) – (9.7), the corresponding vector fields can be written as i = Nji k ek ∂ + (Dα )k ek ∂ , E i = 1, . . . , n, i ∂ej ∂fα (9.24) ∂ ∂ Fα = −(Dα )ki ek + γβα k ek , α = 1, . . . , f ∂ei ∂fβ (summation over repeated indices i, j, k = 1, . . . , n and β = 1, . . . , f applies). As in i , Fα act on smooth functions on the space s∗ dual to Section 3.2 the operators E the Lie algebra s with coordinates (e1 , . . . , en , f1 , . . . , ff ), i.e., on functions of n + p variables I(e1 , . . . , en , f1 , . . . , ff ). The operators Ei , Fα define an infinite dimensional representation of the Lie algebra s on the space C ∞ (s∗ ) of smooth functions on s∗ , i.e., satisfy the same commutation relations as the corresponding elements ei , fα in (9.5) – (9.7).
CHAPTER 10
Solvable Lie Algebras with Abelian Nilradicals The simplest possible structure of the nilradical is Abelian. Thus we start our discussion of solvable extensions of various nilradicals by considering the case of an Abelian nilradical. The results presented in this chapter were obtained in [84]. Our presentation follows the general outline of Chapter 9. 10.1. Basic structural theorems Since the nilradical is Abelian, the structure constants Nij k satisfy Nij k = 0. Equation (9.8) is satisfied trivially. From (9.9) we see that the matrices Dα all commute as in (9.12). Theorem 10.1. Let s be a finite-dimensional solvable nonnilpotent Lie algebra over the field F and let its nilradical n of dimension n be Abelian. We can choose a basis of s such that the commutation relations are ([e1 , fα ], . . . , [en , fα ]) = (e1 , . . . , en ) · Dα , Dα = adn (fα ) ∈ F
n×n
(10.1) (10.2) (10.3)
1 ≤ α ≤ f ≤ n,
,
[ei , ek ] = 0, [fα , fβ ] = γαβ j ej ,
γαβ j ∈ F.
The matrices Dα are linearly nilindependent and commute pairwise (10.4)
[Dα , Dβ ] = 0,
1 ≤ α, β ≤ f.
For f ≥ 3 the matrices Dα and the constants γαβ j satisfy k
(10.5)
k
k
γαβ j (Dδ )j + γδα j (Dβ )j + γβδ j (Dα )j = 0 1 ≤ k ≤ n,
1 ≤ α < β < δ ≤ f.
A classification of the Lie algebras s thus amounts to a classification of the matrices Dα and constants γαβ j under the transformations (9.13) – (9.15). Proof. Equations (10.1) – (10.3) simply express the results about the dimension of the nilradical of a solvable Lie algebra (9.1), and the structure of a solvable extension (9.5) – (9.7) in the particular case when n is Abelian. Equation (10.4) is a consequence of the Jacobi identifies (9.9). Similarly, (10.5) is the same as (9.10). The transformations (9.13) – (9.15) leave the relations (10.4) – (10.5) invariant, but transform the matrices Dα and constants γαβ j in (10.1) – (10.3). This completes the proof. In more detail, if we put (10.6)
ei = Sik ek ,
fα = Gβα fβ + rαa ea 107
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10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
we obtain the commutation relations (10.1) – (10.3) with Dα and γαβ j replaced by (10.7) (10.8)
Dα = Gβα (SDβ S −1 ) j = Gμα Gνβ γμν a + rαb Gνβ (Dν )ab − rβb Gμα (Dμ )ab (S −1 )ja . γαβ
We mention that the transformation formulas for Dα , γαβ j would be more complicated for a non-Abelian nilradical which possesses nonvanishing inner derivations. Theorem 10.1 provides us with a general classification procedure for obtaining all solvable Lie algebras s of the considered type for given values of s = dim s and n = dim n. The procedure is: (1) Classify all Abelian subalgebras Af (n) of gl(n, F) of dimension f = s − n, containing no nilpotent elements, into conjugacy classes under the action of GL(n, F). Choose a representative of each conjugacy class, i.e., use the transformation (10.6) to transform the matrices Dα , α = 1, . . . , f into some chosen canonical form. For f = 1 there is just one matrix D and it can be transformed to its Jordan canonical form. For f = n the matrices (D1 , . . . , Df ) form a basis of a Cartan subalgebra of gl(n, F). As shown below (Theorem 10.6) the algebra s will be decomposable for n > 2, F = C and n > 4, F = R. For 1 < f < n, 4 ≤ n, the problem of classifying the Abelian subalgebras Af (n) ⊂ gl(n, F) is more difficult. Use should be made of known results on maximal Abelian subalgebras of gl(n, F) [57, 74, 122] (see below). (2) Determine the structure constants γαβ j of (10.3). For f = 1 the question does not arise. For f ≥ 2 we can simplify them using the transformation (10.6). We have always at our disposal the transformation (10.6) with S = 1n×n , G = 1f ×f , j i.e., with arbitrary constants rαa . That leads to equivalent structure constants γαβ (10.9)
γαβ = γαβ j + rαa (Dβ )ja − rβa (Dα )ja . j
For some sets of derivations Dα the more general formula (10.6), rather than (10.9), can be used to further simplify and standardize γαβ j . One must however require that the matrices S, G in (10.6) leave the already standardized matrices Dα invariant. From a slightly different viewpoint, cohomological considerations along the lines of Section 8.2, (8.10) with B ≡ 0 can also be employed. The cohomology group encountered here is H 2 (s/n, n; A) where A stands for the representation of the Abelian algebra s/n by matrices Dα on the vector space n. When the cohomology group H 2 (s/n, n; A) is trivial we can always set γαβ j = 0 for all α, β and j. (3) Weed out decomposable Lie algebras among the constructed solvable Lie algebras. We see from Theorem 10.1 and the discussion above that we cannot expect to have a nice closed form result for solvable Lie algebras of arbitrary dimensions. We shall present further partial results that together make the classification of solvable Lie algebras easy, once the dimensions n and f are fixed. We shall concentrate on the case F = C but we will also point out the differences that occur for the field F = R. From (10.7) we see that a good classification procedure is to first use the basis transformations (10.6) to transform the matrices D1 , . . . , Df to some standard form. The constants rαj can then be used to simplify the structure constants γαβ j .
10.1. BASIC STRUCTURAL THEOREMS
109
In particular, for given α, β the structure constants γαβ j can be always annulled when the sum of image spaces of the derivations Dα and Dβ contains the whole nilradical n. The matrices Dα , α = 1, . . . , f , form an Abelian subalgebra Af (n) ⊂ gl(n, F), containing no nilpotent matrices. Each set Af (n) is contained in at least one maximal Abelian subalgebra (MASA) of gl(n, F). A sizable literature exists on MASAs of the classical Lie algebras. For a review of classical results, including the Kravchuk signatures and Kravchuk normal forms of MASAs of sl(n, C), we refer to D.A. Suprunenko and R.I. Tyshkevich [122]. More recent results on MASAs of other classical Lie algebras can be found in [57, 74, 137]. The pertinent results for our purposes are: (1) A MASA of gl(n, F) can always be written as MASA gl(n, F) = FIn ⊕ MASA sl(n, F) . (2) A MASA of sl(n, F) as a set of matrices can be either indecomposable, or decomposable into a direct sum of indecomposable ones. (3) An indecomposable MASA of sl(n, C) is always a maximal Abelian nilpotent subalgebra (MANS) [37, 74, 90, 122, 137]. A MANS is represented by nilpotent matrices in any finite dimensional representation. A MANS is characterized by its Kravchuk signature (10.10)
(λ μ ν),
1 ≤ λ,
1 ≤ ν,
0 ≤ μ,
λ + μ + ν = n,
λ, μ, ν ∈ Z.
All elements X of a MANS of sl(n, C) can be simultaneously written in the Kravchuk normal form ⎛ ⎞ 0λ P Q (10.11) X = ⎝ 0 R S ⎠ , P ∈ Cλ,μ , Q ∈ Cλ,ν , R ∈ Cμ,μ , S ∈ Cμ,ν . 0 0 0ν where the matrix Q is free, i.e., contains λ · ν arbitrary parameters. The matrices R and S are completely determined by the matrix P ; in addition, the matrices R form an ANS of sl(μ, C)
(10.12)
⎛ 0 ⎜ R=⎝ 0
..
.
⎞ ∗ ⎟ ⎠. 0
The matrix P may be free-rowed, i.e., the elements of its first row are free (μ arbitrary parameters) and all the others are determined by the first row. Alternatively for k ≥ 2, non-free-rowed MANS may exist where the first row of P contains some zeroes and new parameters occur in lower rows. The symbols 0λ and 0ν denote square null matrices of the indicated dimension. (4) An indecomposable MASA of sl(n, R) can be either absolutely indecomposable or indecomposable, but not absolutely indecomposable. The absolutely indecomposable ones are MANS and can be written as in (10.12), but with real entries. The nonabsolutely indecomposable MASAs of sl(n, R), become decomposable
110
10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
after complexification. They exist only for n even and have the form [137] ⎞ ⎛ 0 −1 ⎟ ⎜1 0 ⎟ ⎜ ⎟ ⎜ 0 −1 ⎟ ⎜ ⎟ ⎜ 1 0 R⎜ ⎟ ⊕ MANS sl(r, C) , ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎝ 0 −1⎠ 1 0 where r = n/2 and sl(r, C) is represented by the matrices ⎛ ⎞ α11 . . . α1r ⎜ .. .. ⎟ α = aij −bij , a , b ∈ R, ⎝ . ij ij ij . ⎠ bij aij αr1 . . . αrr
r
aii = 0.
i=1
The corresponding MANS of sl(r, C) can again be written in Kravchuk normal form. Let us now return to the commuting matrices Dα of (10.1) – (10.3) and (10.7). They form an Abelian subalgebra of gl(n, F) and hence a subalgebra of some MASA. This MASA cannot be a MANS; as a matter of fact it contains no nilpotent elements at all. Hence all matrices Dα can be simultaneously brought to the same block diagonal form. Moreover each block can be brought to a triangular form. We thus arrive at the following theorem. Theorem 10.2. Let s be a finite-dimensional solvable Lie algebra over C with an Abelian nilradical n. The matrices Dα in (10.1) – (10.3) in Theorem 10.1 can be simultaneously transformed into a block diagonal form ⎞ ⎛ 1 1 Tα (aα ) ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ p p ⎟ ⎜ Tα (aα ) ⎟, (10.13) Dα = ⎜ p+1 ⎟ ⎜ Tα (0) ⎟ ⎜ ⎟ ⎜ .. ⎠ ⎝ .
(10.14)
(10.15)
⎛ j aα ⎜ j j Tα (aα ) = ⎝ 0 p+q
⎟ ⎠ ∈ Cnj ×nj ,
ajα
1 ≤ j ≤ f, f + 1 ≤ j ≤ p, p + 1 ≤ j ≤ p + q,
1 ≤ f ≤ p ≤ n,
1 ≤ α ≤ f,
p + q ≤ n,
∗ ..
.
nj = n,
Tαp+q (0)
⎧ j ⎪ ⎨= δα , j aα ∈ C, ⎪ ⎩ = 0,
⎞
j=1
(10.16)
n1 ≥ n2 ≥ · · · ≥ np ≥ 1,
np+1 ≥ np+2 ≥ · · · ≥ np+q ≥ 0.
For each j, the off-diagonal parts of the corresponding matrices Tαj (ajα ), i.e., {Tαj (ajα )−ajα 1nj ×nj | α = 1, . . . , f }, span an Abelian nilpotent subalgebra of sl(nj , C). In order to obtain all complex solvable Lie algebras of the considered type, we must consider all partitions of n satisfying (10.16). For each partition we must construct all inequivalent Abelian indecomposable subalgebras of sl(nj , C), 1 ≤ j ≤ p + q.
10.1. BASIC STRUCTURAL THEOREMS
111
Remark 10.3. (1) Notice that the Abelian subalgebras of sl(nj , C) need not be maximal. They must however be indecomposable, otherwise they would also appear in some other, finer partition of n. (2) The situation for F = R is conceptually quite similar, however the usual complications arise. Thus, in addition to the nonnilpotent and nilpotent blocks Tjα (ajα ) and Tkα (0), respectively, a further type of nonnilpotent block can occur, namely ⎞ ⎛ j bα −cjα ⎟ ⎜cjα bjα ∗ ⎟ ⎜ ⎟ ⎜ α α α . .. (10.17) Tj (bj , cj ) = ⎜ ⎟ ⎟ ⎜ ⎝ 0 bjα −cjα ⎠ cjα bjα with cjα = 0 for at least one value of α. Each block Tαj (bjα , cjα ) provides either one, or two, nonnilpotent matrices Dα . As an example consider f = 2, n = 4. We can have one real 4 × 4 Jordan block (10.17) ⎞ ⎞ ⎛ ⎛ 0 −1 ˜b1 1 0 ˜b2 c˜2 c˜1 ⎜1 0 −˜ ⎜ c1 ˜b1 ⎟ c2 ˜b2 ⎟ ⎟ , D2 = ⎜0 1 −˜ ⎟ D1 = ⎜ ⎠ ⎝ ⎝ 0 −1 1 0⎠ 1 0 0 1 or two real 2 × 2 Jordan blocks, for instance ⎛ ⎛ ⎞ b1 −1 0 ⎜ 1 b1 ⎜0 ⎟ ⎟ , D2 = ⎜ D1 = ⎜ ⎝ ⎝ b2 0 ⎠ 0 b2
⎞
0 0 b3 1
⎟ ⎟ −1⎠ b3
or a combination of one real 2 × 2 Jordan block (10.17) with blocks Tjα (ajα ) and Tkα (0). In general the matrices Dα will have the form ⎞ ⎛ 1 1 Tα (aα ) ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ p p ⎟ ⎜ T (a ) α α ⎟ ⎜ ⎟ ⎜ p+1 p+1 p+1 ⎟ ⎜ Tα (bα , cα ) ⎟ ⎜ ⎟ ⎜ . Dα = ⎜ .. ⎟ ⎟ ⎜ ⎟ ⎜ p+q (bp+t , cp+t ) T α α α ⎟ ⎜ p+t+1 ⎟ ⎜ T (0) α ⎟ ⎜ ⎟ ⎜ . .. ⎠ ⎝ p+q+t Tα (0) p+i with Tαj (ajα ) as in (10.14), Tαp+i (bp+i α , cα ) as in (10.17). The linear nilindependence of D1 , . . . , Df is assured by setting
ajα = δαj ,
1 ≤ j ≤ min(p, f ),
and by appropriately specifying the entries
1 ≤ α ≤ f ≤ p + 2t
p+j (bp+j α , cα ),
1 ≤ j ≤ t.
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10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
Theorem 10.2 goes quite far towards a classification of the solvable algebras s with Abelian nilradical n, as can be seen from the following example. Example 10.4 (F = C, dim s = s = 8, dim n = n = 5, f = 3). We have 3 matrices D1 , D2 , D3 ∈ C5×5 , hence we need at least 3 blocks to assure linear nilindependence. The allowed partitions, values of p and q and corresponding matrices are (1) 5 = 3 + 1 + 1 (p = 3, q = 0) ⎞
⎛ 1 a1 b1 ⎜ 1 c1 ⎜ 1 D1 = ⎜ ⎜ ⎝ 0
⎟ ⎟ ⎟, ⎟ ⎠
⎞
⎛ 0 a2 b2 ⎜ 0 c2 ⎜ 0 D2 = ⎜ ⎜ ⎝ 1
0
⎟ ⎟ ⎟, ⎟ ⎠
⎞
⎛ 0 a3 b3 ⎜ 0 c3 ⎜ 0 D3 = ⎜ ⎜ ⎝ 0
0
⎟ ⎟ ⎟. ⎟ ⎠ 1
Commutativity requires that the three-dimensional block should represent a MASA of gl(3, C). There are 3 possible Kravchuk signatures, for which we have
(10.18)
(2 0 1)
aα = 0,
(1 0 2)
cα = 0,
(1 1 1)
a α = cα .
α = 1, 2, 3,
We mention that here, and in some cases below, further simplifications are possible. Take, e.g., the Kravchuk signature (1 1 1), i.e., aα = cα . By a change of basis in the nilradical we can transform (a1 , b1 ) into (1, 0), (0, 1), or (0, 0). For a1 = 1, b1 = 0 no further simplifications are possible. For a1 = 0, b1 = 1 we can transform (a2 , b2 ) into (a2 , 0), (0, b2 ) or (0, 0) for a2 = 0; a2 = 0, b2 = 0; or a2 = b2 = 0, respectively. In the last case we can take (a3 , b3 ) into (1, 0), (0, 1), or (0, 0), as the case may be. Thus, the number and range of parameters can be greatly restricted at the price of splitting each case into many subcases. We shall skip this type of discussion below. (2) 5 = 2 + 2 + 1 (p = 3, q = 0) ⎞
⎛
1 a1 ⎜ 1 ⎜ 0 b1 D1 = ⎜ ⎜ ⎝ 0
⎟ ⎟ ⎟, ⎟ ⎠
⎞
⎛ 0 a2 ⎜ 0 ⎜ 1 b2 D2 = ⎜ ⎜ ⎝ 1
0
⎟ ⎟ ⎟, ⎟ ⎠
⎞
⎛
0 a3 ⎜ 0 0 ⎜ 0 b3 D3 = ⎜ ⎜ ⎝ 0
0
⎟ ⎟ ⎟. ⎟ ⎠ 1
(3) 5 = 2 + 1 + 1 + 1 ⎞
⎛
1 a1 ⎜ 1 ⎜ 0 D1 = ⎜ ⎜ ⎝ 0
⎟ ⎟ ⎟, ⎟ ⎠
⎞
⎛ 0 a2 ⎜ 0 ⎜ 1 D2 = ⎜ ⎜ ⎝ 0
b1
⎟ ⎟ ⎟, ⎟ ⎠
⎞
⎛
0 a3 ⎜ 0 ⎜ 0 D3 = ⎜ ⎜ ⎝ 1
b2
p = 3,
q=1
for (b1 , b2 , b3 ) = (0, 0, 0),
p = 4,
q=0
for (b1 , b2 , b3 ) = (0, 0, 0).
⎟ ⎟ ⎟, ⎟ ⎠ b3
10.1. BASIC STRUCTURAL THEOREMS
(4) 5 = 5 × 1 ⎛ 1 ⎜ 0 ⎜ 0 D1 = ⎜ ⎜ ⎝ a1
⎞ ⎟ ⎟ ⎟, ⎟ ⎠
⎞
⎛ 0 ⎜ 1 ⎜ 0 D2 = ⎜ ⎜ ⎝ a2
⎟ ⎟ ⎟, ⎟ ⎠
113
⎛ ⎜ ⎜ D3 = ⎜ ⎜ ⎝
⎞
0
⎟ ⎟ ⎟, ⎟ ⎠
0 1 a3
p = 3,
b1 q=2
b2 for ai = bi = 0,
b3
p = 4,
q=1
for (a1 , a2 , a3 ) = (0, 0, 0), bi = 0,
p = 5,
q=0
for (a1 , a2 , a3 ) = (0, 0, 0), (b1 , b2 , b3 ) = (0, 0, 0).
All 4 above case also occur for F = R, but further ones also exist, for example (5) 5 = 3 + 2 (p = 1, q = 0, t = 1) ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 1 a1 b1 0 a2 b2 0 a3 b3 ⎟ ⎟ ⎟ ⎜ 1 c1 ⎜ 0 c2 ⎜ 0 c3 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟. ⎜ ⎜ ⎜ 1 0 0 , D , D D1 = ⎜ 2 =⎜ 3 =⎜ ⎟ ⎟ ⎟ ⎝ ⎝ ⎝ 0 ⎠ 0 −1⎠ 1 0⎠ 0 1 0 0 1 (Equation (10.18) holds for ai , bi , ci .) (6) 5 = 1 + 2 + 2 (p = 2, q = 0, t = 1) ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ 1 0 0 ⎜ 0 a1 ⎜ 1 a2 ⎜ 0 a3 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ , D2 = ⎜ 0 1 ⎟ , D3 = ⎜ 0 0 ⎟. 0 0 D1 = ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎝ ⎝ ⎝ ⎠ ⎠ b1 b2 b3 −1⎠ 1 b3 b1 b2 (7) 5 = 1 + 2 + 2 (p = 1, q = 1, t = 1) ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 1 0 0 ⎟ ⎟ ⎟ ⎜ 0 0 ⎜ 0 −1 ⎜ 1 0 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟. ⎜ ⎜ ⎜ D1 = ⎜ 0 0 ⎟ , D2 = ⎜ 1 0 ⎟ , D3 = ⎜ 0 1 ⎟ ⎠ ⎠ ⎝ ⎝ ⎝ 0 1 0 a2 0 a3 ⎠ 0 0 0 0 0 0 (8) 5 = 1 + 2 + 2 (p = 1, q = 0, t = 2) ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 1 0 0 ⎟ ⎟ ⎟ ⎜ 0 0 ⎜ 0 −1 ⎜ 1 0 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟, ⎜ ⎜ ⎜ D1 = ⎜ 0 0 ⎟ , D2 = ⎜ 1 0 ⎟ , D3 = ⎜ 0 1 ⎟ ⎠ ⎠ ⎝ ⎝ ⎝ a1 −b1 a2 −b2 a3 −b3 ⎠ b1 a1 b2 a2 b3 a3 (b1 , b2 , b3 ) = (0, 0, 0). (9) 5 = 1 + 2 + 2 (p = 1, q = 0, t = 2) ⎞ ⎞ ⎛ ⎛ 1 0 ⎟ ⎟ ⎜ 0 0 ⎜ 0 −1 ⎟ ⎟ ⎜ ⎜ ⎟ , D2 = ⎜ 1 0 ⎟, 0 0 D1 = ⎜ ⎟ ⎟ ⎜ ⎜ ⎝ ⎝ b1 0 ⎠ b2 0 ⎠ 0 b1 0 b2
⎞ ⎛ 0 ⎟ ⎜ 0 0 ⎟ ⎜ ⎟. 0 0 D3 = ⎜ ⎟ ⎜ ⎝ b3 −1⎠ 1 b3
114
10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
(10) 5 = 1 + 2 + 1 + 1 ⎞ ⎞ ⎛ ⎛ 1 0 ⎟ ⎟ ⎜ 0 0 ⎜ 0 −1 ⎟ ⎟ ⎜ ⎜ ⎟ , D2 = ⎜ 1 0 ⎟, 0 0 D1 = ⎜ ⎟ ⎟ ⎜ ⎜ ⎠ ⎠ ⎝ ⎝ a1 a2 b1 b2 p = 1, q = 2, t = 1 for ai = bi = 0,
⎞
⎛ 0 ⎜ 1 0 ⎜ D3 = ⎜ ⎜ 0 1 ⎝ a3
⎟ ⎟ ⎟, ⎟ ⎠ b3
p = 2,
q = 1,
t = 1 for (a1 , a2 , a3 ) = (0, 0, 0), bi = 0,
p = 3,
q = 0,
t = 1 for (a1 , a2 , a3 ) = (0, 0, 0), (b1 , b2 , b3 ) = (0, 0, 0).
10.2. Decomposability properties of the solvable Lie algebras So far, nothing guarantees that the Lie algebras s described in Theorems 10.1 and 10.2 are indecomposable. Indeed, in general, they may be decomposable into direct sums of lower-dimensional Lie algebras, either solvable with Abelian nilradicals, or Abelian s = s 1 ⊕ s2 ⊕ · · · ⊕ sk .
(10.19)
As described in Chapter 5, this decomposition may be central, which happens when some of the factors are Abelian, or noncentral. Proposition 10.5. Let the algebra s of Theorems 10.1 and 10.2 be such that the set of matrices Dα contains q0 one-dimensional zero blocks: (10.20)
q − q0 + 1 ≤ j ≤ q
Tp+t+j (0) = 0,
and let (10.21)
q0 >
f (f − 1) . 2
Then the algebra is decomposable. Proof. The existence of the q0 zero blocks implies that the center C(s) contains the corresponding q0 elements ej , f − q0 + 1 ≤ j ≤ f . These elements ej do not figure in [s, n] in (10.1). The only way they can be contained in s2 is on the right-hand side of relations (10.3). Only f (f − 1)/2 such relations exist, hence at most that many linearly independent elements of n figure in (10.3). If (10.20) – (10.21) hold then the center C(s) contains at least f (f − 1) >0 2 linearly independent elements, not contained in s2 . By Theorem 5.7, the algebra is decomposable. q1 = q0 −
A noncentral decomposition into two (or, successively, more) solvable Lie algebras occurs if the matrices Dα of (10.13) can be split into two sets satisfying 1 0 0 Dα 0 , Dβ = , Dα = 0 Dβ2 0 0 1 ≤ α ≤ f0 < f,
f0 + 1 ≤ β ≤ f,
Dα1 ∈ Fn1 ×n1 ,
Dβ1 ∈ Fn2 ×n2 ,
n1 + n2 = n.
The two sets of elements S1 = {f1 , . . . , ff0 , e1 , . . . , en1 },
S2 = {ff0 +1 , . . . , ff , en1 +1 , . . . , en1 +n2 }
10.2. DECOMPOSABILITY PROPERTIES OF THE SOLVABLE LIE ALGEBRAS
115
will form two mutually commuting Lie algebras if we also have [fα , fβ ] = 0,
1 ≤ α ≤ f0 ,
f0 + 1 ≤ β ≤ f.
10.2.1. Nilradicals of minimal dimension. Let us consider the case when s 1 n = dim n = dim s = , i.e., n = f . 2 2 In this case the matrices Dα , α = 1, . . . , n form an n-dimensional Abelian subalgebra of gl(n, F) containing no nilpotent elements. This is only possible if the algebra {D1 , . . . , Dn } is actually a Cartan subalgebra of gl(n, F) (a maximal Abelian selfnormalizing subalgebra, i.e., a MASA containing only semisimple elements). Consider first the case of F = C. Over the field of complex numbers all Cartan subalgebras of a semisimple Lie algebra are mutually conjugate. In particular, the Cartan subalgebra of gl(n, C) can be represented by n diagonal matrices and we can put (Dα )ik = δαi δαk , [ei , fα ] = δiα ei . From relations (10.5) we obtain γαβ j = 0 for j = α, β. From (10.8) with G = 1f ×f , S = 1n×n , rαβ = −γαβ β (no summation) we see that we can set γαβ j = 0,
1 ≤ α, β, j ≤ n.
Thus the algebra s is indecomposable for f = n = 1 and is decomposable into a direct sum of two-dimensional algebras for n = f ≥ 2. Now consider the algebra s over the field of real numbers F = R. Cartan subalgebras of the real simple Lie algebras were classified by B. Kostant [63] and M. Sugiura [121]. In particular gl(n, R) has [n/2] + 1 inequivalent classes of Cartan subalgebras. Each Cartan subalgebra can be represented by the matrices: ⎛ ⎞ a1 ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ an1 ⎜ ⎟ + , ⎜ ⎟ b1 −c1 ⎟ , 0 ≤ n2 ≤ n , Cn2 = ⎜ ⎜ ⎟ c1 b1 2 (10.22) ⎜ ⎟ ⎜ ⎟ . . ⎜ ⎟ . ⎜ ⎟ ⎝ bn2 −cn2 ⎠ cn2 bn2 n1 + 2n2 = n. From (10.22) we see that we can choose the matrices Dα to satisfy ⎧ ⎪ 1 ≤ α ≤ n1 ; ⎨δαi δαk , (10.23) (Dα )ik = bα (δαi δαk + δiα+1 δkα+1 ), α = n1 +1, n1 +3, . . . , n1 +2n2 −1; ⎪ ⎩ cα (δiα−1 δkα − δiα δkα−1 ), α = n1 +2, n1 +4, . . . , n1 +2n2 . From relations (10.5) and (10.8) we see that we can set γαβ j = 0 for all values of α, β, j. Let us sum up the results as a theorem. Theorem 10.6. Precisely two indecomposable solvable real Lie algebras s with Abelian nilradical n satisfying (10.24)
s = dim s = 2 dim n = 2n
116
10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
exist, namely: (10.25)
n = 1,
[e, f ] = e
(10.26)
n = 2,
[e1 , f1 ] = e1
[e1 , f2 ] = e2
[e2 , f1 ] = e2
[e2 , f2 ] = −e1
[f1 , f2 ] = 0
[e1 , e2 ] = 0.
Precisely one indecomposable complex solvable Lie algebra satisfying (10.24) exists; it is given by the commutation relations (10.25). Every other solvable Lie algebra satisfying (10.24) is decomposable into a direct sum of the Lie algebras (10.25) for F = C, and into a direct sum of the Lie algebras of (10.25) and (10.26) for F = R. 10.3. Solvable Lie algebras with centers of maximal dimension Consider again a solvable Lie algebra s with an Abelian nilradical n. An important characteristic of s is the dimension of its center C(s). Here we shall establish the maximal possible dimension (10.27)
dM = max[dim C(s)],
compatible with the requirement that the algebra s must be indecomposable. The indecomposability together with (9.3) implies C(s) ⊆ s2 ⊆ n. We also have C(s) = n, otherwise s would be nilpotent. Let us define a matrix A ∈ Ff n×n obtained by stacking together all the matrices Dα of Theorems 10.1 and 10.2 ⎛ ⎞ D1 ⎜ D2 ⎟ ⎜ ⎟ (10.28) A = ⎜ . ⎟ ∈ Ff n×n . ⎝ .. ⎠ Df In view of (10.1) – (10.3) the center C(s) of s is equal to the intersection of kernels of all derivations Dα , 1 ≤ α ≤ f . Therefore, C(s) is the nullspace of the matrix A. In order to maximize the dimension of C(s) we must hence maximize the dimension of the nullspace of A, i.e., minimize the rank of A. If we are working over C we may assume that the matrices Dα are in the form (10.13). Obviously, only the nilpotent blocks Tp+j (0) contribute to the computation of the nullspace. p The rank of A is bounded from below by j=1 nj . Since we are interested in the case of minimal rank of A we can assume without loss of generality that all the blocks Tαj (ajα ), j ≤ p are one-dimensional. Equations (10.13) – (10.14) now imply (10.29)
(Dα )ik = δαi δαk ,
1 ≤ i, k ≤ n where i ≤ f or k ≤ f .
We investigate the possible structures of the remaining blocks which minimize the rank of A and preserve the indecomposability of s. We distinguish two cases depending on the relation between the dimensions n and f . (1) n − f ≤ f (f − 1)/2. In order to maximize the dimension of the nullspace of A in this case we can set all the remaining blocks in Dα equal to zero (10.30)
(Dα )ik = 0,
f + 1 ≤ i ≤ n, 1 ≤ k ≤ n
10.3. SOLVABLE LIE ALGEBRAS WITH CENTERS OF MAXIMAL DIMENSION
117
and put (10.31)
[fα , fβ ] = e˜αβ
where e˜αβ , 1 ≤ α < β ≤ f span the whole vector space spanned by ef +1 , . . . , en . In this case we have C(s) = span{ef +1 , . . . , en } and dM = n − f.
(10.32)
Notice that the Jacobi identities (10.5) are satisfied identically and we have span{[fα , fβ ] | 1 ≤ α < β ≤ f } = C(s). (2) n−f > f (f −1)/2. In this case we can only assume that f (f −1)/2 columns of A consist entirely of zeros which we assume without the loss of generality to be the last ones. If there were more vanishing columns in A, the corresponding nullspace would not be spanned by the commutators [fα , fβ ] and the algebra would necessarily be decomposable. To avoid decomposability the commutation relations between fα , fβ must take the form (10.31) with (10.33)
{˜ eαβ | 1 ≤ α < β ≤ f } = span{en−f (f +1)/2+1 , . . . , en }.
On dimensional grounds, e˜αβ must be linearly independent. That leaves the square block in the matrices Dα between the rows f + 1 and n − f (f − 1)/2 undetermined. Its dimension is f (f + 1) . 2 Let us introduce two integer constants q0 , ν such that N0 = n −
(10.34)
(10.35)
N0 = (f + 1)q0 + ν,
0 ≤ ν ≤ f,
0 ≤ q0 ,
q0 , ν ∈ Z.
