Loops in Group Theory and Lie Theory [Reprint 2011 ed.] 3110170108, 9783110170108, 2001042383

Table of contents :
Preface
Notation
Introduction
Part I. General theory of transitive sections in groups and the geometry of loops
1 Elements of the theory of loops
1.1 Basic facts on loops
1.2 Loops as sections in groups
1.3 Topological loops and differentiable loops
2 Scheerer extensions of loops
3 Nets associated with loops
4 Local 3-nets
5 Loop-sections covered by 1-parameter subgroups and geodesic loops
6 Bol loops and symmetric spaces
7 Bol nets
8 Strongly topological and analytic Bol loops
9 Core of a Bol loop and Bruck loops
9.1 Core of a Bol loop
9.2 Symmetric spaces on differentiable Bol loops
10 Bruck loops and symmetric quasigroups over groups
11 Topological and differentiable Bruck loops
12 Bruck loops in algebraic groups
13 Core-related Bol loops
14 Products and loops as sections in compact Lie groups
14.1 Pseudo-direct products
14.2 Crossed direct products
14.3 Non-classical differentiable sections in compact Lie groups
14.4 Differentiable local Bol loops as local sections in compact Lie groups
15 Loops on symmetric spaces of groups
15.1 Basic constructions
15.2 A fundamental reduction
15.3 Core loops of direct products of groups
15.4 Scheerer extensions of groups by core loops
16 Loops with compact translation groups and compact Bol loops
17 Sharply transitive normal subgroups
Part II. Smooth loops on low dimensional manifolds
18 Loops on 1-manifolds
19 Topological loops on 2-dimensional manifolds
20 Topological loops on tori
21 Topological loops on the cylinder and on the plane
21.1 2-dimensional topological loops on the cylinder
21.2 Non-solvable left translation groups
22 The hyperbolic plane loop and its isotopism class
23 3-dimensional solvable left translation groups
23.1 The loops L(α) and their automorphism groups
23.2 Sharply transitive sections in £2 × ℝ
23.3 Sections in the 3-dimensional non-abelian nilpotent Lie group
23.4 Non-existence of strongly left alternative loops
24 4-dimensional left translation group
25 Classification of differentiable 2-dimensional Bol loops
26 Collineation groups of 4-dimensional Bol nets
27 Strongly left alternative plane left A-loops
28 Loops with Lie group of all translations
29 Multiplicative loops of locally compact connected quasifields
29.1 2-dimensional locally compact quasifields
29.2 Rees algebras Qε
29.3 Mutations of classical compact Moufang loops
Bibliography
Index

Citation preview

de Gruyter Expositions in Mathematics 35

Editors

Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, Columbia University, New York R.O.Wells, Jr., Rice University, Houston

de Gruyter Expositions in Mathematics 1 2

The Analytical and Topological Theory of Semigroups, Κ. H. Hofmann, J. D. Lawson, J. S. Pym (Eds.) Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues

3

The Stefan Problem, A. M. Meirmanov

4

Finite Soluble Groups, K. Doerk, T. O. Hawkes

5

The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin

6

Contact Geometry and Linear Differential Equations, V. E. Nazaikinskii, B. Yu. Sternin

7

Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, Μ. V. Zaicev

V. E. Shatalov,

8

Nilpotent Groups and their Automorphisms, Ε. I. Khukhro

9

Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug

10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini 11

Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon, G. Zhao

12

Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub

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Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Nazarov, B. A. Plamenevsky

14

Subgroup Lattices of Groups, R. Schmidt

15

Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, P. H. Tiep

16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti, A. J. Sommese 17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno 18

Unitary Representation Theory of Exponential Lie Groups, H. Leptin, J. Ludwig

19

Blow-up in Quasilinear Parabolic Equations, A. A. Samarskii, V.A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov

20

Semigroups in Algebra, Geometry and Analysis, Κ. H. Hofmann, J. D. Lawson, Ε. B. Vinberg (Eds.)

21

Compact Projective Planes, H. Salzmann, D. Betten, Τ. Grundhöf er, Η. Hühl, R. Löwen, M. Stroppel

22

An Introduction to Lorentz Surfaces, Τ. Weinstein

23

Lectures in Real Geometry, F. Broglia (Ed.)

24

Evolution Equations and Lagrangian Coordinates, A. M. Meirmanov, V. V. Pukhnachov, S. I. Shmarev

25

Character Theory of Finite Groups, B. Huppert

26

Positivity in Lie Theory: Open Problems, J. Hilgert, J. D. Lawson, K.-H. Neeb, Ε. B. Vinberg (Eds.)

27

Algebra in the Stone-Cech Compactification, N. Hindman, D. Strauss

28

Holomorphy and Convexity in Lie Theory, K.-H. Neeb

29

Monoids, Acts and Categories, M. Kilp, U. Knauer, Α. V. Mikhalev

30

Relative Homological Algebra, Edgar E. Enochs, Overtoun M. G. Jenda

31

Nonlinear Wave Equations Perturbed by Viscous Terms, Viktor P. Maslov, Petr P. Mosolov

32

Conformal Geometry of Discrete Groups and Manifolds, Boris N. Apanasov

33

Compositions of Quadratic Forms, Daniel B. Shapiro

34

Extension of Holomorphic Functions, Marek Jarnicki, Peter Pflug

Loops in Group Theory and Lie Theory by

Peter Τ. Nagy Karl Strambach

W DE _G Walter de Gruyter · Berlin · New York 2002

Authors Peter Τ. Nagy Institute of Mathematics University of Debrecen P.O.B. 12 4010 Debrecen, Hungary [email protected]

Mathematics

Karl Strambach Mathematisches Institut der Universität Erlangen-Nürnberg Bismarckstr. Υ/τ 91054 Erlangen, Germany strambach@mi. uni-erlangen. de

Subject Classification

2000:

22-02, 20-02; 20N05, 20G20, 22E60, 51A20, 51A25, 51H20, 53C30, 53C35, 57S10, 57S15, 57S20 Key words: Loops, quasigroups, Lie groups, Lie transformation groups, Lie algebras, tangent algebra, symmetric spaces, 3-net, 3-web, configurations, collineation groups, Bol loops, Bruck loops, Moufang loops, topological translation planes, quasifields

©

Printed on acid-free paper which falls within the guidelines of the A N S I to ensure permanence and durability.

Library of Congress — Cataloging-in-Publication Data Nagy, Peter Tibor. Loops in group theory and Lie theory / Peter T. Nagy, Karl Strambach. p. cm ISBN 3-11-017010-8 1. Loops (Group theory) 2. Lie groups. I. Strambach, Karl. II. Title. QA 174.2 .N34 2001 512'.2-dc21 2001042383

Die Deutsche Bibliothek — Cataloging-in-Publication Data Nagy, Peter T.: Loops in group theory and Lie theory / by Peter T. Nagy ; Karl Strambach. - Berlin ; New York : de Gruyter, 2002 (De Gruyter expositions in mathematics ; 35) ISBN 3-11-017010-8

© Copyright 2002 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Typesetting using the authors' TgX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Cover design: Thomas Bonnie, Hamburg.

Preface

In this book the theory of loops is viewed as a part of group theory. Special attention is paid to topological, differentiable, and algebraic loops which are treated using Lie groups, algebraic groups and symmetric spaces. The topic of this book requires some patience with computations in non-associative structures; the basic notions needed as well as facts from the general theory of loops are collected in Section 1.1. The reader is expected to have a basic knowledge of group theory and some familiarity with Lie groups, Lie algebras and differential geometry of homogeneous spaces. For the new material complete proofs are given. For the needed known results, the proofs of which require extensive preparation not concerning our topic, precise available references are quoted. We tried to organize the sections of this book in such a way that they are readable mostly independently from each other. But e.g. the sections 5,6, 8,9,10,11 and 13 as well as 2, 9, 10, 14, 15 and 16 treat closely related topics from the same point of view and with similar methods. The second part of this book is mainly devoted to examine the possibilities in small dimensions; the variety of constructed examples shows that a classification is feasible only for loops with some weak associativity condition. The authors are grateful to the Volkswagen-Stiftung (RiP-Program at Oberwolfach), to the Paul Erdös Summer Research Center for Mathematics, to Deutscher Akademischer Austauschdienst and to the Hungarian Ministry of Education for the partial support of this project. Special thanks are due to A. Figula, T. Grundhöfer, Ο. Η. Kegel, G. P. Nagy, A. Schleiermacher and V. Zambelli who read the whole or some parts of this manuscript and gave us many useful suggestions. We also want to mention a workshop at the Palacky University of Olomouc supported by GACR 201/99/0265 of J. Mikes. Finally we thank L. Kozma for the preparation of the book in final form.

Notation

In principle, we try to avoid in the text symbols and abbreviations, in particular if they are not of standard use in the literature. A s usual Z , R, C denote the integer, real and complex numbers, respectively. H means the (real) quaternions and Ο the (real) octonions (Cayley numbers). If ( Μ , ·) is a loop then χ · (y · z) is often denoted by λ: · yzIf G is a group and Μ is a subset of G then (M) is the subgroup generated by M . If Μ is a subset in a topological space S then Μ is the closure of Μ in S. If Η is a subgroup of G then we write Η < G and G/H is the set of left cosets of Η in G. G' means mainly the commutator subgroup of G. For the classical groups we use the standard notation as e.g. GL„(^T), PGL„(A"), SOn{K), P S L 2 ( / 0 , P S p n ( K ) , P S U „ ( / 0 for a suitable field K. For the Lie algebras of the classical groups we use the corresponding notation gln, $on, on, $un, etc. If the algebraic structures M\ and M2 are isomorphic we write Mi = M2. If φ : A —> ß is a map then the image of an element χ e A is usually denoted by φ(χ); but sometimes we write χ φ or χφ. For structures M\ and Mi the structure M\ χ Mi is given as usual by {C*i,*2);e

M\,x2

e M2}.

Let G i and G2 be groups and φ : G\ — G 2 be a homomorphism. Then we distinguish between the following two subgroups of G\ χ Gi'· G l X G is & section with respect to the natural projection π : G G/Η. One may identify L with the factor space G/H and transport the multiplication. Hence, the theory of loops coincides with the theory of triples (G, Η, σ), where G is a group, Η a subgroup containing no normal non-trivial subgroup of G and σ a section G/H —• G such that a(G/H) acts sharply transitively on the left cosets x H , χ e G, and generates G. According to Baer [7], the set a ( G / H ) of representatives is sharply transitive on the factor set G/H if and only if σ (G/H) is a set of representatives in G for every subgroup conjugate to Η. In [101] and [102] it is stressed that it is always possible and fruitful to carry over the loop properties into properties of the section σ . If C denotes the category of topological spaces, differentiable manifolds, or of algebraic varieties, then a loop is a C-loop if its multiplication, left division and right division are C-morphisms. In the theory of triples (G, Η, σ), where G is a Lie group, Η a closed subgroup and σ . G/H ^ G a C-morphism, the triple (G, Η,σ) defines a C-loop multiplication by the rule (χι H) • (X2H) =

σ(χιΗ)χ2Η.

The first attempt to deal with differentiable and analytic loops was to follow the ideas of Sophus Lie and to classify analytic loops by their tangential objects. In the last 30 years this research program has been applied successfully to differentiable Moufang loops (they are automatically analytic, cf. [102]) by Kuz'min, Kerdman and Nagy. By their results the theory of differentiable Moufang loops has been carried up to the level of the theory of Lie groups. Since the Hausdorff-Campbell formula works also for binary Lie algebras, the theory of diassociative analytic local loops may be treated successfully using binary Lie algebras the structure of which has been determined by A. N. Grishkov. After this progress the attention turned to the class of differentiable Bol loops. Here as well the investigations proceeded in the spirit of Sophus Lie. With any analytic Bol loop L there is associated a Bol algebra B(L) such that two Bol loops are locally isomorphic if and only if the corresponding Bol algebras are isomorphic ([91], XII.8.12. Proposition). Hence the analytic local Bol loops can be classified by the Bol algebras. But at this point a crucial difference from the theory of Lie groups and Lie Moufang loops comes to light. Whereas any local Lie group or local Lie Moufang loop may be embedded into a global one, this fact fails to be true for local analytic Bol loops (and hence much more so for analytic local loops in general). Already the classification of local analytic 2-dimensional Bol loops by Ivanov ([61], [62], [63]) and the classification of global differentiable 2-dimensional loops show the great difference between the varieties of differentiable global and local Bol loops.

Introduction

3

Moreover, in Section 17 we exemplify by a class of analytic loops that the infinitesimal behavior of a loop does not determine its global properties. Hence the investigation of global differentiable loops cannot be reduced to that of local loops, and the procedure to investigate suitable sections in Lie groups seems to us the only feasible method for the classification of differentiable global loops for which the group topologically generated by left translations is a Lie group. This restriction seems relatively mild to us since differentiable loops with some weak associativity conditions have this property (e.g. Bol loops, see [91], Proposition XII.2.14 and [102]). With any loop L there is associated an incidence structure, called a 3-net Ν (cf. [8]); if L is differentiable or analytic then Ν is also differentiable or analytic. Conversely, to a 3-net Ν there corresponds a full class of isotopic loops, and with each point of Ν taken as origin we may associate a coordinate loop defining the multiplication for the points of a line through the origin graphically. Two coordinate loops of a net Ν are isomorphic if and only if their points of origin are in the same orbit of the collineation group Θ of TV which preserves the directions ([8], p. 50). Any identity in a loop corresponds to a configuration in the associated 3-net, and configurations in 3-nets yield identities in some coordinate loops. The 3-nets associated with Bol loops are of particular importance. For every line of a distinguished pencil in such a net there exists an involutory collineation fixing this line pointwise. These reflections generate a group Γ acting transitively on the distinguished pencil. If the Bol loop is differentiable then Γ is a Lie group which induces the structure of an affine symmetric space on the distinguished pencil. In an algebraic setting this symmetric space is a left distributive groupoid which is called the core of the Bol loop. These relations allow us to apply the rich theory of symmetric spaces as well as the results on left distributive quasigroups in order to classify wide classes of Bol loops. 3-webs are incidence geometries associated with differentiable local loops; their pencils of lines form 3 foliations. If a 3-web is associated with a differentiable local Bol loop then for every leaf of one of the 3 foliations there exists a local reflection. The group generated by these local reflections induces an affine locally symmetric space on the manifold of leaves of this foliation; this connects the theory of local differentiable Bol loops with the classical theory of locally symmetric spaces. Our contribution has two parts. The first part contains the foundations of our new methods and their applications to the extension theory of Bol loops, to the algebraic theory of symmetric spaces and to the Lie theory of smooth loops. Moreover, we classify strongly 2-divisible finite, differentiable and algebraic Bruck loops, and we clarify the role of compactness of the group topologically generated by the left translations of a compact loop. In the second part we apply our methods to the topological and differentiable loops on manifolds of small dimension. In the Sections 1, 3 and 4 we develop the foundations for our point of view, describe the interactions between the properties of sharply transitive sections and the corresponding loops, discuss the relations between configurations in 3-nets and corresponding identities in the coordinate loops and show what the local version of these concepts is in the case of differentiable objects; at this instant the importance of

4

Introduction

differentiable foliations and 3-webs comes into light. Using the methods of algebraic topology we prove that any topological loop and hence any topological 3-net on a connected topological manifold is orientable. In Section 2 we thoroughly analyze the structure of proper loops which are extensions of groups by groups. In contrast to the obstructions for study of general extensions of loops, the special case of extension theory developed in Section 2 allows a transparent description by group theoretical methods. If we work in the category of loops realized on topological or differentiable manifolds then our basic assumption is that the group G topologically generated by the left translations is a Lie group. In Section 5 we study differentiable loops such that the sharply transitive section corresponding to L is (locally) covered by 1-parameter subgroups. This class (locally) coincides with the opposite loops of geodesic loops with respect to a uniquely determined affine connection with vanishing curvature. Special cases of these loops are the opposite loops of the geodesic (local) loops of reductive homogeneous spaces and the differentiable (local) Bol loops. Sections 6 to 13 are devoted to a thorough study of Bol loops, their analytic and algebraic properties as well as to their relations to the classical theory of affine symmetric spaces. Here we pay attention to the global as well as to the local Bol loops. In Section 6 some representations of local Bol loops are given as sections in Lie groups and their relations to Lie triple systems and Bol algebras. In Section 7 the geometric version of isotopism classes of Bol loops, the Bol nets, are investigated. Of special interest for us is the group of collineations of a Bol net as well as some of its subgroups, for instance the group generated by the Bol reflections. The core of a Bol loop introduced and studied in Section 9 is a symmetric space in the sense of Loos, and it may be used in the theory of groups of exponent 3. (Cf. Corollary 9.7). The closest relation between a local Bol loop L and its local core takes place if the Bol algebra of L is a Lie triple system. For a global Bol loop this is the case if the core is a symmetric quasigroup and the Bol loop L satisfies the automorphic inverse property. Then the Bol loop L is a left Α-loop, and to any element χ there exists precisely one element y with χ — y2. These loops are called strongly 2-divisible Bruck loops. Differentiable strongly 2-divisible loops having the left inverse property are already Bruck loops if they are left Α-loops and satisfy the automorphic inverse property. Differentiable Bol loops are always locally strongly 2-divisible. G. Glauberman proved [36] that any strongly 2-divisible Bruck loop L can be embedded into the group G generated by the left translations of L and that the multiplication of the embedded loop L is induced by the multiplication of G. We generalize this embedding to the class of strongly 2-divisible Bol loops. Moreover, using this construction we classify strongly 2-divisible differentiable connected Bruck loops in Section 11. They correspond in a unique way to pairs (G, σ), where G is a connected Lie group and σ is an involutory automorphism of G such that the subgroup centralized by σ contains no non-trivial normal subgroup of G, the exponential mapping from the (— l)-eigenspace m of σ on the Lie algebra g into G is a diffeomorphism and expm generates G. Our systematic study of strongly 2-divisible Bruck loops in algebraic groups (Section 12),

Introduction

5

as well as of Bruck loops associated with symmetric quasigroups over groups (Section 10) is motivated by the increasing importance of this class of loops in algebra and geometry. One of the reasons for this is the classification problem of sharply 2-transitive permutation groups. Any such group corresponds in a unique way to a so-called near domain which is an algebraic structure (F, + , ·; 0, 1) with two operations: with respect to the addition (F, + ; 0) is a Bruck loop such that χ + χ = 0 implies χ = 0, with respect to the multiplication ( F \ {0}, ·; 1) is a group and the left distributive law holds (cf. [66], [75]). Till now, the only known examples of near domains are the near fields where the additive structure is an abelian group. Although this question motivated many interesting contributions, the existence of strongly 2divisible Bruck loops having a sharply transitive group of automorphisms remains an interesting and important problem (cf. e.g. [76], [77], [68]). For differentiable Bol loops it is no restriction to assume that the group topological^ generated by the left translations is a Lie group. We prove in Section 9 that the category of real analytic Bol loops coincides with the category of connected topological Bol loops for which the group topologically generated by the left translations is a connected, locally compact, locally connected and finite-dimensional topological group (cf. [92]). As a consequence we obtain that any closed subloop of an analytic Bol loop is analytic. For differentiable Moufang loops the group topologically generated by all left and right translations is a Lie group. In constrast to this, there are examples of differentiable Bol loops having a Lie group as the group topologically generated by all left and right translations as well as those for which this is not the case (cf. Sections 7 and 22). In Section 13 we consider local Bol loops associated with the same symmetric space and ask under what conditions two such local Bol loops are locally isotopic. If the symmetric space is compact and irreducible then the answer is affirmative if one excludes the symmetric space on the 7-sphere as well as the Grassmannian manifold of 3-planes in the 8-dimensional real vector space. We prove here also a generalization of a theorem of A. Fomenko [29] that if the group G topologically generated by the left translations of a simply connected differentiable (local) Bol loop L is reductive then L is the direct product of a reductive Lie group by a direct product of proper Bol loops Li not having any connected non-trivial normal subloop. Moreover, the symmetric space associated with L(- is irreducible and G decomposes in an analogous way as L into factors. In Section 14 we give for loops generalizations of the semidirect products of groups. These constructions allow us to find examples of compact differentiable connected loops having a compact Lie group as the group topologically generated by the left translations. These examples of loops are realized on products on 7-spheres, 7-dimensional real projective spaces and of spaces of compact Lie groups; in them the associativity law is strongly violated. But it is not possible to discover examples with good associativity properties besides the Scheerer extensions studied in Sections 2 and 15. The reason for this are the theorems which give a full classification of compact differentiable Bol loops. These loops are repeated extensions of groups by groups and

6

Introduction

by Moufang loops; the theory of these extensions is developed in Sections 2 and 15. It follows in particular that every connected compact differentiable simple Bol loop is a Moufang loop. These results proved in Section 16 belong to the main achievements of the first part of our work. The following theorem has the same quality: Every compact connected topological loop is a Moufang loop if the group topologically generated by all left and right translations is a compact Lie group. According to [55] the topological loops such that the group topologically generated by all their left and right translations is a compact Lie group are precisely those which have an invariant uniformity. In contrast to this, at the end of Section 14, we prove that there are many local Bol loops having an interpretation as local sections in compact Lie groups. The result in [121] shows that the assumption of compactness for the group G topologically generated by the left translations of a topological loop is very restrictive. If G is a compact quasi-simple Lie group then it must be locally isomorphic to SOs(K). If G is not quasi-simple, then Scheerer describes in [121] the global sections in G. It was his description which put us in the position to classify compact differentiable Bol loops. The group topologically generated by the left translations of such a loop is a reductive compact Lie group having at least 2 quasi-simple factors; in this case one of these two factors is locally isomorphic to SOs(M). It seems to us that the most interesting objects for the study of compact topological or differentiable loops which are not Bol loops but which have a compact connected Lie group as the group topologically generated by the left translations are the loops homeomorphic to the (η — 1)-dimensional projective space or to the (η — l)-sphere, where η e {4, 8} since several natural algebraic or analytic assumptions force these loops to be classical. Till now no sharply transitive continuous section in the group SOs (M) respectively S04.(R) is known which does not correspond to the Moufang loop or to the group of octonions or of quaternions of norm one, respectively. But for η = 4 or η — 8 we have been able to find loops diffeomorphic to the (η — 1)-dimensional projective space, respectively to the (η — 1)-dimensional sphere such that the groups topologically generated by the left, right and by all translations coincide; they are Lie groups isomorphic to PSL„ (M), or to SL„(R), respectively (cf. Theorem 29.3). In Section 17 we deal with topological loops homeomorphic to R", admitting a sharply transitive subgroup Ν of the group topologically generated by the left translations and give examples in which the group G topologically generated by all left and right translations is a nilpotent Lie group; in these cases Ν is not noraial in G. If in contrast to this we assume that Ν is a normal subgroup of the group G topologically generated by all left and right translations then G (which must be a Lie group) is not nilpotent. Whereas any 2-dimensional loop in this class must be a group, in dimension 3 there are already interesting examples of proper loops of this type. The main result in this section is the statement that for any simple compact Lie algebra g the multiplication χ ο y = χ + y + [χ, >>] defines a loop L on g such that the group G topologically generated by the left translations coincides with the group generated by all translations of L and that G is isomorphic to a semidirect product G = Τ χ Η. The normal subgroup Τ in this semidirect product is the group Kn of all translations