We can now maximize the dimension of the nullspace of A by choosing the remaining blocks in Dα to be of the type T f +j (0), satisfying T f +j (0) ∈ F(f +1)×(f +1) ,
j = 1, . . . , q0 ,
T f +q0 +1 (0) ∈ Fν×ν ,
ν 2;
the precise structure of the blocks will be specified below. As we shall see this is the structure which allows to maximize the dimension of the nullspace of A while keeping the algebra indecomposable. If the blocks T f +j (0) were of a larger size we could not accomplish the indecomposability in view of (10.40). For ν = 1 we must add one further nonzero 1 × 1 block on the diagonal of Dα instead of T f +q0 +1 (0). We thus have for ν ≥ 2 ⎞ ⎛ α E ⎟ ⎜ Tα1 (0) ⎟ ⎜ ⎟ ⎜ . .. ⎟ ⎜ (10.36) Dα = ⎜ ⎟ q0 ⎟ ⎜ Tα (0) ⎟ ⎜ q +1 0 ⎠ ⎝ Tα (0) 0d×d (10.37)
E α ∈ Cf ×f ,
(E α )ik = δiα δkα ,
d=
f (f − 1) . 2
118
10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
In (10.36) the blocks Tαj (0), Tαq0 +1 (0) take the following form which maximizes the nullspace ⎞ ⎛ (j) 0 . . . 0 bα 1 ⎜ . .. ⎟ .. ⎟ ⎜ . .. . ⎟ ∈ C(f +1)×(f +1) , 1 ≤ j ≤ q0 , Tαj (0) = ⎜ (10.38) ⎜ (j) ⎟ ⎝ 0 bα f ⎠ 0 ⎛ (q +1) ⎞ 0 . . . 0 bα 10 ⎜ .. ⎟ .. ⎜ . . ⎟ (10.39) Tαq0 +1 (0) = ⎜ ⎟ ∈ Cν×ν . (q0 +1) ⎠ ⎝ 0 b α ν−1
0 In this way, each nullspace of A. The condition that the matrices ⎛ (j) b11 . . . ⎜ . (j) ⎜ M = ⎝ .. (j) b1f . . .
block Tαj (0) corresponds to a f -dimensional subspace of the that the algebra s be indecomposable, i.e., C(s) ⊂ s2 , requires ⎞ (j) bf 1 .. ⎟ ⎟ . ⎠, (j) bf f
j = 1, . . . , q0 ,
M (q0 +1)
⎛ (q +1) b110 ⎜ . =⎜ ⎝ .. (q0 +1) b1 ν−1
⎞ (q +1) . . . bf 10 .. ⎟ ⎟ . ⎠ (q0 +1) . . . bf ν−1
all have maximal rank (10.40)
rank M (j) = f,
j = 1, . . . , q0 ,
rank M (q0 +1) = ν − 1 < f.
It is easy to verify that the Jacobi identities (10.5) are satisfied. The dimension of the center is ⎧ f (f − 1) ⎪ ⎨ + f q0 + ν − 1, for ν ≥ 1, 2 dim C(s) = ⎪ ⎩ f (f − 1) + f q , for ν = 0. 0 2 When we eliminate q0 using (10.34) and (10.35) we obtain ⎧ 1 ⎪ ⎪ ⎨ 2(f + 1) 2f n − f (f + 1) − 2(f + 1 − ν) , ν ≥ 1, dim C(s) = 1 ⎪ ⎪ ⎩ 2f n − f (f + 1) , ν = 0. 2(f + 1) The matrices Dα of (10.36) – (10.38) can be subjected to a simultaneous similarity transformation Dα = G−1 Dα G. We choose G in the form ⎞ ⎛ 1f ×f ⎟ ⎜ G1 ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ . G=⎜ ⎟, ⎟ ⎜ G q0 ⎟ ⎜ (10.41) ⎠ ⎝ G(q0 +1) 1f (f −1)/2×f (f −1)/2 (j) (q0 +1) M M Gj = , j = 1, . . . , q0 , , Gq0 +1 = 1 1
10.3. SOLVABLE LIE ALGEBRAS WITH CENTERS OF MAXIMAL DIMENSION
119
(q0 +1) ∈ F(ν−1)×(ν−1) is a square matrix containing (ν − 1) linearly indewhere M pendent columns of M (q0 +1) . Thus we transform the rows in the matrix (10.38) into ones satisfying (j)
bα i = δαi ,
(10.42)
(q +1) bα i0
=
α = 1, . . . , f, j = 1, . . . , q0 α = 1, . . . , ν − 1.
δαi ,
The results of this section can now be summed up as a theorem. Theorem 10.7. Let s be a solvable nonnilpotent indecomposable Lie algebra of dimension s = n + f with an Abelian nilradical n of dimension n with f < n. Let C(s) be the center of s. Define a nonnegative integer ν ≤ f by the following relation f (f + 1) ν = n− (mod f + 1). 2 The maximal possible dimension dM of the center (10.43) ⎧ ⎪ ⎪ n−f ⎪ ⎪ ⎪ ⎪ ⎨ 1 [2f n − f (f + 1)] dM = 2(f + 1) ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ [2f n − f (f + 1) + 2(ν − f − 1)] 2(f + 1)
C(s) is given by f (f − 1) ; 2 f (f − 1) if n − f > , ν = 0; 2 f (f − 1) if n − f > , ν ≥ 1. 2 if n − f ≤
The algebra s, over the fields F = C, or F = R, can be realized as in Theorems 10.1 and 10.2, with the matrices Dα realized as in (10.36) – (10.39), satisfying (10.42). For ν = 1 the nilpotent block of dimension ν × ν is replaced by aqα0 +1 on the diagonal of Dα with (aq10 +1 , . . . , aqf0 +1 ) = (0, 0, . . . , 0). The structure constants γαβ j in (10.3) must be such that the commutation relations [fα , fβ ] generate the entire subspace (10.33). Remark 10.8. (1) The construction of the matrices Dα , described above, is always possible and guarantees the maximal dimension of C(s). It is however not necessarily unique. What is unique is the rank of the matrix A, i.e., the value dM in (10.43). (2) Over the field F = R the linear nilindependence of the matrices Dα can be arranged as in (10.36) – (10.38). Alternatively, in the left top corner of Dα we can replace any pair
0 ,
1 0
1
by
0 1 0 , 1 0 1
−1 0
to obtain a different Lie algebra s with the same center C(s) (these algebras are inequivalent over R, equivalent over C).
120
10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
Let us consider an example with s = 15, f = 3, n = 12. We then have N0 = 6, q0 = 1, ν = 2. We have ⎛ 1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ D1 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ 0 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ D2 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ 0 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ D3 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
0 0 0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0 0 0
a1 0 0 0 0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
1 0 0 0 0 0
0 0 0 0
0 0 0 0
0 1 0 0 0 0
a2 0 0 0 0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
0 1 0 0 0 0
0 0 0 0
0 0 0 0
0 0 1 0 0 0
a3 0 0 0 0
(a1 , a2 , a3 ) = (1, 1, a) or (1, 0, 0) (up to permutations)
and condition (10.33) is satisfied by putting
[f1 , f2 ] = e10 ,
[f2 , f3 ] = e11 ,
[f3 , f1 ] = e12 .
10.4. SOLVABLE LIE ALGEBRAS WITH 1 NONNILPOTENT ELEMENT
121
10.4. Solvable Lie algebras with one nonnilpotent element and an n-dimensional Abelian nilradical Any such algebra has commutation relations [f1 , ei ] =
n
1 ≤ i ≤ n,
Dik ek ,
k=1
1 ≤ i, j ≤ n.
[ei , ej ] = 0,
˜ are isomorphic if and only if we have Two algebras given by D, D ˜ = λGDG−1 , D
λ ∈ F, λ = 0, G ∈ SL(n, F).
The matrix D can be transformed into Jordan canonical form over the respective field F. We shall first put the blocks that are indecomposable over the field C and order them by decreasing size. In the case of the field R we first put the blocks that are indecomposable over both R and C and below them blocks that are indecomposable over R but not absolutely decomposable (i.e., they become decomposable after field extension). These blocks are also ordered by decreasing size. ˜ that are related by The algebras given by D and D (1) permutations of blocks of equal size and Jordan type, and/or (2) multiplication by a nonzero constant are isomorphic. We can have the following cases for matrix D: (1) Diagonalizable over F. In this case (a) Order eigenvalues |a1 | ≥ |a2 | ≥ · · · ≥ |an | > 0 (if an = 0 then the algebra is decomposable). (b) Divide by the largest one and get ⎛ ⎜ ⎜ D=⎜ ⎝
⎞
1
⎟ ⎟ ⎟. ⎠
a2 ..
. an
If 1 > |a2 | > · · · > |an | > 0 then D is unique. (c) If we have |a2 | = · · · = |ak | = 1, k ≤ n set aj = eiφj , 0 ≤ φj < 2π, 1 ≤ j ≤ k and order (10.44)
0 = φ 1 ≤ φ2 ≤ · · · ≤ φ k .
˜ of such form are isomorphic if the numbers φ1 , Two algebras given by D, D φ2 , . . . , φk can be transformed into each other by an overall change of phase φj → φj + α mod 2π, j = 1, . . . , k accompanied by a suitable permutation of indices j so that (10.44) holds. (d) Order all the remaining elements by increasing phases if their magnitudes are equal.
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10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
(2) Real matrix D diagonalizable over C but not over R. We put ⎞ ⎛ α1 ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ α k ⎟ ⎜ ⎟ ⎜ β γ 1 1 ⎟. ⎜ D=⎜ ⎟ −γ β 1 1 ⎟ ⎜ ⎟ ⎜ . .. ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ βl γl −γl βl where k + 2l = n, γi > 0, αi = 0 (γi = 0 is not allowed by the assumed block structure and the sign of γi can be changed by the permutation of the elements in the nilradical; if any of the constants αi satisfies αi = 0 then the algebra is decomposable). Order the blocks and choose the overall multiple so that α1 = |α1 | ≥ α2 ≥ · · · ≥ αk > −|α1 |, 1 = γ1 ≥ γ2 ≥ · · · ≥ γl > 0. If γi = γi+1 then order βi > βi+1 . (3) Nondiagonalizable over C (a) In the case of algebras over C proceed as follows (i) Order Jordan blocks by size ⎛ ⎞ D1 ⎜ ⎟ D2 ⎜ ⎟ D=⎜ ⎟, . . ⎝ ⎠ . Dk where dim Di = ni , ni ≥ ni+1 , ki=1 ni = n, and the blocks are of the form ⎞ ⎛ ai 1 0 ... 0 ⎜ ai 1 . . . 0 ⎟ ⎟ ⎜ ⎟ ⎜ .. .. Di = ⎜ ⎟. . . ⎟ ⎜ ⎝ ai 1 ⎠ ai If ni = 1 then ai = 0 (otherwise the algebra would be decomposable). If all ni ≥ 2 then at least one ai = 0 (otherwise the algebra would be nilpotent). If ai = 0 for some i then the characteristic series are different from the generic case. (ii) Let n1 = · · · = nj > nj+1 . Order |a1 | ≥ |a2 | ≥ · · · ≥ |aj | and apply the rules of the diagonalizable case if |ai | = |ai+1 |. (iii) Order the remaining blocks of the same size by decreasing absolute value of ai . In the case of equal absolute values order by increasing phases. (b) In the case of algebras over R the block diagonal form of D ⎛ ⎞ D1 ⎜ ⎟ D2 ⎜ ⎟ D=⎜ ⎟, . .. ⎝ ⎠ Dk may contain two types of blocks.
10.5. SOLVABLE LIE ALGEBRAS WITH 2 NONNILPOTENT ELEMENTS
Type I
⎛ α ⎜ ⎜ ⎜ DI = ⎜ ⎜ ⎝
1 α
0 1 .. .
... ... .. . α
123
⎞ 0 0⎟ ⎟ ⎟ ⎟. ⎟ 1⎠ α
Type I blocks with α = 0 are allowed if dim DI ≥ 2 but the characteristic series for algebras with α = 0 differ from those with α = 0. Type II
⎛
β ⎜−γ ⎜ ⎜ ⎜ ⎜ DII = ⎜ ⎜ ⎜ ⎜ ⎝
γ β .. .
1 0 .. .
0 1 .. .
... ... .. .
β −γ
γ β
1 0 β −γ
⎞ 0 0⎟ ⎟ ⎟ ⎟ ⎟ , 0⎟ ⎟ ⎟ 1⎟ γ⎠ β
γ > 0.
(i) Put type I blocks first (if any occur). If all the blocks are of type I then at least one αi must satisfy αi = 0. Order both types of blocks by decreasing size. • Order blocks of type I of same size by decreasing value of αi . • Order blocks of type II of same size by decreasing value of γi . If γi = γi+1 order them by decreasing value of βi . (ii) If the first block is of type I set α1 = 1 using the overall multiplying factor. (iii) If the first block is of type II set γ1 = 1 and β1 ≥ 0. If β1 = 0 then set the first nonzero βi positive. If more than one block of Type II occurs with ni = ni+1 and γi = γi+1 then order them by decreasing values of βi . 10.5. Solvable Lie algebras with two nonnilpotent elements and n-dimensional Abelian nilradical Any such algebra g has commutation relations [f1 , ei ] = [f2 , ei ] =
n
(D1 )ki ek , 1 ≤ i ≤ n,
k=1 n
(D2 )ki ek , 1 ≤ i ≤ n,
k=1
[ei , ej ] = 0,
1 ≤ i, j ≤ n,
[f1 , f2 ] ∈ C(g), where the matrices D1 , D2 ∈ Fn×n are linearly nilindependent and commute. Two 1, D 2 are isomorphic if and only if we have algebras given by D1 , D2 and D (10.45)
1 = G(x1 D1 + x2 D2 )G−1 , D
2 = G(y1 D1 + y2 D2 )G−1 D
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10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
x1 x2 ∈ GL(2, F), G ∈ SL(n, F). y1 y2 Using the SL(n, F) part of the equivalence (10.45) we can put simultaneously D1 and D2 into block diagonal form ⎛ ⎛ ⎞ ⎞ A1 B1 ⎜ ⎜ ⎟ ⎟ A2 B2 ⎜ ⎜ ⎟ ⎟ , D2 = ⎜ (10.46) D1 = ⎜ ⎟ ⎟ . . .. .. ⎝ ⎝ ⎠ ⎠ Ak Bk k where dim Ai = dim Bi = ni , n1 ≥ n2 ≥ · · · ≥ nk , i=1 ni = n and span{Ai , Bi } are indecomposable Abelian subalgebras of gl(ni , F) meaning that Ai and Bi cannot be simultaneously block diagonalized into smaller blocks. There remain some SL(n − 2, F) transformations at our disposal, namely (1) Permutations of the same size blocks (simultaneous in both D1 and D2 ), and (2) Changes of basis inside the blocks, i.e., conjugation by G of the form where
G ∈ GL(n1 , F) × · · · × GL(nk , F). This amounts to a classification of two-dimensional Abelian subalgebras of gl(ni , F) for 1 ≤ i ≤ k. Therefore the form (10.46) still doesn’t fix the representant of the equivalence class with respect to (10.45) uniquely. For example, let us consider the case of D1 and D2 both diagonal, ⎛ ⎛ ⎞ ⎞ a1 b1 ⎜ ⎜ ⎟ ⎟ a2 b2 ⎜ ⎜ ⎟ ⎟ , D = D1 = ⎜ ⎜ ⎟ ⎟. 2 .. .. ⎝ ⎝ ⎠ ⎠ . . an
bn
By permutation of the basis elements of the nilradical together with the GL(2, F) part of the equivalence (10.45) we may transform D1 , D2 into the form ⎞ ⎞ ⎛ ⎛ a1 b1 ⎟ ⎟ ⎜ ⎜ .. .. ⎟ ⎟ ⎜ ⎜ . . ⎟ ⎟ ⎜ ⎜ , D (10.47) D1 = ⎜ = ⎟ ⎟ ⎜ 2 a b n−2 n−2 ⎟ ⎟ ⎜ ⎜ ⎠ ⎠ ⎝ ⎝ 0 1 1 0 with |a1 | ≤ |a2 | ≤ · · · ≤ |an−2 | ≤ 1, |bi | ≤ |an−2 |. If a1 = b1 = 0 then by indecomposability we find [f1 , f2 ] = ce1 , c = 0 which by an obvious rescaling of e1 can be put in the form [f1 , f2 ] = e1 . If |a1 | + |b1 | = 0 then C(g) = 0 and we have [f1 , f2 ] = 0. The choice of D1 , D2 in the form of (10.47) fixes all the continuous symmetries in the equivalence (10.45). Unfortunately, in general there are still some discrete
10.6. INVARIANTS OF SOLVABLE ALGEBRAS WITH ABELIAN NILRADICALS
125
symmetries present which further complicate the choice of the representative of the equivalence class. For example if n = 4 then we may use 1 x1 x2 b2 −a2 = , Δ = a1 b2 − a2 b1 y1 y2 Δ −b1 a1 accompanied by a permutation of the basis elements in the nilradical to show that the algebras given by ⎛ ⎛ ⎞ ⎞ a1 b1 ⎜ ⎜ ⎟ ⎟ a2 b2 ⎟ , D2 = ⎜ ⎟ D1 = ⎜ ⎝ ⎝ ⎠ ⎠ 0 1 1 0 and
⎛ −b2 /Δ ⎜ D1 = ⎜ ⎝
⎞ a2 /Δ 0
⎟ ⎟, ⎠ 1
⎛ ⎜ D2 = ⎜ ⎝
b1 /Δ
⎞ ⎟ ⎟ ⎠
−a1 /Δ 1 0
are isomorphic. Consequently we may impose an additional constraint on our parameters so that |Δ| = |a1 b2 − a2 b1 | ≤ 1. This fixes the choice of D1 , D2 uniquely if sharp inequalities hold in all the inequalities encountered. If there is any equality, e.g., |Δ| = 1, we have several choices of D1 , D2 allowed for isomorphic algebras and one of them must be chosen. 10.6. Generalized Casimir invariants of solvable Lie algebras with Abelian nilradicals Let us now apply the methods of Chapter 3 to calculate the generalized Casimir invariants of solvable Lie algebras s with Abelian nilradicals. The Lie brackets of these algebras were given in Theorem 10.1. The vector fields (9.24) of Section 9.5 in this case are i = (Dα )k ek ∂ , E i = 1, . . . , n, i ∂fα ∂ ∂ + γβα k ek , α = 1, . . . , f. Fα = −(Dα )ki ek ∂ei ∂fβ The function I(e1 , . . . , en , f1 , . . . , ff ) will be an invariant of the coadjoint representation of s if it satisfies the following linear first order partial differential equations (10.48)
i I = 0, E
(10.49)
Fα I = 0, α = 1, . . . , f.
i = 1, . . . , n,
Our aim is to find a complete set of elementary solutions of equations (10.49) – (10.48), i.e., a functionally independent set of generalized Casimir invariants.
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10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
10.6.1. General form of the generalized Casimir invariants and their number. Theorem 10.9. The solvable Lie algebra s over the field C satisfying the commutation relations (10.1) – (10.3) has exactly m=n−f
(10.50)
functionally independent generalized Casimir invariants. They depend only on the variables ei , corresponding to the elements of the Abelian nilradical n, (10.51)
Ij = Ij (e1 , . . . , en ),
j = 1, . . . , m.
Proof. The general invariant I(e1 , . . . , en , f1 , . . . , ff ) must satisfy (10.49), (10.48), in particular (10.52)
i I = E
f n
(Dα )ki ek
α=1 k=1
∂ I = 0, ∂fα
i = 1, . . . , n.
The matrices Dα have the block diagonal form of Theorem 10.2, Equation (10.13). Consequently, particular vector fields in (10.52) take the form
(10.53)
1 = −e1 ∂ , E ∂f1 n +n +···+n +1 E 1 2 f −1
n +1 = −en +1 ∂ , . . . , E 1 1 ∂f2 ∂ = −en1 +n2 +···+nf −1 +1 . ∂ff
The form of the vector fields in (10.53) implies that the invariants of the algebra s, and hence in particular the elementary invariants Ij , do not depend on f1 , . . . , ff , as indicated in (10.51). Now consider (10.49) which simplifies to (10.54)
Fα I(e1 , . . . , en ) =
n
(Dα )ki ek
i,k=1
∂ I = 0, ∂ei
α = 1, . . . , f.
Equation (10.54) can be rewritten in matrix form as ⎛ ⎞ ⎛ ⎞ ∂I/∂e1 0 ⎜ .. ⎟ ⎜ .. ⎟ (10.55) M ⎝ . ⎠ = ⎝.⎠ . ∂I/∂en
0
From (10.54) we see that the matrix M has maximal rank at generic points of the space n∗ with the coordinates (e1 , . . . , en ). Thus, we have f independent equations for a function of n variables. The number of independent solutions is hence n − f , as stated in (10.50). This completes the proof. Notice that the structure constants γαβ j of (10.3) play no role in the calculation of the Casimir invariants. Theorem 10.9 is valid also over the field F = R. The proof is quite analogous, but involves the usual complications due to the fact that the field R is not algebraically closed and hence the normal forms of commuting matrices can be more complicated. We omit the details of the proof. The generalized Casimir invariants (10.51) are thus a complete set of n − f functionally independent solutions of (10.55) (or equivalently of (10.54)). The
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127
actual form of the invariants depends on the dimension f of the factor algebra s/n and on the specific form of the matrices Dα . The invariants are of course basis dependent. Let us now consider some special cases in appropriate bases. They bring out all the characteristic properties of the general case. 10.6.2. Diagonal structure matrices. The simplest case, for any values of f and n, occurs when all the structure matrices Dα are diagonal (for an appropriate choice of the basis in the nilradical n). By linear combinations of the elements fα ⊂ F we can transform the structure matrices to the form ⎛ ⎛ ⎞ ⎞ 1 0 ⎜ 0 ⎜ .. ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ . ⎜ ⎜ ⎟ ⎟ . .. ⎜ ⎜ ⎟ ⎟ 0 ⎜ ⎜ ⎟ ⎟ ⎟ , . . . , Df = ⎜ ⎟. 0 1 (10.56) D1 = ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ a a 11 f1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ . . .. .. ⎝ ⎝ ⎠ ⎠ a1n−f af n−f The n − f invariants can be chosen to be (10.57)
−af k
1k −a2k Ik = ef +k e−a e2 · · · ef 1
,
k = 1, . . . , n − f.
Thus, Ik are all rational if the eigenvalues aαk are rational. 10.6.3. Casimir invariants in the case f = 1. Let us consider the Lie algebra s over the field C for the case f = 1. The algebra s has a basis (e1 , . . . , en , f1 ), where f1 is the only element of the basis not contained in the nilradical. The commutation relations in this case are (10.58)
[f1 , ei ] =
n
Dik ek ,
[ei , ek ] = 0.
k=1
In view of Theorem 10.9 any invariant of such a Lie algebra s will depend only on e1 , . . . , en and will be a solution of a single 1st order linear PDE (10.59)
n i,k=1
Dik ek
∂ I(e1 , . . . , en ) = 0. ∂ei
The solution of (10.59) is precisely the task that was studied in Section 3.2.2. The vector field n ∂ F1 = Dik ek ∂ei i,k=1
which represents the nonnilpotent element f1 , has the form (3.11) after identification of variables ej ≡ xj and the matrix D once reduced to its Jordan canonical form coincides with the matrix F of (3.18). The complete set of elementary invariants of the algebra s = span{e1 , . . . , en , f1 } is therefore given by Theorem 3.4 upon replacing the variables xj by ej .
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10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
10.6.4. Casimir invariants for low-dimensional nilradicals. In the previous Section 10.6.3 we considered the case of the largest possible nilradical of a solvable nonnilpotent Lie algebra, namely the case f = 1. Now we shall consider the opposite situation, when the factor algebra s/n is large, namely f = n, n − 1 and n − 2. The case f = n is only possible (over C) if the algebra s satisfies dim s = 2, or is decomposable into a direct sum of two-dimensional solvable Lie algebras as was shown in Theorem 10.6. The algebra s in this case has no Casimir invariants at all (see (10.50) of Theorem 10.9). The case f = n − 1. We have n − 1 linearly nilindependent commuting matrices D1 , . . . , Dn−1 ∈ Cn×n . Hence, they form a subalgebra of a decomposable MASA of gl(n, C). Only 2 decompositions of n are allowed by the condition of linear nilindependence, namely n one-dimensional blocks, i.e., all matrices Di are diagonalizable, or 1 two-dimensional block and (n − 2) one-dimensional ones. In this case we have the following result. Theorem 10.10. Let s be a complex indecomposable solvable Lie algebra of dimension 2n − 1 with an Abelian nilradical of dimension n. Then s has precisely one generalized Casimir invariant that, in an appropriate basis, has one of the following two forms (1) If the matrices Dα of (10.1) are simultaneously diagonalizable, we have 1 a1 a2 an−1 (10.60) I= (e e · · · en−1 ). en 1 2 (2) If the matrices Dα are not simultaneously diagonalizable, we have e2 (10.61) I= − ln(eb11 eb32 · · · ebnn−1 ). e1 In the first case the constants aα ∈ C are eigenvalues of the structure matrices Dα as in (10.56). In the second case the constants bα ∈ C are the off-diagonal elements of the structure matrices. Proof. It suffices to notice that in the first case all matrices Dα can be written as (10.62)
Dα = diag(δα1 , . . . , δαn−1 , aα ),
aj ∈ C.
Reading off the operators Fα of (10.48) in this case, we see that they all annihilate the invariant (10.60). In the second case we choose . - 0 δα1 (10.63) Dα = diag , δα2 , . . . , δαn−1 , bα ∈ C, α = 1, . . . , n − 1. bα δα1 Again, all the corresponding operators Fα will annihilate the “logarithmic” invariant I of (10.61). The case f = n − 2. The matrices Dα ∈ Cn×n again form a decomposable MASA of gl(n, C) and four different decompositions are allowed. Let us state the results as a theorem. Theorem 10.11. Let s be a complex indecomposable solvable Lie algebra of dimension 2n − 2 with an Abelian nilradical of dimension n. The structure
10.6. INVARIANTS OF SOLVABLE ALGEBRAS WITH ABELIAN NILRADICALS
129
matrices {Dα } and invariants {I1 , I2 } have, in the appropriate basis, one of the following forms (1) Decomposition n = n × 1 (10.64)
Dα = diag(δα1 . . . δαn−2 , aα , bα ),
aα , bα ∈ C, α = 1, . . . , n − 2.
Then (10.65)
I1 =
1 an−2 (ea1 ea2 · · · en−2 ), en−1 1 2
I2 =
1 b1 b2 bn−2 (e e · · · en−2 ). en 1 2
(2) Decomposition n = 2 + (n − 2) × 1 , + cα 0 (10.66) , bα3 , . . . , bαn−1 , aα , Dα = diag yα cα 1 (10.67) , cα = 0, 2 ≤ α ≤ n − 2, aα ∈ C, c1 = 0 (10.68) (10.69)
bαj = δjα+1 bαj = δjα+1
j = 3, . . . , n − 1 for c1 = 1, j = 3, . . . , n for c1 = 0,
yα ∈ C, at least one yα satisfies yα = 0. For c1 = 1 we have 1 a1 a2 an−2 (e e · · · en−1 ), en 1 3 e2 yn−2 − ln(ey11 ey32 · · · en−1 ). I2 = e1 I1 =
(10.70)
For c1 = 0 we have (10.71)
I1 = e1 , I2 = e2 − e1 ln(ey31 ey42 · · · eynn−2 ),
(3) Decomposition n = 2 + 2 + (n − 4) × 1 , + 0 0 δ δα1 , α2 , δα3 , . . . , δαn−2 (10.72) Dα = diag yα δα1 zα δα2
(10.73) (10.74)
yα , zα ∈ C, at least one yα and one zβ satisfies yα = 0, zβ = 0. In this case the invariants are e2 I1 = − ln(ey11 ey32 ey53 · · · eynn−2 ) e1 e4 − ln(ez11 ez32 ez53 · · · eznn−2 ). I2 = e3
(4) Decomposition n = 3 + (n − 3) × 1 ⎡⎛ ⎞ ⎤ δα1 ⎠ , δα2 , . . . , δαn−2 ⎦ (10.75) Dα = diag ⎣⎝ uα δα1 yα zα δα1 The complex constants uα , yα and zα satisfy one of the following conditions • uα = zα for all α, uα = zα = 0 for at least one value of α. • uα = 0 for all α, yβ = 0, zγ = 0 for at least one value of β and γ. • zα = 0 for all α, yβ = 0, uγ = 0 for at least one value of β and γ.
130
(10.76)
(10.77)
10. SOLVABLE LIE ALGEBRAS WITH ABELIAN NILRADICALS
The invariants in this basis are e2 I1 = − ln eu1 1 eu4 2 · · · eunn−2 e1
2 e2 e3 e2 1 − un−2 + zn−2 + I2 = yn−2 e1 e1 2 e1 (u y −u y ) (u y −u y ) (un−2 yn−3 −un−3 yn−2 ) . + ln e1 n−2 1 1 n−2 e4 n−2 2 2 n−2 · · · en−1
Proof. The proof is the same in all cases. The operators Fα of (10.48) are read off from the matrices Dα . It is then easy to verify that we have Fα I1 = Fα I2 = 0 for all α in all cases. It is obvious from the form of I1 and I2 that they are functionally independent. The conditions on the constants uα , yα and zα in the fourth case are necessary to assure commutativity of the matrices Dα . They correspond to Kravchuk signatures (1 1 1), (2 0 1) and (1 0 2), respectively. 10.6.5. Summary. The main results of this section can be summed up as follows: (1) A solvable Lie algebra s, over the field C, of dimension n + f , with an Abelian nilradical of dimension n, has precisely n − f generalized Casimir invariants. All of them are functions of the variables ei , representing the elements of the nilradical in (s∗ )∗ (Theorem 10.9). (2) The generalized Casimir invariants in general involve logarithms of polynomials in ei , as well as rational and irrational functions of ei (Theorems 3.4, 10.10 and 10.11). (3) Explicit expressions for the generalized Casimir operators were given for the cases when the dimension of the nilradical is equal, or close to its minimal, or alternatively to itsmaximal possible value. (4) The generalized Casimir operators are rational functions only when all the structure matrices Dα of (10.1) are diagonal and the eigenvalues satisfy certain rationality conditions. Logarithmic expressions occur as soon as any of the structure matrices contains nontrivial Jordan blocks. To illustrate the complications arising over the field of real numbers, consider the four-dimensional Lie algebra s4,5 of Section 17.2, expressed as in Theorem 10.1 with ⎞ ⎛ α β 1 ⎠ α, β ∈ R (10.78) D1 = ⎝ −1 β Obviously, D1 is diagonalizable over C, but not over R. In agreement with Theorem 10.9 (valid also for F = R) we have two invariants, both depending on (e1 , e2 , e3 ) only. However, their form is: (10.79)
I1 =
(e22 + e23 )α e2β 1
,
I2 = ln e1 + α arctan
e2 e3
(or, equivalently, I˜2 = e1 exp (α arctan e2 /e3 ), as presented in Section 17.2). Thus, in addition to rational and irrational functions and logarithms, we obtain inverse trigonometric functions. In Part 4 of this book we classify all complex and real Lie algebras of dimension dim s ≤ 6 and present a basis for the (generalized) Casimir invariants in each case.
CHAPTER 11
Solvable Lie Algebras with Heisenberg Nilradical The purpose of this chapter is to construct all indecomposable solvable Lie algebras s that have the (2m + 1)-dimensional Heisenberg algebra h(m) as their nilradical. The main results presented in this chapter were obtained in [106]. 11.1. The Heisenberg relations and the Heisenberg algebra The Heisenberg algebras h(m) with their basis written conventionally in the form (11.1)
(h, p1 , . . . , pm , q1 , . . . , qm )
is of primordial importance in quantum mechanics. In this context the extension s of the algebra h(m) by further operators f1 , . . . , ff is a question of the algebra of quantum mechanical observables. The Heisenberg relations (11.2)
[ˆ xa , pˆb ] = iδab ,
1 ≤ a, b ≤ m
underlie all of quantum mechanics. In standard quantum theory x ˆa are the coordinates of a particle and pˆa = −i∂xa are the components of its linear momentum operator. An abstract theory of the Heisenberg relations exists and is called “umbral calculus” [38, 103]. Other realizations exist, for instance pˆa can be represented by difference operators ∇a and x ˆa by coordinates on a lattice multiplied by an appropriate shift operator [40, 69, 70]. This can be used to develop quantum theory on a lattice while preserving continuous symmetries like Galilei, Poincaré or conformal invariance. Standard quantum mechanics and quantum mechanics on a lattice can be viewed as the theory of polynomials or convergent power series in the operators realizing the Lie brackets (11.2). The Heisenberg algebra h(m) is also a subalgebra of the quantum mechanical Galilei algebra [67, 132] (or extended Galilei algebra). The operators pˆa in this interpretation generate space translations, qˆa generate Galilei boosts and h changes the phase of the wavefunction. The algebra h(m) is also a subalgebra of the symmetry algebra of the heat equation, [86], of the nonlinear Schrödinger equation with any nonlinearity F (|ψ|) depending only on the absolute value of the wavefunction ψ and of many other partial differential equations occurring in nonrelativistic physical theories [49, 76]. In this context, extensions of the Heisenberg algebra h(m) are part of a study of physical theories with symmetries going beyond translations and Galilei boosts. An algebra that plays an important role in the microscopic theory of collective motions in nuclei is a semidirect product of the symplectic Lie algebra with the Heisenberg algebra as an ideal [96, 97, 104, 105] (11.3)
wsp(2m, R) = sp(2m, R) h(m). 131
132
11. SOLVABLE LIE ALGEBRAS WITH HEISENBERG NILRADICAL
As we shall show below, all solvable indecomposable Lie algebras s obtained as extensions of the Heisenberg algebra h(m) are closely related to subalgebras of wsp(2m, R) (or its complex version wsp(2m, C)). 11.2. Classification of solvable Lie algebras with nilradical h(m) 11.2.1. Basic classification theorem. Let us consider the Heisenberg algebra h(m) in its standard basis (h, p1 , . . . , pm , q1 , . . . , qm ) with commutation relations [pi , qj ] = δij h,
(11.4)
[pi , pj ] = [qi , qj ] = 0,
[pi , h] = [qi , h] = 0,
1 ≤ i, j ≤ m.