Introduction

7

of the affine space over the vector space g, where Κ — R, unless g is the unitary Lie algebra su„(C) or the symplectic Lie algebra su„(H); in the last two cases Κ is the field of complex numbers or of quaternions, respectively; the group Η is isomorphic to the connected component of the group GL/(AT), where I is the dimension over Κ of the minimal representation of g, and acts irreducibly on g. A further statement which is valid for any finite-dimensional loop L having a Lie group G as the group topologically generated by its left translations is Theorem 20.1. There we describe explicitly the structure of G if L is homeomorphic to the «-dimensional torus. We want to stress that, in general, for coverings G of Lie groups which are topologically generated by the left translations of a connected topological loop there are no topological loops having G as the group topologically generated by their left translations. We confirm this phenomenon in Section 19 showing that no proper covering of the group PSL2(M) can be a group topologically generated by the left translations of a 2-dimensional topological loop. Many of the examples of loops admitting a sharply transitive normal subgroup Ν in the group generated by left translations may be realized in the category of algebraic loops and algebraic groups. From the topological point of view the simplest topological loops L are those which are realized on one-dimensional topological manifolds and which have a locally compact group G as the group topologically generated by their left translations. We shall deal with this class in Section 18. We show that for any proper loop L of this type the group G is a covering of the group PSL2OR), and we describe all loops L which are coverings of differentiable loops homeomorphic to the circle and have only trivial centre. In this class there are proper loops satisfying the left inverse property; but any topological left Α-loop as well as any monassociative loop in this class is already one of the 2 one-dimensional Lie groups. The main part of Sections 19 to 22 is devoted to topological loops on 2-dimensional manifolds having a locally compact group G as the group topologically generated by their left translations. Any such loop is homeomorphic to the torus, to the cylinder or to the plane. The abundance of the examples of these loops constructed there shows that a complete classification of such loops cannot be expected. But they may be described explicitly if they are coverings of loops realized on the torus or the cylinder and if G is not solvable. This is shown in Sections 19 and 20. If the group G topologically generated by the left translations of a 2-dimensional topological loop L is a quasi-simple Lie group then it is isomorphic to the group PSL2(K). For this group we give explicit descriptions of some sharply transitive sections and show that there exists one among them consisting only of parabolic elements. Moreover, if the group topologically generated by the left translations of a topological connected 2-dimensional loop is a non-solvable Lie group then the group topologically generated by all left and right translations cannot be a Lie group. In contrast to this we give in Section 28 examples of 2-dimensional connected differentiable loops having a solvable Lie group topologically generated by their left translations for which the group topologically generated by all left and right translations is also a Lie group.

8

Introduction

The exceptional status of the hyperbolic plane loop and of its isotopy class is based upon the fact that it is associated with the hyperbolic plane geometry and generalizes in a direct way the vector group of the euclidean plane. We treat the hyperbolic plane loop in Section 22. In particular, we characterize it there within the class of strongly left alternative 2-dimensional connected topological loops by the group topologically generated by the left translations, as well as within the 2-dimensional connected topological Bruck loops by the fixed point free action of the inner mapping group. In the second part of our work we aim at a classification of connected differentiable Bol loops of dimension 2 and the determination of the full collineation group of all 4-dimensional differentiable 3-nets which have a Bol loop among their coordinate loops. This goal is achieved in Sections 23 and 24. In view of the results of Section 8 this is also the classification of connected 2-dimensional topological Bol loops having a locally compact group topologically generated by their left translations. The determination of the 2-dimensional differentiable Bol loops L is based on the fact that with any such L there is associated a 2-dimensional symmetric space and that these spaces as well as the groups Σ generated by their reflections are classified ([34]). Since the groups Σ are related by means of the collineation groups of the corresponding nets to the groups G generated by the left translations of the coordinate loops which are Bol loops, the groups Σ allow us to determine possible candidates for the groups G. It turns out that the connected Lie groups G are of dimension 3 or 4. This fact motivated us to classify, in Section 23, all differentiable strongly left alternative connected loops having a 3-dimensional Lie group as the group topologically generated by their left translations. Furthermore we characterize the left A-loops among them; beside the hyperbolic plane loop they correspond, up to isomorphism, to the 3-dimensional connected Lie groups having precisely two 1-dimensional normal subgroups. The 2-dimensional differentiable connected Bol loops are dominated by Bruck loops: every such Bol loop is isotopic to a Bruck loop. There exist precisely two 2-dimensional connected differentiable Bruck loops. Both of them are closely related to metric plane geometries and their groups of motions. One of them is the geodesic loop of the hyperbolic plane. The other one, which is also homeomorphic to R 2 , is the geodesic loop of the symmetric space realized on the manifold of lines of positive slope in the pseudo-euclidean affine plane. The left, right as well as the middle nucleus of the hyperbolic plane loop is trivial, the left and the middle nucleus of the pseudo-euclidean plane loop is trivial but its right nucleus is isomorphic to ®L It is remarkable that the behavior of the right translations of the 2-dimensional differentiable connected Bol loops L differs fundamentally from that of left translations fundamentally: the group topologically generated by the right translations of L cannot be a Lie group. The 4-dimensional differentiable Bol nets attracted our interest since they carry all information about the isotopism classes of 2-dimensional differentiable Bol loops. We calculated the groups topologically generated by the Bol reflections and determined

Introduction

9

the full collineation groups of these Bol nets. As a consequence of the knowledge of the full collineation groups of 4-dimensional differentiable Bol nets we obtained explicitly the automorphism groups of all 2-dimensional differentiable connected Bol loops. Since the strongly left alternative differentiable left Α-loops can be locally represented as geodesic loops of reductive homogeneous spaces (with respect to their canonical connection) and this class of homogeneous spaces is an immediate generalization of the class of symmetric spaces, we enumerate in Section 27 all 2-dimensional differentiable connected strongly left alternative left Α-loops and give there elegant representations for them. But the class of differentiable left Α-loops is not contained in the class of differentiable strongly left alternative loops. We illustrate this in Section 23 giving 2-dimensional examples of differentiable left Α-loops which are not strongly left alternative. Examples constructed in Sections 14 and 17 show that for a topological loop L the assumption that the group Γ topologically generated by all left and right translations of L be a Lie group is too weak for a classification; this class contains plenty of loops differing from each other fundamentally from an algebraic point of view. In Section 28 we examine the efficiency of this condition in connection with the low dimensionality of topological loops. For 1-dimensional connected topological loops L the assumption that the group Γ topologically generated by all left and right translations be a Lie group is too restrictive since any such loop must be a group (Theorem 18.18). We show in Section 28 that also the structure of Lie groups Γ which are the groups topologically generated by all left and right translations of 2-dimensional connected topological loops is very limited: they are semidirect products of the vector group R" with R and have 1-dimensional centre. But in contrast to the 1-dimensional case, there are plenty of proper differentiable 2-dimensional connected loops with the above property; they can be described, but not completely classified. In the last section we remember that a rich source of examples for topological loops realized on the topological product of R and the η-sphere Sn with η e {1, 3, 7} are the multiplicative loops Q* of quasifields Q coordinatizing locally compact connected topological translation planes. If in Q the left distributive law holds then the group topologically generated by the left translations of the loop Q* homeomorphic to R χ Sn is a closed connected subgroup of the group GL„+i(R). We show in Section 29 that for 2-dimensional proper loops Q* the group topologically generated by the left translations is the connected component of the group GL2(R), whereas the group topologically generated by all left and right translations cannot be a Lie group. The groups topologically generated by the left translations as well as by all left and right translations which we calculated for the multiplicative loops of Rees algebras and which are always Lie groups show that the situation for the multiplicative loops of four- and eight-dimensional locally compact connected quasifields becomes much more complicated. The theory of sharply transitive sections σ : G/H —> G in Lie groups presented in this book can also be seen as a contribution to theoretical kinematics in the sense of

10

Introduction

Karger [65] and Bottema [12]. If we consider G as a group of motions on the space G/H then the image a(G/H) has an interpretation as a sharply transitive manifold of motions acting on the left coset space G/H by left translations. Recently the theory of analytic loops and quasigroups enjoys an increasing interest in theoretical physics (cf. [40], [41], [85]). A concrete example is the relativistic addition of velocities which is a Bruck loop multiplication on the manifold {υ G W1· IMI2 < c2} (cf. [131], [132]).

Parti General theory of transitive sections in groups and the geometry of loops

In the first part we treat in a broader context properties of such global and local sections in groups which can be seen as left translations of global and local loops, respectively. Groups of main interest to us are the Lie groups, but we deal also with abstract, topological, and algebraic groups. Caused by the lack of associativity in loops it is sometimes advantageous to use geometrical methods, namely the study of 3-nets; these are incidence structures associated with loops. We show that for the treatment of differentiable loops having a neighbourhood of the unit element simply covered by 1-parameter subgroups the methods of differential geometry are powerful tools. Among the loops, apart from Moufang loops, the Bol loops have the strongest associativity properties. The classification of differentiable Bol loops is one of the main problems in the theory of loops given by smooth sections in a Lie group. We describe the differentiable Bol loops by symmetric spaces. This method allows us to classify smooth as well as algebraic Bruck loops, i.e. Bol loops satisfying the automorphic inverse property. Generalizing in various ways the notion of the semidirect product to loops we construct loop sections in non-simple compact Lie groups. Till now we did not dispose of an extension theory in the category of Bol loops. We develop such a theory for the simplest case that the Bol loops are extensions of groups by groups. The starting point for this was a description of global sections in compact Lie groups by H. Scheerer; the motivation was the classification of compact differentiable Bol loops. A differentiable version of our abstract extension theory permits to reach this goal: proper differentiable compact Bol loops are classified by so called Scheerer extensions. In particular there does not exist any compact differentiable simple proper Bol loop. Another result in the same direction is the fact that any connected locally compact topological loop for which the group topologically generated by all its left and right translations is a compact Lie group must be a Moufang loop. We conclude the general theory considering loops having a sharply transitive normal subgroup in the group topologically generated by their left translations. We show that there are plenty of examples of such smooth and algebraic loops. The most important class among them seems to be the class of smooth loops defined on semisimple compact real Lie algebras (L, [ . , . ] ) by χ Ο _Y = χ + y + [JC, y], for any x,y e L.