For future convenience let us denote (11.5)
p = (p1 , . . . , pm ),
q = (q1 , . . . , qm ).
We wish to extend this algebra to an indecomposable solvable Lie algebra s of dimension 2m + 1 + f having h(m) as its nilradical. This means we wish to add f further linearly nilindependent elements to h(m). Let us denote them f1 , . . . , ff . The commutation relations of s involving the new elements fα will have the form ⎛ ⎞ 2aα ρα1 ρα2 T ⎠, aα 1m×m + Aα Bα ([fα , h], [fα , p], [fα , q]) = (h, p, q) · ⎝ σα1 T (11.6) σα2 Cα aα 1m×m + Dα aα ∈ F,
ρα1 , ρα2 , σα1 , σα2 ∈ F1×m ,
Aα , Bα , Cα , Dα ∈ Fm×m .
The superscript T denotes transposition and the constants aα were split off from Aα and Dα for future convenience. Further, we have (11.7)
[fα , fβ ] = rαβ h + μαβ i pi + ναβ i qi
for some constants rαβ , μαβ i , ναβ i ∈ F. First, we proceed to simplify the Lie brackets (11.6) as much as is possible. We perform a change of basis fα = fα + ρα1,i qi − ρα2,i pi . This allows us to set ρα1,i and ρα2,i to zero in (11.6). Let us now impose the Jacobi identities. From the triplets (fα , pi , h) and (fα , qi , h) we obtain that σα1 = σα2 = 0 in (11.6). From the triplets (fα , pi , pj ), (fα , pi , qj ) and (fα , qi , qj ) we find that the matrices Aα , Bα , Cα , Dα ∈ Fm×m must satisfy (11.8)
Dα = −ATα ,
It follows that the matrices (11.9)
Bα = BαT ,
Aα Xα = Cα
Bα −ATα
Cα = CαT .
belong to the symplectic Lie algebra sp(2m, F), i.e., the satisfy. 0 1m×m T . (11.10) Xα K + KXα = 0, K = −1m×m 0 By taking suitable linear combinations of the elements fα we can arrange to have a1 = 1 or 0 and a2 = · · · = af = 0.
11.2. CLASSIFICATION OF SOLVABLE LIE ALGEBRAS WITH NILRADICAL h(m)
133
The Jacobi identities for the triplets (fα , fβ , pi ) and (fα , fβ , qi ) imply that μαβ i = ναβ i = 0. Further they imply that the matrices Xα must commute. For a1 = 0 they must also be linearly nilindependent, otherwise the nilradical would be larger than h(m). For a1 = 1 the matrices X2 , . . . , Xf must be linearly nilindependent for the same reason, though X1 may be nilpotent or even vanish. This imposes a restriction on the number of elements f that can be added. Indeed, the number of commuting linearly nilindependent matrices Xα ∈ sp(2m, F) is less or equal to the rank m of sp(2m, F) [90]. The remaining Jacobi identities for (fα , fβ , fγ ) (for f ≥ 3) imply that for a1 = 1 we have rαβ = 0 whenever α = 1 and β = 1. Redefining fα as fα = fα − r1α h we obtain r1α = 0 as well. For a1 = 0 the Jacobi identities (fα , fβ , fγ ) do not restrict the values of rαβ . The obtained commutation relations can be further simplified by transformations that respect the commutation relations in the nilradical h(m) and the simplifications already achieved. We put ξ = ξ · G.
ξ = ( p, q),
(11.11)
The commutation relations (11.4) are written as (11.12)
1 ≤ a < b ≤ 2m.
[ξa , ξb ] = Kab h,
The transformation G in (11.11) must then satisfy GT · K · G = K
(11.13)
with K as in (11.10), i.e., G ∈ Sp(2m, F) belongs to the symplectic Lie group. A further allowed transformation is the scaling (11.14)
pi = λpi ,
qi = λqi ,
h = λ2 h,
λ ∈ F, λ = 0.
We have thus proven the following result. Theorem 11.1. Every indecomposable solvable Lie algebra s (over the field F = C or F = R), containing the Heisenberg algebra h(m) as its nilradical, can be written in a canonical basis (h, p1 , . . . , pm , q1 , . . . , qm , f1 , . . . , ff ) with the commutation relations (11.4), supplemented by
(11.15)
= (h, ξ) · Mα , ([fα , h], [fα , ξ]) 0 2aα , Mα = 0 aα 12m×2m + Xα [fα , fβ ] = rαβ h,
α, β = 1, . . . , f.
The vector ξ is defined as ξ = (p1 , . . . , pm , q1 , . . . , qm ). The constants aα satisfy a1 ∈ {0, 1},
a2 = · · · = af = 0.
The matrices X1 , . . . , Xf ∈ sp(2m, F) satisfy (11.10) and (11.16)
[Xα , Xβ ] = 0.
For a1 = 0, or a1 = 1 the sets {X1 , . . . , Xf }, or {X2 , . . . , Xf } are linearly nilindependent, respectively.
134
11. SOLVABLE LIE ALGEBRAS WITH HEISENBERG NILRADICAL
The constants rαβ satisfy (11.17)
rαβ = 0,
1 ≤ α, β ≤ f,
for a1 = 1,
rαβ = −rβα ∈ F,
1 ≤ α, β ≤ f,
for a1 = 0.
The dimension of the solvable Lie algebra s is (11.18)
dim s = 2m + 1 + f,
0 < f ≤ m + 1.
The maximal value f = m + 1 is achieved precisely if we have a1 = 1, X1 = 0 and span{X2 , . . . , Xm+1 } is a Cartan subalgebra of sp(2m, F). We then also have rαβ = 0 for all α, β. Two algebras s and s characterized by (a1 , Xα , rαβ ) and (a1 , Xα , rαβ ) are equivalent if, after an allowed transformation we have (11.19)
a1 = a1 ,
Xα = Xα ,
rαβ = rαβ .
Allowed transformations are the symplectic transformations (11.11), the scaling (11.14) and linear combinations of fα , respecting the form of (11.15). A classification of the solvable algebras s thus reduces to a classification of Abelian subalgebras of sp(2m, F) containing no nilpotent elements. Elements of sp(2m, C) and sp(2m, R) have been classified [12, 42], as have maximal Abelian subalgebras [90]. 11.3. The lowest dimensional case m = 1 The Heisenberg algebra h(1) is three dimensional with basis (h, p, q). Following Theorem 11.1, we can extend h(1) either by one element f1 , or by two commuting elements f1 , f2 . Let us consider these two cases separately. (1) dim s = 4 (f = 1). The algebra sp(2, C) has two types of elements, nilpotent and nonnilpotent. The algebra sp(2, R) has three types: nilpotent, compact and noncompact. The algebra s is completely characterized by one matrix M , figuring in the commutation relation (11.15). For F = C precisely three possibilities occur, namely ⎞ ⎛ 2 0 ⎠ , b ≥ 0, M (1) = ⎝ 1 + b 0 1−b ⎞ ⎞ ⎛ ⎛ (11.20) 2 0 1 1⎠ , 1 0 ⎠. M (3) = ⎝ M (2) = ⎝ 0 1 0 −1 For F = R two additional inequivalent possibilities occur, characterized by the matrices ⎞ ⎞ ⎛ ⎛ 2 0 1 −α⎠ , α > 0, (11.21) M (4) = ⎝ M (5) = ⎝ 0 −1⎠ . α 1 1 0 Over C, M (4) is equivalent to M (1) , M (5) to M (3) . The corresponding algebras are presented in Part 4, Section 17.3. M (1) gives rise to two classes of algebras therein, namely s4,8 and s4,11 which differ by the dimensions of ideals in the characteristic series.
11.4. THE CASE m = 2
135
(2) dim s = 5 (f = 2). Since in this case we must have a1 = 1 on dimensional grounds, we have (11.22)
[f1 , f2 ] = 0
and the action of f1 and f2 on h(1) is characterized Up to a change of basis there is only one possibility ⎞ ⎛ ⎛ 2 0 1 0⎠ , M2 = ⎝ (11.23) M1 = ⎝ 0 1
by a pair of matrices M1 , M2 . over F = C ⎞ 1 0
0 ⎠. −1
Over F = R two inequivalent possibilities occur, namely (11.23) and ⎞ ⎞ ⎛ ⎛ 2 0 1 0⎠ , M2 = ⎝ 0 −1⎠ . (11.24) M1 = ⎝ 0 1 1 0 The corresponding algebras are presented in Part 4, Section 18.6.
11.4. The case m = 2 The Heisenberg algebra h(2) is five-dimensional; we have 0 < f ≤ 3, i.e., we can add one, two or three elements fα . (1) dim s = 6. Construction of these algebras amounts to the determination of canonical forms of elements of sp(4, C/R) modulo scaling. This was done in [42] and the resulting algebras are presented in Section 19.7 in Part 4. They were also presented in [106, Table A1]. For F = C, the list contains 12 classes of such algebras. For F = R, there are 25 classes in the list. (2) dim s = 7. These algebras require the classification of 2-dimensional Abelian subalgebras of sp(4, C/R). The resulting matrices Aα were presented in [106, Table A2]. Since the results are quite lengthy we do not reproduce them here. (3) dim s = 8. We must have a1 = 1, a2 = a3 = 0. The commutation relations for f1 , f2 , f3 therefore read [fα , fβ ] = 0,
1 ≤ α, β ≤ 3.
The action of fα on h, p1 , p2 , q1 , q2 is given by three matrices Mα as in (11.15). For F = C the Cartan subalgebra of sp(4, C) is unique and we have ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 2 0 0 ⎟ ⎟ ⎟ ⎜ 1 ⎜ 1 ⎜ 0 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ (1) (1) (1) ⎟ ⎟ ⎟. ⎜ ⎜ ⎜ 1 0 1 (11.25) M1 = ⎜ ⎟ , M2 = ⎜ ⎟ , M3 = ⎜ ⎟ ⎠ ⎠ ⎠ ⎝ ⎝ ⎝ 1 −1 0 1 0 −1 For sp(4, R) we have four inequivalent Cartan subalgebras [90], so in addition to the algebra s characterized by the matrices (11.25) we have three more cases,
136
11. SOLVABLE LIE ALGEBRAS WITH HEISENBERG NILRADICAL
characterized by ⎛ 2 ⎜ 1 ⎜ (2) 1 (11.26) M1 = ⎜ ⎜ ⎝ 1
⎞ ⎟ ⎟ ⎟ , M (2) 2 ⎟ ⎠
⎛ 0 ⎜ 1 ⎜ 1 =⎜ ⎜ ⎝ −1
⎞ ⎟ ⎟ ⎟, ⎟ ⎠
−1 ⎞ ⎛ 2 0 ⎟ ⎟ ⎜ 1 ⎜ 1 ⎟ ⎟ ⎜ ⎜ (3) ⎟ , M (3) = ⎜ ⎟, 1 0 (11.27) M1 = ⎜ 2 ⎟ ⎟ ⎜ ⎜ ⎝ ⎝ 1 ⎠ −1 ⎠ 1 0 ⎞ ⎞ ⎛ ⎛ 2 0 ⎟ ⎜ 1 ⎜ 0 0 1 0⎟ ⎟ ⎟ ⎜ ⎜ (4) ⎟ , M (4) = ⎜ 0 0 0 0⎟ , 1 (11.28) M1 = ⎜ 2 ⎟ ⎟ ⎜ ⎜ ⎝ ⎝ −1 0 0 0⎠ 1 ⎠ 1 0 000 ⎛
1
⎞
⎞ ⎛ 0 ⎟ ⎜ 0 1 ⎟ ⎜ (2) ⎟, ⎜ M3 = ⎜ −1 0 ⎟ ⎝ 0 1⎠ −1 0 ⎞ ⎛ 0 ⎜ 0 0 0 0⎟ ⎟ ⎜ (3) ⎟ M3 = ⎜ ⎜ 0 0 0 1⎟ , ⎝ 0 0 0 0⎠ 0 −1 0 0 ⎞ ⎛ 0 ⎜ 0 0 0 0⎟ ⎟ ⎜ (4) ⎟ M3 = ⎜ ⎜ 0 0 0 1⎟ . ⎝ 0 0 0 0⎠ 0 −1 0 0
The low-dimensional cases m = 1 and m = 2 show that explicit detailed classifications become rather long and cumbersome for 2 ≤ f ≤ m already for m = 2 and we go no further here. On the other hand for f = m + 1 the results are very simple: a1 = 1, rαβ = 0, X1 = 0 and the matrices Xα , 2 ≤ α ≤ m + 1 form a Cartan subalgebra of sp(2m, F). There is only one conjugacy class of Cartan subalgebras for sp(2m, C) (as for any complex simple Lie algebra). The real algebra sp(2m, R) has (m + 2)2 /4 different classes of Cartan subalgebras for m even and (m + 1)(m + 3)/4 for m odd [54, 63, 90, 121]. For f = 1 the results are also quite simple both for F = C and F = R and reduce to a classification of elements of sp(2m, F) [42]. 11.5. Generalized Casimir invariants We shall realize the coadjoint representation of s in a space of differentiable functions of 2m + 1 + f variables which we denote (h, p1 , . . . , pm , q1 , . . . , qm , f1 , . . . , ff ). The algebra s of Theorem 11.1 will be realized by differential operators, the form of which can be read off from the Lie brackets (11.4) and (11.15). Namely, we have (11.29) = −2a1 h ∂ , H ∂f1 ∂ ∂ ∂ − (Aα )ki pk + (Cα )ki qk − a1 p i , Pi = h ∂qi ∂fα ∂f1 i = −h ∂ − (Bα )ki pk − (Aα )ik qk ∂ − a1 qi ∂ , Q ∂pi ∂fα ∂f1 ∂ ∂ + (Aα )ki pk + (Cα )ki qk + aα pi Fα = 2aα h ∂h ∂pi ∂ ∂ + (Bα )ki pk − (Aα )ik qk + aα qi + rαβ h , ∂qi ∂fβ i, k = 1, . . . , m, α = 1, . . . , f, a1 = 0, 1, aα = 0, α ≥ 2, rαβ = 0 for a1 = 1
11.5. GENERALIZED CASIMIR INVARIANTS
137
(summation over repeated indices is understood). The matrices Aα , Bα = BαT and Cα = CαT are the same as in the Lie brackets (11.15). The invariants of s are obtained as functionally independent solutions of the system of first-order linear partial differential equations = 0, (11.30) HI
Pi I = 0,
i I = 0, Q
Fα I = 0, i = 1, . . . , m,
α = 1, . . . , f
where I = I(h, p1 , . . . , pm , q1 , . . . , qm , f1 , . . . , ff ). Let us treat the two cases a1 = 1 and a1 = 0 separately. Theorem 11.2. An indecomposable solvable Lie algebra s with the Lie brackets (11.4) and (11.15) and a1 = 1 (and hence rαβ = 0) has precisely (f − 1) generalized Casimir invariants. They can be written in the form 1 2hfα + 2(Aα )ij pi qj + (Cα )ij qi qj − (Bα )ij pi pj , α = 2, . . . , f (11.31) Iα−1 = h (repeated indices i, j = 1, . . . , m are summed over). to the function Proof. We have a1 = 1. Applying H I(h, p1 , . . . , pm , q1 , . . . , qm , f1 , . . . , ff ) we find that I is independent of f1 . Applying the remaining operators corresponding i of (11.29), successively to I and using the to the nilradical h(m), namely Pi and Q method of characteristics, we find that I can depend only on h and Zα , 2 ≤ α ≤ f where (11.32)
Zα = 2hfα + 2(Aα )ij pi qj + (Cα )ij qi qj − (Bα )ij pi pj .
Let us now apply the operators Fα (α ≥ 2) to I(h, Z2 , . . . , Zf ). We have (11.33) Fα Zβ = 2(Aα Aβ − Aβ Aα + Bα Cβ − Bβ Cα )ik qi pk + 2(Aβ Bα − Aα Bβ )ik pi pk + 2(Cα Aβ − Cβ Aα )ik qi qk where α, β ≥ 2 and matrix multiplication of Aα with Aβ etc. is assumed. The commutativity conditions (11.16) imply that the first term in (11.33) vanishes. The symmetric parts of the second and third term also vanish as a result of the same commutativity relations. Finally, the antisymmetric parts, when contracted with the symmetric tensor pi pj (qi qj , respectively) also vanish. Hence we have Fα Zβ = 0 for α, β ≥ 2. Finally, we apply F1 , and obtain f ∂I ∂I + (11.34) F1 I(h, Z2 , . . . , Zf ) = 2 h = 0. Zβ ∂h ∂Zβ β=2
From (11.34) we conclude that the solution I of (11.30) is an arbitrary function of the invariants (11.31) and this proves the theorem. Theorem 11.3. An indecomposable solvable Lie algebra s with the Lie brackets (11.4) and (11.15) and a1 = 0 has N = (f + 1 − rank R) functionally independent Casimir invariants (11.35)
I1 = h,
Iμ =
f α=1
aμα Zα ,
2≤μ≤N
138
11. SOLVABLE LIE ALGEBRAS WITH HEISENBERG NILRADICAL
where R = (rαβ )fα,β=1 ,
RT = −R
and (11.36)
Zα = 2hfα + 2(Aα )ij pi qj + (Cα )ij qi qj − (Bα )ij pi pj ,
1≤α≤f
(Zα are the same as in (11.32), the only difference is in the range of index α). The constants aμα are determined from the condition that the column vectors aμ span the nullspace of the matrix R R · aμ = 0,
(11.37)
2 ≤ μ ≤ N.
Proof. From the form of the operators (11.29) with a1 = 0 we see directly i I = 0 imply that h is an invariant. The equations Pi I = 0 and Q I = I(h, Z1 , . . . , Zf ) with Zα as in (11.36), and 1 ≤ α ≤ f . Following the same reasoning as in the proof of Theorem 11.2 we find that (11.38) Fα Zβ = 2h2 rαβ . It follows that elementary solutions of (11.30) are linear combinations (11.35) of f Zα with constant coefficients aμα such that β=1 rαβ aμβ = 0. This completes the proof. 11.5.1. Examples. To illustrate Theorem 11.2, let us consider the eight-dimensional Lie algebra s characterized by the matrices of (11.25). Applying the theorem we obtain 1 1 (11.39) I1 = (hf2 + p1 q1 ), I2 = (hf3 + p2 q2 ). h h To illustrate Theorem 11.3, let us consider the case of the algebras s with m arbitrary and f = 3. The matrix R in this case is ⎛ ⎞ 0 r12 r13 0 r23 ⎠ (11.40) R = ⎝−r12 −r13 −r23 0 and its rank is either 0 or 2. For rank R = 0 we have r12 = r13 = r23 = 0 and we obtain 4 invariants (11.41)
h,
Z1 ,
Z2 ,
Z3
with Zα as in (11.36). When rank R = 2 the two invariants are (11.42)
I1 = h,
I2 = r23 Z1 − r13 Z2 + r12 Z3 .
As a further illustration of Theorem 11.3, consider the Lie algebras s with m arbitrary and f = 4. The 4 × 4 matrix antisymmetric matrix R can have rank 0, 2 or 4. The invariants are respectively (11.43)
rank R = 0
h, Z1 , Z2 , Z3 , Z4 ,
(11.44)
rank R = 2
h, r23 Z1 − r13 Z2 + r12 Z3 , r24 Z1 − r14 Z2 + r12 Z4 ,
(11.45)
rank R = 4
h.
Notice that the invariants presented in (11.44) are valid for a generic antisymmetric rank-2 matrix R and vanish for some particular choices of R; when that happens
11.5. GENERALIZED CASIMIR INVARIANTS
139
a different set of solutions of(11.37) must be substituted into (11.35). An example of such a nongeneric case is obtained if we have r34 = 0, r12 = r13 = r14 = r23 = r24 = 0. Equations (11.38) imply ∂I ∂I = =0 ∂Z3 ∂Z4 so the Casimir invariants are h, Z1 and Z2 .
CHAPTER 12
Solvable Lie Algebras with Borel Nilradicals Let us now consider an infinite class of nilpotent Lie algebras which have a close relationship with simple Lie algebras. Every complex simple Lie algebra g has a uniquely defined (up to equivalence) maximal solvable subalgebra, its Borel subalgebra b(g). In turn, the Borel subalgebra (like every solvable Lie algebra) has a unique maximal nilpotent ideal, its nilradical NR b(g) . We shall call these nilpotent algebras “Borel nilradicals.” For each simple complex Lie algebra g the Borel nilradical is the Lie algebra consisting of all positive root spaces. It is of course the nilradical only of the Borel subalgebra b(g), not of g itself (g being simple has no ideals at all). For real Lie algebras the maximal solvable subalgebra is in general not unique. The maximal solvable subalgebras of all real classical algebras were classified in [88, 89, 93]. For the split real form of the complex simple Lie algebra there exists a distinguished maximal solvable subalgebra that has the same basis as the Borel subalgebra but considered over R instead of C. We shall also call their nilradicals “Borel nilradicals” (over R), sometimes adding the adjective “split” to highlight its relation to the split real form. We shall treat all simple Lie algebras simultaneously (the classical and exceptional complex ones and the split real ones) using bases for the nilradicals provided by the positive roots. This simultaneous treatment is made possible by the fact that all outer derivations of these nilradicals are known, due to the work of G. F. Leger and E. M. Luks [67]. Our analysis of solvable extensions of Borel nilradicals based on this knowledge was originally presented in [118] which we shall closely follow here. An example of a Borel nilradical is the algebra of strictly upper triangular matrices t(l + 1), the Borel nilradical of Al ≡ sl(l + 1, F). This case was considered in [123].
12.1. Outer derivations of nilradicals of Borel subalgebras Let g be a simple complex Lie algebra, g0 its Cartan subalgebra, l = rank g = dim g0 . As in Section 2.3 let us denote by Δ the set of all roots, by Δ+ the set of all positive roots and by ΔS = {α1 , . . . , αl } the set of simple roots. Let gλ denote the root subspace of the root λ, spanned by the root vector eλ . We use the notation , for the scalar product induced by the Killing form on g on the real vector subspace of (g0 )∗ spanned by the simple roots. Let Sβ denote the Weyl reflection with respect to the root β, Sβ (α) = α − 2
α, β β, β, β 141
α ∈ Δ.
142
12. SOLVABLE LIE ALGEBRAS WITH BOREL NILRADICALS
As recalled in Section 2.3 every (semi)simple complex Lie algebra g contains a unique (up to isomorphisms) maximal solvable subalgebra, its Borel subalgebra b(g). It contains the Cartan subalgebra and all positive root vectors (12.1)
b(g) = g0 ({gλ | λ ∈ Δ+ }).
The properties of root systems imply that the Borel subalgebra is indeed a solvable subalgebra of g with the nilradical (12.2) NR b(g) = {gλ | λ ∈ Δ+ }. For the sake of brevity we shall call the nilpotent Lie algebra NR b(g) the Borel nilradical. It is always possible to realize the simple Lie algebra g in such a manner that its Borel subalgebra b(g) is represented by upper triangular matrices. For sl(l + 1, C), b(sl(l+1, C)) is simply the set of all traceless (l+1)×(l+1) upper triangular matrices and its nilradical is the set of traceless strictly upper triangular matrices. The same holds true also for its split, i.e., maximally noncompact, real form sl(l + 1, R). All other series Bl , Cl , Dl of classical complex Lie algebras and their split real forms can be realized by matrices X ∈ Cn×n or X ∈ Rn×n satisfying the equation (12.3)
XK + KX T = 0
where the matrix K is chosen as follows Bl = so(2l + 1, C), so(l + 1, l) (12.4)
Cl = sp(2l, C), sp(2l, R) Dl = so(2l, C), so(l, l)
where
K = J2l ,
⎛ ⎜ ⎜ Jm = ⎜ ⎝ 1
1 . ..
K = J2l+1 , 0 Jl , K= −Jl 0
⎞ 1 ⎟ ⎟ ⎟ ∈ Fm×m . ⎠
The corresponding Borel subalgebra b(g) is represented by triangular ma upper trices satisfying the condition (12.3). The nilradical NR b(g) is represented by strictly upper triangular matrices satisfying the condition (12.3). Let αi , i = 1, . . . , l = rank g denote the simple roots, ΔS = {αi }li=1 and let l l + mi αi ∈ Δ , mi ≥ m . (12.5) g m = gλ λ = i=1
i=1
The root subspaces gα = span{eα }, α ∈ ΔS generate the entire NR b(g) = {gλ | λ ∈ Δ+ } through commutators [gλ , gμ ] = gλ+μ
whenever λ, μ, λ + μ ∈ Δ+
and this implies that the ideals in the lower central series of the nilradical NR b(g) of the Borel subalgebra are ) *m = gm . NR b(g)
12.1. OUTER DERIVATIONS OF NILRADICALS OF BOREL SUBALGEBRAS
143
The center z of NR b(g) is one-dimensional and is spanned by eλ where λ is the highest root of g, i.e., the unique root such that no root of the form λ + αj exists. The center z coincides with the last nonvanishing ideal in the lower central series. All derivations of the nilradical n = NR b(g) were found in [67] and the result is summed up in Proposition 12.1. Proposition 12.1. Let g be a complex simple Lie algebra of rank l, g0 its Cartan subalgebra, ΔS = {α1 , . . . , αl } the set of simple roots and n = NR b(g) . The algebra Der(n) of derivations of the nilradical n = NR b(g) of the Borel subalgebra of a complex simple Lie algebra g satisfies • Der(n) = Out(n) Inn(n), • Inn(n) = {ad x | x ∈ n} is the ideal of inner derivations, • dim Out(n) = 2l, i | i = 1, . . . , l} where the derivations Di , D i are de• Out(n) = span{Di , D fined as follows. The derivations Di act diagonally in the basis of n consisting of positive root vectors eα , α ∈ Δ+ Di (eα ) = mi eα ,
α=
l
mj αj ∈ Δ+ .
j=1
i are nilpotent outer derivations which act on simple root vectors as D i (eβ ) = eγ , where γ = Sαi (λ), if β = αi , (12.6) D 0, if β = αj , j = i i on eα , α ∈ Δ+ \ΔS follows where λ is the highest root of g. The action of D from the definition of a derivation (2.26). For future reference, we list the highest roots corresponding to all simple Lie algebras in Table 12.1. For the sake of brevity, we shall write Si (λ) instead of Sαi (λ) and introduce nonnegative integer constants si Si (λ) = λ − si αi ,
i = 1 . . . , l (no summation).
We notice that for g = Al only two constants si , namely s1 and sl , are nonvanishing and equal to one; for all other simple algebras only one si is nonvanishing and turns Table 12.1. The highest roots of simple Lie algebras. Al :
l
Dl :
αi λ = α1 + 2 lj=2 αi λ = 2 l−1 j=1 αi + αl λ = α1 + 2 l−2 j=2 αi + αl−1 + αl
E6 :
λ = α1 + 2α2 + 3α3 + 2α4 + α5 + 2α6
E7 :
λ = 2α1 + 3α2 + 4α3 + 3α4 + 2α5 + α6 + 2α7
E8 :
λ = 2α1 + 4α2 + 6α3 + 5α4 + 4α5 + 3α6 + 2α7 + 3α8
F4 :
λ = 2α1 + 3α2 + 4α3 + 2α4
G2 :
λ = 2α1 + 3α2
Bl : Cl :
λ=
j=1
144
12. SOLVABLE LIE ALGEBRAS WITH BOREL NILRADICALS
Table 12.2. Dimensions of the nilradicals NR b(g) and nonvanishing constants si in the equation Si (λ) = λ − si αi . Al Bl Cl Dl E6 E7 E8 F4 G2
l(l + 1) dim NR b(Al ) = 2 dim NR b(Bl ) = l2 dim NR b(Cl ) = l2 dim NR b(Dl ) = l(l − 1) dim NR b(E6 ) = 36 dim NR b(E7 ) = 63 dim NR b(E8 ) = 120 dim NR b(F4 ) = 24 dim NR b(G2 ) = 6
s1 = sl = 1, s2 = 1, s1 = 2, s2 = 1, s6 = 1, s1 = 1, s7 = 1, s1 = 1, s1 = 1.
out to beequal to 1 or 2, see Table 12.2. We also list the dimensions of the nilradicals NR b(g) in Table 12.2. i act on the vectors eβ when β ∈ Δ+ \ΔS . We proceed Let us determine how D by induction, using the Leibniz property of derivations (2.26). First, we evaluate i ([eα , eα ]) = [D i (eα ), eα ] + [eα , D i (eα )]. D j j j k k k i . Without loss of When both i = j and i = k the result is 0 by definition of D generality we take i = j and i = k and obtain i ([eα , eα ]) = [D i (eα ), eα ] = [eS (λ) , eα ]. D i i k k k i Were the result nonvanishing, we would have Si (λ) + αk ∈ Δ+ , i.e., also Si (Si (λ) + αk ) = λ + αk − 2
αk , αi αi ∈ Δ+ . αi , αi
Since 2αk , αi /αi , αi is an off-diagonal element of the Cartan matrix, it is a nonpositive integer. That means we would have a root larger than λ, contradicting the maximality of the highest root λ. Therefore, i ([eα , eα ]) = 0 D j k i (eβ ) = 0 for all roots β = for any j, k = 1, . . . , l. Next, let us assume that D l l mj αj such that 2 ≤ j=1 mj ≤ M and consider Di ([eαj , eβ ]) with β = j=1 l l j=1 mj αj , j=1 mj = M . When j = i we again obtain 0 using the Leibniz property (2.26). When i = j we have i ([eα , eβ ]) = [eS (λ) , eβ ]. D i i Using the Table 12.2 we find that for all simple algebras except Cl we have either Si (λ) = λ − αi or Si (λ) = λ.
12.1. OUTER DERIVATIONS OF NILRADICALS OF BOREL SUBALGEBRAS
145
Consequently, no root of the form Si (λ) + β, where β = lj=1 mj αj , exists when l j=1 mj ≥ 2. Therefore the Lie bracket [eSi (λ) , eβ ] vanishes. In the special case of the algebras Cl we have S1 (λ) = λ − 2α1 and we must perform a more detailed analysis. Let us express eβ as a multiple Lie bracket of simple root vectors eαj , eβ = eαj1 , . . . , [eαjM −1 , eαjM ] . . . M where β = K=1 αjK and use the Jacobi identity sufficiently many times: [eS1 (λ) , eβ ] = [eS1 (λ) , eαj1 ], [. . . , [eαjM −1 , eαjM ] . . . + · · · + eαj1 , . . . , eαjM −1 , [eS1 (λ) , eαjM ] . . . . 1 the corresponding commutator of eαjK with eS1 (λ) vanishes. Whenever jK = When jK = 1, we have [eS1 (λ) , eα1 ] = ceλ−α1 for some integer constant c. The only positive root vector which does not commute with eλ−α1 is eα1 ; therefore, any further commutator with any eαj , j = 1 gives zero. Finally, the commutator of eλ−α1 with eα1 can in our computation arise only through eS1 (λ) , [eα1 , eα1 ] which vanishes immediately. Therefore, in all cases we have shown that i ([eα , eβ ]) = 0 D j
l l when β = j=1 mj αj and j=1 mj = M , concluding the induction step. To sum up, we have just shown that for any simple complex Lie algebra g i of the algebra NR b(g) give zero whenever they act on eβ , the derivations D β ∈ Δ+ \ΔS . Let us assume from now on that l > 2. Then we always have Si (λ) ∈ / ΔS for all i = 1, . . . , l and consequently (12.7)
i ◦ D j (eα ) = 0 D k
for every αk ∈ ΔS . The previous calculation allows us to conclude that (12.7) must hold for any α ∈ Δ+ , i.e., we have j = 0 i ◦ D D for any i, j = 1, . . . , l. j , The derivations Di commute among each other and act diagonally on D (12.8)
j ] ∈ span{D j }. [Di , D
To conclude, we have just seen that under the assumption that l isgreater i span a Lie subalgebra Out NR b(g) of the than 2, the 2l outer derivations D , D i algebra of all derivations Der NR b(g) . This algebra can be further decomposed into a semidirect sum of an l-dimensional Abelian ideal spanned by the nilpotent i and an l-dimensional Abelian subalgebra spanned by Di . derivations D The cases l = 1 and l = 2 are discussed in the following section, above Theorem 12.2.