Section 1

Elements of the theory of loops and the relations with group theory

First, we introduce the basic notions and give needed facts. In particular, we lay the foundation for the investigation of loops as sections in groups. In the last part we analyze simple properties of topological and differentiable loops, which are defined on the factor space of Lie groups using continuous or differentiable sections and prove some characteristic results for them. For example we show that any topological loop on a manifold is orientable and classify for differentiable loops the structure of factor loops. Moreover, we clarify in detail the relations among invariants of topological as well as of differentiable loops having the same universal covering loop.

1.1

Basic facts on loops

A set Q with a binary operation (JE, y) Μ* Χ Ο y is called a quasigroup if for any given a, b Ε Q the equations a ο y = b and χ ο a — b have precisely one solution which we denote by y = a\b and χ = b/a. The left translations λχ : y χ οy :Q ^ Q and the right translations ρχ \ y y ο χ \ Q —> Q are bijections of Q, and (JE, y) x\y = ^ λ " 1 , respectively (x,y) i-»· y/x = yQxx are further binary operations on Q. If a quasigroup Q has an element 1 with 1 O J C = X O 1 = J C then it is called a loop and 1 is the unit element of Q. Two quasigroups (Q\,°) and (Q2, *) are called isotopic if there are three bijections α, β, γ : Q\ —>• (?2 such that a(x)*ß(y)

= γ (χ ο y)

holds for any x,y e Q\. If β = γ then the quasigroups (Q 1,0) and (Ö2, *) are called left isotopic, if a — γ then they are called right isotopic. If the bijection γ is the identity map i then the isotopism (a, ß, i) is called a principal isotopism. Any isotopism (α, β, γ) can be decomposed into the product of a principal isotopism and an isomorphism (cf. [107], III. 1.4 Theorem, p. 58). If (α, ß, i) is a principal isotopism from the loop (L, o) onto the loop (L, *) defined on the same set L then there exist elements a, b e (L, o) such that (α, β, i) = ( ρ " 1 , λ^ 1 , i) where ρα and Xt> are right and left translations of (L, o), respectively. The loop (L, *) is called principal left isotopic to (L, o) if b = 1. It is called principal right isotopic to (L, o) if a = 1.

14

1 Elements of the theory of loops

If (L, ·) is a loop then the operation (;t,;y)i-».x:*;y = y - ;t defines again a loop (L, *) which is called the opposite loop of L. For any quasigroup (Q, ·) and arbitrary elements a, b e Q the multiplication x * y = χ ja • b\y determines an isotopic loop on the set Q having b • a as the unit element. The kernel of a homomorphism a : (L, o) —» (Z/, *) of a loop L into a loop L' is a normal subloop Ν of L, i.e. a subloop of L such that χ Ο Ν = Ν Ο χ,

(χ ο Ν) Ο y = χ ο (Ν ο y)

and

χ ο (y ο Ν) = (χ ο y) ο Ν

hold for all χ, y e L. Conversely, if TV is a normal subloop of L then the factor loop L/N is a homomorphic image of L (cf. [18], pp. 60—61;[107], pp. 30-31.). Hence, the normal subloops of L give (up to isomorphism) the epimorphisms of L onto loops. A loop L is called simple if {1} and L are its only normal subloops. If in a loop L an identity F ( x i , . . . , xm) = Η (χι,..., xm) holds, where F and Η are monomials of the variables x i , . . . , xm, then this identity is also satisfied in any subloop of L as well as in any loop which is a homomorphic image of L. A bijection a : L —• L of a loop (L, ·) is called a left (respectively right) pseudoautomorphism of L if there exists an element c € L such that (c · ck(a:)) · ) = c • a(x • j )

(respectively «(λ) · ( a ( y ) • c) = a(x • y) • c )

holds for all x, y e L. The element c is called a companion of a . A left (right) pseudo automorphism a satisfies α(1) = 1. Let C be the category of topological spaces, C°°-dijferentiable manifolds or analytical manifolds. A quasigroup Q is an C-quasigroup if Q is an object in the category G and the mappings (jc, y) h> χ ο )i, (jt, y) i-^· x\y, (x, y) ι-* y/x : Q2 —• Q are C-morphisms. Let L be an object in the category C. Let ·, \ and / be C-morphisms from open domains of L χ L such that the following conditions hold: O/y) - y = x,

y • (yV)

=

(xy)/y

= χ,

y\(yx)

= *

if the left side of the identities is defined. The object L is called a local quasigroup in the category C with respect to the operations ·, \ and /. A local quasigroup L in the category C is called a local loop in C if there is an element e of L such that xe = ex — χ je = for all χ e L. The left, right and middle nucleus, respectively, of a loop L are the subgroups of L which are defined in the following way: Ni = {u; ux • y = u • xy, x, y e L}, and

Nr = [w; χ • yw = xy · w, x, y e L}

Nm — {υ; xv • y = χ · vy, x, y e L}.

1.1 Basic facts on loops

15

The intersection Ν — N[ (Ί Nr Π Nm is called the nucleus of L. If L is a loop in the category C then N[, Nr, Nm and Ν are closed subgroups of L. The centre Ζ of a loop L is the largest subgroup of the nucleus Ν οϊ L such that zx = xz for all χ e L, ζ e Z. Any subgroup of Ζ is a normal subgroup of L, called central subgroup of L. A loop L has the left inverse property if there exists a bijection χ χλ : L —> L x such that x (xy) = y for every x, y € L. Similarly a loop L is said to have the right inverse property if there exists a bijection χ ι-> xQ : L —» L such that (yx)xQ = y for every x, y e L. If a loop satisfies the left and the right inverse property then it is called a loop with the inverse property. If a loop L satisfies the left or right inverse property then one has evidently λ χ — x° and we denote this element by x _ 1 . A loop L satisfying the identity (x · = x(y • xz) respectively z(x;y · x) — (zx • y)* for all x, y, ζ € L is called a left respectively right Bol loop. If one of these identities is satisfied in a local loop then it is called a local left respectively local right Bol loop. In the following we use the term Bol loop for a left Bol loop. Every loop isotopic to a Bol loop is also a Bol loop (cf. [107], IV.6.15 Theorem). Any such loop has the left inverse property which means that χ " 1 χ = jcjc - 1 = e and x~1 (jcy) = y for all x,y e L, where e is the unit of L. Indeed, from xz = [x(e/x · λ:)]ζ = x[e/x{xz)] it follows that x~l = e/x. Moreover, in any Bol loop L the identity xm(xn · y) = xm+ny with jc m + 1 = xmx is satisfied for all integers m, η and x,y e L (cf. [105], IV. 6.5. Theorem). The loops satisfying the left and the right Bol identity are called Moufang loops. A loop L is left (or right) alternative if χ (xy) = x2y (or (yx)x — yx2) for all x, y e L. Lemma 1.1. A loop L is a Moufang loop if and only if one of the following conditions holds: (i) L is a Bol loop satisfying the right inverse property; (ii) in L the identity (xy) · (zx) = [x(>'z)]jc is fulfilled; (iii) in L the identity (xy) • (zx) = x[(_yz)x] is fulfilled; (iv) L is a right alternative Bol loop. Proof. If L is a Moufang loop then it satisfies condition (i). Let now L be a Bol loop with right inverse property. Using the right and the left inverse property we obtain from the identity [yfo0-1](*>0

= y

the relation y ( x y ) - 1 = x~l and hence (jc^) - 1 = This means that the mapping ι : ζ m>is an anti-automorphism of L. Applying t : L —> L to the left Bol identity we obtain the right Bol identity.

16

1 Elements of the theory of loops

A Moufang loop L satisfies the left as well as the right inverse property. Hence the mapping ι : ζ ι-»· is an anti-automorphism of L. Using the right Bol identity we obtain (jcm-1)-1 [(jcu)jc] = (mjc_1)[(jcu)x] = (uv)x or equivalently (;cu)jt = (xm_1)[(MI;)X] . Putting u~l = s and uv = ί we have [•x(s?)]·* = (xs)(tx)

.

Conversely, let L be a loop satisfying this identity. For s = x\e we have [x(;c\e · t)]x = tx and hence x(x\e • t) = t for all x, t e L. This means that L has the left inverse property. The identity [xCst)]* = (xs)(tx) with s = e gives ( x t ) x = x(tx). For t — e/x we have x[(s • = [(5 · e/x)x\x = xs or (s • ejx)x

—s

which yields the right inverse property. With u = s~l, OtsHi.*) has the shape (xv)x = (λ:μ _1 )[(μι;)λ:]

or

ν = st the identity [jt(si)]* =

(mx _1 )[(xi»)x] = (uv) χ

since in L the mapping ι : ζ ι-> z~x is an anti-automorphism. For w = ux~l we obtain ιυ[(χιΟ*] = [(ιυχ)υ]Λ: which is the right Bol identity. Applying to the right Bol identity the anti-automorphism i : L —> L we obtain the left Bol identity. Using the anti-automorphism ι : ζ η · Γ 1 we obtain that the identities (ii) and (iii) are equivalent. A Moufang loop satisfies the right Bol identity and hence it is right alternative. Conversely, a right alternative Bol loop L fulfills the right inverse property since = [jc(3?jc2)]jc—1 = [jc((yjc)x)]x _1 = jc(jjt). Hence from (i) it follows that L is a Moufang loop. • Let a : L —> L be a left pseudo-automorphism with a companion c of a loop L which has the left and the right inverse property. Since a(e) — e we have (ca(ji:))a(jc - 1 ) = c or a(x~l) = a ( j c ) - 1 . Then applying the anti-automorphism ί : ζ ι—> to the relation (c · α(χ))ο;(}') = ca(xy) we see that a is also a right pseudo-automorphism with a companion c~1. Hence for Moufang loops we can speak of pseudo-automorphisms.

1.2 Loops as sections in groups Lemma 1.2. In a Moufang loop L the mapping T(s) : u i-> s any s £ L a pseudo-automorphism with a companion s~3.

17 1

(us) : L —>· L is for

Proof. In the identity (ii) of the previous lemma we put χ = s - 1 , y = us, ζ = s~l(vs~l) and obtain, using the identity = (xy)x (cf. Lemma 1.1 (ii)) as well as the right Bol identity, [s-HusMs-Hvs-^y^ls-^iusKs-'ivs-1))]}*-* =

{s-l[(us)((s-iv)s-1)]}s-]

=

{s-l[([(us)s-l]v)s-l]}s-'

= {^-'[(kiOJ- 1 ]}*- 1 . 1 For ν = e we have (μ5·)]5·—3 = [,s _1 (m.s -1 )].s -1 for all u e L. With this relation the previous identity gives

(MJ>] {

1

(υί)] j - 3 } = {s - 1 [(ui;)s]}s- 3

which proves the assertion.