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12. SOLVABLE LIE ALGEBRAS WITH BOREL NILRADICALS
12.2. Solvable extensions of the Borel nilradicals NR b(g) Let us now study the structure of any solvable Lie algebra with the nilradical NR b(g) , l = rank g > 2. fact that there are only l linearly nilindependent derivations Di in From the Out NR b(g) we conclude that the maximal number ofnonnilpotent basis ele ments in any solvable Lie algebra s with the nilradical NR b(g) is l. One algebra with this number of nonnilpotent basis elements is already known, namely the Borel subalgebra b(g) of the simple Lie algebra g. The following argument will show that it is the only one (up to isomorphisms). 12.2.1. Solvable extensions of the Borel nilradicals of maximal dimension. Let us assume that we have a solvable Lie algebra s with the nilradical n = NR b(g) and l = rank g nonnilpotent basis elements fi . They define l outer i such that D i = ad(fi )|n . Using the translinearly nilindependent derivations D formation (9.13) we may choose the basis vectors fi so that i = Di + D
(12.9)
l
j ωji D
j=1
j are the derivations defined above. We proceed to show that we can where Di , D i i lie in the subalgebra Out NR b(g) of Der NR b(g) put ωj = 0, ∀i, j. Since D j ] ∈ Inn NR b(g) must hold, we find that i, D and at the same time [D j ] = 0. k, D [D
(12.10)
This requirement together with (12.8) in turn implies i ] + ω k [D i , Dj ] = 0. ωij [Dk , D i
(12.11)
for every i, j, k = 1, . . . , l such that k = j (no summation over i). For any given i , D˜ı ] = 0. Consequently, the value of ω˜ı together i we can find ˜ı such that [D i i ] completely determines all with the root system specifying the Lie brackets [Dk , D ωij for j = ˜ı. Altogether, we still have one undetermined parameter ω˜iı for each i = 1, . . . , l. We proceed to eliminate these parameters through a suitable choice of automorphism in (9.14). i )2 = 0 and consequently φ˜i (ti ) ≡ exp(ti D i ) = 1 + ti D i . Under the We have (D j transformation (9.14) the derivations Dj , Dj , D change as follows i , Dj ], Dj → Dj + ti [D (12.12)
j = Dj + D
l
j , j → D D i , Dj ] + k → Dj + ti [D ωkj D
k=1
l
k. ωkj D
k=1
˜ı which transforms Now we consider step by step each i = 1, . . . , l. We take D i , D˜ı ] = 0. nontrivially under the transformation (12.12) by definition of ˜ı, due to [D ˜ ı We use it to set ωi = 0 after the transformation. Equation (12.11) then implies that after the transformation all ωij = 0. The automorphisms φ˜i (ti ) commute amongst each other, i.e., we can independently set to zero the constants ωij belonging to
12.2. SOLVABLE EXTENSIONS OF THE BOREL NILRADICALS NR(b(g))
147
different values of i without changing the others. Therefore we have found that our j can be brought to the form derivations D j = Dj D ˜ ˜ ˜ through a conjugation by a suitable automorphism Φ = φ1 (t1 ) ◦ · · · ◦ φl (tl ) of NR b(g) . As we have just seen, the action of the nonnilpotent basis vectors fi on the nilradical NR b(g) is the same as in the Borel algebra. In order to show the uniqueness of the maximal solvable extension of NR b(g) it remains to be shown that we can always accomplish [fi , fj ] = 0. The kernel of the adjoint representation ad : n → gl(n) of the nilradical n is the center z of the algebra n which in our case is spanned by eλ . Consequently we have [fi , fj ] = γij eλ ,
γij = −γji
which is the inverse image of the relation [ad(fi )|n , ad(fj )|n ] = 0 (cf. (12.10)). Let us write the highest root in terms of simple roots as λ=
l
λj αj , λj ∈ N.
j=1
The Jacobi identity (fi , fj , fk ) implies that (12.13)
λi γjk + λj γki + λk γij = 0.
Let us perform a transformation fi → fi + τi eλ which induces a change of the constants γij γij → γij + λi τj − λj τi . In this way we may set to zero for example all γ1j , j = 2, . . . , l by selecting τ1 = 1 and choosing τj so that γ1j + λ1 τj − λj = 0 (recall that all λj ≥ 1). After such a transformation, (12.13) becomes λ1 γjk = 0,
j, k = 1
and implies that all γjk must now vanish. Therefore, we have found a basis (eα , fi | α ∈ Δ+ , i = 1, . . . , l) in our solvable algebra such that [fi , eβ ] = mi eβ , [fi , fj ] = 0.
where β =
l
mj αj ,
j=1
To sum up, we have found that for any complex simple Lie algebra g such that rank g > 2 the maximal solvable Lie algebra with the nilradical NR b(g) is unique and isomorphic to the Borel subalgebra b(g) of g. The same is true also when rank g = 1 or rank g = 2, i.e., g = sl(2), sl(3), so(5) or G2 . In the case of g = G2 the proof given above can be taken over without modifications, since Si (λ) ∈ Δ+ \ΔS for all simple roots αi . In the cases of g = sl(2), sl(3), so(5) this is no longer true but the result follows from the investigation of Abelian nilradicals (g = sl(2)) in Chapter 10, Heisenberg nilradicals (g = sl(3)) in Section 11.3, and the nilradical n4,1 of Section 13.1 (g = so(5)). Thus, we have proven the following theorem:
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12. SOLVABLE LIE ALGEBRAS WITH BOREL NILRADICALS
Theorem 12.2. Let g be a complex simple Lie algebra, b(g) its Borel subalgebra and n = NR the nilradical of b(g). The solvable Lie algebra with b(g) the nilradical NR b(g) of the maximal dimension dim n + rank g is unique and isomorphic to the Borel subalgebra b(g) of g. 12.2.2. Solvable extensions of the split real form of the nilradical NR b(g) . As was already noted, the maximal solvable subalgebra of a real form of a complex Lie algebra is not necessarily unique. Let us consider the split real form g and its maximal solvable subalgebra that consists of the maximally noncompact Cartan subalgebra (that remains diagonalizable over R) and all positive root spaces. That means that it is spanned (over R) by all hα and eα in the Weyl – Chevalley basis where α ∈ Δ+ . Let us call it the split Borel subalgebra and also denote it by b(g). The results of [67] also apply to real split Borel subalgebras. Our Theorem 12.2 holds for the split real form of a complex simple Lie algebra g with one exception: when g = sl(3, R) there are several such maximal solvable algebras (see Section 11.2). The proof for the cases rank g > 2 or g = G2 is the same as above, the proof for the case g = so(3, 2) (the split real form of so(5, C)) follows from Section 13.1 below. 12.2.3. Solvable extensions of the Borel nilradicals of less than maximal dimension. After investigating the case of the maximal dimension solvable extension in the previous Sections let us consider the cases where the number of nonnilpotent elements of s is smaller than the rank of g, i.e., q < l. In such a case, we have derivations a = D
(12.14)
l
j , σja Dj + ωja D
a = 1, . . . , q
j=1
a = ad(fa )|n . representing the elements fa in the adjoint representation of s on n, D a The q × l matrix σ = (σj ) must have maximal rank, i.e., q, in view of the nilinde a . However we can no longer set σ a equal to the Kronecker delta δ a pendence of D j j as was the case for q = l. b ] = 0, remains and implies a, D The condition (12.10), i.e., [D l
k = 0 (σja ωkb − σjb ωka ) λj − δjk (1 + sj ) D
j,k=1
k gives a which after separation of coefficients of linearly independent derivations D 1 set of 2 q(q − 1)l conditions (12.15)
l
(σja ωkb − σjb ωka ) λj − δjk (1 + sj ) = 0,
k = 1, . . . , l
j=1
for 2q·l unknown constants σja , ωkb . When all ωkb = 0, any choice of σja solves (12.15). Let us now find out how the constants σja , ωja transform under transformations Φ of the form (9.14), i.e., a change of basis in the nilradical. When Φ belongs to the connected component of Aut(n) generated by exponentiation of Der(NR b(g) ), the transformation (9.14) will not change the constants σja , owing to the fact that j ] ∈ span{D j }. We mention that certain permutations of the [Di , Dj ] = 0, [Di , D constants σja may become possible under parity transformations belonging to other
12.2. SOLVABLE EXTENSIONS OF THE BOREL NILRADICALS NR(b(g))
149
components of Aut(n). E.g., the Dynkin diagram and the root system of sl(l + 1, C) has the reflection symmetry αj ↔ αl+1−j which gives rise to an automorphism of a n interchanging σja with σl+1−j . The constants ωka transform as follows: • under automorphisms exp(uj Dj ) the constants ωka get scaled by a factor k ] = djk D k. exp(uj djk ) where [Dj , D j ) = 1 + tj D j the constants ω a • under automorphisms φ˜j (tj ) = exp(tj D k are shifted to l a a a a σj λj − σk (1 + sk ) tk (12.16) ω k = ωk − j=1
j , l σ a Dk . as follows from the evaluation of the commutator D k=1 k Let us assume that ωka = 0 for certain a and k. We may add a suitable multiple a to all D b , b = a so that after the addition we have ω b = 0 for all b = a. of D k Next, we check whether lj=1 σja λj − σka (1 + sk ) = 0. If it holds, we can use the a transformation (12.16) to set also ωka = 0. If not, we may attempt to modify D b , b = a. If by addition of any multiple of D l
(12.17)
σjb λj − σkb (1 + sk ) = 0
j=1
a and next eliminate ω a = 0, holds for at least one b = 1, . . . , q, we can first change D k i.e., now we have ωkb = 0 for all b, including b = a. Since the automorphisms φ˜k (tk ) for different k commute we may repeat the procedure for all values of k. Consequently, for any k = 1, . . . , l we may set all ωka = 0 whenever the condition (12.17) is satisfied for at least one a ∈ {1, . . . , q}. Let us find out for how many values of k we can always transform the parameters a ωk into ωka = 0. The matrix Q with entries Qjk = (λj − (1 + sk )δjk ) has the form ⎛ ⎞ λ1 − (1 + s1 ) λ1 λ1 ... λ1 ⎜ ⎟ λ2 λ2 − (1 + s2 ) λ2 ... λ2 ⎜ ⎟ ⎜ ⎟ λ3 λ3 λ3 − (1 + s3 ) . . . λ3 Q=⎜ ⎟. ⎜ ⎟ .. .. .. .. ⎝ ⎠ . . . . λl
λl
λl
...
λl − (1 + sl )
Subtracting the second column from the first, the third from the second, etc., and using λk > 0 and si ≥ 0 we find that it has always the maximal rank l, i.e., l a a σ λ − σ (1 + s ) can be Q is a regular matrix. The expressions Rka = j k k j=1 j written as elements of the matrix product R = σ · Q. Whenever the k-th column (Rka )a=1,...,q of the matrix R is nonvanishing we can eliminate all parameters ωka a . The number of indices k such that the column (Ra )a=1,...,q from our derivations D k does not vanish is bounded from below by the rank of matrix R, i.e., also by the rank of matrix σ, which is equal to q. Therefore, we have at most l − q values of the index k such that some parameters ωka are nonvanishing. Now, after the simplifications introduced above, we revisit the commutativity condition (12.15). We observe that (12.15) can be written as a sum of terms, all of
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12. SOLVABLE LIE ALGEBRAS WITH BOREL NILRADICALS
them containing the expression ωkd
(12.18)
l
σjc λj − δjk (1 + sj )
j=1
where c = a and d = b or vice versa. If the condition (12.17) is satisfied we have set ωkd = 0; otherwise, lj=1 σjc λj − δjk (1 + sj ) = 0. Either way, all terms of the form (12.18) vanish and consequently the commutativity condition (12.15) is always satisfied. Finally, we consider the Lie brackets between elements fa , fb . Equation (9.7) takes the form (12.19)
[fa , fb ] = γab eλ ,
a, b = 1, . . . , q.
The constants γab in (12.19) can be modified by transformations of the form fa → fa = fa + τa eλ . a , a = 1, . . . , q but may Such a transformation does not change the derivations D a such modify the Lie brackets [fa , fb ]. Indeed, when there exists at least one D l j a that κ = j=1 λ σj = 0, we have a (eλ ) = κeλ = 0. [fa , eλ ] = D Now we may choose τa = 0 and τb = −γab /κ, a = b so that we obtain γab κeλ = 0, [fa , fb ] = γab eλ − κ i.e., we have set γab = 0 for the given a and all b = 1, . . . , q. The Jacobi identity (fa , fb , fc ) c (eλ ) + γbc D a (eλ ) + γca D b (eλ ) = 0 γab D reduces to a (eλ ) = κγbc eλ = 0, ∀ b, c = 2, . . . , q γbc D and implies that all constants γbc = 0 vanish. Therefore, the basis elements fa in s can be chosen so that they commute whenever the condition l
λj σja = 0
j=1
l holds for at least one a. On the other hand, if j=1 λj σja = 0 for all a, the Jacobi identity (fa , fb , fc ) is satisfied trivially. Thus, we have proven the following theorem Theorem 12.3. Let g be a complex simple Lie algebra or its split real form, b(g) its (split) Borel subalgebra. Let g be different from sl(3, R). Any solvable extension s of the nilradical NR b(g) by q nonnilpotent elements fa , a = 1, . . . , q ≤ a and a constant q × q antisymrank g is defined by q commuting derivations D a metric matrix γ = (γab ). The derivations D determine the Lie brackets a (eα ), a = 1, . . . , q, α ∈ Δ+ [fa , eα ] = D and take the form a = ad(fa )|n = D
l j=1
j , σja Dj + ωja D
a = 1, . . . , q,
12.2. SOLVABLE EXTENSIONS OF THE BOREL NILRADICALS NR(b(g))
151
where the matrix σ = (σja ), a = 1, . . . , q, j = 1, . . . , l, has the maximal possible rank q. For any given value of k all parameters ωka are equal to zero when the condition (12.20)
l
σja λj − σka (1 + sk ) = 0
j=1
is satisfied for at least one a ∈ {1, . . . , q}. The condition (12.20) is always satisfied for at least q values of the index k, i.e., there are at most l − q values of k such that some of the parameters ωka are nonvanishing. The matrix γ = (γab ) defines the Lie brackets [fa , fb ] = γab eλ , When
l
a, b = 1, . . . , q.
λj σja = 0
j=1
holds for at least one a ∈ {1, . . . , q}, the constants γab are all equal to 0, i.e., [fa , fb ] = 0. Similarly as for Theorem 12.2, Theorem 12.3 was proven above under the assumption that rank g > 2. The derivation is the same when g = G2 . When g = sl(2, C) or g = sl(2, R) the result is trivial. When g = sl(3, C) the result follows from the consideration of Heisenberg nilradicals in Section 11. When g = sl(3, R) the results of Theorem 12.3 do not hold as was already indicated in Section 12.2.2 (for more details, see Section 11.3). In the cases of g = so(5, C) and g = so(3, 2) it follows from the investigation of the nilradical n4,1 in Section 13.1. The conditions in Theorem 12.3 are sufficient, i.e., any set of constants σja , ωja and γab satisfying the properties listed in the theorem gives rise to a solvable extension of the nilradical NR b(g) . On the other hand, the description presented in Theorem 12.3 is not unique, i.e., different choices of σja , ωja and γab may lead to a by any isomorphic algebras. As already noted, we may replace the derivations D linearly independent combination of them thus changing all the parameters σja , ωja and γab . Also we may employ the scaling automorphisms to change the values of ωja and γab . We remark that by virtue of indecomposability of the Borel nilradicals, all solvable Lie algebras described in Theorem 12.3 are indecomposable. 12.2.4. Solvable extensions of dimension nNR +1. The results contained in Theorem 12.3 can be further refined in the particular case q = 1. Any sin 1 of the form (12.14) with nonvanishing vector (σ 1 ) defines an gle derivation D j (nNR + 1)-dimensional solvable extension of the nilradical n. Let us investigate what restrictions we may impose on the parameters ωk1 through a choice of a suitable transformation (9.14). In the transformation (12.16) we choose tk so that ω k1 = 0 whenever the expresl 1 1 sion j=1 σj λj − σk (1 + sk ) is nonvanishing. Otherwise, the parameter ωk1 remains nonvanishing after conjugation by any automorphism from exp(Der(NR b(g) )). We already know that there can be at most l − 1 nonvanishing parameters ωk1 = 0. On the other hand, for any choice of up to l − 1 indices ku , u = 1, . . . , l − 1
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12. SOLVABLE LIE ALGEBRAS WITH BOREL NILRADICALS
we can find constants σj1 such that l
σj1 λj − σk1u (1 + sku ) = 0,
u = 1, . . . , l − 1
j=1
and consequently the parameters ω k1u cannot be set to zero by conjugation by any automorphism from the connected part of the automorphism group Aut(n). That means that there are up to l − 1 nondiagonal nonremovable parameters in the 1 (whose positions may be changed by transformations belonging to derivation D other components of Aut(n) if the root system of g allows them). The nonvanishing parameters ωk1 can be scaled to 1 over the field of complex numbers. Over the field of real numbers, the usual problems with square roots may arise, so in some cases we can only normalize ωk1 to ωk1 = ±1. In order to show this, let us consider the automorphisms φˆj (vj ), vj = 0 defined by l mj ˆ φj (vj )eα = vj eα , α = mj αj ∈ Δ+ j=1
which generalize the inner automorphisms φj (uj ) = exp(uj Dj ) to include also scaling by negative numbers. Under the transformation φˆj (vj ) the constant ωk1 d gets scaled by the factor vj jk where djk = λj − δjk (1 + sk ), λ −δjk (1+sk )
ωk1 → vj j
ωk1 .
Combining l such commuting automorphisms together, ˆ φ(v) = φˆ1 (v1 ) ◦ · · · ◦ φˆl (vl ),
(12.21) we transform
3l ωk1
(12.22)
→
λj j=1 vj vk1+sk
ωk1 .
Let us consider what can be accomplished using (12.22). Let us first deal with all k such that sk = 0 (and ωk1 = 0 by assumption). We set vk = ωk1 . We still have at our disposal at least one constant vi such that si = 0. We can use it to set 3l λj j=1 vj = 1 over the field of complex numbers. Over the field of real numbers 3l 3l λ λ we can set j=1 vj j = 1 when λi is odd and j=1 vj j = ±1 when λi is even. Thus all nonvanishing constants ωk1 such that sk = 0 were brought by a suitable automorphism (12.21) to 1 over the field of complex numbers and to = ±1 over the field of real numbers (where the value of is the same for all k). Assuming that the above described rescaling was already performed, we want to scale ωi1 with si = 0 to a convenient value. We recall that at most l − 1 parameters ωj1 are nonvanishing. Therefore, if ωi1 = 0 we have at least one k, k = i such that ˆ ωk1 = 0. We choose an automorphism φ(v) such that vj = 1 whenever j = i, k,
and vk is chosen so that
3l
vi1+si = ωi1
λj j=1 vj = 1. ±ωi1 when si
Over the field of real numbers we may 3l λ = is odd and j=1 vj j = ±1 when λk is accomplish only ˆ even. The automorphism φ(v) constructed in this way does not affect the values vi1+si
12.3. SOLVABLE LIE ALGEBRAS WITH TRIANGULAR NILRADICALS
153
ωj1 set to 1 in the previous step (or scales them by a common minus factor when 3l λj 1 j=1 vj = −1) and allows us to scale ωi with si = 0 to 1 (±1, respectively). In the case of the algebra Al the procedure is repeated once again for the second index ˆı such that sˆı = 0. To sum up, we have the following theorem. Theorem 12.4. Let g and b(g) satisfy the conditions of Theorem 12.3. Any solvable extension of the nilradical NR b(g) by one nonnilpotent element is up to isomorphism defined by a single derivation (12.23)
= ad(f1 )|n = D
l
j σj Dj + ωj D
j=1
chosen so that the first nonvanishing parameter σj is equal to one. The parameter ωk vanishes whenever lj=1 σj λj − σk (1 + sk ) = 0. At most l − 1 parameters ωk are nonvanishing. They are all equal to 1 over the field of complex numbers. Over the field of real numbers they are equal to ±1 and all parameters ωk with k such that sk = 0 have the same sign. 12.3. Solvable Lie algebras with triangular nilradicals 12.3.1. The structure of the algebras of strictly upper triangular matrices and their derivations. Let us now apply the results introduced above in Sections 12.1, 12.2 to the case of the nilradical of the Borel subalgebra of the Lie algebra sl(l+1) over the fields of complex and real numbers. The algebra sl(l+1, R) is the split real form of sl(l + 1, C); therefore, the results of the previous sections apply in both cases. We shall perform some of the calculations explicitly to illustrate the general arguments of the previous subsections. The Borel subalgebra of the simple matrix Lie algebra sl(l + 1, F) over the field F of complex or real numbers can be naturally identified with the algebra of upper triangular (l + 1) × (l + 1) matrices. A basis for this subalgebra can be chosen in the form ti , eik ∈ F(l+1)×(l+1) , where (12.24)
(ti )aa = δia − δi+1,a ,
1 ≤ i ≤ l,
(eik )ab = δia δkb ,
(ti )ab = 0,
a = b,
1 ≤ i < k ≤ l + 1.
The vectors ti span the Cartan subalgebra of sl(l + 1). Let φi be functionals on g0 = span{ti | i = 1, . . . , l} defined as φi : φi (tj ) = δij − δi,j−1 ,
1 ≤ i, j ≤ l
and φl+1 = −
l
φi .
i=1
The vectors eik are common eigenvectors of ad(tj ) and consequently they are root vectors. The root corresponding to the vector eik is αik = φi − φk .
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12. SOLVABLE LIE ALGEBRAS WITH BOREL NILRADICALS
In the chosen ordering of roots these roots αik , 1 ≤ i < k ≤ l + 1 form the set of positive roots Δ+ . Their negatives αki = −αik correspond to root vectors which are obtained by matrix transposition from eik . The nilradical t(l + 1) of the Borel subalgebra of the Lie algebra sl(l + 1) is spanned by the vectors eik and can be naturally identified with the algebra t(l + 1) of strictly upper triangular (l + 1) × (l + 1) matrices, henceforth called triangular nilradical. Its Lie brackets are [eik , eab ] = δka eib − δbi eak
(12.25) and its dimension is
n ≡ dim t(l + 1) = 12 l(l + 1).
(12.26)
The simple roots, i.e., those positive roots which cannot be written as a sum of two positive roots, are αi ≡ αi,i+1 = φi − φi+1 ,
(12.27)
1 ≤ i ≤ l.
The basis vectors ti of the Cartan subalgebra were chosen so that they coincide with the vectors hαi of (2.34) up to a constant multiple, hαi = 2(l + 1)ti . Any positive root αik is expressed in terms of the simple roots as (12.28)
αik =
k−1
αj .
j=i
The highest root is λ = α1,l+1 =
(12.29)
l
αj .
j=1
There are only two simple roots α such that λ − α is again a root, namely α = α1 and α = αl . Consequently, we have Sα1 (λ) =
l
αj = α2,l+1 ,
j=2
Sαi (λ) = λ,
Sαl (λ) =
l−1
αj = α1,l ,
j=1
2 ≤ i ≤ l − 1.
From the knowledge of the root system of sl(l+1) we can immediately construct a basis of outer derivations of the nilpotent algebra t(l + 1) using the results of [67] recalled earlier (cf. (12.6)). There are 2l linearly independent outer derivations of i , 1 ≤ i ≤ l: t(l + 1) which can be chosen as Di , D Di (ejk ) = ejk when j ≤ i ≤ k − 1, Di (ejk ) = 0 otherwise, 1 (e12 ) = e2,l+1 , D 1 (eik ) = 0 otherwise, D Dl (el,l+1 ) = e1,l , Dl (eik ) = 0 otherwise, i (ej,j+1 ) = δij e1,l+1 and D i (ejk ) = 0, when 2 ≤ i ≤ l − 1 we have D 1 ≤ j < k − 1 ≤ l. i are nilpotent We observe that the derivations Di are nonnilpotent whereas D i together with the inner derivations span the nilradical of Der t(l+1) . operators. D Let us denote by Out t(l + 1) the subalgebra of Der t(l + 1) spanned by the outer i. derivations Di , D • • • •
12.3. SOLVABLE LIE ALGEBRAS WITH TRIANGULAR NILRADICALS
155
The main results on solvable Lie algebras with triangular nilradicals were obtained in [123]. They were generalized to any Borel nilradical in [118] and were reproduced in Sections 12.1 and 12.2 above. For the sake of completeness we present the original result of [123] here as a proposition. Proposition 12.5. Every solvable Lie algebra s with a triangular nilradical t(l + 1) can be transformed to a canonical basis (eik , fα ), α = 1, . . . , f , 1 ≤ i < k ≤ l + 1, with commutation relations (12.25) and [fα , eik ] = (12.30) dα ik,pq epq p with(DifferentialGeometry): with(LieAlgebras): with(LinearAlgebra): (we have suppressed the output by using colon instead of semicolon to end commands). Next, we build the structure of a real Lie algebra Alg by introducing its Lie brackets, basis vectors and the name for the algebra. > ALG := LieAlgebraData([[e1, e2] = 4*e2-2*e5, [e1, e3] = -4*e3-2*e6+2*e8, [e1, e5] = 2*e5, [e1, e6] = -e6+e8, [e1, e8] = e6-e8, [e2, e3] = e1-4*e4-2*e5-e6+4*e7+e8, [e2, e6] = e4+e5-e7, [e2, e8] = -e4-e5+e7, [e3, e5] = 2*e4+e6-2*e7-e8, [e4, e6] = e7, [e4, e8] = e7, [e5, e6] = e4-e7, [e5, e8] = -e4+e7, [e6, e7] = -e7, [e7, e8] = e7], [e1, e2, e3, e4, e5, e6, e7, e8], Alg); ALG := [[e1 , e2 ] = 4 e2 − 2 e5 , [e1 , e3 ] = −4 e3 − 2 e6 + 2 e8 , [e1 , e5 ] = 2 e5 , [e1 , e6 ] = −e6 + e8 , [e1 , e8 ] = e6 − e8 , [e2 , e3 ] = e1 − 4 e4 − 2 e5 − e6 + 4 e7 + e8 , [e2 , e6 ] = e4 + e5 − e7 , [e2 , e8 ] = −e4 − e5 + e7 , [e3 , e5 ] = 2 e4 + e6 − 2 e7 − e8 , [e4 , e6 ] = e7 , [e4 , e8 ] = e7 , [e5 , e6 ] = e4 − e7 , [e5 , e8 ] = −e4 + e7 , [e6 , e7 ] = −e7 , [e7 , e8 ] = e7 ] We declare the algebra to Maple as an object to be used by the DifferentialGeometry package and verify that it satisfies Jacobi identities. > DGsetup(ALG, verbose); The following vector fields have been defined and protected: [e1 , e2 , e3 , e4 , e5 , e6 , e7 , e8 ] The following differential 1-forms have been defined and protected: [θ1 , θ2 , θ3 , θ4 , θ5 , θ6 , θ7 , θ8 ] Lie algebra: Alg Alg> Query("Jacobi"); true Next, we check whether the algebra is indecomposable and if not we decompose it into a direct sum. Alg> Query("Indecomposable"); false Alg> Decompose(factoralgebras = true);
15.2. COMPUTER-ASSISTED IDENTIFICATION OF A GIVEN LIE ALGEBRA
⎡⎡
221
⎤ 0 0 0 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ , [e1 , e2 , e3 , e4 − e7 , e5 , e6 − e8 , e6 + e8 , e7 ], 1 1⎥ 0 0 0 0 0 − ⎥ 2 2⎥ 1 ⎥ 1 ⎥ 0 0 0 0 0 ⎦ 2 2 0 0 0 1 0 0 1 0 [[[e1 , e2 ] = 4 e2 − 2 e5 , [e1 , e3 ] = −4 e3 − 2 e6 , [e1 , e5 ] = 2 e5 , [e1 , e6 ] = −2 e6 , [e2 , e3 ] = e1 − 4 e4 − 2 e5 − e6 , [e2 , e6 ] = 2 e4 + 2 e5 , [e3 , e5 ] = 2 e4 + e6 , [e5 , e6 ] = 2 e4 ], [[e1 , e2 ] = −2 e2 ]]] From the last part of the output we find that the algebra Alg decomposes into the direct sum of a 6-dimensional ideal spanned by the vectors e1 , e2 , e3 , e4 − e7 , e5 , e6 − e8 and a 2-dimensional ideal spanned by e6 + e8 and e7 . Before proceeding we check that these are indeed ideals. Alg> Query([e1, e2, e3, e4-e7, e5, e6-e8], "Ideal"), Query([e6+e8, e7], "Ideal"); true, true We declare these two subalgebras as new objects in Maple and investigate their properties, e.g., their radicals and Levi decompositions. We denote the basis vectors in SubAlg1 by f1 , . . . , f6 and the basis of SubAlg2 by h1 , h2 . Alg> DGsetup(LieAlgebraData([e1,e2,e3,e4-e7,e5,e6-e8], SubAlg1), [seq(f || k, k = 1..6)], [alpha]); Lie algebra: SubAlg1 SubAlg1> ChangeFrame(Alg): DGsetup(LieAlgebraData([e6+e8, e7], SubAlg2), [h1, h2], [beta]); Lie algebra: SubAlg2 SubAlg2> ChangeFrame(SubAlg1); Radical(); 1 ⎢⎢0 ⎢⎢ ⎢⎢0 ⎢⎢ ⎢⎢0 ⎢⎢ ⎢⎢0 ⎢⎢ ⎢⎢ ⎢⎢0 ⎢⎢ ⎢⎢ ⎢⎢ ⎣⎣0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
[f6 , f5 , f4 ] SubAlg1> LeviDecomposition(); 1 f4 , f3 − f4 ]] 2 We have found that the radical of SubAlg1 is spanned by f4 , f5 and f6 whereas the Levi factor can be chosen as span of f1 − 4f4 − 2f5 − f6 , f2 + 12 f4 and f3 − f4 . To find to which 3–dimensional real simple algebra is the Levi factor isomorphic, we calculate its Killing form and find its signature. SubAlg1> K := Killing([f1-4*f4-2*f5-f6, f2+(1/2)*f4, f3-f4]); ⎤ ⎡ 40 0 0 K := ⎣ 0 0 10⎦ 0 10 0 SubAlg1> convert(map(sign, Eigenvalues(K)), list); [[f4 , f5 , f6 ], [f1 − 4 f4 − 2 f5 − f6 , f2 +
[−1, 1, 1] The Killing form is indefinite, so the algebra is noncompact. Hence the Levi factor is isomorphic to the algebra sl(2, R).
222
15. STRUCTURE OF THE LISTS OF LOW-DIMENSIONAL LIE ALGEBRAS
In order to identify the radical of SubAlg1 we find the dimensions of its characteristic series. SubAlg1> US = map(nops, Series([f4, f5, f6], "Upper")); CS = map(nops, Series([f4, f5, f6], "Lower")); DS = map(nops, Series([f4, f5, f6], "Derived")); US = [1, 3] CS = [3, 1, 0] DS = [3, 1, 0] Thus the radical of the 6-dimensional ideal SubAlg1 is nilpotent and isomorphic to the Heisenberg algebra n3,1 . Together with the known information about the Levi factor we identify the subalgebra SubAlg1 as sl(2) n3,1 . The second ideal SubAlg2 is solvable and nonnilpotent, as is easily found using SubAlg1> ChangeFrame(SubAlg2); Radical(); Nilradical(); [h2 , h1 ] [h2 ] i.e., it is isomorphic to s2,1 . To sum up, we have found that the given Lie algebra Alg has the structure (sl(2) n3,1 ) ⊕ s2,1 .
(15.1)
An explicit change of basis performing the decomposition (15.1) is obtained via a composition of the changes of bases found above: SubAlg2> newbas := subs({f1 = e1, f2 = e2, f3 = e3, f4 = e4-e7, f5 = e5, f6 = e6-e8, h1 = e6+e8, h2 = e7}, [f1-4*f4-2*f5-f6, f2+(1/2)*f4, f3-f4, f4, f5, f6, h1, h2]); 1 1 newbas := [e1 − 4 e4 − 2 e5 − e6 + 4 e7 + e8 , e2 + e4 − e7 , e3 − e4 + e7 , 2 2 e4 − e7 , e5 , e6 − e8 , e6 + e8 , e7 ] SubAlg2> ChangeFrame(Alg); DGsetup(LieAlgebraData(newbas, DecomposedAlg), [s1,s2,s3,r1,r2,r3,q1,q2],[omega], verbose); The following vector fields have been defined and protected: [s1 , s2 , s3 , r1 , r2 , r3 , q1 , q2 ] The following differential 1-forms have been defined and protected: [ω1 , ω2 , ω3 , ω4 , ω5 , ω6 , ω7 , ω8 ] Lie algebra: DecomposedAlg The structure of the Lie algebra Alg in this new basis is given by the following table of Lie brackets: DecomposedAlg> MultiplicationTable("LieTable"); ⎡ ⎤ | s1 s2 s3 r1 r2 r3 q1 q2 −−
⎢ s1 ⎢ s2 ⎢ ⎢ s3 ⎢ r1 ⎢ r2 ⎢ ⎣ r3
q1 q2
−− −− | 0 | −4 s2 | 4 s3 | 0 | −2 r1 − 2 r2 | 4 r1 + 2 r3 | 0 | 0
−− 4 s2 0 −s1 0 0 −2 r1 − 2 r2 0 0
−− −4 s3 s1 0 0 −2 r1 − r3 0 0 0
−− −− 0 2 r1 + 2 r2 0 0 0 2r1 + r3 0 0 0 0 0 −2 r1 0 0 0 0
−− −4 r1 − 2 r3 2 r1 + 2 r2 0 0 2 r1 0 0 0
−− 0 0 0 0 0 0 0 2 q2
−− 0 ⎥ ⎥ 0 ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎦ −2 q2 0
15.2. COMPUTER-ASSISTED IDENTIFICATION OF A GIVEN LIE ALGEBRA
223
which makes the decomposition (15.1) into (15.2) sl(2, R) = span{s1 , s2 , s3 },
n3,1 = span{r1 , r2 , r3 },
s2,1 = span{q1 , q2 }
clearly visible. For more examples and tutorials the reader may consult the website http: //digitalcommons.usu.edu/dg/.
CHAPTER 16
Lie Algebras up to Dimension 3 16.1. One-dimensional Lie algebra • n1,1 US = [1] e1
CS = [1, 0]
e1 0
DS = [1, 0] dim Der = 1
Casimir invariant: e1
16.2. Solvable two-dimensional Lie algebra with the nilradical n1,1 e1 0
e1 • s2,1
US = [0] e2
e1 e1
CS = [2, 1] DS = [2, 1, 0]
Casimir invariant: none
16.3. Nilpotent three-dimensional Lie algebra • n3,1 US = [1, 3] e1 e2
e1 0
e2 0 0
e3 0 e1
CS = [3, 1, 0] DS = [3, 1, 0] dim Der = 6
Casimir invariant: e1 225
226
16. LIE ALGEBRAS UP TO DIMENSION 3
16.4. Solvable three-dimensional Lie algebras with the nilradical 2n1,1
e1
• s3,1
e1 0
e2 0 US = [0]
e3
e1 e1
e2 ae2
CS = [3, 2]
DS = [3, 2, 0] where the values of the parameter a are over the field C: 0 < |a| ≤ 1, if |a| = 1 then arg(a) ≤ π; over the field R: 0 < |a| ≤ 1. Casimir invariant: ea1 e2 • s3,2 US = [0] e1 e2 CS = [3, 2] e3 e1 e1 + e2 DS = [3, 2, 0] Casimir invariant: e2 e1 exp − e1 • s3,3 (over the field C: isomorphic to s3,1 ) US = [0] e3 where α ≥ 0. Casimir invariant:
e1 αe1 − e2
e2 e1 + αe2
CS = [3, 2] DS = [3, 2, 0]
e1 (e21 + e22 ) exp 2α arctan e2
16.5. Simple three-dimensional Lie algebras • sl(2, F) e1 e2
e1 0
e2 2e1 0
e3 −e2 2e3
US = [0] CS = [3] DS = [3]
Casimir invariant: 4e1 e3 + e22 • so(3, R) (over the field C: isomorphic to sl(2, C)) e1 e2
e1 0
e2 e3 0
e3 −e2 e1
Casimir invariant: e21 + e22 + e23
US = [0] CS = [3] DS = [3]
CHAPTER 17
Four-Dimensional Lie Algebras 17.1. Nilpotent four-dimensional Lie algebra • n4,1 e1 0
e1 e2 e3
e2 0 0
e3 0 0 0
US = [1, 2, 4]
e4 0 e1 e2
CS = [4, 2, 1, 0] DS = [4, 2, 0] dim Der = 7
Casimir invariants: 2e1 e3 − e22
e1 ,
17.2. Solvable four-dimensional algebras with the nilradical 3n1,1
e1 e2
e1 0
e2 0 0
e3 0 0
• s4,1 US = [1, 2] e4
e1 0
e2 e1
Casimir invariants: e1 , • s4,2
e3 e3
CS = [4, 2, 1] DS = [4, 2, 0]
e2 e3 exp − e1 US = [0]
e4
e1 e1
Casimir invariants:
e2 e1 + e2
e3 e2 + e3
DS = [4, 3, 0]
e2 e1 exp − e1
2e1 e3 − e22 , e21
• s4,3
CS = [4, 3]
US = [0] e4
e1 e1
e2 ae2
e3 be3
CS = [4, 3] DS = [4, 3, 0]
where the values of the parameters a, b are 0 < |b| ≤ |a| ≤ 1, 227
(a, b) = (−1, −1)
228
17. FOUR-DIMENSIONAL LIE ALGEBRAS
If one or both equalities hold further restrictions on the parameters are necessary in order to avoid redundancies, as discussed in Section 10.4. Casimir invariants: ea1 eb1 , e2 e3 • s4,4 US = [0] e1 e2 e3 CS = [4, 3] e4 e1 e1 + e2 ae3 DS = [4, 3, 0] where a = 0. Casimir invariants:
e2 e1 exp − e1
ea1 , e3
• s4,5 (over the field C: isomorphic to s4,3 ) US = [0] e4
e1 αe1
e2 βe2 − e3
where α > 0. Casimir invariants: (e22 + e23 )α e2β 1
,
e3 e2 + βe3
CS = [4, 3] DS = [4, 3, 0]
e2 e1 exp α arctan e3
17.3. Solvable four-dimensional Lie algebras with the nilradical n3,1
e1 e2
e1 0
e2 0 0
e3 0 e1
• s4,6 US = [1] e4
e1 0
e2 e2
e3 −e3
CS = [4, 3] DS = [4, 3, 1, 0]
Casimir invariants: e1 ,
e2 e3 + e1 e4
• s4,7 (over the field C: isomorphic to s4,6 ) US = [1] e4
e1 0
e2 −e3
e3 e2
CS = [4, 3] DS = [4, 3, 1, 0]
Casimir invariants: e1 ,
e22 + e23 − 2e1 e4
17.4. SOLVABLE 4-DIMENSIONAL LIE ALGEBRAS WITH THE NILRADICAL 2n1,1
229
• s4,8 US = [0] e4
e1 (1 + a)e1
e2 e2
e3 ae3
CS = [4, 3] DS = [4, 3, 1, 0]
where the values of the parameter a are over the field C: 0 < |a| ≤ 1, if |a| = 1 then arg(a) < π; over the field R: −1 < a ≤ 1, a = 0. Casimir invariant: none • s4,9 (over the field C: isomorphic to s4,8 ) US = [0] e4
e1 2αe1
e2 αe2 − e3
e3 e2 + αe3
CS = [4, 3] DS = [4, 3, 1, 0]
where α > 0. Casimir invariant: none • s4,10 US = [0] e4
e1 2e1
e2 e2
e3 e2 + e3
CS = [4, 3] DS = [4, 3, 1, 0]
Casimir invariant: none • s4,11 US = [0] e4
e1 e1
e2 e2
e3 0
CS = [4, 2] DS = [4, 2, 0]
Casimir invariant: none 17.4. Solvable four-dimensional Lie algebras with the nilradical 2n1,1
• s4,12
e1 e2 e1 0 0 (over the field C: decomposable) e3 e4
e1 e1 −e2
e2 e2 e1
e3 0 0
Casimir invariant: none
US = [0] CS = [4, 2] DS = [4, 2, 0]
CHAPTER 18
Five-Dimensional Lie Algebras 18.1. Nilpotent five-dimensional Lie algebras • n5,1 e1 e2 e3 e4
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e2
US = [2, 5] CS = [5, 2, 0] DS = [5, 2, 0] dim Der = 13
Casimir invariants: e1 ,
e2 e3 − e1 e4
e2 ,
• n5,2 e1 e2 e3 e4
e1 0
e2 0 0
e3 0 0 0
e4 0 0 e2 0
e5 0 0 e1 e3
US = [2, 3, 5] CS = [5, 3, 2, 0] DS = [5, 3, 0] dim Der = 10
Casimir invariants: e1 ,
e2 ,
e23 + 2e2 e5 − 2e1 e4
• n5,3 e1 e2 e3 e4
e1 0
e2 0 0
e3 0 0 0
e4 0 e1 0 0
e5 0 0 e1 0
US = [1, 5] CS = [5, 1, 0] DS = [5, 1, 0] dim Der = 15
Casimir invariants: e1 • n5,4 e1 e2 e3 e4
e1 0
e2 0 0
e3 0 0 0
e4 0 0 e1 0
e5 0 e1 0 e2
Casimir invariants: e1 231
US = [1, 3, 5] CS = [5, 2, 1, 0] DS = [5, 2, 0] dim Der = 10
232
18. FIVE-DIMENSIONAL LIE ALGEBRAS
• n5,5 e1 0
e1 e2 e3 e4
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 e1 e2 e3
US = [1, 2, 3, 5] CS = [5, 3, 2, 1, 0] DS = [5, 3, 0] dim Der = 9
Casimir invariants: e1 ,
2e1 e3 − e22 ,
e32 + 3e4 e21 − 3e2 e3 e1
e2 0 0
e5 0 e1 e2 e3
• n5,6 e1 0
e1 e2 e3 e4
e3 0 0 0
e4 0 0 e1 0
US = [1, 2, 3, 5] CS = [5, 3, 2, 1, 0] DS = [5, 3, 0] dim Der = 8
Casimir invariants: e1 18.2. Solvable five-dimensional Lie algebras with the nilradical 4n1,1 e1 0
e1 e2 e3
• s5,1
e2 0 0
e3 0 0 0
e4 0 0 0 US = [1, 2, 3]
e5
e1 0
e2 e1
e1 ,
e22
e3 e2
e4 e4
Casimir invariants: − 2e1 e3 ,
CS = [5, 3, 2, 1] DS = [5, 3, 0] e2 e4 exp − e1
• s5,2 US = [1, 2] e5
e1 0
e2 e1
e3 e3
e4 e3 + e4
Casimir invariants: e1 ,
e1 e4 − e2 e3 , e3
CS = [5, 3, 2] DS = [5, 3, 0] e2 e3 exp − e1
• s5,3 US = [1, 2] e5
e1 0
e2 e1
e3 e3
e4 ae4
where 0 < |a| ≤ 1. If |a| = 1 then arg(a) ≤ π.
CS = [5, 3, 2] DS = [5, 3, 0]
18.2. SOLVABLE ALGEBRAS WITH THE NILRADICAL 4n1,1
Casimir invariants:
• s5,4
ea3 e2 e1 , , e3 exp − e4 e1 (over the field C: isomorphic to s5,3 ) US = [1, 2] e5
e1 0
where 0 ≤ α. Casimir invariants:
e2 e1
e3 αe3 − e4
e4 e3 + αe4
e2 , (e23 + e24 ) exp −2α e1
e1 ,
CS = [5, 3, 2] DS = [5, 3, 0]
e2 e3 + arctan e1 e4
• s5,5 US = [0] e5
e1 e1
e2 e1 + e2
Casimir invariants: 2e1 e3 − e22 , e21 • s5,6
e3 e2 + e3
e4 e3 + e4
3e21 e4 − 3e1 e2 e3 + e32 , e31
CS = [5, 4] DS = [5, 4, 0] e2 e1 exp − e1 US = [0]
e5
e1 e1
e2 e1 + e2
e3 ae3
e4 e3 + ae4
CS = [5, 4] DS = [5, 4, 0]
where 0 < |a| ≤ 1. If |a| = 1 then arg(a) ≤ π. Casimir invariants: e1 e4 − e2 e3 ea1 e2 , , e1 exp − e1 e3 e3 e1 • s5,7 US = [0] e1 e1
e5
e2 e1 + e2
e3 e2 + e3
where a = 0. Casimir invariants: 2e1 e3 − e22 , e21
ea1 , e4
e4 ae4
CS = [5, 4] DS = [5, 4, 0]
e2 e1 exp − e1
• s5,8 (over the field C: isomorphic to s5,6 ) US = [0] e5
e1 αe1 − e2
where 0 ≤ α.
e2 e1 + αe2
e3 e1 + αe3 − e4
e4 e2 + e3 + αe4
CS = [5, 4] DS = [5, 4, 0]
233
234
18. FIVE-DIMENSIONAL LIE ALGEBRAS
Casimir invariants: e1 e4 − e2 e3 e1 e3 + e2 e4 2 2 , , e1 + e2 exp −2α e21 + e22 e21 + e22
e1 e3 + e4 e2 e1 + arctan 2 2 e1 + e2 e2
• s5,9 US = [0] e1 e1
e5
e2 ae2
e3 be3
e4 ce4
CS = [5, 4] DS = [5, 4, 0]
where the values of the parameters a, b, c are 0 < |c| ≤ |b| ≤ |a| ≤ 1. If one or more equalities hold further restrictions on the parameters are necessary in order to avoid redundancies, as discussed in Section 10.4. Casimir invariants: ea1 eb1 ec1 , , e2 e3 e4 • s5,10 US = [0] e1 e1
e5
e2 e1 + e2
e3 ae3
e4 be4
CS = [5, 4] DS = [5, 4, 0]
where 0 < |b| ≤ |a|, if |b| = |a| then arg(a) ≤ arg(b). Casimir invariants: ea1 eb1 e2 , , e1 exp − e3 e4 e1 • s5,11 (over the field C: isomorphic to s5,9 ) US = [0] e1 αe1
e5
e2 βe2
e3 γe3 − e4
where α > 0, β = 0, α ≥ β ≥ −α. Casimir invariants: eβ1 , eα 2
(e23 + e24 )β , e2γ 2
e4 e3 + γe4
CS = [5, 4] DS = [5, 4, 0]
e3 e1 exp α arctan e4
• s5,12 (over the field C: isomorphic to s5,10 ) US = [0] e5
e1 e1
e2 e1 + e2
where β > 0. Casimir invariants: e23 + e24 , e2α 1
e3 αe3 − βe4
e4 βe3 + αe4
e2 , e1 exp − e1
CS = [5, 4] DS = [5, 4, 0]
e2 e3 β + arctan e1 e4
18.3. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ n1,1
235
• s5,13 (over the field C: isomorphic to s5,9 ) US = [0] e5
e1 αe1 − e2
e2 e1 + αe2
e3 βe3 − γe4
e4 γe3 + βe4
CS = [5, 4] DS = [5, 4, 0]
where 0 < γ ≤ 1, 0 ≤ α. If α = 0 then 0 ≤ β. If γ = 1 then |α| ≤ |β|. Casimir invariants: (e21 + e22 )β e3 e1 e1 2 2 . , arctan − γ arctan , (e1 + e2 ) exp 2α arctan (e23 + e24 )α e4 e2 e2 When α → 0 the first and third invariant become dependent and one of them should be replaced by e1 2 2 (e3 + e4 ) exp 2β arctan . e2 18.3. Solvable five-dimensional Lie algebras with the nilradical n3,1 ⊕ n1,1
e1 e2 e3
e1 0
e2 0 0
e3 0 e1 0
e4 0 0 0
• s5,14 US = [1, 2, 3] e5
e1 0
e2 0
e3 e2
e4 e4
CS = [5, 3, 2, 1] DS = [5, 3, 0]
Casimir invariant: e1 • s5,15 US = [1, 2] e5
e1 0
e2 e2
e3 −e3
e4 e1
CS = [5, 3] DS = [5, 3, 1, 0]
Casimir invariant: e1 • s5,16 (over the field C: isomorphic to s5,15 ) US = [1, 2] e5
e1 0
e2 −e3
e3 e2
e4 e1
Casimir invariant: e1
CS = [5, 3] DS = [5, 3, 1, 0]
236
18. FIVE-DIMENSIONAL LIE ALGEBRAS
• s5,17 US = [1] e1 0
e5
e2 e2
e3 −e3
e4 ae4
CS = [5, 4] DS = [5, 4, 1, 0]
where the values of the parameter a are over the field C: 0 ≤ Re(a), if Re(a) = 0 then 0 < Im(a); over the field C: 0 < a. Casimir invariant: e1 • s5,18 US = [1] e1 0
e5
e2 −e2
e3 e3 + e4
e4 e4
CS = [5, 4] DS = [5, 4, 1, 0]
Casimir invariant: e1 • s5,19 (over the field C: isomorphic to s5,17 ) US = [1] e1 0
e5
e2 −e3
e3 e2
e4 αe4
CS = [5, 4] DS = [5, 4, 1, 0]
where α > 0. Casimir invariant: e1 • s5,20 US = [1] e5
e1 e1
e2 e2
e3 e4
e4 0
CS = [5, 3, 2] DS = [5, 3, 0]
Casimir invariant: e4 • s5,21 US = [0] e5
e1 2e1
e2 e2 + e3
e3 e3 + e4
Casimir invariant:
e4 e4
CS = [5, 4] DS = [5, 4, 1, 0]
e24 e1
• s5,22
US = [0] e5
e1 (a + 1)e1
e2 e2
e3 ae3
e4 be4
CS = [5, 4] DS = [5, 4, 1, 0]
where the values of the parameters a, b are over the field C: 0 < |a| ≤ 1, b = 0, if |a| = 1 then arg(a) < π;
18.3. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ n1,1
over the field R: −1 < a ≤ 1, a, b = 0. Casimir invariant: ea+1 4 eb1 • s5,23 US = [0] e5
e1 (a + 1)e1
e2 ae2
e3 e3 + e4
where a = 0, −1. Casimir invariant:
e4 e4
CS = [5, 4] DS = [5, 4, 1, 0]
ea+1 4 e1
• s5,24
US = [0] e5
e1 2e1
e2 e2 + e3
e3 e3
e4 ae4
CS = [5, 4] DS = [5, 4, 1, 0]
where a = 0. Casimir invariant: • s5,25
ea1 e24 (over the field C: isomorphic to s5,22 ) US = [0] e1 2αe1
e5
e2 αe2 − e3
e3 e2 + αe3
where α = 0, β > 0. Casimir invariant:
e4 βe4
CS = [5, 4] DS = [5, 4, 1, 0]
e2α 4 eβ1
• s5,26
US = [0] e5
e1 (a + 1)e1
e2 e2
e3 ae3
e4 e1 + (a + 1)e4
CS = [5, 4] DS = [5, 4, 1, 0]
where the values of the parameter a are over the field C: 0 < |a| ≤ 1, if |a| = 1 then arg(a) < π; over the field R: −1 < a ≤ 1, a = 0. Casimir invariant: e4 e1 exp −(a + 1) e1 • s5,27 US = [0] e5
e1 2e1
e2 e2 + e3
e3 e3
e4 e1 + 2e4
CS = [5, 4] DS = [5, 4, 1, 0]
237
238
18. FIVE-DIMENSIONAL LIE ALGEBRAS
Casimir invariant:
e4 e1 exp −2 e1
• s5,28 (over the field C: isomorphic to s5,26 ) US = [0] e5
e1 2αe1
e2 αe2 + e3
e3 −e2 + αe3
where α > 0. Casimir invariant:
e4 e1 + 2αe4
CS = [5, 4] DS = [5, 4, 1, 0]
e4 e1 exp −2α e1
• s5,29 US = [0] e5
e1 e1
e2 0
e3 e3 + e4
Casimir invariant:
e4 e4
CS = [5, 3] DS = [5, 3, 0]
e4 e1
• s5,30
US = [0] e5
e1 e1
e2 e2
e3 0
e4 ae4
where a = 0. Casimir invariant:
CS = [5, 3] DS = [5, 3, 0]
ea1 e4
• s5,31
US = [0] e5
e1 e1
Casimir invariant:
e2 e2
e3 0
e4 e1 + e4
CS = [5, 3] DS = [5, 3, 0]
e4 e1 exp − e1
• s5,32 US = [0] e5 Casimir invariant:
e1 e1
e2 0
e3 e3 + e4
e4 e1 + e4
e4 e1 exp − e1
CS = [5, 3] DS = [5, 3, 0]
18.4. SOLVABLE ALGEBRAS WITH THE NILRADICAL n4,1
239
18.4. Solvable five-dimensional Lie algebras with the nilradical n4,1 e1 0
e1 e2 e3
e2 0 0
e3 0 0 0
e4 0 e1 e2
• s5,33 US = [1] e1 0
e5
e2 −e2
e3 −2e3
e4 e4
CS = [5, 4] DS = [5, 4, 2, 0]
Casimir invariant e1 • s5,34 US = [0] e5
e1 3e1
e2 2e2
e3 e3
Casimir invariant:
e4 e3 + e4
CS = [5, 4] DS = [5, 4, 2, 0]
(2e1 e3 − e22 )3 e41
• s5,35
US = [0] e5
e1 (a + 2)e1
e2 (a + 1)e2
where a = 0, −2. Casimir invariant:
e3 ae3
e4 e4
CS = [5, 4] DS = [5, 4, 2, 0]
(2e1 e3 − e22 )2+a 2(1+a)
e1
• s5,36
US = [0] e5
e1 2e1
e2 e2
Casimir invariant:
e3 0
e4 e4
CS = [5, 3] DS = [5, 3, 1, 0]
2e1 e3 − e22 e1
• s5,37
US = [0] e5 Casimir invariant:
e1 e1
e2 e2
e3 e3
e4 0
2e1 e3 − e22 e21
CS = [5, 3] DS = [5, 3, 0]
240
18. FIVE-DIMENSIONAL LIE ALGEBRAS
• s5,38 US = [0] e1 e1
e5
e2 e2
e3 e1 + e3
e4 0
CS = [5, 3] DS = [5, 3, 0]
where the value of the parameter is over the field C: = 1; over the field R: = ±1. Casimir invariant: 2e1 e3 − e22 e−2 exp 1 e21 18.5. Solvable five dimensional Lie algebras with the nilradical 3n1,1 e1 0
e1 e2
• s5,39 e4 e5
e1 0 0
e2 e2 0
e3 0 e3
e2 0 0 e4 0 e1
e3 0 0 US = [1] CS = [5, 3, 2] DS = [5, 3, 0]
Casimir invariant: • s5,40
e1 (over the field C: isomorphic to s5,39 ) e4 e5
e1 0 0
e2 e2 −e3
e3 e3 e2
e4 0 e1
US = [1] CS = [5, 3, 2] DS = [5, 3, 0]
Casimir invariant: e1 • s5,41 e4 e5
e1 e1 0
e2 0 e2
e3 ae3 be3
e4 0 0
US = [0] CS = [5, 3] DS = [5, 3, 0]
where the values of the parameters a, b are 0 < |b| ≤ |a| ≤ 1. If one or both equalities hold further restrictions on the parameters are necessary in order to avoid redundancies, as discussed in Section 10.4. Casimir invariant: ea1 eb2 e3 • s5,42 US = [0] e e e e 1
e4 e5
ae1 e1
2
3
4
e2 0
e3 e2
0 0
CS = [5, 3] DS = [5, 3, 0]
18.7. LEVI DECOMPOSABLE ALGEBRA
241
ea2 e3 exp e1 e2 (over the field C: isomorphic to s5,41 )
Casimir invariant: • s5,43
e4 e5 where α + β = 0. Casimir invariant: 2
e1 αe1 βe1
e2 e2 −e3
e3 e3 e2
US = [0]
e4 0 0
CS = [5, 3] DS = [5, 3, 0]
2
e21 e2 exp 2β arctan (e22 + e23 )α e3
18.6. Solvable five-dimensional Lie algebras with the nilradical n3,1
e1 e2
• s5,44 e4 e5
e1 e1 0
e2 e2 e2
e1 0
e3 0 −e3
e2 0 0
e4 0 0
e3 0 e1 US = [0] CS = [5, 3] DS = [5, 3, 1, 0]
Casimir invariant: • s5,45
e1 e5 + e2 e3 e1 (over the field C: isomorphic to s5,44 ) e4 e5
e1 2e1 0
e2 e2 e3
Casimir invariant:
e3 e3 −e2
e4 0 0
US = [0] CS = [5, 3] DS = [5, 3, 1, 0]
2e1 e5 + e22 + e23 e1
18.7. Five-dimensional Levi decomposable Lie algebra • sl(2, F) 2n1,1 e1 e2 e3 e4
e1 0
e2 2e1 0
e3 −e2 2e3 0
e4 e5 e4 0 0
e5 0 −e5 e4 0
Casimir invariant: e1 e24 − e2 e4 e5 − e3 e25
US = [0] CS = [5] DS = [5]
CHAPTER 19
Six-Dimensional Lie Algebras 19.1. Nilpotent six-dimensional Lie algebras • n6,1
e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 0 e2 0
e6 0 0 0 e3 e1
e2 ,
e3 ,
e1 e4 + e2 e6 − e3 e5
e3 0 0 0
e4 0 0 0 0
US = [3, 6] CS = [6, 3, 0] DS = [6, 3, 0] dim Der = 18
Casimir invariants: e1 , • n6,2
e1 e2 e3 e4 e5
e1 0
e2 0 0
e5 0 0 0 −e1 0
e6 0 0 e1 e2 0
US = [2, 6] CS = [6, 2, 0] DS = [6, 2, 0] dim Der = 17
Casimir invariants: e1 ,
e2
• n6,3 (over the field R only, decomposable over the field C)
e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e2 0
e6 0 0 −e2 e1 0
Casimir invariants: e1 , 243
e2
US = [2, 6] CS = [6, 2, 0] DS = [6, 2, 0] dim Der = 16
244
19. SIX-DIMENSIONAL LIE ALGEBRAS
• n6,4 e1 0
e1 e2 e3 e4 e5
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 0 0
e6 0 0 e2 e1 e3
US = [2, 4, 6] CS = [6, 3, 2, 0] DS = [6, 3, 0] dim Der = 13
Casimir invariants: e1 ,
e2
• n6,5 e1 0
e1 e2 e3 e4 e5
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e2 e2 0
e6 0 0 e1 0 e3
US = [2, 4, 6] CS = [6, 3, 2, 0] DS = [6, 3, 0] dim Der = 12
Casimir invariants: • n6,6
e1 , e2 (over the field C: isomorphic to n6,5 ) e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e2 0
e6 0 0 −e2 e1 e3
e1 ,
e2
US = [2, 4, 6] CS = [6, 3, 2, 0] DS = [6, 3, 0] dim Der = 12
Casimir invariants: • n6,7 e1 0
e1 e2 e3 e4 e5
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 0 0 0
e6 0 0 e1 e2 e3
US = [2, 4, 6] CS = [6, 3, 1, 0] DS = [6, 3, 0] dim Der = 15
Casimir invariants: e1 ,
e2 ,
e1 e4 − e2 e3 ,
e3 0 0 0
e4 0 0 0 0
2e1 e5 − e23
• n6,8 e1 e2 e3 e4 e5
e1 0
e2 0 0
e5 0 0 0 −e1 0
e6 0 0 e1 e2 e3
US = [2, 4, 6] CS = [6, 3, 1, 0] DS = [6, 3, 0] dim Der = 14
19.1. NILPOTENT SIX-DIMENSIONAL LIE ALGEBRAS
Casimir invariants: e1 ,
e2
• n6,9 e1 0
e1 e2 e3 e4 e5
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 0 e2 0
e6 0 0 e1 0 e3
US = [2, 4, 6] CS = [6, 3, 1, 0] DS = [6, 3, 0] dim Der = 13
Casimir invariants: e1 ,
e2
e5 0 0 0 e2 0
e6 0 0 e1 e3 e4
e1 ,
e2
• n6,10 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
US = [2, 3, 4, 6] CS = [6, 4, 3, 1, 0] DS = [6, 4, 0] dim Der = 11
Casimir invariants: • n6,11 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e3 0
e6 0 e1 0 0 0
US = [1, 4, 6] CS = [6, 2, 1, 0] DS = [6, 2, 0] dim Der = 14
Casimir invariants: e1 ,
2e1 e4 − e23
• n6,12 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e3 0
e6 0 e1 0 0 e2
Casimir invariants: e1,
2e1 e4 − e23
US = [1, 3, 6] CS = [6, 3, 1, 0] DS = [6, 3, 0] dim Der = 12
245
246
19. SIX-DIMENSIONAL LIE ALGEBRAS
• n6,13 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e3 0
e6 0 e1 0 e2 0
US = [1, 3, 6] CS = [6, 3, 1, 0] DS = [6, 3, 0] dim Der = 11
where the values of the parameter are over the field C: = 1; over the field R: = ±1. Casimir invariants: e1 , 2e1 e4 − e22 − e23 • n6,14 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e3 0
e6 0 e1 0 e2 e4
US = [1, 3, 4, 6] CS = [6, 4, 3, 1, 0] DS = [6, 4, 0] dim Der = 10
where the values of the parameter are over the field C: = 1; over the field R: = ±1. Casimir invariants: e1 , 2e1 e4 − e22 − e23 • n6,15 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e3 0
e6 0 e1 0 e1 e4
US = [1, 3, 4, 6] CS = [6, 3, 2, 1, 0] DS = [6, 3, 0] dim Der = 11
Casimir invariants: e1 ,
2e1 (e2 − e4 ) + e23
• n6,16 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e3 0
e6 0 e1 0 0 e4
Casimir invariants: e1 ,
2e1 e4 − e23
US = [1, 3, 4, 6] CS = [6, 3, 2, 1, 0] DS = [6, 3, 0] dim Der = 12
19.1. NILPOTENT SIX-DIMENSIONAL LIE ALGEBRAS
• n6,17 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e2 0
e6 0 e1 e2 e3 0
US = [1, 2, 4, 6] CS = [6, 3, 2, 1, 0] DS = [6, 3, 0] dim Der = 10
Casimir invariants: e1 ,
3e21 e4 − 3e1 e2 e3 + e32
• n6,18 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 e1 0
e5 0 0 e2 e3 0
e6 0 e1 0 0 e4
US = [1, 2, 3, 4, 6] CS = [6, 4, 3, 2, 1, 0] DS = [6, 4, 1, 0] dim Der = 9
Casimir invariants: 2e1 e5 − 2e2 e4 + e23
e1 , • n6,19 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 e1 0
e5 0 0 e2 e3 0
e6 0 e1 0 e2 e4
US = [1, 2, 3, 4, 6] CS = [6, 4, 3, 2, 1, 0] DS = [6, 4, 1, 0] dim Der = 8
Casimir invariants: e1 ,
6e21 e5 − 6e1 e2 e4 + 3e1 e23 + 2e32
• n6,20 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 0 0 0
e6 0 e1 e2 e3 e4
US = [1, 2, 3, 4, 6] CS = [6, 4, 3, 2, 1, 0] DS = [6, 4, 0] dim Der = 11
Casimir invariants: e1 ,
2e1 e3 − e22 ,
3e21 e4 − 3e1 e2 e3 + e32 ,
8e31 e5 + 4e1 e22 e3 − 8e21 e2 e4 − e42
247
248
19. SIX-DIMENSIONAL LIE ALGEBRAS
• n6,21 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 0 e1 0
e6 0 e1 e2 e3 e4
US = [1, 2, 3, 4, 6] CS = [6, 4, 3, 2, 1, 0] DS = [6, 4, 0] dim Der = 10
Casimir invariants: 2e1 e3 − e22
e1 , • n6,22 e1 e2 e3 e4 e5
e1 0
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e2 0
e6 0 e1 e2 e3 e4
US = [1, 2, 3, 4, 6] CS = [6, 4, 3, 2, 1, 0] DS = [6, 4, 0] dim Der = 9
Casimir invariants: 3e4 e21 − 3e1 e3 e2 + e32
e1 ,
19.2. Solvable six-dimensional Lie algebras with the nilradical 5n1,1
e1 e2 e3 e4
e1 0
e2 0 0
e3 0 0 0
e3 e1
e4 e2
e5 e5
e4 0 0 0 0
e5 0 0 0 0
• s6,1 US = [2, 4] e1 0
e6
e2 0
Casimir invariants: e1 e4 − e2 e3 ,
e1 ,
e2 ,
e1 0
e2 e1
e3 e2
e22 ,
3e21 e4
CS = [6, 3, 1] DS = [6, 3, 0] e3 e5 exp − e1
• s6,2 US = [1, 2, 3, 4] e6
e4 e3
e5 e5
CS = [6, 4, 3, 2, 1] DS = [6, 4, 0]
Casimir invariants: e1 ,
2e1 e3 −
− 3e1 e2 e3 +
e32 ,
e2 e5 exp − e1
19.2. SOLVABLE ALGEBRAS WITH THE NILRADICAL 5n1,1
• s6,3 US = [1, 2, 3] e1 0
e6
e2 e1
e3 e2
e4 e4
e5 e4 + e5
CS = [6, 4, 3, 2] DS = [6, 4, 0]
Casimir invariants: e1 ,
2e1 e3 − e22 ,
e1 e5 − e2 e4 , e4
e1 0
e4 e4
e2 e4 exp − e1
• s6,4 US = [1, 2, 3] e6
e2 e1
e3 e2
e5 ae5
CS = [6, 4, 3, 2] DS = [6, 4, 0]
where 0 < |a| ≤ 1. If |a| = 1 then arg(a) ≤ π. Casimir invariants: ea4 e2 2 e1 , 2e1 e3 − e2 , , e4 exp − e5 e1 • s6,5 (over the field C: isomorphic to s6,4 ) US = [1, 2, 3] e6
e1 0
e2 e1
e3 e2
where 0 ≤ α. Casimir invariants: e1 ,
2e1 e3 − e22 ,
e4 αe4 − e5
e5 e4 + αe5
CS = [6, 4, 3, 2] DS = [6, 4, 0]
e2 , (e24 + e25 ) exp −2α e1
e2 e4 + arctan e1 e5
• s6,6 US = [1, 2] e6
e1 e1
e2 e1 + e2
e3 e2 + e3
e4 0
e5 e4
Casimir invariants: e4 ,
e1 e5 − e2 e4 , e1
2e1 e3 − e22 , e21
CS = [6, 4, 3] DS = [6, 4, 0] e2 e1 exp − e1
• s6,7 US = [1, 2] e6
e1 ae1
e2 e1 + ae2
e3 0
e3 ,
e1 e4 − e2 e3 , e1
e4 e3
where a = 0. Casimir invariants: ea5 , e1
e5 e5
CS = [6, 4, 3] DS = [6, 4, 0]
e2 e5 exp − e1
249
250
19. SIX-DIMENSIONAL LIE ALGEBRAS
• s6,8 US = [1, 2] e1 0
e6
e2 e1
e3 e3
e4 ae4
e5 be5
CS = [6, 4, 3] DS = [6, 4, 0]
where the values of the parameters a, b are 0 < |b| ≤ |a| ≤ 1. If one or both equalities hold further restrictions on the parameters are necessary in order to avoid redundancies, as discussed in Section 10.4. Casimir invariants: ea3 eb3 e2 e1 , , , e3 exp − e4 e5 e1 • s6,9 (over the field C: isomorphic to s6,8 ) US = [1, 2] e1 0
e6
e2 e1
e3 αe3
where 0 < α. Casimir invariants: (e24 + e25 )α
e1 ,
e2β 3
e4 βe4 − e5
e5 e4 + βe5
e2 , e3 exp −α e1
,
e23
CS = [6, 4, 3] DS = [6, 4, 0]
e4 exp 2α arctan e5
• s6,10 US = [0] e6
e1 e1
e2 e1 + e2
e3 e2 + e3
e4 e3 + e4
e5 e4 + e5
Casimir invariants: e2 − e5 , e1 • s6,11
2e1 e3 − e22 , e21
3e21 e4 − 3e1 e2 e3 + e32 , e31
CS = [6, 5] DS = [6, 5, 0] e2 e1 exp − e1 US = [0]
e6
e1 e1
e2 e1 + e2
e3 e2 + e3
e4 ae4
e5 e4 + ae5
where a = 0. Casimir invariants: 2e1 e3 − e22 , e21
e1 e5 − e2 e4 , e1 e4
ea1 , e4
CS = [6, 5] DS = [6, 5, 0]
e2 e1 exp − e1
• s6,12 US = [0] e6 where a = 0.