•

Lemma 1.3. Let (L, *) be a Bol loop and Μ be a subset of L with Μ * Μ c Μ and Μ-1 = Μ. Then Μ is a subloop of L. Proof Since L has the left inverse property the solution χ — a~[ * b of the equation a * χ = b is contained in Μ for a, b e Μ. The solution of the equation χ * a = b can be expressed as x = b/a = a~l * [(a * b) * a~])] since {α - 1 * [(α * b) * a~1]} * a = a~l * {(a * b) * ( a - 1 * a)} — b.

•

A local loop L for which the unit element e e L has a neighbourhood U such that for every x, y, ζ € U the identity · xz) = (x • yx)z holds is called a local (left) Bol loop. Analogously we can define the local right Bol loops and local Moufang loops, too.

1.2

Loops as sections in groups

Let G be a group, Η a subgroup of G and π : G G/H the natural mapping χ i-> xH. A mapping σ : G/H —• G is called a section if the composed map jr οσ : G/H —> G/H is the identity. The image a(G/H) forms a system of representatives for the left cosets of Η in G. Every loop L can be reconstructed from the set Λ of its left translations; moreover Λ is the image of a section σ : G/Ge -> G in the transformation group G generated by the left translations, where Ge is the stabilizer of the identity e € L in G. Namely, let G be a group, let Η be a subgroup of G and σ : G/H —»Gbea section satisfying the following conditions:

18

1 Elements of the theory of loops

1. σ(Η) = 1 and a{G/H)

generates G.

2. a(G/H) operates sharply transitively on G/H, which means that to any χ Η and yH there exists precisely one ζ e a(G/H) with zxH = yH. Then the multiplication given by λ: * y — G, where Η is a subgroup of G, is called sharply transitive if the image a(G/H) satisfies the above conditions 1 and 2. If σ : G/H —>• G satisfies only the condition 2, then we say that σ acts sharply transitively. Proposition 1.5. Let σ : G/H —> G be a sharply transitive section of the group G corresponding to the loop multiplication on G/Η given by(xH)*(yH) = o{xH)yH. Then the loop (G/H, *) is isomorphic to the loop (a(G/H), o) defined on a{G/H) by χ ο y = a(xyH) with respect to the isomorphism σ : G/H —• a(G/H). Proof. We have σ(χΗ)οσ{γΗ)

= σ ((σ (χ Η)σ (y Η)) Η) =

σ(σ(χΗ)Η*σ(γΗ)Η).

• R. Baer [7] gave the following characterization of sections σ : G/H -> G which act sharply transitively on G/H. Proposition 1.6. Let σ : G/H —>• G be a section with σ(Η) = 1 eG. Then a(G/H) acts sharply transitively on G/H if and only ifa(G/H) forms a system of representatives in G for the cosets of each conjugate subgroup Η8 =gHg~l, g&G. Proof. Suppose first that the set a(G/H) acts sharply transitively on G/H. Then in any coset xH, χ e G there is precisely one element of a(G/H). The relation l σ{αΗ) e b(gHg~ ) is satisfied for given a, b, g e G if and only if the equation a(aH)gH = bgH holds. Since σ acts sharply transitively there exists for any g,b e G precisely one solution σ(αΗ) e a(G/H) of this equation. Conversely, we suppose that a(G/H) is a system of representatives for the cosets of each conjugate subgroups gHg~x, g e G. This means that in the coset b(gHg~l)

1.2 Loops as sections in groups

19

there is precisely one element σ(αΗ). From this it follows that σ{αΗ) is the unique solution of the equation a(aH)gH — bgH. • Lemma 1.7. Let Ν be a normal subloop of a loop L and let G be the group generated by the left translations of L. Then the subgroup Μ generated by the left translations {λ„; η e Ν) is a normal subgroup of G. Proof We show that for every left translation λχ, χ ε L, the product λ ^ ' λ ^ λ ^ , η e Ν, is a left translation by an element of N. Indeed, λ~'λ„λ Λ (ζ) = jc\[n(xz)] = *\[(·*·ζ)πι] = = zn2 = ητ,ζ = λ„ 3 ζ for suitable n\, «2, «3 e N. • A loop L satisfies the left inverse property if and only if for any χ e L the left translation λχ : y h» xy : L —>• L fulfills ( λ * ) - 1 = λ χ \ ι , where 1 is the unit of L. Similarly a loop L satisfies the right inverse property if for any right translation qx the equation ( ρ * ) - 1 = Q\/x holds. In a loop satisfying one of these two properties we have := \/x = JC\1. A loop L is a Bol loop if and only if for any JC, y e L the product λχλ},λχ is again a left translation, namely λ^λ^λ* = λ*^*). In particular one has λχλ\/χλχ = λχ and -1 1 hence λ\/χ = ( λ * ) . It follows that \/x — λι/*(1) = λ " ^ ) = ΛΓ\1 = χ'1. The left nucleus Νι of a loop L with the left inverse property consists of the elements η e L for which ληΑ — A, where Λ is the set of left translations of L. The middle nucleus Nm of L consists of the elements m e L such that Αλ,η — A. For a loop L satisfying the left inverse property it follows from the relation ( λ „ Λ ) - 1 = Λ - 1 λ " 1 = Λλ„-ι that the left and the middle nucleus coincide. Similarly for a loop L with the right inverse property we have Nm = Nr. Hence for loops L with the inverse property one has TV/ = Nm = Nr (cf. [97]). Lemma 1.8.

(a) The nucleus Ν of a Moufang loop L is a normal subgroup of L.

(b) In a commutative Moufang loop L the mapping Y : Χ Μ» JC3 is a centralizing endomorphism of L, which means that y(L) is in the nucleus Ν of L. Proof, (a) Since the nucleus Ν of a Moufang loop L satisfies ( x y ) N — x(yN) and ( x N ) y = x(Ny) for any χ, y e L the subgroup Ν is normal in L if and only if xN = Nx οτχ~χΝχ = Ν for any jc e L. According to Lemma 1.2 the mapping T(x) is a pseudo-automorphism of L. For any pseudo-automorphism a with a companion c and for any η e N, x, y e L we have a(wjc)[o;(_y)c] = a(n •

=

a(n)\a(xy)c\

= a(n)[a(x)(a(y)c)] . Putting a(y)c = 1 we obtain a(nx) x, y e L we get

= a(n)a(x)

and hence for any η e TV and

[α(«)α(χ)] [aOOc] = a(n)[a(j:)(a(y)c)]

20

1 Elements of the theory of loops

Since the mappings je ^ α ( χ ) and y a(y)c are bijections it follows that a (n) e Ν for any η e Ν. Hence the pseudo-automorphism T(x) leaves the nucleus Ν invariant. (b) Using the identity (ii) of Lemma 1.1 for a commutative Moufang loop L we have (xy)3 = [OoOCy*)]^)

= [(xy2)x](xy)

=

[*(*/)]ϋυ0

= [jc(jc;y3)]jc = jc[jc(jcy3)] = x3y3 . Therefore the mapping γ : jc η» χ 3 is an endomoφhism of L. For a commutative Moufang loop L the mapping T(x~v) : y j t ( j j t _ 1 ) is the identity map which is a pseudo-automorphism with a companion jc3 (cf. Lemma 1.2). Hence y(zx3)

= T{x~l)y

• (T(x~l)z-x3)

= T(x~l)(yz)x3

=

(yz)x3

and consequently x3 is contained in the nucleus Ν of L.

•

Lemma 1.9. Let (L, ·) be a loop and let (L, *) be the principal isotopic loop given by the multiplication χ * y — xja • b\y, where a, b € L are fixed elements. Then x · y = x */b * where */ and \* are the left division and right division in (L, *), respectively. The groups generated by the left translations, by the right translations and by all translations of(L, ·) and (L, *) respectively, coincide. Proof. We have jc\*z = b[(x/a)\z]

and ζ */y = [z/(b\y)]a.

Therefore

χ */b * a\*y = jca * by = jc · y, which proves the first assertion. The group generated by the left translations λ* = λ χ / α λ ^ of (L, *) is contained in the group generated by the left translations of (L, ·). Similarly, the left translations λχ — λ* ^ λ * " 1 of (L, ·) are contained in the group generated by the left translations of (L, *). Analogous arguments show the statement for the group generated by the right translations of (L, ·) and (L, *). • Lemma 1.10. Let L\ and L2 be isotopic loops and let N\, Nlr and Nlm be the left, right and middle nucleus of Li (i — 1,2), respectively. Then Nj = N f , N} = N} 2 and N]m = Nm " " " — · Proof. The isotopic loops L\ and L2 are isomoφhic to the principal isotopic loops (L, ·) and (L, *), respectively, where the operation * is defined on L by x * y = xja • b\y ,

(a,b G L) .

It follows from Lemma 1.9 that χ • y = χ */a * b^y .

21

1.2 Loops as sections in groups

We denote by A and A* = Λλ^ 1 = ^ χ e L] the set of left translations of (L, ·) and of (L, *), respectively. The left nuclei N[ of (L, ·) and N* of (L, *) are given by the set Ni = {n e L; λ„Α = A}, respectively N* = {n' e L; λ*,A* = A*}. For any η e Ni we have Ki baA*

=

! Mn-ba)/a^b 1 A* = A* 1 = λ „ λ * λ ^ Λ * = λ ^ Α λ - = Αλ^ 1 = A*

and hence η • ba e N*. Conversely, for any n' e N* we have λη'*(α*^)Α = A since xy = x*/b * a\*y. Hence n' * (a * b) = η' * 1 = n'/a • b\ 1 e /V/. In particular, if η' — η • ba then n'/a • b\ 1 e Afy. Since the mappings η η · ba : Ni ^ and η' ι — n ' / a • b\ 1 : N* Ν ι are injective we conclude N* = Ni • ba. The mapping η η • ba : Ni Nf is an isomorphism of groups since for m, η € Λ// we have (m · ba) * (n · ba) = (m • ba)/a · b\(n • ba) = mb • b\(n • ba) = m(n • ba) = mn · ba . The claim for the right nuclei Nr and N* can be proved considering the opposite loops of (L, ·) and (L, *). The middle nucleus Nm of (L, ·), respectively N* of (L, *) is given by the set Nm = {« € L; Λλ η = A}, respectively N* = {n' e L; Α*λ*, = A*}. For any η e Nm we have Α λ

* 1ηα = A*kbna/akbl

=

= Αλ^'λ^,λ,,λ^1

= A*

and hence bna e N^. Similarly, for any η' e Ν£ we havea*n'*b = (b\n')/a e Nm. From this it follows N* — bNma. The mapping η ι->- bna : Nm —>• Ν£ is a group isomorphism since for any m,n e Nm one has (bma) * (bna) = bm -na = b- mn · a.