e1 ae1
e2 e1 + ae2
e3 e2 + ae3
e4 e3 + ae4
e5 e5
CS = [6, 5] DS = [6, 5, 0]
19.2. SOLVABLE ALGEBRAS WITH THE NILRADICAL 5n1,1
Casimir invariants: 2e1 e3 − e22 , e21 • s6,13
3e21 e4 − 3e1 e2 e3 + e32 , e31
ea5 , e1
251
e2 e1 exp −a e1 US = [0]
e6
e1 ae1
e2 e1 + ae2
e3 e2 + ae3
e4 e4
e5 be5
CS = [6, 5] DS = [6, 5, 0]
where a = 0, 0 < |b| ≤ 1. If |b| = 1 then arg(b) ≤ π. Casimir invariants: 2e1 e3 − e22 ea4 ea5 e2 , , , e1 exp −a e21 e1 e1 eb1 • s6,14 US = [0] e6
e1 ae1
e2 e1 + ae2
e3 be3
e4 e3 + be4
e5 e5
CS = [6, 5] DS = [6, 5, 0]
where the values of the parameters a, b are 0 < |b| ≤ |a|. If |a| = |b| then arg(a) ≤ arg(b). Casimir invariants: e1 e4 − e2 e3 ea5 eb5 e2 , , , e5 exp − e1 e3 e1 e3 e1 • s6,15 (over the field C: isomorphic to s6,13 ) US = [0] e6
e1 αe1
e2 e1 + αe2
where 0 < α. Casimir invariants: 2e1 e3 − e22 , e21
e3 e2 + αe3
(e24 + e25 )α e2β 1
,
e4 βe4 − e5
e5 e4 + βe5
e2 , e1 exp −α e1
CS = [6, 5] DS = [6, 5, 0]
e2 e4 + arctan e1 e5
• s6,16 (over the field C: isomorphic to s6,14 ) US = [0] e6
e1 αe1 − e2
e2 e1 + αe2
e3 e1 + αe3 − e4
where 0 < β. Casimir invariants: e1 e4 − e2 e3 (e21 + e22 )β , , 2 2 e1 + e2 e2α 5 • s6,17
e4 e2 + e3 + αe4
e1 e3 + e2 e4 , e5 exp −β e21 + e22
e5 βe5
e6
e2 ae2
e3 be3
e4 ce4
e5 de5
DS = [6, 5, 0]
e1 e3 + e2 e4 e1 + arctan e21 + e22 e2
US = [0] e1 e1
CS = [6, 5]
CS = [6, 5] DS = [6, 5, 0]
252
19. SIX-DIMENSIONAL LIE ALGEBRAS
where the values of the parameters a, b, c, d are 0 < |d| ≤ |c| ≤ |b| ≤ |a| ≤ 1. If one or more equalities hold further restrictions on the parameters are necessary in order to avoid redundancies, as discussed in Section 10.4. Casimir invariants: ea1 eb1 ec1 ed1 , , , e2 e3 e4 e5 • s6,18 US = [0] e6
e1 ae1
e2 e1 + ae2
e3 e3
e4 be4
e5 ce5
CS = [6, 5] DS = [6, 5, 0]
where the values of the parameters a, b, c are 0 < |c| ≤ |b| ≤ 1, a = 0. If one or both equalities hold further restrictions on the parameters are necessary in order to avoid redundancies, as discussed in Section 10.4. Casimir invariants: ea3 ea4 ea5 e2 , , , e exp −a 1 e1 ec1 e1 eb1 • s6,19 (over the field C: isomorphic to s6,17 ) US = [0] e6
e1 αe1
e2 βe2
e3 γe3
where 0 < |γ| ≤ |β| ≤ α. Casimir invariants: eα eα 2 3 , , eγ1 eβ1
e4 δe4 − e5
(e24 + e25 )α , e2δ 1
e5 e4 + δe5
CS = [6, 5] DS = [6, 5, 0]
e4 e1 exp α arctan e5
• s6,20 (over the field C: isomorphic to s6,18 ) US = [0] e6
e1 αe1
e2 e1 + αe2
e3 βe3
where 0 < α, β = 0. Casimir invariants: eα (e24 + e25 )α 3 , , e2γ eβ1 1
e4 γe4 − e5
e5 e4 + γe5
e2 , e1 exp −α e1
CS = [6, 5] DS = [6, 5, 0]
e2 e4 + arctan e1 e5
• s6,21 (over the field C: isomorphic to s6,17 ) US = [0] e6
e1 αe1 − e2
e2 e1 + αe2
e3 βe3 − γe4
e4 γe3 + βe4
where 0 < δ, 0 < γ ≤ 1. If γ = 1 then α ≤ β. Casimir invariants: (e21 + e22 )δ (e23 + e24 )δ e1 e3 , , γ arctan − arctan , 2α 2β e5 e e 2 4 e5
e5 δe5
CS = [6, 5] DS = [6, 5, 0]
e1 e5 exp δ arctan e2
19.3. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ 2n1,1
19.3. Solvable six-dimensional Lie algebras with the nilradical n3,1 ⊕ 2n1,1
e1 e2 e3 e4
e1 0
e2 0 0
e3 0 e1 0
e3 e4
e4 0
e5 e5
e1 ,
e4
e4 0 0 0 0
e5 0 0 0 0
• s6,22 US = [2, 4] e6
e1 0
e2 0
CS = [6, 3, 1] DS = [6, 3, 0]
Casimir invariants: • s6,23 US = [2, 3, 4] e6
e1 0
e2 e3
e3 e4
e4 0
e5 e5
CS = [6, 4, 3, 1] DS = [6, 4, 0]
Casimir invariants: e1 ,
e4
• s6,24 US = [2, 3] e6
e1 0
e2 e2
e3 −e3
e4 0
e5 e4
CS = [6, 4, 3] DS = [6, 4, 1, 0]
Casimir invariants: e1 ,
e4
• s6,25 (over the field C: isomorphic to s6,24 ) US = [2, 3] e6
e1 0
e2 −e3
e3 e2
e4 e5
e5 0
CS = [6, 4, 3] DS = [6, 4, 1, 0]
Casimir invariants: e1 ,
e5
e4 e4
e5 e1
• s6,26 US = [1, 4] e6
e1 0
e2 0
e3 0
Casimir invariants: e1 ,
CS = [6, 2, 1] DS = [6, 2, 0]
e5 e4 exp − e1
253
254
19. SIX-DIMENSIONAL LIE ALGEBRAS
• s6,27 US = [1, 3, 4] e6
e1 0
e2 0
e3 e4
e4 e1
e5 e5
CS = [6, 3, 2, 1] DS = [6, 3, 0]
e4 e5 exp − e1
Casimir invariants: e1 , • s6,28
US = [1, 3, 4] e6
e1 0
e2 0
e3 e2
e4 e1
e5 e5
CS = [6, 3, 2, 1] DS = [6, 3, 0]
e4 e5 exp − e1
Casimir invariants: e1 , • s6,29
US = [1, 2, 3, 4] e6
e1 0
e2 e3
e3 e4
e4 e1
e5 e5
CS = [6, 4, 3, 2, 1] DS = [6, 4, 0]
e1 ,
e4 e5 exp − e1
e3 −e3
e4 e5
Casimir invariants: • s6,30
US = [1, 2, 3] e6
e1 0
e2 e2
e5 e1
CS = [6, 4, 3] DS = [6, 4, 1, 0]
Casimir invariants: • s6,31
e1 , 2e1 e4 − e25 (over the field C: isomorphic to s6,30 ) US = [1, 2, 3] e6
e1 0
e2 −e3
e3 e2
e4 e5
e5 e1
CS = [6, 4, 3] DS = [6, 4, 1, 0]
Casimir invariants: 2e1 e4 − e25
e1 , • s6,32
US = [1, 2, 3] e6
e1 0
e2 0
e3 e2
e4 e4
e5 ae5
CS = [6, 4, 3, 2] DS = [6, 4, 0]
where the value of the parameter a is 0 < |a| ≤ 1. If |a| = 1 then arg(a) ≤ π. Casimir invariants: ea4 e1 , e5
19.3. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ 2n1,1
• s6,33 US = [1, 2, 3] e1 0
e6
e2 0
e3 e2
e4 e4
Casimir invariants:
e5 e4 + e5
CS = [6, 4, 3, 2] DS = [6, 4, 0]
e5 e4 exp − e4
e1 ,
• s6,34 (over the field C: isomorphic to s6,32 ) US = [1, 2, 3] e6
e1 0
e2 0
e3 e2
where α ≥ 0. Casimir invariants: e1 ,
e4 αe4 − e5
e5 e4 + αe5
CS = [6, 4, 3, 2] DS = [6, 4, 0]
e4 (e24 + e25 ) exp 2α arctan e5
• s6,35 US = [1, 2] e1 0
e6
e2 ae2
e3 −ae3
where arg(a) < π, a = 0. Casimir invariants: e1 ,
e4 e4
e5 e1
CS = [6, 4] DS = [6, 4, 1, 0]
e5 e4 exp − e1
• s6,36 US = [1, 2] e6
e1 0
e2 −e2
e3 e3 + e4
Casimir invariants: e1 ,
e4 e4
e5 e1
CS = [6, 4] DS = [6, 4, 1, 0]
e5 e4 exp − e1
• s6,37 (over the field C: isomorphic to s6,35 ) US = [1, 2] e6
e1 0
e2 −e3
e3 e2
where α = 0. Casimir invariants: e1 ,
e4 αe4
e5 e1
CS = [6, 4] DS = [6, 4, 1, 0]
e5 e4 exp −α e1
255
256
19. SIX-DIMENSIONAL LIE ALGEBRAS
• s6,38 US = [1, 2] e6
e1 (a + 1)e1
e2 e2
e3 ae3
e4 0
e5 e4
CS = [6, 4, 3] DS = [6, 4, 1, 0]
where 0 < |a| ≤ 1. If |a| = 1 then arg(a) < π. Casimir invariants: e5 e4 , e1 exp −(a + 1) e4 • s6,39 US = [1, 2] e1 2e1
e6
e2 e2
e3 e2 + e3
e4 0
e5 e4
CS = [6, 4, 3] DS = [6, 4, 1, 0]
e5 e1 exp −2 e4 (over the field C: isomorphic to s6,38 )
Casimir invariants:
e4 ,
• s6,40
US = [1, 2] e6
e1 2αe1
e2 αe2 − e3
e3 e2 + αe3
where α > 0. Casimir invariants:
e4 0
e5 e4
CS = [6, 4, 3] DS = [6, 4, 1, 0]
e5 e1 exp −2α e4
e4 , • s6,41
US = [1, 2] e1 e1
e6
e2 e2
e3 e4
Casimir invariants: e5 , • s6,42
e4 e5
e5 0
CS = [6, 4, 3, 2] DS = [6, 4, 0]
e4 e1 exp − e5 US = [1, 2]
e6
e1 e1
e2 e2
e3 0
Casimir invariants: e4 , • s6,43
e4 0
e5 e4
CS = [6, 3, 2] DS = [6, 3, 0]
e5 e1 exp − e4 US = [1]
e6
e1 0
e2 e2 + e4
e3 −e3 + e5
e4 e4
e5 −e5
CS = [6, 5] DS = [6, 5, 1, 0]
19.3. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ 2n1,1
257
Casimir invariants: e1 ,
e4 e5
• s6,44 (over the field C: isomorphic to s6,43 ) US = [1] e1 0
e6
e2 −e3 + e4
e3 e2 + e5
e4 −e5
e1 ,
e24 + e25
e5 e4
CS = [6, 5] DS = [6, 5, 1, 0]
Casimir invariants: • s6,45 US = [1] e1 0
e6
e2 ae2
e3 −ae3
e4 e4
e5 be5
CS = [6, 5] DS = [6, 5, 1, 0]
where a = 0, arg(a) < π, 0 < |b| ≤ 1. If |b| = 1 then arg(b) ≤ π. Casimir invariants: eb4 e1 , e5 • s6,46 (over the field C: isomorphic to s6,45 ) US = [1] e1 0
e6
e2 −e3
e3 e2
e4 αe4
e5 βe5
CS = [6, 5] DS = [6, 5, 1, 0]
where the values of the parameters α, β are 0 < |β| ≤ |α|. If |α| = |β| then β ≤ α. Casimir invariants: eβ4 e1 , eα 5 • s6,47 US = [1] e6
e1 0
e2 −e2
e3 e3 + e4
e4 e4
where a = 0. Casimir invariants: e1 , • s6,48
e5 ae5
CS = [6, 5] DS = [6, 5, 1, 0]
ea4 e5 US = [1]
e6
e1 0
e2 e2
e3 −e3
where a = 0. Casimir invariants: e1 ,
e4 ae4
e5 e4 + ae5
e5 e4 exp −a e4
CS = [6, 5] DS = [6, 5, 1, 0]
258
19. SIX-DIMENSIONAL LIE ALGEBRAS
• s6,49 (over the field C: isomorphic to s6,48 ) US = [1] e6
e1 0
e2 −e3
e3 e2
e4 αe4 + e5
where α = 0. Casimir invariants:
e5 αe5
CS = [6, 5] DS = [6, 5, 1, 0]
e4 e5 exp −α e5
e1 , • s6,50
US = [1] e6
e1 0
e2 −e2
e3 e3 + e4
Casimir invariants:
e4 e4 + e5
e5 e5
CS = [6, 5] DS = [6, 5, 1, 0]
e4 e5 exp − e5
e1 ,
• s6,51 (over the field C: isomorphic to s6,45 ) US = [1] e6
e1 0
e2 αe2
e3 −αe3
e4 βe4 − e5
e5 e4 + βe5
where α > 0, β ≥ 0. Casimir invariants: e1 ,
(e24
+
CS = [6, 5] DS = [6, 5, 1, 0]
e4 2β arctan e5
e25 ) exp
• s6,52 (over the field C: isomorphic to s6,45 ) US = [1] e6
e1 0
e2 −e3
e3 e2
e4 αe4 − βe5
where α ≥ 0, β > 0. Casimir invariants: e1 ,
(e24
+
e25 )β
e5 βe4 + αe5
CS = [6, 5] DS = [6, 5, 1, 0]
e4 exp 2α arctan e5
• s6,53 US = [1] e6
e1 e1
e2 e2 + e4
e3 e5
Casimir invariants: e5 ,
e4 e4
e5 0
e4 e1
CS = [6, 4, 3] DS = [6, 4, 0]
19.3. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ 2n1,1
• s6,54 US = [1] e1 ae1
e6
e2 ae2
e3 e5
e4 e4
e5 0
where a = 0. Casimir invariants:
CS = [6, 4, 3] DS = [6, 4, 0]
ea4 e1
e5 , • s6,55
US = [1] e1 e1
e6
e2 e2 + e4
e3 e5
Casimir invariants: e5 , • s6,56
e4 e1 + e4
e5 0
CS = [6, 4, 3] DS = [6, 4, 0]
e4 e1 exp − e1 US = [1]
e6
e1 e1
e2 e2
e3 e5
Casimir invariants: e5 , • s6,57
e4 e1 + e4
e5 0
CS = [6, 4, 3] DS = [6, 4, 0]
e4 e1 exp − e1 US = [0]
e1 2ae1
e6
e2 ae2 + e3
where a = 0. Casimir invariants:
e3 ae3 + e4
e24 , e1
• s6,58
e4 ae4
e5 e5
CS = [6, 5] DS = [6, 5, 1, 0]
e2a 5 e1 US = [0]
e6
e1 (a + 1)e1
e2 e2 + e4
e3 ae3 + e5
e4 e4
e5 ae5
CS = [6, 5] DS = [6, 5, 1, 0]
where 0 < |a| ≤ 1. If |a| = 1 then arg(a) < π. Casimir invariants: e4 e5 ea+1 4 , e1 e1 • s6,59 US = [0] e6
e1 2e1
e2 e2 + e3
e3 e3 + e4
e4 e4 + e5
e5 e5
CS = [6, 5] DS = [6, 5, 1, 0]
259
260
19. SIX-DIMENSIONAL LIE ALGEBRAS
Casimir invariants:
e25 , e1
• s6,60
e4 e1 exp −2 e5 US = [0]
e1 2e1
e6
e2 e2 + e3
e3 e3 + e4
Casimir invariants:
e24 , e1
• s6,61
e4 e4
e5 e1 + 2e5
CS = [6, 5] DS = [6, 5, 1, 0]
e5 e1 exp −2 e1 US = [0]
e6
e1 2e1
e2 e2 + e3 + e4
Casimir invariants:
e3 e3 + e5 e25 , e1
• s6,62
e4 e4 + e5
e5 e5
CS = [6, 5] DS = [6, 5, 1, 0]
e4 e1 exp −2 e5 US = [0]
e6
e1 (a + 1)e1
e2 e2
e3 ae3
e4 (1 + a)e4 + e5
e5 e1 + (1 + a)e5
CS = [6, 5] DS = [6, 5, 1, 0]
where 0 < |a| ≤ 1. If |a| = 1 then arg(a) < π. Casimir invariants: 2e1 e4 − e25 e5 , e1 exp −(a + 1) e21 e1 • s6,63 US = [0] e6
e1 2e1
e2 e2
e3 e2 + e3
e4 2e4 + e5
Casimir invariants: 2e1 e4 − e25 , e21
e5 e1 + 2e5
CS = [6, 5] DS = [6, 5, 1, 0]
e5 e1 exp −2 e1
• s6,64 (over the field C: isomorphic to s6,62 ) US = [0] e6
e1 2αe1
e2 αe2 − e3
e3 e2 + αe3
e4 2αe4 + e5
where α > 0. Casimir invariants: 2e1 e4 − e25 , e21
e5 e1 + 2αe5
e5 e1 exp −2α e1
CS = [6, 5] DS = [6, 5, 1, 0]
19.3. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ 2n1,1
261
• s6,65 (over the field C: isomorphic to s6,58 ) US = [0] e6
e1 2αe1
e2 αe2 − e3 + e4
e3 e2 + αe3 + e5
where α > 0. Casimir invariants: e24 + e25 , e1
e4 αe4 − e5
e5 e4 + αe5
CS = [6, 5] DS = [6, 5, 1, 0]
e4 e1 exp 2α arctan e5
• s6,66 US = [0] e6
e1 (a + b)e1
e2 ae2
e3 be3
e4 e4
e5 ce5
CS = [6, 5] DS = [6, 5, 1, 0]
where the values of the parameters a, b, c are 0 < |b| ≤ |a|, 0 < |c| ≤ 1. If |a| = |b| then arg(a) ≤ arg(b), a = −b. If |c| = |1| then arg(c) ≤ π. Casimir invariants: ea+b ea+b 4 5 , e1 ec1 • s6,67 US = [0] e1 (a + 1)e1
e6
e2 ae2
e3 e3 + e4
e4 e4
e5 be5
CS = [6, 5] DS = [6, 5, 1, 0]
where a = −1, 0, b = 0. Casimir invariants: ea+1 4 , e1
ea+1 5 eb1
• s6,68
US = [0] e6
e1 2ae1
e2 ae2
e3 e2 + ae3
e4 e4
e5 be5
CS = [6, 5] DS = [6, 5, 1, 0]
where the values of the parameters a, b are a = 0, 0 < |b| ≤ 1. If |b| = 1 then arg(b) ≤ π. Casimir invariants: e2a e2a 4 5 , e1 eb1 • s6,69 (over the field C: isomorphic to s6,66 ) US = [0] e6
e1 2αe1
e2 αe2 − e3
e3 e2 + αe3
e4 βe4
e5 γe5
CS = [6, 5] DS = [6, 5, 1, 0]
where the values of the parameters α, β, γ are α = 0, 0 < |γ| ≤ |β|. If |γ| = |β| then γ ≤ β.
262
19. SIX-DIMENSIONAL LIE ALGEBRAS
Casimir invariants: e2α 4 eβ1
• s6,70
,
e2α 5 eγ1 US = [0]
e6
e1 2e1
e2 e2
e3 e2 + e3
e4 ae4
where a = 0. Casimir invariants:
CS = [6, 5] DS = [6, 5, 1, 0]
e5 e1 exp −2 e4
e24 , ea1
• s6,71
e5 e4 + ae5
US = [0] e1 (a + b)e1
e6
e2 ae2
e3 be3
e4 e4
e5 e1 + (a + b)e5
CS = [6, 5] DS = [6, 5, 1, 0]
where 0 < |b| ≤ |a|, a = −b. If |a| = |b| then arg(a) ≤ arg(b). Casimir invariants: ea+b e5 4 , e1 exp −(a + b) e1 e1 • s6,72 US = [0] e6
e1 (a + 1)e1
e2 ae2
e3 e3 + e4
where a = −1, 0. Casimir invariants: ea+1 4 , e1
e4 e4
e5 e1 + (a + 1)e5
CS = [6, 5] DS = [6, 5, 1, 0]
e5 e1 exp −(a + 1) e1
• s6,73 US = [0] e6
e1 2ae1
e2 ae2
e3 e2 + ae3
where a = 0. Casimir invariants: e2a 5 , e1
• s6,74
e4 e1 + 2ae4
e5 e5
CS = [6, 5] DS = [6, 5, 1, 0]
e4 e1 exp −2a e1 US = [0]
e6
e1 (a + 1)e1
e2 e2
e3 ae3
e4 be4
e5 e4 + be5
where 0 < |a| ≤ 1, b = 0. If |a| = 1 then arg(a) < π.
CS = [6, 5] DS = [6, 5, 1, 0]
19.3. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ 2n1,1
Casimir invariants:
e5 e1 exp −(a + 1) e4
ea+1 4 , eb1
• s6,75
US = [0] e6
e1 (a + 1)e1
e2 ae2
e3 e3 + e4
where a = 0, −1. Casimir invariants:
e4 e4 + e5
e5 e5
CS = [6, 5] DS = [6, 5, 1, 0]
e4 e1 exp −(a + 1) e5
ea+1 5 , e1
• s6,76 (over the field C: isomorphic to s6,71 ) US = [0] e6
e1 2αe1
e2 αe2 − e3
where α > 0, β = 0. Casimir invariants:
• s6,77
e3 e2 + αe3
e2α 4
,
e4 βe4
e5 e1 + 2αe5
CS = [6, 5] DS = [6, 5, 1, 0]
e5 e4 exp −β e1
eβ1 (over the field C: isomorphic to s6,74 ) US = [0]
e6
e1 2αe1
e2 αe2 − e3
where α > 0, β = 0. Casimir invariants:
• s6,78
e3 e2 + αe3
e2α 5
e4 βe4 + e5
e5 βe5
CS = [6, 5] DS = [6, 5, 1, 0]
e4 e5 exp −β e5
, eβ1 (over the field C: isomorphic to s6,66 ) US = [0]
e6
e1 (α + β)e1
e2 αe2
e3 βe3
e4 γe4 − e5
e5 e4 + γe5
CS = [6, 5] DS = [6, 5, 1, 0]
where 0 < |β| ≤ |α|, β = −α, γ > 0. Casimir invariants: (e24 + e25 )α+β e4 , e exp (α + β) arctan 1 e5 e2γ 1 • s6,79 (over the field C: isomorphic to s6,68 ) US = [0] e6
e1 2αe1
e2 αe2
where α = 0, β ≥ 0.
e3 e2 + αe3
e4 βe4 − e5
e5 e4 + βe5
CS = [6, 5] DS = [6, 5, 1, 0]
263
264
19. SIX-DIMENSIONAL LIE ALGEBRAS
Casimir invariants: (e24 + e25 )α eβ1
e4 e1 exp 2α arctan e5
,
• s6,80 (over the field C: isomorphic to s6,66 ) US = [0] e6
e1 2αe1
e2 αe2 − e3
e3 e2 + αe3
e4 βe4 − γe5
where α, γ > 0. Casimir invariants: (e24 + e25 )α eβ1
,
e5 γe4 + βe5
CS = [6, 5] DS = [6, 5, 1, 0]
e4 exp 2α arctan e5
eγ1
• s6,81 US = [0] e1 e1
e6
e2 0
e3 e3 + e4
where a = 0. Casimir invariants:
e4 e4
e4 , e1
• s6,82
e5 ae5
CS = [6, 4] DS = [6, 4, 0]
ea1 e5 US = [0]
e6
e1 e1
e2 0
e3 e3 + e4
Casimir invariants: e4 , e1
• s6,83
e4 e4
e5 e1 + e5
CS = [6, 4] DS = [6, 4, 0]
e5 e1 exp − e1 US = [0]
e6
e1 e1
e2 0
e3 e3 + e4
Casimir invariants: e5 , e1
• s6,84
e4 e4 + e5
e5 e5
CS = [6, 4] DS = [6, 4, 0]
e4 e1 exp − e5 US = [0]
e6
e1 e1
Casimir invariants:
e2 e2
e3 0
e4 e4 + e5
2e1 e4 − e25 , e21
e5 e1 + e5 e5 e1 exp − e1
CS = [6, 4] DS = [6, 4, 0]
19.3. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ 2n1,1
• s6,85 US = [0] e1 e1
e6
e2 0
e3 e3 + e4
e4 e4 + e5
Casimir invariants:
CS = [6, 4] DS = [6, 4, 0]
e5 e1 exp − e1
2e1 e4 − e25 , e21
• s6,86
e5 e1 + e5
US = [0] e1 ae1
e6
e2 ae2
e3 0
e4 e4
e5 be5
CS = [6, 4] DS = [6, 4, 0]
where a = 0, 0 < |b| ≤ 1. If |b| = 1 then arg(b) ≤ π. Casimir invariants: ea4 ea5 , e1 eb1 • s6,87 US = [0] e6
e1 ae1
e2 ae2
e3 0
e4 e4
where a = 0. Casimir invariants:
e5 e1 + ae5
CS = [6, 4] DS = [6, 4, 0]
e5 e1 exp −a e1
ea4 , e1 • s6,88
US = [0] e6
e1 e1
e2 0
e3 e3 + e4
where a = 0. Casimir invariants: ea1 , e5
e4 e1 + e4
e5 ae5
CS = [6, 4] DS = [6, 4, 0]
e4 e1 exp − e1
• s6,89 US = [0] e6
e1 e1
e2 e2
e3 0
e4 ae4
where a = 0. Casimir invariants: ea1 , e4
e5 e4 + ae5
e5 e1 exp − e4
CS = [6, 4] DS = [6, 4, 0]
265
266
19. SIX-DIMENSIONAL LIE ALGEBRAS
• s6,90 (over the field C: isomorphic to s6,86 ) US = [0] e6
e1 αe1
e2 αe2
e3 0
e4 βe4 − e5
where α = 0, β ≥ 0. Casimir invariants: (e24 + e25 )α e2β 1
e5 e4 + βe5
CS = [6, 4] DS = [6, 4, 0]
e4 e1 exp α arctan e5
,
19.4. Solvable six-dimensional Lie algebras with the nilradical n4,1 ⊕ n1,1 e1 0
e1 e2 e3 e4
e2 0 0
e3 0 0 0
e4 0 e1 e2 0
e5 0 0 0 0
• s6,91 US = [1, 2, 4] e6
e1 0
e2 0
e3 e1
Casimir invariants: e25
e1 , • s6,92
e4 0
e5 e5
CS = [6, 3, 2, 1] DS = [6, 3, 0]
2 e2 − 2e1 e3 exp e21 US = [1, 2, 3, 4]
e6
e1 0
e2 0
e3 0
e4 e3
e5 e5
CS = [6, 4, 3, 2, 1] DS = [6, 4, 0]
Casimir invariants: 2e1 e3 − e22
e1 , • s6,93
US = [1, 2, 3, 4] e6
e1 0
e2 0
e3 e1
e4 e3
e5 e5
CS = [6, 4, 3, 2, 1] DS = [6, 4, 0]
where = 1 over the field C, = ±1 over the field R. Casimir invariants: 2 e2 − 2e1 e3 e1 , e2 exp 5 e21 • s6,94 US = [1, 2] e6
e1 0
e2 −e2
e3 −2e3
e4 e4
e5 e1
CS = [6, 4] DS = [6, 4, 2, 0]
19.4. SOLVABLE ALGEBRAS WITH THE NILRADICAL n4,1 ⊕ n1,1
Casimir invariants: (2e1 e3 −
e1 ,
e5 2 e1
e22 ) exp
• s6,95 US = [1] e1 0
e6
e2 −e2
e3 −2e3
e4 e4 + e5
e5 e5
CS = [6, 5] DS = [6, 5, 2, 0]
Casimir invariants: e1 ,
(2e1 e3 − e22 )e25
• s6,96 US = [1] e1 0
e6
e2 −e2
e3 −2e3
e4 e4
e5 ae5
CS = [6, 5] DS = [6, 5, 2, 0]
where a = 0. Casimir invariants: (2e1 e3 − e22 )a e25
e1 , • s6,97
US = [1] e6
e1 0
e2 −e2
e3 −2e3 + e5
Casimir invariants: e1 ,
e4 e4
e5 −2e5
CS = [6, 5] DS = [6, 5, 2, 0]
2e1 e3 − e22 e5 exp e1 e5
• s6,98 US = [1] e6
e1 2e1
e2 e2
Casimir invariants:
e3 e5
e4 e4
e5 0
CS = [6, 4, 3] DS = [6, 4, 1, 0]
e5 ,
2 e2 − 2e1 e3 e1 exp e1 e5
e2 e2
e3 e3
• s6,99 US = [1] e6
e1 e1
e4 e5
e5 0
Casimir invariants: e5 ,
2e1 e3 − e22 e21
CS = [6, 4, 3] DS = [6, 4, 0]
267
268
19. SIX-DIMENSIONAL LIE ALGEBRAS
• s6,100 US = [1] e1 e1
e6
e2 e2
e3 e1 + e3
e4 e5
e5 0
CS = [6, 4, 3] DS = [6, 4, 0]
where = 1 over the field C, = ±1 over the field R. Casimir invariants: 2 e2 − 2e1 e3 2 e5 , e1 exp e21 • s6,101 US = [0] e1 e1
e6
e2 0
e3 −e3
e4 e4 + e5
Casimir invariants: 2e1 e3 − e22 , • s6,102
e5 e1 + e5
CS = [6, 5] DS = [6, 5, 2, 0]
e5 e1 exp − e1 US = [0]
e6
e1 3e1
e2 2e2
e3 e3
where a = 0. Casimir invariants:
e4 e3 + e4
e5 ae5
(2e1 e3 − e22 )3 , e41
• s6,103
CS = [6, 5] DS = [6, 5, 2, 0]
e35 ea1 US = [0]
e6
e1 3e1
e2 2e2
Casimir invariants:
e3 e3 + e5 e35 , e1
• s6,104
e21
e4 e3 + e4
e5 e5
CS = [6, 5] DS = [6, 5, 2, 0]
2 e2 − 2e1 e3 exp 3 e1 e5 US = [0]
e6
e1 3e1
e2 2e2
e3 e3
e4 e3 + e4
Casimir invariants: (2e1 e3 − e22 )3 , e41
• s6,105
e5 e1 + 3e5
CS = [6, 5] DS = [6, 5, 2, 0]
e5 e1 exp −3 e1 US = [0]
e6
e1 (a + 2)e1
e2 (a + 1)e2
e3 ae3
e4 e4
e5 be5
CS = [6, 5] DS = [6, 5, 2, 0]
19.4. SOLVABLE ALGEBRAS WITH THE NILRADICAL n4,1 ⊕ n1,1
where a = 0, −2, b = 0. Casimir invariants:
(2e1 e3 − e22 )a+2
ea+2 5 , eb1
2(1+a)
e1
• s6,106
US = [0] e6
e1 (a + 2)e1
e2 (a + 1)e2
where a = −2, 0. Casimir invariants:
e3 ae3
e4 e4 + e5
e5 e5
CS = [6, 5] DS = [6, 5, 2, 0]
(2e1 e3 − e22 )a+2
ea+2 5 , e1
2(a+1)
e1
• s6,107
US = [0] e6
e1 (a + 2)e1
e2 (a + 1)e2
where a = −2, 0. Casimir invariants: ea+2 5 , ea1
e21
e3 ae3 + e5
e4 e4
e5 ae5
CS = [6, 5] DS = [6, 5, 2, 0]
2e1 e3 − e22 exp −(a + 2) e1 e5
• s6,108 US = [0] e6
e1 (a + 2)e1
e2 (a + 1)e2
e3 ae3
e4 e4
where a = −2, 0. Casimir invariants: (2e1 e3 − e22 )a+2 2(1+a)
,
e1
e5 e1 + (a + 2)e5
CS = [6, 5] DS = [6, 5, 2, 0]
e5 e1 exp −(a + 2) e1
• s6,109 US = [0] e6
e1 2e1
e2 e2
Casimir invariants:
e3 0
e4 e4 + e5
e5 e5
2e1 e3 − e22 , e1
• s6,110
CS = [6, 4] DS = [6, 4, 1, 0] e25 e1 US = [0]
e6 where a = 0.