• In the following theorem we show that the isomorphisms and isotopisms between the loops L \ and Lj defined on the same set and having the same group G generated by the left translations can be described by the relations between the stabilizers of the unit of L ι and of the unit of Lj in G and by the properties of the sets of left translations of L\ and L2. Theorem 1.11. Let L be a loop and G be the group generated by the left translations λχ,χ e L, of L. We denote by Η the stabilizer of I e L in G and by σ : L —• G the map χ 1—> Χχ. (i) Let a : G —» G be an automorphism of G, let σ : G/H —> G, where o(gH) — for g 6 G, and let an '• G/H —> G/a(H) be the mapping given by kH i-> a(k)a(H). Then the section σ = α ο σ ο adetermines a loop multiplication on the cosets ga(H), g e G, with the identity a(H), and

22

1 Elements of the theory of loops

this loop on G/a(H) is isomorphic to the loop L on G/H. If L and. L' are isomorphic loops having the same group G of l e f t translations and the same stabilizer Η of 1 € L, L' then there is an automorphism of G leaving Η invariant and mapping the l e f t translations of L onto the l e f t translations of L'. Conversely, if a is an automorphism of the group G generated by the left translations ofL which leaves the stabilizer Η of 1 e L in G and the set {λ^; χ e L} of l e f t translations invariant then a induces by α (λχ) = automorphism ) an a : L —• L of the loop L. (ii) Let nb '• G L be the mapping defined by Kb(g) = g(b), (b e L). The map cib · L —> G defined by Ob(x) = o{x/b) (b e L) is a section with respect to nb '· G —> L. This section defines a loop multiplication χ ·φ) y on L with identity b e L and it is principal l e f t isotopic to the original multiplication of L: x -(b) y = x/b • y. The stabilizer subgroup of the new identity b in G is gHg~{ where g is an element of G satisfying g ( l ) = b. (iii) IfAdg denotes the inner automorphism χ ι-* gxg~l of G and σ : G/H —» G is the section given by left translations of L then σ — Ad s σ Ad~1 determines together with Η a loop multiplication which is l e f t isotopic to that ofL, i.e. there is ab G L such that this loop is isomorphic to one in which the multiplication is given by y) x/b • y. The seta (G/H) is equal to ga'(G/H)g~l. (iv) The map σ/, : L —• G defined by Ob (χ) = σ(χ)σφ)~ι (b £ L) is a section with respect to Kb '. G L : g i—• g(b). This section defines a loop multiplication χ -(b) y on L with identity b e L which is principal right isotopic to the original multiplication ofL: χ ·ψ) y = χ • a(b)~ly. The stabilizer subgroup of the new identity b in G is gHg~] where g ( l ) = b. (v) Let L' be a loop isotopic to L. Then there exists a loop L" isomorphic to L' such that the group generated by the left translations of L" is the group G, the stabilizer H" of \" € L" in G is conjugate to Η and the section corresponding to L" is obtained from the one corresponding to L by the constructions described in (ii) and (iv). In particular if the loops L and L' have the same group G generated by the l e f t translations then there exists an automorphism a : G —> G such that the image a (H) of the stabilizer Η of 1 e L in G is the stabilizer of the identity V ε L' in G. Proof

(i) The multiplication determined by σ on the cosets of a(H) is given by

[ga(H)} * [ka(H)] = 5(ga{H))a(ku(H))a(H)

.

The isomorphism is given by the mapping an '• uH Μ* a(u)a(H)

a(uH • vH) = a(a(uH)a(vH)H) = (α ο σ ο

= (a ο a)(uH)(a

1 )(α(w)a(//))(«

= a(a(u)a(H)]a(a(v)a(H))a(H)

οσ ο

ο

= a(uH).

Then

σ)(νΗ)α(Η)

α~^1){α(ν)α(Η))α(Η) = a(u)a(H)

* α(ν)α(Η)

.

1.2 Loops as sections in groups

23

Conversely, if L and L' are isomorphic then this isomorphism induces a bijective mapping between the sets of left translations preserving the group multiplication. (ii) The assertion is an immediate consequence of the fact that the multiplication x/b-y gives a loop which is principal left-isotopic to L and the corresponding section is χ M>· o{x/b). (iii) If g e G is an arbitrary element in the group generated by the the left translations of the loop L then the section σ of L and the subgroup gHg~l determine the isotopic loop with identity g ( l ) and the section σ(χ) = σ ( χ / # ( 1 ) ) . If we apply to σ the inner automorphism A d ~ ! then the section σ = A d " 1 σ Adg and the stabilizer Η = Adgl ( g Η ) determine a loop isotopic to L. The image of the section σ can be obtained by the application of A d " 1 to the image of the section σ . (iv) The assertion follows from the fact that χ • b\y gives an isotopic loop and the corresponding section is σ(χ)σφ)~Κ (ν) Since all loops L' isotopic to L are isomorphic to a loop L" the multiplication of which is given by χ ja • b\y for suitable a,b e L the statement is clear. • Corollary 1.12. Let Gi (i = 1, 2) be a group, Hi (i = 1, 2) be a subgroup of Gi containing no normal non-trivial subgroup of Gi and let σ, : Gil Hi —> Gi be a sharply transitive section on G JHi such that σ, ( G , / / / , ) generates Gi. We denote by Li the loop defined on Gi/Hi by the section ai. (i) The loop Lj is principal left isotopic to L \ if and only ifG2 — G ι, Hi — Ad g H\ for a suitable g e G\ anda\(G\/H\) = criiGi/Hi). (ii) The loop Li is principal right isotopic to L\ if and only if Gi = G \, Hi — Ad g Η ι for a suitable g € G\ and cri(kg~x (gHg~x)) = σ\ (kH)[a\(gH)]~l for all k e G. (iii) The loop Li is principal isotopic to L\ if and only if Gi = G\, Hi = Adg H\ andai{kf-{{fHf~x)) = σχ {kH)[ox ( / / / ) ] " 1 for suitable g, f e G and for all k e G. (iv) The loop Li is isomorphic to L\ if and only if there exists an isomorphism a : G\ —» Gi such that a(H\) = Hi and σι = α ο σι ο α^1 where ctH\ '• Gi/Hi —Gi/Hi is the mapping given by ctnx{kH\) — a(k)Hi, k 6 Gi. (ν) The loop Li is left isotopic to L\ if and only if there exists an isomorphism a : Gi — G 2 and an element g e G1 such that Hi — a ο A d g ( / / i ) and ai(Gi/H2) = a oa\(G\/H\). Proof (i) If L1 and Li are principal left isotopic loops then every left translation of Li is a left translation of L\ and conversely. Hence o\{G\/H\) — oi{Gi/Hi) and Gi — Gi. The assertion (i) follows now from Theorem 1.11 (ii).

24

1 Elements of the theory of loops

(ii) If L\ and L2 are principal right isotopic then the set 02^2!H2) of left translaλ tions of L2 is equal to the set a\(G\/Η\)λ~ for a suitable λ e o\(G\/H\) (cf. Theorem 1.11 (iv)). Since the group generated by a\{G\/H\) coincides with the group generated by o\(G\/Η\)λ~χ we have G2 — G\. Theorem 1.11 (iv) implies that for principal right isotopic loops L\ and L2 the conditions in (ii) are satisfied. Conversely, let L\ and L2 be loops satisfying the conditions of (ii). We denote G — G\— G2. Because of H2 = gH\g~x the homogeneous space G/H2 parameterizes the same set as G/H\ if we choose as origin the coset gH\. Clearly the stabilizer of gH\ in G is the group H2. The point kH\ (k e G) corresponds in G/H2 to the coset kg~l (gH\g~x) = kg~lH2. It follows that L2 is principal right isotopic to L\ if and only if a2(kg-lH2) = σι (kH^fa (gHx)]"'. (iii) The assertion follows from (i) and (ii) since any principal isotopism is a product of a principal left and of a principal right isotopism. Let β : L\ —• L2 be an isomorphism. If Λ ^ denotes the left translation by JC in Li (ι = 1, 2) then we have = β ~ ] . Since products of left translations and of their inverses in L\ and L2 are in one-to-one correspondence to each other the loop isomorphism β determines uniquely a group isomorphism a : G1 —> G j . If an element g of G \ fixes the identity of L \ then clearly the element a(g) fixes the identity of Zv2- Since β : (Li, ·) (L2, *) is an isomorphism we have ß(kH\)

* β(ΙΗ\)

= ß((kH\)

· (IH\))

for all k, I e G. This relation is equivalent to [a2(ß(kHi))](ß(lHi))

= ß G2jH2 is a loop isomorphism. (v) Any left isotopism is a product of a principal left isotopism γ and an isomorphism α. According to (i) the principal left isotopism γ yields the data G\ = Gi, H[ = Adg Hx for a suitable g e G1 and a\{G\/H\) = a[(G\/H[). Using (iv) we obtain G2 = « ( C j ) = a ( G i ) , H2 — οι{Η[) — a ο Ad g Η ι and

1.2 Loops as sections in groups

25

a2{G2/H2) = α ο σ ( ο a ~ ! ( G 2 / / / 2 ) = α ο a[(G\/H[) = a ο a , ( G , / / / i ) . Conversely, if the conditions of (v) are satisfied then the element g e G\ determines a principal left isotopism which is associated with the data G\ = G\, H[ = Ad g H\ and σι ( G i / # i ) = a[(G\/H[). We have G 2 = a(G\), H2 = a(H[) and a2{G2/H2)

= α ο a[{G\/H[)

=αοσ[ο

aH){G2/H2).

•

Proposition 1.13. Let Κ be a group, Η a subgroup of Κ and σ : (Κ/Η) —> Κ a section with σ{Η) = 1 e Κ acting sharply transitively on the factor space Κ/Η = {xH\ χ Ε Κ }. Then the multiplication (xH) * (yH) = a(xH)yH defines a loop (L, *) on K/H. Let U be the subgroup of Κ generated by σ(Κ/Η). Then the mapping σ' : U/{U Π Η) —>• U defined by a'(u(U Π Η)) = a(uH) yields a sharply transitive section on the factor space U/((/ Π Η) and the loop L' corresponding to σ' is isomorphic to L. The group G generated by the left translations of L is isomorphic to the group U/N, where Ν is a normal subgroup of U contained in Η which is maximal with respect to this property. The stabilizer G\ of 1 6 L in G is isomorphic to the group (U ΓΊ H)/N. Proof. Clearly the multiplication of (L, *) has the identity element Η and L = {xH\ χ 6 K) = {yH; y e σ(Κ/Η)} = [uH; u e U). The element σ(aH)~lbH is the solution of the equation (aH)*(xH) = bH. The existence o f t h e unique solution of the equation (xH)*(aH) — bH is guaranteed by the sharply transitive action of the χ section σ . The group Κ is the disjoint union Κ = Η · As C U we have U — U Γ) Κ = U Π {{JX€CJ(K/H)(U CIXH)) = \Jx€a{K,H)x{U Π Η). The mapping φ : uH u(U Π Η) with u € U is an isomorphism of the loop L onto the loop L' since φ{μΗ * υΗ) = φ{σ(μΗ)υΗ)

= a(uH)v(U

= a'(u(U η H))v(U

Π Η)

η Η) = u(U Π Η)* v(U η Η).

The elements u e U fixing every element of L form a normal subgroup Ν of U which is contained in U Π Η and which is maximal with respect to this property. Clearly we have Gi = (U η Η)/Ν. • Remark 1.14. Let (L, *) be a loop as defined in the previous proposition. (a) L has the left inverse property if and only if [σ(Κ/Η)]~ιΝ

=

σ(Κ/Η)Ν.