e1 2e1
e2 e2
e3 0
e4 e4
e5 ae5
CS = [6, 4] DS = [6, 4, 1, 0]
269
270
19. SIX-DIMENSIONAL LIE ALGEBRAS
Casimir invariants:
2e1 e3 − e22 , e1
• s6,111
e25 ea1 US = [0]
e1 2e1
e6
e2 e2
e3 0
e4 e4
e5 e1 + 2e5
Casimir invariants: 2e1 e3 − e22 , e1
CS = [6, 4] DS = [6, 4, 1, 0]
e5 e1 exp −2 e1
• s6,112 US = [0] e6
e1 e1
e2 e2
e3 e3
where a = 0. Casimir invariants:
e4 0
e5 ae5
2e1 e3 − e22 , e21
• s6,113
CS = [6, 4] DS = [6, 4, 0]
ea1 e5 US = [0]
e6
e1 e1
e2 e2
e3 e3 + e5
Casimir invariants: e5 , e1
• s6,114
e21
e4 0
e5 e5
CS = [6, 4] DS = [6, 4, 0]
2 e2 − 2e1 e3 exp e1 e5 US = [0]
e6
e1 e1
Casimir invariants: • s6,115
e2 e2
e3 e3
e4 0
e5 e1 + e5
CS = [6, 4] DS = [6, 4, 0]
2e1 e3 − e22 , e21
e5 e1 exp − e1
e3 e3 + e5
e5 e1 + e5
US = [0] e6
e1 e1
e2 e2
e4 0
CS = [6, 4] DS = [6, 4, 0]
where = 1 over the field C, = ±1 over the field R. Casimir invariants: e5 2e1 e3 − e22 − e25 , e1 exp − e21 e1
19.5. SOLVABLE ALGEBRAS WITH THE NILRADICAL n5,1
• s6,116 US = [0] e1 e1
e6
e2 e2
e3 e1 + e3
e4 0
e5 ae5
CS = [6, 4] DS = [6, 4, 0]
where a = 0 and = 1 over the field C, = ±1 over the field R. Casimir invariants: 2 ea1 e2 − 2e1 e3 2 , e1 exp e5 e21 19.5. Solvable six-dimensional Lie algebras with the nilradical n5,1 e1 0
e1 e2 e3 e4
e2 0 0
e3 0 0 0
e4 0 0 0 0
e5 0 0 e1 e2
• s6,117 US = [2] e1 0
e6
e2 0
e3 e3
e4 e4
e5 −e5
CS = [6, 5] DS = [6, 5, 2, 0]
Casimir invariants: e1 ,
e2
• s6,118 US = [1, 2] e6
e1 0
e2 e1
e3 e3
Casimir invariants:
e4 e3 + e4
e5 −e5
CS = [6, 5] DS = [6, 5, 2, 0]
e2 (e1 e4 − e2 e3 ) exp − e1
e1 , • s6,119
US = [1, 2] e6
e1 0
e2 e2
e3 0
e4 e4
Casimir invariants: e1 , • s6,120
e5 e3
CS = [6, 4, 3, 2] DS = [6, 4, 0]
e1 e4 − e2 e3 e2 US = [1, 2]
e6
e1 0
e2 e2
e3 −e1
e4 e2 + e4
e5 e3
CS = [6, 4, 3, 2] DS = [6, 4, 0]
271
272
19. SIX-DIMENSIONAL LIE ALGEBRAS
e2 e3 − e1 e4 e22 exp e1 e2
Casimir invariants: e1 , • s6,121
US = [1, 2] e1 0
e6
e2 e2
e3 0
e4 e4
Casimir invariants:
e5 0
CS = [6, 3, 2] DS = [6, 3, 0]
e1 e4 − e2 e3 e2
e1 , • s6,122
US = [1, 2] e1 0
e6
e2 e2
e3 −e1
e4 e2 + e4
e5 0
CS = [6, 3, 2] DS = [6, 3, 0]
e2 e3 − e1 e4 e22 exp e1 e2
Casimir invariants: e1 , • s6,123
US = [1] e6
e1 0
e2 2e2
e3 −e3
e4 e4
Casimir invariants:
e5 e4 + e5
CS = [6, 5] DS = [6, 5, 2, 0]
(e1 e4 − e2 e3 )2 e2
e1 , • s6,124
US = [1] e6
e1 0
e2 e2
e3 −ae3
e4 (1 − a)e4
e5 ae5
CS = [6, 5] DS = [6, 5, 2, 0]
where a = 0, 1. Casimir invariants: e1 ,
(e1 e4 − e2 e3 )ea−1 2
• s6,125 US = [1] e6
e1 0
e2 e2
e3 e2 + e3
Casimir invariants: e1 , • s6,126
e4 2e4
e5 −e5
CS = [6, 5] DS = [6, 5, 2, 0]
e1 e4 − e2 e3 e2 exp e22 US = [1]
e6
e1 0
e2 e2
e3 −e3
e4 0
e5 e5
CS = [6, 4] DS = [6, 4, 1, 0]
19.5. SOLVABLE ALGEBRAS WITH THE NILRADICAL n5,1
273
Casimir invariants: e1 ,
e1 e4 − e2 e3
• s6,127 US = [1] e1 0
e6
e2 e2
e3 −e3
Casimir invariants:
e4 e1
e5 e5
CS = [6, 4] DS = [6, 4, 1, 0]
e2 e3 − e1 e4 e2 exp e21
e1 , • s6,128
US = [0] e1 2e1
e6
e2 ae2
e3 e3
where a = 0, 1. Casimir invariants:
e4 (a − 1)e4
e5 e3 + e5
(e1 e4 − e2 e3 )2 , e1 e22
• s6,129
CS = [6, 5] DS = [6, 5, 2, 0]
ea1 e22 US = [0]
e1 3e1
e6
e2 2e2
e3 e2 + 2e3
Casimir invariants:
e5 e4 + e5
CS = [6, 5] DS = [6, 5, 2, 0]
e1 e4 − e2 e3 e1 exp 3 e22
e32 , e21
• s6,130
e4 e4
US = [0] e6
e1 2e1
e2 e1 + 2e2
e3 e3
e4 e3 + e4
Casimir invariants: (e1 e4 − e2 e3 )2 , e31
e5 e4 + e5
CS = [6, 5] DS = [6, 5, 2, 0]
e2 e1 exp −2 e1
• s6,131 US = [0] e6
e1 e1
e2 ae2
e3 (1 − b)e3
e4 (a − b)e4
e5 be5
CS = [6, 5] DS = [6, 5, 2, 0]
where the values of the parameters a, b are 0 < |a| ≤ 1, b = 0, 1, a. If |a| = 1 then arg(a) ≤ π. Casimir invariants: ea1 , (e1 e4 − e2 e3 )eb−a−1 1 e2
274
19. SIX-DIMENSIONAL LIE ALGEBRAS
• s6,132 US = [0] e6
e1 e1
e2 ae2
e3 (2 − a)e3
where a = 0, 1, 2. Casimir invariants:
e4 e1 + e4
e5 (a − 1)e5
CS = [6, 5] DS = [6, 5, 2, 0]
e2 e3 − e1 e4 e1 exp e21
ea1 , e2 • s6,133
US = [0] e6
e1 e1
e2 e1 + e2
e3 (1 − a)e3
e4 e3 + (1 − a)e4
where a = 0, 1. Casimir invariants: (e1 e4 −
e2 e3 )ea−2 , 1
e5 ae5
CS = [6, 5] DS = [6, 5, 2, 0]
e2 e1 exp − e1
• s6,134 (over the field C: isomorphic to s6,131 ) US = [0] e6
e1 e2 e3 e4 e5 αe1 − e2 e1 + αe2 (α − β)e3 − e4 e3 + (α − β)e4 βe5
where 0 < β. Casimir invariants: (e1 e4 − e2 e3 )2α (e21 + e22 )β−2α ,
CS = [6, 5] DS = [6, 5, 2, 0]
e1 e4 − e2 e3 e1 exp −β arctan e21 + e22 e2
• s6,135 US = [0] e6
e1 e1
e2 2e2
e3 0
e4 e4
e5 e4 + e5
CS = [6, 4] DS = [6, 4, 1, 0]
Casimir invariants: e21 , e2 • s6,136
e1 e4 − e2 e3 e21 US = [0]
e6
e1 2e1
e2 e2
e3 e3
e4 0
e5 e3 + e5
Casimir invariants: e22 , e1
e1 e4 − e2 e3 e1
CS = [6, 4] DS = [6, 4, 1, 0]
19.5. SOLVABLE ALGEBRAS WITH THE NILRADICAL n5,1
• s6,137 US = [0] e1 2e1
e6
e2 e2
e3 e2 + e3
Casimir invariants:
e5 e3 + e5
CS = [6, 4] DS = [6, 4, 1, 0]
e1 e4 − e2 e3 e1 exp 2 e22
e22 , e1
• s6,138
e4 0
US = [0] e1 e1
e6
e2 2e2
Casimir invariants:
e3 0
e4 e1 + e4
CS = [6, 4] DS = [6, 4, 1, 0]
e2 e3 − e1 e4 e1 exp e21
e21 , e2
• s6,139
e5 e5
US = [0] e6
e1 e1
e2 ae2
e3 0
where a = 0, 1. Casimir invariants:
e4 (a − 1)e4
e5 e5
e1 e4 − e2 e3 , e2
• s6,140
CS = [6, 4] DS = [6, 4, 1, 0]
ea1 e2 US = [0]
e6
e1 e1
e2 ae2
e3 e3
e4 ae4
e5 0
CS = [6, 4] DS = [6, 4, 0]
where 0 < |a| ≤ 1. If |a| = 1 then arg(a) ≤ π. Casimir invariants: e1 e4 − e2 e3 ea1 , e1 e2 e2 • s6,141 US = [0] e6
e1 e1
e2 e2
e3 e3
e4 e1 + e4
e5 0
CS = [6, 4] DS = [6, 4, 0]
e2 e2 e3 − e1 e4 , e1 exp e1 e21 (over the field C: isomorphic to s6,146 with a = 1)
Casimir invariants:
• s6,142
US = [0] e6
e1 e1
e2 e2
e3 −e2 + e3
e4 e1 + e4
e5 0
CS = [6, 4] DS = [6, 4, 0]
275
276
19. SIX-DIMENSIONAL LIE ALGEBRAS
e2 e3 − e1 e4 e1 exp e21 + e22
Casimir invariants: e2 , e1
• s6,143
US = [0] e1 e1
e6
e2 e1 + e2
Casimir invariants:
e3 e3
e4 e3 + e4
CS = [6, 4] DS = [6, 4, 0]
e2 e1 exp − e1
e1 e4 − e2 e3 , e21
• s6,144
e5 0
US = [0] e6
e1 e1
e2 e1 + e2
e3 e2 + e3
e4 e3 + e4
Casimir invariants:
e5 0
CS = [6, 4] DS = [6, 4, 0]
e2 e1 exp − e1
3e4 e21 − 3e1 e2 e3 + e32 , e31
• s6,145 (over the field C: isomorphic to s6,140 ) US = [0] e6
e1 αe1 − e2
e2 e1 + αe2
where 0 ≤ α. Casimir invariants: e1 e4 − e2 e3 , e21 + e22 • s6,146
e3 αe3 − e4
(e21
+
e4 e3 + αe4
e5 0
CS = [6, 4] DS = [6, 4, 0]
e1 2α arctan e2
e22 ) exp
US = [0] e6
e1 e1
e2 ae2
e3 e1 + e3
e4 −e2 + ae4
e5 0
CS = [6, 4] DS = [6, 4, 0]
where 0 < |a| ≤ 1. If |a| = 1 then arg(a) ≤ π. Casimir invariants: ea1 e1 e4 − e2 e3 , e21 exp e2 e1 e2 • s6,147 (over the field C: isomorphic to s6,146 ) US = [0] e6
e1 αe1 − e2
e2 e1 + αe2
e3 e2 + αe3 − e4
Casimir invariants: 2e1 e4 − 2e2 e3 − e1 e2 2 2 , (e1 + e2 ) exp 2α e21 + e22
e4 e3 + αe4
e5 0
CS = [6, 4] DS = [6, 4, 0]
2e1 e4 − 2e2 e3 − e1 e2 e2 + arctan 2 2 e1 + e2 e1
19.6. SOLVABLE ALGEBRAS WITH THE NILRADICAL n5,2
• s6,148 US = [0] e1 e1
e6
e2 e1 + e2
Casimir invariants:
e3 0
e4 e3
e1 e4 − e2 e3 , e1
• s6,149
e5 e5
CS = [6, 4, 3] DS = [6, 4, 1, 0]
e2 e1 exp − e1 US = [0]
e1 e1
e6
e2 e2
Casimir invariants:
e3 0
e4 0
e5 e5
CS = [6, 3] DS = [6, 3, 0]
e1 e4 − e2 e3 e1
e2 , e1
19.6. Solvable six-dimensional Lie algebras with the nilradical n5,2 e1 0
e1 e2 e3 e4
e2 0 0
e3 0 0 0
e4 0 0 e2 0
e5 0 0 e1 e3
• s6,150 US = [1] e1 0
e6
e2 3e2
e3 e3
Casimir invariants:
e4 2e4
e5 −e5
CS = [6, 5] DS = [6, 5, 3, 0]
(2e1 e4 − 2e2 e5 − e23 )3 e22
e1 , • s6,151
US = [0] e6
e1 e1
e2 −e2
Casimir invariants: e1 e2 ,
e3 0
e4 e2 − e4
e5 e5
CS = [6, 5] DS = [6, 5, 3, 0]
−2e1 e4 + 2e2 e5 + e23 e21 exp e1 e2
• s6,152 (over the field C: isomorphic to s6,151 ) US = [0] e6 where = ±1.
e1 e2
e2 −e1
e3 0
e4 −e5
e5 e2 + e4
CS = [6, 5] DS = [6, 5, 3, 0]
277
278
19. SIX-DIMENSIONAL LIE ALGEBRAS
Casimir invariants: e1 e2 + 2e1 e4 − 2e2 e5 − e23 − (e21 + e22 ) arctan
e21 + e22 ,
e2 e1
• s6,153 US = [0] e6
e1 3e1
e2 e1 + 3e2
e3 2e3
e4 e4 + e5
Casimir invariants:
e5 e5
CS = [6, 5] DS = [6, 5, 3, 0]
e2 e1 exp −3 e1
(2e1 e4 − 2e2 e5 − e23 )3 , e41
• s6,154 (over the field C: isomorphic to s6,155 ) US = [0] e6
e1 3αe1 + e2
e2 −e1 + 3αe2
e3 2αe3
Casimir invariants: (2e1 e4 − 2e2 e5 − e23 )3 , (e21 + e22 )2 • s6,155
e4 αe4 − e5
e5 e4 + αe5
CS = [6, 5] DS = [6, 5, 3, 0]
e2 (e21 + e22 ) exp 6α arctan e1 US = [0]
e6
e1 (2a + 1)e1
where 0 < |a| ≤ 1, a = Casimir invariants:
e2 (a + 2)e2
e3 (a + 1)e3
e4 e4
e5 ae5
CS = [6, 5] DS = [6, 5, 3, 0]
If |a| = 1 then arg(a) ≤ π.
− 12 .
e2a+1 2 , ea+2 1
(2e1 e4 − 2e2 e5 − e23 )2a+1 e2a+2 1
• s6,156 US = [0] e6
e1 e1
e2 2e2
Casimir invariants: e21 , e2
• s6,157
e3 e3
e4 e4
e5 0
CS = [6, 4] DS = [6, 4, 1, 0]
2e1 e4 − 2e2 e5 − e23 e2 US = [0]
e6
e1 e1
e2 2e2
e3 e3
e4 e1 + e4
where the value of the parameter is over the field C: = 1; over the field R: = ±1.
e5 0
CS = [6, 4] DS = [6, 4, 1, 0]
19.7. SOLVABLE ALGEBRAS WITH THE NILRADICAL n5,3
Casimir invariants:
2e1 e4 − 2e2 e5 − e23 e21 exp − e21
e21 , e2
19.7. Solvable six-dimensional Lie algebras with the nilradical n5,3 e1 0
e1 e2 e3 e4
e2 0 0
e3 0 0 0
e4 0 e1 0 0
e5 0 0 e1 0
• s6,158 US = [1, 3] e1 0
e6
e2 0
e3 e3
e4 0
e5 −e5
CS = [6, 3] DS = [6, 3, 1, 0]
Casimir invariants: • s6,159
e1 , e1 e6 + e3 e5 (over the field C: isomorphic to s6,158 ) US = [1, 3] e1 0
e6
e2 0
e3 e5
e4 0
e5 −e3
e1 ,
2e1 e6 + e23 + e25
e3 e3
e4 e2
CS = [6, 3] DS = [6, 3, 1, 0]
Casimir invariants: • s6,160 US = [1, 2, 3] e1 0
e6
e2 0
e5 −e5
CS = [6, 4, 3] DS = [6, 4, 1, 0]
Casimir invariants: • s6,161
e1 , 2e1 e6 + 2e3 e5 − e22 (over the field C: isomorphic to s6,160 ) US = [1, 2, 3] e1 0
e6
e2 0
e3 e5
e4 e2
e5 −e3
CS = [6, 4, 3] DS = [6, 4, 1, 0]
Casimir invariants: e1 ,
2e1 e6 − e22 + e23 + e25
• s6,162 US = [1] e6
e1 0
e2 e2
e3 ae3
e4 −e4
e5 −ae5
CS = [6, 5] DS = [6, 5, 1, 0]
where 0 < |a| ≤ 1 and arg(a) < π. If |a| = 1 then arg(a)
0. Casimir invariants: (e21 + e22 )β e4 , exp −2β e2α e3 3
US = [0]
e5 0 0
CS = [6, 4] DS = [6, 4, 0]
e1 e3 exp β arctan e2
• s6,224 (over the field C: isomorphic to s6,214 , resp. s6,215 when β = 0) e5 e6
e1 αe1 − e2 γe1
e2 e1 + αe2 γe2
e3 βe3 e3
e4 βe4 e3 + e4
US = [0]
e5 0 0
CS = [6, 4] DS = [6, 4, 0]
where α ≥ 0, β 2 + γ 2 = 0. If α = 0 then β ≥ 0. Casimir invariants: e2α e4 e4 e1 3 , e3 exp − + β arctan exp 2(βγ − α) (e21 + e22 )β e3 e3 e2 When β → 0 these invariants become dependent and one of them should be replaced by e2γ e1 3 exp 2(βγ − α) arctan e21 + e22 e2 • s6,225 (over the field C: isomorphic to s6,220 ) e5 e6
e1 −e2 e1
e2 e1 e2
e3 e1 − e4 αe1 − βe2 + e3
where α ≥ 0. Casimir invariants: e2 e3 − e1 e4 , (e21 + e22 )β exp 2 2 e1 + e22
e4 e2 + e3 βe1 + αe2 + e4 -
(e21
+
e22 )α
e5 0 0
US = [0] CS = [6, 4] DS = [6, 4, 0]
e1 e3 + e2 e4 e1 exp −2 + arctan e21 + e22 e2
• s6,226 (over the field C: isomorphic to s6,213 ) e5 e6
e1 −e2 e1
e2 e1 e2
e3 αe3 − βe4 γe3
e4 βe3 + αe4 γe4
e5 0 0
US = [0] CS = [6, 4] DS = [6, 4, 0]
.
19.12. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ n1,1
293
where 0 < β ≤ 1, α ≥ 0. If β = 1 then |γ| ≤ 1. Casimir invariants: e4 e2 e23 + e24 e2 arctan − β arctan , exp −2α arctan e3 e1 (e21 + e22 )γ e1 • s6,227 (over the field C: isomorphic to s6,214 , resp. s6,215 when β = 0) e1 −e2 e1
e5 e6
e2 e1 e2
e3 αe3 βe3
where α ≥ 0, α + β = 0. Casimir invariants: e4 e1 + arctan , e3 e2 2
e4 e3 + αe4 βe4
US = [0]
e5 0 0
CS = [6, 4] DS = [6, 4, 0]
2
e23 e1 exp 2α arctan (e21 + e22 )β e2
• s6,228 (over the field C: isomorphic to s6,213 ) e5 e6
e1 αe1 − e2 γe1
e2 e1 + αe2 γe2
e3 βe3 δe3 − e4
e4 βe4 e3 + δe4
e5 0 0
US = [0] CS = [6, 4] DS = [6, 4, 0]
where 0 ≤ β ≤ γ, 0 < γ. If β = 0 then α ≥ 0. If β = γ then α ≤ δ. Casimir invariants: - . e2 e4 2 2 , + γ arctan (e1 + e2 ) exp −2 α arctan e1 e3 exp 2β arctan(e2 /e1 ) + 2δ arctan(e4 /e3 ) e23 + e24 19.12. Solvable six-dimensional Lie algebras with the nilradical n3,1 ⊕ n1,1
e1 e2 e3
e1 0
e2 0 0
e3 0 e1 0
e4 0 0 0
• s6,229 e5 e6
e1 0 0
e2 e2 0
e3 −e3 0
e4 0 e4
e5 0 −e1
US = [1] CS = [6, 4] DS = [6, 4, 1, 0]
e1 e5 + e2 e3 e1 , e4 exp e21 (over the field C: isomorphic to s6,229 )
Casimir invariants: • s6,230
e5 e6
e1 0 0
e2 e3 0
e3 −e2 0
e4 0 e4
e5 0 e1
US = [1] CS = [6, 4] DS = [6, 4, 1, 0]
294
19. SIX-DIMENSIONAL LIE ALGEBRAS
Casimir invariants: e24
e1 ,
2e1 e5 + e22 + e23 exp − e21
• s6,231 e1 e1 e1
e5 e6
e2 e2 0
Casimir invariants:
e3 0 e3
e4 0 0
US = [1]
e5 0 e4
CS = [6, 4, 3] DS = [6, 4, 1, 0]
e1 e6 − e1 e5 − e2 e3 e1 exp e1 e4
e4 ,
• s6,232 (over the field C: isomorphic to s6,231 ) e1 2e1 0
e5 e6
e2 e2 e3
Casimir invariants:
e3 e3 −e2
e4 0 0
e5 0 e4
US = [1] CS = [6, 4, 3] DS = [6, 4, 1, 0]
2e1 e6 + e22 + e23 e1 exp e1 e4
e4 , • s6,233 e1 0 e1
e5 e6
e2 e2 0
e3 −e3 e3
e4 0 e1 + e4
Casimir invariants:
e5 0 0
US = [0] CS = [6, 4] DS = [6, 4, 1, 0]
e1 e5 + e2 e3 , e1
e4 e1 exp − e1
e2 e2 0
e5 0 0
• s6,234 e5 e6
e1 e1 e1
e3 0 e3
e4 ae4 be4
US = [0] CS = [6, 4] DS = [6, 4, 1, 0]
where |a| ≤ |b|, b = 0. If |a| = |b| then arg(a) ≤ arg(b). Casimir invariant: none • s6,235 e5 e6
e1 2e1 0
e2 e2 e3
e3 e3 0
e4 ae4 e4
e5 0 0
Casimir invariant: none
US = [0] CS = [6, 4] DS = [6, 4, 1, 0]
19.12. SOLVABLE ALGEBRAS WITH THE NILRADICAL n3,1 ⊕ n1,1
• s6,236 e1 e1 0
e5 e6
e2 ae2 e2
e3 (1 − a)e3 −e3
e4 e4 e1
e5 0 0
US = [0]
e5 0 0
US = [0]
CS = [6, 4] DS = [6, 4, 1, 0]
where Im a ≥ 0. If a ∈ R then a ≥ 1. Casimir invariant: none • s6,237 e1 (1 + a)e1 e1
e5 e6
e2 ae2 e2
e3 e3 e4
e4 e4 0
CS = [6, 4] DS = [6, 4, 1, 0]
Casimir invariant: none • s6,238 e1 e1 e1
e5 e6
e2 e2 0
e3 0 e3 + e4
e4 0 e4
e5 0 0
US = [0] CS = [6, 4] DS = [6, 4, 1, 0]
Casimir invariant: • s6,239
none (over the field C: isomorphic to s6,234 ) e1 2e1 0
e5 e6
e2 e2 e3
e3 e3 −e2
e4 αe4 βe4
e5 0 0
US = [0] CS = [6, 4] DS = [6, 4, 1, 0]
where α + β = 0, β ≥ 0. Casimir invariant: 2
2
• s6,240
none (over the field C: isomorphic to s6,236 ) e1 2e1 0
e5 e6
e2 e2 e3
e3 e3 −e2
e4 2e4 e1
e5 0 0
US = [0] CS = [6, 4] DS = [6, 4, 1, 0]
Casimir invariant: • s6,241 e5 e6
none (over the field C: isomorphic to s6,236 )
e1 2αe1 2e1
e2 αe2 − e3 e2
e3 e2 + αe3 e3
e4 2αe4 e1 + 2e4
where α ≥ 0. Casimir invariant: none
e5 0 0
US = [0] CS = [6, 4] DS = [6, 4, 1, 0]
295
296
19. SIX-DIMENSIONAL LIE ALGEBRAS
19.13. Solvable six-dimensional Lie algebra with the nilradical n4,1
e1 e2 e3
e1 0
e2 0 0
e3 0 0 0
e4 0 e1 e2
• s6,242 e1 2e1 e1
e5 e6
e2 e2 e2
e3 0 e3
e4 e4 0
US = [0]
e5 0 0
CS = [6, 4] DS = [6, 4, 2, 0]
Casimir invariant: none 19.14. Simple six-dimensional Lie algebra • so(1, 3) (over the field C: isomorphic to sl(2, C) ⊕ sl(2, C)) e1 0
e1 e2 e3 e4 e5
e2 e3 0
e3 −e2 e1 0
e4 0 −e6 e5 0
e5 e6 0 −e4 −e3 0
e6 −e5 e4 0 e2 −e1
US = [0] CS = [6] DS = [6]
Casimir invariants: e1 e4 + e2 e5 + e3 e6 ,
e21 + e22 + e23 − e24 − e25 − e26
19.15. Six-dimensional Levi decomposable Lie algebras • sl(2, F) n3,1 e1 e2 e3 e4 e5
e1 0
e2 2e1 0
e3 −e2 2e3 0
e4 0 0 0 0
e5 e6 e5 0 0 0
e6 0 −e6 e5 0 e4
US = [1] CS = [6] DS = [6]
Casimir invariants: e4 ,
4e1 e3 e4 − 2e1 e25 + e22 e4 + 2e2 e5 e6 + 2e3 e26
• sl(2, F) 3n1,1 e1 e2 e3 e4 e5
e1 0
e2 2e1 0
e3 −e2 2e3 0
e4 0 −2e4 e5 0
e5 2e4 0 −2e6 0 0
e6 −e5 2e6 0 0 0
US = [0] CS = [6] DS = [6]
19.15. LEVI DECOMPOSABLE ALGEBRAS
Casimir invariants: 2e1 e6 + e2 e5 + 2e3 e4 ,
4e6 e4 + e25
• so(3, R) 3n1,1 (over the field C: isomorphic to sl(2, C) 3n1,1 ) e1 e2 e3 e4 e5
e1 0
e2 e3 0
e3 −e2 e1 0
e4 0 −e6 e5 0
e5 e6 0 −e4 0 0
e6 −e5 e4 0 0 0
US = [0] CS = [6] DS = [6]
Casimir invariants: e1 e4 + e2 e5 + e3 e6 ,
e24 + e25 + e26
• sl(2, F) s3,1,a=1 e1 e2 e3 e4 e5
e1 0
e2 2e1 0
e3 −e2 2e3 0
e4 e5 e4 0 0
e5 0 −e5 e4 0 0
Casimir invariant: none
e6 0 0 0 e4 e5
US = [0] CS = [6, 5] DS = [6, 5]
297
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Index
epimorphism, 71
ad-diagonalizable, 18 algorithm direct sum decomposition, 56 Levi decomposition, 63 nilradical, 75 associative algebra semisimple, 48 automorphism, 15
filiform algebra, 175 adapted basis, 175 model, 175 N-graded, 190 special, 176 Galilei algebra, 131 extended, 131
Borel nilradical, 142 Borel subalgebra, 142
Heisenberg algebra, 131 highest root, 143 hypercenter, 12, 73
Cartan matrix, 19 Cartan subalgebra, 17 Casimir invariant, 23 generalized, 23, 24, 41, 218 Casimir operator, 23 center, 12 higher, 12 central automorphism, 51 central decomposition, 51 centralizer, 13 characteristic sequence, 46 characteristically nilpotent, 100 coboundary, 21, 94, 101 cochain, 20 cochain complex, 21 cocycle, 21, 94, 101 twisted, 45 cohomology, 21, 95, 101, 108 cohomology operator, 21 commuting algebra, 47 composition series, 71
ideal, 11 characteristic, 15 idempotent, 47 orthogonal, 47 trivial, 47 invariant form, 16 invariant subspace, 14 Killing form, 16, 41, 97 Kravchuk normal form, 93 Kravchuk signature, 93 Levi decomposition, 3, 63, 203, 241, 296 nontrivial, 203 Levi extension, 204 Levi factor, 6, 17, 63 Lie algebra, 11 absolutely indecomposable, 53, 56 decomposable, 47, 50, 56 indecomposable, 54 nilpotent, 12, 89, 225, 227, 231, 243 perfect, 11, 63 rank, 18, 40 semisimple, 13 simple, 11, 17, 226, 296 solvable, 11, 99, 225–227, 232, 248 symplectic, 131 linearly nilindependent elements, 99 linearly nilindependent matrices, 99
decomposable matrix, 47 degree of nilpotency, 12 derivation, 15, 42 (α, β, γ)-, 42 inner, 15 outer, 15 derived algebra, 11 division ring, 48 Dixmier invariant, 41 Dynkin diagram, 19, 40
Maple 305
306
computer algebra system, 219 LieAlgebras package, 219 method of characteristics, 24 method of moving frames, 32 minimal polynomial, 54 nilpotent element, 99 nilradical, 13, 71, 99 Abelian, 102, 107 Borel, 141 filiform, 175 Heisenberg, 131 normalizer, 13 point transformation, 4 radical, 3, 12, 17, 63, 64, 75 Jacobson, 48 rank of nilpotent algebra, 46 representation, 14 adjoint, 14 faithful, 14 fully reducible, 14 irreducible, 14 reducible, 14 root, 18 positive, 18 simple, 18 root subspace, 18 root system, 18 semisimple element, 18 series characteristic, 12, 40 derived (DS), 11, 218 lower central (CS), 12, 218 upper central (US), 12, 218 solvable extension, 100 split real form, 18, 142 subalgebra, 11 theorem Cartan’s criteria, 16, 39 Levi, 3, 17 Schur lemma, 14 triangular nilradical, 154 Weyl group, 19 Weyl – Chevalley basis, 18
INDEX
Published Titles in This Series ˇ 33 Libor Snobl and Pavel Winternitz, Classification and Identification of Lie Algebras, 2014 32 Pavel Bleher and Karl Liechty, Random Matrices and the Six-Vertex Model, 2014 31 Jean-Pierre Labesse and Jean-Loup Waldspurger, La Formule des Traces Tordue d’apr` es le Friday Morning Seminar, 2013 30 Joseph H. Silverman, Moduli Spaces and Arithmetic Dynamics, 2012 29 Marcelo Aguiar and Swapneel Mahajan, Monoidal Functors, Species and Hopf Algebras, 2010 28 Saugata Ghosh, Skew-Orthogonal Polynomials and Random Matrix Theory, 2009 27 Jean Berstel, Aaron Lauve, Christophe Reutenauer, and Franco V. Saliola, Combinatorics on Words, 2008 26 Victor Guillemin and Reyer Sjamaar, Convexity Properties of Hamiltonian Group Actions, 2005 25 Andrew J. Majda, Rafail V. Abramov, and Marcus J. Grote, Information Theory and Stochastics for Multiscale Nonlinear Systems, 2005 24 Dana Schlomiuk, Andre˘ıA. Bolibrukh, Sergei Yakovenko, Vadim Kaloshin, and Alexandru Buium, On Finiteness in Differential Equations and Diophantine Geometry, 2005 23 J. J. M. M. Rutten, Marta Kwiatkowska, Gethin Norman, and David Parker, Mathematical Techniques for Analyzing Concurrent and Probabilistic Systems, 2004 22 Montserrat Alsina and Pilar Bayer, Quaternion Orders, Quadratic Forms, and Shimura Curves, 2004 21 Andrei Tyurin, Quantization, Classical and Quantum Field Theory and Theta Functions, 2003 20 Joel Feldman, Horst Kn¨ orrer, and Eugene Trubowitz, Riemann Surfaces of Infinite Genus, 2003 19 18 17 16
L. Lafforgue, Chirurgie des grassmanniennes, 2003 G. Lusztig, Hecke Algebras with Unequal Parameters, 2003 Michael Barr, Acyclic Models, 2002 Joel Feldman, Horst Kn¨ orrer, and Eugene Trubowitz, Fermionic Functional Integrals and the Renormalization Group, 2002 15 Jos´ e I. Burgos Gil, The Regulators of Beilinson and Borel, 2002 14 Eyal Z. Goren, Lectures on Hilbert Modular Varieties and Modular Forms, 2002 13 Michael Baake and Robert V. Moody, Editors, Directions in Mathematical Quasicrystals, 2000 12 Masayoshi Miyanishi, Open Algebraic Surfaces, 2000 11 Spencer J. Bloch, Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves, 2000 10 James D. Lewis and B. Brent Gordon, A Survey of the Hodge Conjecture, Second Edition, 1999 9 Yves Meyer, Wavelets, Vibrations and Scalings, 1998 8 7 6 5
Ioannis Karatzas, Lectures on the Mathematics of Finance, 1997 John Milton, Dynamics of Small Neural Populations, 1996 Eugene B. Dynkin, An Introduction to Branching Measure-Valued Processes, 1994 Andrew Bruckner, Differentiation of Real Functions, 1994
4 David Ruelle, Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval, 1994 3 V. Kumar Murty, Introduction to Abelian Varieties, 1993 2 M. Ya. Antimirov, A. A. Kolyshkin, and R´ emi Vaillancourt, Applied Integral Transforms, 1993 1 Dan Voiculescu, Kenneth J. Dykema, and Alexandru Nica, Free Random Variables, 1992
ISBN:978-0-8218-4355-0
9
80821 843550
CRMM/33
AMS 011 the Web www.ams.org
Classification and Identification of Lie Algebras ˇ Libor Snobl and Pavel Winternitz Department of Physics Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague
Analytic and algebraic methods in physics XII, November 2, 2013 the meeting dedicated to the 75th birthday of professor Miloslav Havl´ıˇcek ˇ Libor Snobl
Classification and Identification of Lie Algebras
Outline
1
The monograph: Classification and Identification of Lie Algebras
2
Why to be interested in identification and classification of Lie algebras?