(b) L is a Bol loop if and only if the product aba belongs to σ(Κ/Η)Ν a, be σ(Κ/Η).

for any

26

1 Elements of the theory of loops

Lemma 1.15. Let G be a group, Η be a subgroup and σ : G/H —> G a section corresponding to a loop L on G/H such that G is the group generated by the left translations of L. The loop L is a Bol loop if and only if the subset a(G/H) c G is closed with respect to the operation (jc, y) χ + y = xy~xxfor all x, y e a(G/H). Proof. L is a Bol loop if and only if for all left translations λ χ , λγ the product λχλγλχ and λ " 1 are again left translations. Since the set σ (G/H) consists of left translations the assertion follows. • Proposition 1.16. Let (L, *) be a loop and G be the group generated by the set A of left translations of L. If Η is the stabilizer of 1 e L in G then we denote by π : G —»• Η the mapping which assigns to every element g = kh with λ e A, h e Η the element h. For a subgroup U of the group G the map u i-» u (1) : U —> U(l) c L is a homomorphism if and only ifv~]n(U)v C Η for all ν e U. Proof Let σ : G/H —• G be the section corresponding to the loop L and let A = a{G/H). We have (uH) * (vH) = o(uH)vH = un(u)~xvH = uvH if and only if n(u)~lv = vh with a suitable h e Η for any u,v e U. • Proposition 1.17. Let L be a loop and L\, L2 be subloops of L. Let G,G\ and G2 be the groups generated by the left translations of L,L\ and L2, respectively. Let Η, H\ and H2 be the stabilizers of the identity of L, L\ and L2 in G, G \ and G2, respectively. The loop L is the direct product of the loops L\ and L2 if and only if G = G\ χ G2, Η — H\ χ H2 and for any left translation λ of L there exist uniquely determined left translations λ] of L\ and λ2 of L2 such that λ = λ\λ2 and any product λ\λ2 of left translations λ,· of Li (i = 1, 2) is a left translation of L. Proof. If L = L\ χ L2 then the decompositions of G, Η and λ follow immediately. Conversely, from the unique decomposition λχ = λγλζ = λζλγ for any χ e L with y e L\ and ζ € L2 we obtain L\ Π L2 = {1} and yz = zy. Moreover, for χ e L\, y e L2 the left translation kxy has the unique decomposition kxy = λχλγ and hence jc · yz = xy · ζ for any λ ε LI, y, ζ e L2. If χ, y e L\ and ζ e L2 then χy • ζ = ζ • xy = x • zy = χ • yz since λζλχ = λχλζ and zL\ = L\z. • Proposition 1.18. Let Lbea loop and G be the group generated by the left translations of L. We denote by Η the stabilizer of I 6 L in G. If G and Η are the direct products G = G[ χ G2 and Η = H\ χ H2 with Hi C G, (i — 1,2) then L is the product of two loops L\ and L2 such that L, is isomorphic to a loop Li having Gi as the group generated by the left translations of Li and Hi as the corresponding stabilizer subgroup (i = 1,2). Let σ = (σι, σ2) : G/H —G\ χ G2 be the section corresponding to the loop L with σ,· : G/H Gi (i = 1, 2). The loop L\ is a normal subloop of L if and only if 02(81 H\. 82H2) = σ2(Ηι,

g2H2)

1.2 Loops as sections in groups

27

for all gi G G\, g2 £ G2. The loop L2 is a normal subloop if and only if a\(g\H\,g2H2)

= σι (gi Hi,

H2).

Moreover, every element of Li commutes with every element of L2 if and only if σι(Ηι,82Η2) = 1 anda2(giHu H2) = \ for all gi e Gi,g2 e G2. Proof We define the functions ρ: (Gi χ H2)/(Hi

χ H2)

G{

and

r : (Hi χ G2)f(H\

χ H2)

by Q(giHi, H2) = a\(g\H\, H2) and x(H\,g2H2) = a2(Hi,g2H2). the subsets L j = { ( g i H u H2)\ gi e G i ) and L2 = {(Hi,g2H2)·, subloops of L. Indeed, (gl Hi, H2) * (kiHi,

H2) = a(giHi,H2)(kiHi,

G2

We prove that g2 e G2} are

H2) = (e(giHu

H2)hHuH2)

is an element of Li since a(giH\, H2) e (giHi, H2) and hence L1 is closed with respect to the multiplication. For the solution (xH\,yH2) of the equation (χΗι, yH2) * (gi Hi, H2) = (kHx,

H2)

we have (ai(xHi,yH2)giHi,a2(xHi,yH2)H2) and using the relation (σι(χΗ\,

=

yH2), σ2(χΗι,

a2(xHi,yH2)H2

yH2))

(kiHuH2),

e (xH 1, yH2) we get

= yH2 — H2

and hence y e H2. The solution (χΗι, yH2) of the equation (gi Hi, H2) *(xH 1, yH2) = satisfies (σι(8ιΗι,Η2)χΗι,σ2(8ιΗι, = (e(giHu

H\, H2)

H2)yH2)

H2)xHi,o2(giHi,H2)yH2)

= (kxH{,

H2)

where a2(giHi, H2) e H2. Hence y e H2 which shows that Li is a subloop. Similarly we can see that h2 is also a subloop. Clearly Li Π L2 — {1}. Any element (xHi,yH2) e L can be decomposed as a product of Li and L2 since (χΗι, yH2) = (χΗι, H2) * (Hi, σ2(χΗ\, H2)~lyH2). This decomposition is commutative if and only if σ2(χΗι, H2) = σι (Hi, yH2) = 1. Because of (Gi χ H2)/H2

= Gi,

(Hi χ G2)/Hi

^ G2,

(Hi χ H2)/H2 (Hi χ H2)/HI

^

Hu

= H2

the sections ρ and r define loops Li and L2 which have Gi and G2 as the groups generated by the left translations. L\ is isomorphic to the loop L1.

28

1 Elements of the theory of loops

L ι is a normal subloop of L if and only if the mapping φ : (gi Hu g2H2) η* g2H2 : L ->· L2 is an epimorphism. This means φ[(8ΐΗι,82Η2)

* (kiHi,k2H2)]

= (g2H2) *

= τ(Hu

g2H2)k2H2.

But we have (σι (gi H\, g2H2)ki Hi ,a2(giHi,

g2H2)k2H2)

= a2(giH\,

g2H2)k2H2

andhence k^1 τ (Hi, g2H2)~]a2(gi Hi, g2H2)k2 e #2 for any gi e G\,g2,k2 where x(H\,g2H2) and a2(giHu g2H2) € g2H2. Therefore Li is a normal subloop of L if and only if the elements k^T(Hi,

l g2H2r a2(giHi,

€ G2,

g2H2)k2

generate a subgroup of H2 which is normal in G. This is equivalent to a2(gj Hi, g2H2) = σ2(Ηι, g2H2) since Η does not contain any non-trivial normal subgroup of G. The assertion for L2 can be proved in an analogous way. • Proposition 1.19. Let G = G ι χ G2 be the direct product of the groups G\ and G2 and Η = Hi χ H2 with Hi C G; 0 = 1, 2) be a subgroup ofG. I f a : G / Η —» G is a sharply transitive section such that σ (G/Η) = M\ χ M2 with Mi C Gi (i = 1, 2) then the loop L corresponding to σ is isomorphic to the direct product of the loops Li and L2, where Li is defined in Gi/Hi with respect to the section σ, the image σι (Gi / Hi) of which is Mi (i = 1,2). Proof The sharp transitivity of σ means that for any (gi, g2), (k\, k2) e G = G ι χ G2 there exist precisely one pair (mi, m2) e M\ χ M2 and one pair (hi, h2) e Hi χ H2 satisfying the system of equations (*)

migi=kihi

(i = 1,2).

We want to show that the sections σ,· (i — 1, 2) are also sharply transitive, i.e. both of the equations m ( g, = (i = 1, 2) have unique solutions. Let the pair (mi, ki) be the solution of the first equation and (m2, k2) be the solution of the second equation. Then the quadruple (mi,ki,m2, k2) is the unique solution of the system (*) of equations. • Proposition 1.20. Let L be a loop and Μ be a normal subgroup of L contained in the intersection of the left nucleus Νι with the middle nucleus Nm of L. If G is the group generated by the left translations of L then the set M* — {km; m e M] of left translations is a normal subgroup of G and the group G' generated by the left translations of the factor loop L/M is isomorphic as a permutation group to the factor group G/M*.

29

1.3 Topological loops and differentiable loops

Proof. First, we show that Μ* is a normal subgroup of G. Since Μ is contained in Νι Π Nm we have Xxtn = λχλ m dfi\dXmx — XmXx for all χ G L andm £ Μ. In particular λ/ηη = λ η λ η and λ " 1 = Xm~\ hold for all m, η e M. Therefore M* is a subgroup of G. The group M* is a normal subgroup of G if and only if λχΜ* = Μ*λχ is fulfilled for all left translations λ χ of L. This is true since λ χ Μ * = Xxm = λ-Μχ = Μ*λχ for all χ e L. •

1.3 Topological loops and differentiable loops If L is a locally compact topological loop then the group G generated by the left translations of L can be provided with a natural topology, called the Arens topology (cf. [8], §6, [51], IX.2) such that G becomes a topological transformation group on L, the group Ge a closed subgroup and σ a continuous section. The topological closure G of the group G in the group of all auto-homeomorphisms of L with respect to its natural (Arens) topology is the group topologically generated by the left translations. In the same manner we can consider the group topologically generated by the right translations or the group topologically generated by all left and right translations. If L is a compact or locally compact and locally connected the Arens topology of the homeomorphism group Θ of L (and hence the topology of any subgroup of Θ) coincides with the compact open topology (cf. [6]). For every topological loop L which has a locally compact group G as the group (topologically) generated by its left translations there exists a continuous section σ : G/H G, where Η is the stabilizer of 1 e L in G such that σ and the closed subgroup Η of G satisfy the conditions 1.) and 2.) of a loop section. Conversely, if G is a locally compact connected group, Η is a connected subgroup of G containing no non-trivial normal subgroup of G and σ . G/H G is a continuous sharply transitive section then the multiplication (χ Η, yH) t-> a(xH)yH defines in G/H a loop (L, ·) such that this multiplication " ·" and the division ( x H , yH) λ~ι„γΗ = σ(χΗΓιγΗ are continuous. Moreover, (L, ·) is a topological loop if and only if for all xH, yH the mapping ( x H , y H ) ι-* yH/xH defined by ( y Η / χ H ) x Η = y H is continuous. Hence, in locally compact connected groups G the sharply transitive continuous sections σ : G/H —> G, for which the set a{G/H) (topologically) generates G, characterize loops defined on locally compact connected spaces such that the multiplication and the left division are continuous. We call these loops locally compact almost topological loops. Proposition 1.21. A locally compact almost topological loop L is a topological loop if one of the following conditions is satisfied: (i) L is compact; (ii) L is locally connected and the right translations of L are

homeomorphisms;