3
What can be found in our book
4
Conclusions
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Outline
1
The monograph: Classification and Identification of Lie Algebras
2
Why to be interested in identification and classification of Lie algebras?
3
What can be found in our book
4
Conclusions
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Outline
1
The monograph: Classification and Identification of Lie Algebras
2
Why to be interested in identification and classification of Lie algebras?
3
What can be found in our book
4
Conclusions
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Outline
1
The monograph: Classification and Identification of Lie Algebras
2
Why to be interested in identification and classification of Lie algebras?
3
What can be found in our book
4
Conclusions
ˇ Libor Snobl
Classification and Identification of Lie Algebras
The monograph: Classification and Identification of Lie Algebras Title: Classification and Identification of Lie Algebras ˇ Authors: Libor Snobl and Pavel Winternitz To appear in CRM Monograph Series, vol. 33, published by the American Mathematical Society in the collaboration with the Centre de Recherches Math´ematiques. ISBN-10: 0-8218-4355-9 ISBN-13: 978-0-8218-4355-0 In press, expected publication date: February 24, 2014
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Why to be interested in identification and classification of Lie algebras? Example: Consider two systems of PDEs, namely shallow water equations in the flat infinite basin UT + UUX + VUY + HX = 0, VT + UVX + VVY + HY = 0, HT + (UH)X + (VH)Y = 0 (1) and in the circular paraboloidal basin subjected to a Coriolis force due to the movement of the fluid inside the basin together with the Earth ω2 − f 2 2 (x + y 2 ), 8 vt + uvx + vvy + (Z + h)y = −fu, ht + (uh)x + (vh)y = 0 (2) and compute their algebras of infinitesimal point symmetries. ut + uux + vuy + (Z + h)x = fv ,
ˇ Libor Snobl
Z (x, y ) =
Classification and Identification of Lie Algebras
Why to be interested in identification and classification of Lie algebras? Example: Consider two systems of PDEs, namely shallow water equations in the flat infinite basin UT + UUX + VUY + HX = 0, VT + UVX + VVY + HY = 0, HT + (UH)X + (VH)Y = 0 (1) and in the circular paraboloidal basin subjected to a Coriolis force due to the movement of the fluid inside the basin together with the Earth ω2 − f 2 2 (x + y 2 ), 8 vt + uvx + vvy + (Z + h)y = −fu, ht + (uh)x + (vh)y = 0 (2) and compute their algebras of infinitesimal point symmetries. ut + uux + vuy + (Z + h)x = fv ,
ˇ Libor Snobl
Z (x, y ) =
Classification and Identification of Lie Algebras
Symmetry algebras of shallow water equations, flat basin
One finds two seemingly different 9–dimensional symmetry algebras gA and gB spanned by the following vector fields, respectively PT = ∂T , PX = ∂X , PY = ∂Y , GX = T ∂X + ∂U , GY = T ∂Y + ∂V , D1 = T ∂T + X ∂X + Y ∂Y , D2 = −T ∂T + U∂U + V ∂V + 2H∂H , L1 = −Y ∂X + X ∂Y − V ∂U + U∂V , Π = T 2 ∂T + TX ∂X + TY ∂Y + (X − TU)∂U +(Y − TV )∂V − 2TH∂H
ˇ Libor Snobl
Classification and Identification of Lie Algebras
(3)
Symmetry algebra of shallow water equations, flat basin, gA
PT PX PY GX GY D1 D2 L1 Π
PT 0 0 0 −PX −PY −PT PT 0 −D1 + D2
PX 0 0 0 0 0 −PX 0 −PY −GX
PY 0 0 0 0 0 −PY 0 PX −GY
GX PX 0 0 0 0 0 −GX −GY 0
ˇ Libor Snobl
GY PY 0 0 0 0 0 −GY GX 0
D1 PT PX PY 0 0 0 0 0 −Π
D2 −PT 0 0 GX GY 0 0 0 Π
L1 0 PY −PX GY −GX 0 0 0 0
Π D1 − D2 GX GY 0 0 Π −Π 0 0
Classification and Identification of Lie Algebras
Symmetry algebra of shallow water equations, paraboloidal basin, Coriolis force P0 = ∂t ,
D = x∂x + y ∂y + u∂u + v ∂v + 2h∂h ,
Y1 = cos(R1 t)∂x − sin(R1 t)∂y − R1 sin(R1 t)∂u − R1 cos(R1 t)∂v , Y2 = sin(R1 t)∂x + cos(R1 t)∂y + R1 cos(R1 t)∂u − R1 sin(R1 t)∂v , Y3 = cos(R2 t)∂x + sin(R2 t)∂y − R2 sin(R2 t)∂u + R2 cos(R2 t)∂v , Y4 = sin(R2 t)∂x − cos(R2 t)∂y + R2 cos(R2 t)∂u + R2 sin(R2 t)∂v , R = y ∂x − x∂y + v ∂u − u∂v , 1 cos(ωt) (x∂x + y ∂y − u∂u − v ∂v + f (y ∂u − x∂v ) − 2h∂h ) + 2 1 sin(ωt) f (y ∂x − x∂y + v ∂u − u∂v ) − ω 2 (x∂u + y ∂v ) + 2∂t , + 2ω 1 K2 = − sin(ωt) (x∂x + y ∂y − u∂u − v ∂v + f (y ∂u − x∂v ) − 2h∂h ) 2 1 + cos(ωt) f (y ∂x − x∂y + v ∂u − u∂v ) − ω 2 (x∂u + y ∂v ) + 2∂t 2ω
K1 =
(4)
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Symmetry algebra of shallow water equations, paraboloidal basin, Coriolis force, gB where1
P0
P0 0
K1 K2 D R Y1 Y2
−ωK2 ωK1 0 0 f +ω Y 2 2 Y1 − f +ω 2 ω−f Y4 2 f −ω Y3 2
Y3 Y4
1 R1 = (ω + f ), 2 K1 ωK2 0 + f2 R) 0 0 1Y 2 3 − 12 Y4
1 (P 0 ω
1Y 2 1
− 12 Y2
K2 −ωK1 −1 (P0 ω
+ f2 R) 0 0 0 − 12 Y4 − 12 Y3 − 12 Y2 − 12 Y1
1 R2 = (ω − f ). 2 D 0
R 0
Y1 Y2 − f +ω 2
Y2 f +ω Y 1 2 1Y 2 4 1Y 2 3
Y3 f −ω Y4 2 − 21 Y1 1Y 2 2
Y4 ω−f Y3 2 1Y 2 2 1Y 2 1
0 0 0 0 Y1 Y2
0 0 0 0 −Y2 Y1
− 12 Y3 1Y 2 4 −Y1 Y2 0 0
Y3
Y4
Y4
−Y3
0
0
0
0
0
0
0
0
−Y2 −Y1 0 0
−Y3 −Y4 0 0
1
D. Levi, M.C. Nucci, C. Rogers, P. Winternitz 1989 J. Phys. A 22 4743–4767 ˇ Libor Snobl
Classification and Identification of Lie Algebras
−Y4 Y3 0 0
Symmetry algebra of shallow water equations, paraboloidal basin, Coriolis force, gB where1
P0
P0 0
K1 K2 D R Y1 Y2
−ωK2 ωK1 0 0 f +ω Y 2 2 Y1 − f +ω 2 ω−f Y4 2 f −ω Y3 2
Y3 Y4
1 R1 = (ω + f ), 2 K1 ωK2 0 + f2 R) 0 0 1Y 2 3 − 12 Y4
1 (P 0 ω
1Y 2 1
− 12 Y2
K2 −ωK1 −1 (P0 ω
+ f2 R) 0 0 0 − 12 Y4 − 12 Y3 − 12 Y2 − 12 Y1
1 R2 = (ω − f ). 2 D 0
R 0
Y1 Y2 − f +ω 2
Y2 f +ω Y 1 2 1Y 2 4 1Y 2 3
Y3 f −ω Y4 2 − 21 Y1 1Y 2 2
Y4 ω−f Y3 2 1Y 2 2 1Y 2 1
0 0 0 0 Y1 Y2
0 0 0 0 −Y2 Y1
− 12 Y3 1Y 2 4 −Y1 Y2 0 0
Y3
Y4
Y4
−Y3
0
0
0
0
0
0
0
0
−Y2 −Y1 0 0
−Y3 −Y4 0 0
1
D. Levi, M.C. Nucci, C. Rogers, P. Winternitz 1989 J. Phys. A 22 4743–4767 ˇ Libor Snobl
Classification and Identification of Lie Algebras
−Y4 Y3 0 0
The Lie algebra gB of (4)
The Lie algebra gB of (4) has the radical (maximal solvable ideal) R(gB ) = span{D, R, Y1 , Y2 , Y3 , Y4 }, the nilradical (maximal nilpotent ideal) NR(gB ) = span{Y1 , Y2 , Y3 , Y4 }, and the Levi factor (semisimple subalgebra complementing the radical) f p = span{P0 + R, K1 , K2 }. 2
ˇ Libor Snobl
Classification and Identification of Lie Algebras
The Lie algebra gB of (4)
The Lie algebra gB of (4) has the radical (maximal solvable ideal) R(gB ) = span{D, R, Y1 , Y2 , Y3 , Y4 }, the nilradical (maximal nilpotent ideal) NR(gB ) = span{Y1 , Y2 , Y3 , Y4 }, and the Levi factor (semisimple subalgebra complementing the radical) f p = span{P0 + R, K1 , K2 }. 2
ˇ Libor Snobl
Classification and Identification of Lie Algebras
The Lie algebra gB of (4)
The Lie algebra gB of (4) has the radical (maximal solvable ideal) R(gB ) = span{D, R, Y1 , Y2 , Y3 , Y4 }, the nilradical (maximal nilpotent ideal) NR(gB ) = span{Y1 , Y2 , Y3 , Y4 }, and the Levi factor (semisimple subalgebra complementing the radical) f p = span{P0 + R, K1 , K2 }. 2
ˇ Libor Snobl
Classification and Identification of Lie Algebras
The Lie algebra gB of (4), continued
In the adjoint representation of gB the element D acts on the nilradical NR(gB ) diagonally as a multiple of a unit matrix whereas R acts on it as a rotation. Both elements commute with the Levi factor. From the indefinite signature of the Killing form of the Levi factor p it follows that p is isomorphic to the simple algebra sl(2, R). The adjoint action of the Levi factor p on the nilradical NR(gB ) corresponds to a direct sum of two 2–dimensional irreducible representations of sl(2, R).
ˇ Libor Snobl
Classification and Identification of Lie Algebras
The Lie algebra gB of (4), continued
In the adjoint representation of gB the element D acts on the nilradical NR(gB ) diagonally as a multiple of a unit matrix whereas R acts on it as a rotation. Both elements commute with the Levi factor. From the indefinite signature of the Killing form of the Levi factor p it follows that p is isomorphic to the simple algebra sl(2, R). The adjoint action of the Levi factor p on the nilradical NR(gB ) corresponds to a direct sum of two 2–dimensional irreducible representations of sl(2, R).
ˇ Libor Snobl
Classification and Identification of Lie Algebras
The Lie algebra gA of (3) Somewhat surprisingly, the Lie algebra gA of (3) has the same structure. When expressed in suitable bases which make the structure transparent, the two algebras gA and gB turn out to be isomorphic as Lie algebras. Namely, the Lie brackets expressed in following two bases of gA and gB , respectively, e1 = PT , e2 = D1 − D2 , e3 = −Π, e4 = −(D1 + D2 ), e5 = L1 , e6 = PY , e7 = PX , e8 = GY , e9 = GX , f 1 P0 + R + K2 , e˜2 = −2K1 , e˜1 = − ω 2 1 f e˜3 = P0 + R + K2 , e˜4 = −D, e˜5 = R, e˜6 = Y1 − Y3 , ω 2 e˜7 = Y2 + Y4 , e˜8 = −Y2 + Y4 , e˜9 = Y1 + Y3 imply the same structure constants ˇ Libor Snobl
Classification and Identification of Lie Algebras
The Lie algebra gA of (3) Somewhat surprisingly, the Lie algebra gA of (3) has the same structure. When expressed in suitable bases which make the structure transparent, the two algebras gA and gB turn out to be isomorphic as Lie algebras. Namely, the Lie brackets expressed in following two bases of gA and gB , respectively, e1 = PT , e2 = D1 − D2 , e3 = −Π, e4 = −(D1 + D2 ), e5 = L1 , e6 = PY , e7 = PX , e8 = GY , e9 = GX , 1 f e˜1 = − P0 + R + K2 , e˜2 = −2K1 , ω 2 1 f e˜3 = P0 + R + K2 , e˜4 = −D, e˜5 = R, e˜6 = Y1 − Y3 , ω 2 e˜7 = Y2 + Y4 , e˜8 = −Y2 + Y4 , e˜9 = Y1 + Y3 imply the same structure constants ˇ Libor Snobl
Classification and Identification of Lie Algebras
The Lie algebra gA of (3) Somewhat surprisingly, the Lie algebra gA of (3) has the same structure. When expressed in suitable bases which make the structure transparent, the two algebras gA and gB turn out to be isomorphic as Lie algebras. Namely, the Lie brackets expressed in following two bases of gA and gB , respectively, e1 = PT , e2 = D1 − D2 , e3 = −Π, e4 = −(D1 + D2 ), e5 = L1 , e6 = PY , e7 = PX , e8 = GY , e9 = GX , 1 f e˜1 = − P0 + R + K2 , e˜2 = −2K1 , ω 2 1 f e˜3 = P0 + R + K2 , e˜4 = −D, e˜5 = R, e˜6 = Y1 − Y3 , ω 2 e˜7 = Y2 + Y4 , e˜8 = −Y2 + Y4 , e˜9 = Y1 + Y3 imply the same structure constants ˇ Libor Snobl
Classification and Identification of Lie Algebras
The Lie algebras gA ' gB
e1 e2 e3 e4 e5 e6 e7 e8 e9
e1 e2 e3 0 2e1 −e2 −2e1 0 2e3 e2 −2e3 0 0 0 0 0 0 0 0 e6 −e8 0 e7 −e9 −e6 −e8 0 −e7 −e9 0
e4 e5 e6 e7 0 0 0 0 0 0 −e6 −e7 0 0 e8 e9 0 0 e6 e7 0 0 e7 −e6 −e6 −e7 0 0 −e7 e6 0 0 −e8 −e9 0 0 −e9 e8 0 0
ˇ Libor Snobl
e8 e9 e6 e7 e8 e9 0 0 e8 e9 e9 −e8 0 0 0 0 0 0 0 0
Classification and Identification of Lie Algebras
Isomorphisms between vector field realizations
That does not by itself imply that the two sets of vector fields (3) and (4) are related to each other by a point transformation but it is a necessary condition for it and a hint that such a transformation may exist. Indeed, using computer algebra we find a locally invertible map Φ : R6 [t, x, y , u, v , h] → R6 [T , X , Y , U, V , H] which transforms the algebra of vector fields (3) into (4). Explicitly, the transformation Φ reads
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Isomorphisms between vector field realizations
That does not by itself imply that the two sets of vector fields (3) and (4) are related to each other by a point transformation but it is a necessary condition for it and a hint that such a transformation may exist. Indeed, using computer algebra we find a locally invertible map Φ : R6 [t, x, y , u, v , h] → R6 [T , X , Y , U, V , H] which transforms the algebra of vector fields (3) into (4). Explicitly, the transformation Φ reads
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Isomorphisms between vector field realizations
That does not by itself imply that the two sets of vector fields (3) and (4) are related to each other by a point transformation but it is a necessary condition for it and a hint that such a transformation may exist. Indeed, using computer algebra we find a locally invertible map Φ : R6 [t, x, y , u, v , h] → R6 [T , X , Y , U, V , H] which transforms the algebra of vector fields (3) into (4). Explicitly, the transformation Φ reads
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Mapping the algebra of vector fields gA to gB
T H U
V
ω 1 f f t , X = cos t x − sin t y , 2 2 2 2 sin ω2 t ω 2 1 f f sin t , Y =− t x + cos t y , = C h sin ω 2 2 2 2 sin 2 t 1 = −2 sin ω2 t cos f2 t u + 2 sin ω2 t sin f2 t v + 2ω + sin ω2 t sin f2 t f + cos f2 t cos ω2 t ω x+ (5) ω f ω f + sin 2 t cos ( 2 t) f − sin 2 t cos 2 t ω y , 1 = 2 sin ω2 t sin f2 t u + 2 sin ω2 t cos f2 t v + 2ω + sin ω2 t cos f2 t f − sin f2 t cos ω2 t ω x− − sin ω2 t sin f2 t f + cos f2 t cos ω2 t ω y , =
cot
where C is an integration constant. ˇ Libor Snobl
Classification and Identification of Lie Algebras
Equivalence of the two shallow water equations
What is more, for a particular choice of the parameter C , namely 1 C = 2, ω the two shallow water equations (1) and (2) are mapped one into the other by the change of dependent and independent variables (5). Thus, mathematically they are locally equivalent although their physical interpretation is different; any solution of one of them gives rise to a (local) solution of the other. This equivalence of equations (1) and (2) would be very difficult, if not impossible, to discover without understanding the structure of the two Lie algebras involved. ˇ Libor Snobl
Classification and Identification of Lie Algebras
Equivalence of the two shallow water equations
What is more, for a particular choice of the parameter C , namely 1 C = 2, ω the two shallow water equations (1) and (2) are mapped one into the other by the change of dependent and independent variables (5). Thus, mathematically they are locally equivalent although their physical interpretation is different; any solution of one of them gives rise to a (local) solution of the other. This equivalence of equations (1) and (2) would be very difficult, if not impossible, to discover without understanding the structure of the two Lie algebras involved. ˇ Libor Snobl
Classification and Identification of Lie Algebras
Equivalence of the two shallow water equations
What is more, for a particular choice of the parameter C , namely 1 C = 2, ω the two shallow water equations (1) and (2) are mapped one into the other by the change of dependent and independent variables (5). Thus, mathematically they are locally equivalent although their physical interpretation is different; any solution of one of them gives rise to a (local) solution of the other. This equivalence of equations (1) and (2) would be very difficult, if not impossible, to discover without understanding the structure of the two Lie algebras involved. ˇ Libor Snobl
Classification and Identification of Lie Algebras
What can be found in our book
The intention of the authors: The purpose of the monograph is to serve as a tool for practitioners of Lie algebra and Lie group theory, i.e., for those who apply Lie algebras and Lie groups to solve problems arising in science and engineering. The main motif: how to transform a randomly obtained basis of a Lie algebra into a “canonical basis” in which all basis independent features of the Lie algebra are directly visible. The book is based on material that was previously dispersed in journal articles, many of them written by one or both of the authors together with collaborators. ˇ Libor Snobl
Classification and Identification of Lie Algebras
What can be found in our book
The intention of the authors: The purpose of the monograph is to serve as a tool for practitioners of Lie algebra and Lie group theory, i.e., for those who apply Lie algebras and Lie groups to solve problems arising in science and engineering. The main motif: how to transform a randomly obtained basis of a Lie algebra into a “canonical basis” in which all basis independent features of the Lie algebra are directly visible. The book is based on material that was previously dispersed in journal articles, many of them written by one or both of the authors together with collaborators. ˇ Libor Snobl
Classification and Identification of Lie Algebras
What can be found in our book
The intention of the authors: The purpose of the monograph is to serve as a tool for practitioners of Lie algebra and Lie group theory, i.e., for those who apply Lie algebras and Lie groups to solve problems arising in science and engineering. The main motif: how to transform a randomly obtained basis of a Lie algebra into a “canonical basis” in which all basis independent features of the Lie algebra are directly visible. The book is based on material that was previously dispersed in journal articles, many of them written by one or both of the authors together with collaborators. ˇ Libor Snobl
Classification and Identification of Lie Algebras
The layout of the book
The book is divided into four parts: General Theory Recognition of a Lie Algebra Given by Its Structure Constants Nilpotent, Solvable and Levi Decomposable Lie Algebras Low-Dimensional Lie Algebras
ˇ Libor Snobl
Classification and Identification of Lie Algebras
General Theory
an introductory review of definitions and notions, a more detailed introduction into the computation of invariants of the coadjoint representation of a Lie algebra (a.k.a. generalized Casimir invariants), including the method of moving frames (Cartan, Fels, Olver, Boyko, Patera, Popovych).
ˇ Libor Snobl
Classification and Identification of Lie Algebras
General Theory
an introductory review of definitions and notions, a more detailed introduction into the computation of invariants of the coadjoint representation of a Lie algebra (a.k.a. generalized Casimir invariants), including the method of moving frames (Cartan, Fels, Olver, Boyko, Patera, Popovych).
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Recognition of a Lie Algebra Given by Its Structure Constants various invariant characteristics of a Lie algebra which can be helpful in its identification are reviewed. Among others dimensions of ideals in the characteristic series, the number and structure of generalized Casimir invariants, (α, β, γ)–derivations and twisted cocycles (Hrivn´ak, Novotn´y) etc. the algorithms for establishing decomposability (and explicit decomposition), Levi decomposition and the nilradical. These chapters are in essence a revision and correction of the paper D. Rand, P. Winternitz and H. Zassenhaus 1988 Linear algebra and its applications 109 197–246, supplemented by numerous new examples. ˇ Libor Snobl
Classification and Identification of Lie Algebras
Recognition of a Lie Algebra Given by Its Structure Constants various invariant characteristics of a Lie algebra which can be helpful in its identification are reviewed. Among others dimensions of ideals in the characteristic series, the number and structure of generalized Casimir invariants, (α, β, γ)–derivations and twisted cocycles (Hrivn´ak, Novotn´y) etc. the algorithms for establishing decomposability (and explicit decomposition), Levi decomposition and the nilradical. These chapters are in essence a revision and correction of the paper D. Rand, P. Winternitz and H. Zassenhaus 1988 Linear algebra and its applications 109 197–246, supplemented by numerous new examples. ˇ Libor Snobl
Classification and Identification of Lie Algebras
Nilpotent, Solvable and Levi Decomposable Lie Algebras
general structure of nilpotent algebras and approaches to their classification, general structure of a solvable algebra, in particular in relation with its nilradical, classification of solvable Lie algebras with the nilradicals of the given specific form including the determination of their generalized Casimir invariants, structure of Levi decomposable algebras.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Nilpotent, Solvable and Levi Decomposable Lie Algebras
general structure of nilpotent algebras and approaches to their classification, general structure of a solvable algebra, in particular in relation with its nilradical, classification of solvable Lie algebras with the nilradicals of the given specific form including the determination of their generalized Casimir invariants, structure of Levi decomposable algebras.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Nilpotent, Solvable and Levi Decomposable Lie Algebras
general structure of nilpotent algebras and approaches to their classification, general structure of a solvable algebra, in particular in relation with its nilradical, classification of solvable Lie algebras with the nilradicals of the given specific form including the determination of their generalized Casimir invariants, structure of Levi decomposable algebras.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Nilpotent, Solvable and Levi Decomposable Lie Algebras
general structure of nilpotent algebras and approaches to their classification, general structure of a solvable algebra, in particular in relation with its nilradical, classification of solvable Lie algebras with the nilradicals of the given specific form including the determination of their generalized Casimir invariants, structure of Levi decomposable algebras.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Types of nilradicals considered in arbitrary finite dimensions
Nilradicals with low degree of nilpotency (J.C. Ndogmo, J. Rubin, P. Winternitz) The algebras investigated in this class are the Abelian and Heisenberg algebras. These algebras possess large algebras of derivations that have well-understood properties. E.g., for an Abelian nilradical, any linear transformation is a derivation and any regular linear map is an automorphism. Consequently, the construction of solvable extensions is reduced to the consideration of Abelian subalgebras in gl(n) and their equivalence. Similarly, for Heisenberg algebras h(n), the task is reduced to the study of Abelian subalgebras of sp(2n). ˇ Libor Snobl
Classification and Identification of Lie Algebras
Types of nilradicals, continued
Nilradicals of Borel subalgebras of simple Lie algebras ˇ (L. Snobl, S. Tremblay, P. Winternitz) Nilpotent algebras in this class have a very particular structure given by the corresponding root diagram. Consequently, all derivations of such algebras can be found in explicit form using cohomological arguments. This was done by G.F. Leger and E.M. Luks. A prime example of a nilradical in this class is the algebra of strictly upper triangular matrices.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Types of nilradicals, continued
Nilradicals with high degree of nilpotency The structure of Lie brackets of such algebras usually significantly restricts the algebra of derivations. Therefore the algebras of derivations can often be written down explicitly in arbitrary dimension and similarly for the automorphisms. Many explicit lists of solvable algebras with nilradicals in this class are known (J.M. Ancochea, R. Campoamor–Stursberg, L. Garcia Vergnolle, D. Kar´asek, ˇ L. Snobl, P. Winternitz and others), we describe in detail our results concerning three such classes of nilradicals and indicate briefly the results of the others.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Levi Decomposable Algebras
general structure of Levi decomposable algebras, the important role of the nilradical, convenient description of Levi decomposable algebras with nilpotent radicals, structural explanation of the low–dimensional classifications obtained previously by an explicit calculation (Turkowski), description of Levi decomposable algebras with nonnilpotent radicals, a straightforward application: classification of all Levi decomposable algebras up to dimension 7
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Levi Decomposable Algebras
general structure of Levi decomposable algebras, the important role of the nilradical, convenient description of Levi decomposable algebras with nilpotent radicals, structural explanation of the low–dimensional classifications obtained previously by an explicit calculation (Turkowski), description of Levi decomposable algebras with nonnilpotent radicals, a straightforward application: classification of all Levi decomposable algebras up to dimension 7
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Levi Decomposable Algebras
general structure of Levi decomposable algebras, the important role of the nilradical, convenient description of Levi decomposable algebras with nilpotent radicals, structural explanation of the low–dimensional classifications obtained previously by an explicit calculation (Turkowski), description of Levi decomposable algebras with nonnilpotent radicals, a straightforward application: classification of all Levi decomposable algebras up to dimension 7
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Levi Decomposable Algebras
general structure of Levi decomposable algebras, the important role of the nilradical, convenient description of Levi decomposable algebras with nilpotent radicals, structural explanation of the low–dimensional classifications obtained previously by an explicit calculation (Turkowski), description of Levi decomposable algebras with nonnilpotent radicals, a straightforward application: classification of all Levi decomposable algebras up to dimension 7
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Low-Dimensional Lie Algebras
A complete list of Lie algebras up to dimension 6 (up to our best knowledge and effort :-), including the identification of the radical, nilradical, Levi factor, the dimensions of their upper, lower and derived series and the generalized Casimir invariants. Thus it contains inter alia the results of J. Patera, R.T. Sharp, P. Winternitz and H. Zassenhaus 1976 J. Math. Phys. 17 986. Substantial effort was invested into the identification of isomorphisms. Thus we are rather confident that any Lie algebra up to dimension 6 is isomorphic to precisely one entry in our tables.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Low-Dimensional Lie Algebras
A complete list of Lie algebras up to dimension 6 (up to our best knowledge and effort :-), including the identification of the radical, nilradical, Levi factor, the dimensions of their upper, lower and derived series and the generalized Casimir invariants. Thus it contains inter alia the results of J. Patera, R.T. Sharp, P. Winternitz and H. Zassenhaus 1976 J. Math. Phys. 17 986. Substantial effort was invested into the identification of isomorphisms. Thus we are rather confident that any Lie algebra up to dimension 6 is isomorphic to precisely one entry in our tables.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Low-Dimensional Lie Algebras, continued
The lists are ordered in such a way as to make the identification of any given low-dimensional Lie algebra written in an arbitrary basis as simple as possible, i.e. they ordered by structural properties of the algebras (rather than by the way they were originally obtained, as was the case in the older classifications, e.g. Morozov, Mubarakzyanov, Turkowski). The lists are significantly more refined compared to the above mentioned ones, i.e. special values of parameters implying different properties are split off as particular cases. E.g. in dimension 6 there are 242 indecomposable classes of solvable nonnilpotent Lie algebras.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Low-Dimensional Lie Algebras, continued
The lists are ordered in such a way as to make the identification of any given low-dimensional Lie algebra written in an arbitrary basis as simple as possible, i.e. they ordered by structural properties of the algebras (rather than by the way they were originally obtained, as was the case in the older classifications, e.g. Morozov, Mubarakzyanov, Turkowski). The lists are significantly more refined compared to the above mentioned ones, i.e. special values of parameters implying different properties are split off as particular cases. E.g. in dimension 6 there are 242 indecomposable classes of solvable nonnilpotent Lie algebras.
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Conclusions
We have demonstrated the importance of proper identification and classification of Lie algebras for practical problems. We have indicated what kind of results can/cannot be found in our monograph once it is published (expected February 2014).
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Conclusions
We have demonstrated the importance of proper identification and classification of Lie algebras for practical problems. We have indicated what kind of results can/cannot be found in our monograph once it is published (expected February 2014).
ˇ Libor Snobl
Classification and Identification of Lie Algebras
Thank you for your attention
ˇ Libor Snobl
Classification and Identification of Lie Algebras