1 Elements of the theory of loops

30 (iii) L is a topological

manifold;

(iv) the stabilizer Η of I € L in G is compact. Proof. If the condition (ii) is satisfied then the Arens topology of the homomorphism group Hom(L) of L coincides with the compact-open topology and hence the mapping Λ: Qx : L Hom(L) is continuous (cf. [24], Chapter XII, Sec. 10, 10.4, p. 275). In this case the mapping y) x/y — ρ~χ{χ) is the composition of the continuous map (x, y) i-^- (Λ;, ρ " 1 ) and of the evaluation map (*, ρ " 1 ) ι-»· ρ " 1 (Λ:) and hence the right division Qc, >>) x/y is continuous. If the condition (i) is satisfied then the assertion follows from (ii) since every bijective continuous map of a compact space is a homeomorphism. Now we assume that L is a topological manifold. Let U be a compact neighbourhood in L. Then Qx ( U ) = £/ χ is also a compact neighbourhood in L (cf. [24], Chapter XVII, Sec. 3, pp. 358-359). Therefore each bijective continuous map ρ χ : L —• L is a homeomorphism and the condition (iii) implies the assertion. If the condition (iv) satisfied then the claim is proved in [56], Theorem 3 (e). • From the previous proposition it follows the Corollary 1.22. Every almost topological loop defined on a homogeneous G/H, where G is a connected Lie group, is a topological loop.

space

There is a bijection between connected topological loops having a Lie group as the group topologically generated by the left translations and the triples (G, Η, σ), where G is a connected Lie group, Η is a closed subgroup containing no non-trivial normal subgroup of G and σ : G / H —• G is a continuous sharply transitive section such that the set a(G/H) generates topologically G. Lemma 1.23. Let L be a locally compact almost topological loop such that the group G topologically generated by the left translations {λ χ ; χ e L} is locally compact with respect to the Arens topology and this topology coincides with the compact open topology . Then G is homeomorphic to the topological product L χ Η as well as to Α χ Η, where A is the set of left translations of L and Η is the stabilizer of 1 G L in G. Moreover the locally compact subspace A is closed in G. In particular if G is connected then L and Η are connected. Conversely, if L is connected then G and Η are connected, too. Proof. The group G acts transitively and effectively as topological transformation group on L and hence the mapping g i-> g ( l ) : G — L is open (cf. [120], 96.8). But then also the restriction of this mapping to A is open and hence the map χ ι-> λχ : L —• A is a homeomorphism (cf. [24], Chap. XI, Sec. 10). Now the continuous map g h> (Ag(i), : G —• Α χ Η has as its inverse ( λ χ , h) i-»· Xxh : Α χ Η —» G. The second part of the assertion follows from the fact that if L is connected then the connected set {λ χ ; χ e L) generates a connected group. •

1.3 Topological loops and differentiable loops

31

Let L be a loop defined on the factor space G/H with respect to a sharply transitive continuous section σ : G/H —> G in the locally compact locally connected group G topologically generated by the left translations of L. Clearly, L is a topological loop if and only if the mapping {xH, yH) ι-»· ζ € a(G/H), where zxH = yH, is continuous. Definition 1.24. A connected topological loop L defined on a differentiable manifold is called almost differentiable (of the class Cr, r e { 1 , . . . , oo, a>}) if the multiplication (x, y) ι — χ • y and the left division (jc, y) i-»· x\y of L are differentiable (of the class Cr). Proposition 1.25. Let L be an almost differentiable n-dimensional {local) loop. The there exists a neighbourhood U of 1 e L such that the local loop induced by L on U is differentiable. Proof. The function (z, y) h-> z/y is the unique solution of the equation (z/y) • y = z. According to the implicit function theorem the solution z/y is a differentiable function of the variables y and ζ if the tangent map of the function χ ι-> χ • y — ργχ has the rank n. Since the map q\ = id and the rank of (ρ ν )* is locally constant there exists a neighbourhood U 3 1 in which (z, y) i-> z/y is differentiable. • Proposition 1.26. Let (L, •) be an almost differentiable loop and let (L, o) be a loop principal isotopic to L such that the multiplication "o" is given by χ oy = χ/a · b\y. Then (L, o) is almost differentiable if and only if the right translation map ρ : L —» L in (L, ·) is a diffeomorphism. Proof We have (x,y) Qäl(x) • λ^'ΟΟ = χ ο y and hence the multiplication is differentiable if and only if the map ρ ~1 is differentiable. For the left division in (L, o) we have (X, y) (λχ)_1(ζ) = (λχ/αλ"1)-1^) = Μλχ/αΓΗζ) where λ° is the left translation in (L, o). Hence the left division in (L, o) is differentiable if and only if the map χ Ι-> χ/α = ρ~1 (JC) is differentiable. • Lemma 1.27. Let L be a connected group G topologically generated by its mapping σ \ χ λχ . L G is a{L) = {λ χ ; χ e L) of the section σ in G.

almost differentiable loop such that the left translations is a Lie group. Then the a differentiable embedding, i.e. the image is an embedded differentiable submanifold

Proof. Let xo be a point of L. We have to prove that the mapping σ is differentiable at First we show that for λΛ0 there exists a neighbourhood UxXQ in G and a finite set of elements z\, . . . , zr in L such that for the pointwise stabilizer Θ of z\,..., zr in G one h a s ( U r l U k ) Π Θ = {1}.

32

1 Elements of the theory of loops

Let zi be an arbitrary element of L. If Θ, is the pointwise stabilizer of the points z\,... ,n and d i m ( 0 , ) > 0 then we choose as Zi+\ a point which is not fixed under the connected component of Θ,·. Clearly d i m ( 0 , + i ) < d i m ( 0 ( ) . Since G is finite dimensional there exist points z\,... ,zr such that Θ = Θ,- is a discrete Lie subgroup of G. Hence there exists a neighbourhood W of 1 e G such that W Π Θ = {1}, and we choose UxXQ satisfying U^ ϋχΧϋ C W. If for u, u' € U\tn one has uzi — u'zi for i = 1, ..., r the u'~Xu(zi) = Zi for i = 1 , . . . ,r and hence u = u'. Now, in the neighbourhood U\XQ the function σ(χ) = λχ is the solution of the system of equations λχΖί = χ Zi (i = 1, ..., r). Since the mapping g m>- (gzi,..., gzr) : U), Lr is a differentiable injection the solution λχ is differentiable by the implicit function theorem. • There is a bijection between almost differentiable loops having a Lie group as the group topologically generated by the left translations and the triples (G, Η,σ), where G is a connected Lie group, Η is a closed subgroup containing no non-trivial normal subgroup of G and σ : G/H G is a differentiable sharply transitive section such that the submanifold o{GjH) generates topologically the group G. Let L be an almost differentiable loop defined on the factor space Gj Η with respect to a sharply transitive differentiable section σ : G/H—> G in the connected Lie group G topologically generated by the left translations. Clearly, L is a differentiable loop if and only if the mapping (xH, yH) h-> ζ : G/H χ G/H —>· a(G/H) determined by zxH = yH is differentiable. The following two propositions show that the class of connected differentiable Bol loops coincides with the class if almost differentiable Bol loops having a Lie group as the group topologically generated by the left translations. Hence the connected differentiable Bol loops can be studied as differentiable sections in Lie groups. Proposition 1.28. Let G be a connected Lie group and Η be a connected closed subgroup containing no normal subgroup φ {1} of G. Let σ : G/H G be a differentiable sharply transitive section such that σ (G/H) generates G topologically and for all a,b 6 a(G/H) also aba e a(G/H). Then there exists a connected differentiable Bol loop L having a(G/H) as the manifold A of its left translations and G as the group topologically generated by A. Proof. Since the section σ is sharply transitive there exists a loop L defined on G/H such that a(G/H) consists of the left translations of L (cf. Definition 1.4). It follows from Remark 1.14 (b) that L is a Bol loop such that the multiplication "o" and the left division are differentiable operators. Since in a Bol loop b/a = a~l o[(aob)oa~l] (cf. the proof of Lemma 1.3) the assertion follows. • Proposition 1.29. Let L be a connected topological Bol loop and let G be the group topologically generated by the left translations of L. Then the following conditions are equivalent:

33

1.3 Topological loops and differentiable loops

(a) L is a differentiable loop. (b) The group G is a Lie group and the mapping σ : χ ι-»· λ* : L —> G is a differentiable embedding. Proof. If L is a differentiable Bol loop then G is a Lie group; this will be proved in Theorem 7.3 (iii) (cf. also [90], pp. 414-^4-16). From Lemma 1.27 it follows that σ : L —> G is a differentiable embedding. Now we assume that the condition (b) is satisfied. Then taking for Η the stabilizer of 1 e L in G we can apply the previous proposition. Hence the assertion is proved.D Definition 1.30. For a loop L the group generated by the mappings kx,y = k~ykxky is called the inner mapping group 3{L) of L. If L is a topological loop then the inner mapping group 3{L) of L is the closure of 3(L) in the group topologically generated by the left translations of L. The next lemma comes from [91], XII.6.3 Proposition. Lemma 1.31. Let L be a (topological) loop and G the group (topologically) generated by the left translations of L. Then the stabilizer Η of 1 e L in G is the group 3{L) (resp.3(L)). Proof. Clearly, the inner mapping group is contained in the stabilizer group Η. Conversely, we show that any element g e G can be written as g = kx • h with χ e L, h e 3{L). Indeed, we have the following identities: ^a^b = λab '

λαλ^) = kab • ka^ ,

= ka\b · (λ^ kaka\b)

1

= ka\b ·

,

λ"1 = λα\ι · (λαλα\ι)_1 = λα\ι ·

for all a, b e L. Because of kakb λ

—Xab-h\

α

Xakb

— ka\b · h-2 1 1

= kakb\ 1 · /l3 = ^a-b\ 1 ' h4 = λ^λ^ι ·

— λα\φ\ΐ)

· he

for suitable ft, e 3(L) (i = 1 , . . . , 6) induction shows that the element g — λ^ 1 ... λ ^ 1 can be written in the form g = λχ • h with χ e L,h e 3(L). • For connected and locally simply connected topological loops there exist universal covering loops (cf. [47], [49], [51], IX.l). The loops L for which the simply connected loop L is the universal covering correspond to discrete central subgroups of L.

1 Elements of the theory of loops

34

Definition 1.32. The connected and locally simply connected topological loops L \ and L2 are called covering related loops if their universal covering loops are isomorphic. Lemma 1.33. Let L \ and L2 be covering related topological loops which are homeomorphic to topological manifolds. If in L1 an identity F(x\,..., xm) = H(x\,..., xm) holds, where F and Η are monomials in the variables x\,..., xm, then this identity is satisfied also in A2· Proof (Cf. [88], p. 18.) We can assume that Lj, j = 1, 2, have the same universal covering loop L. The elements of L are homotopy classes of continuous curves with initial point 1 e L and with the same end point. The covering homomorphism L L\ assigns to every class of curves their end points. Let JC, (t ) be a curve which represents the element χ ι e L for i — 1 , . . . , m. Then from F(*i(f),

...,xm(t))

=

H(xx{t),...,xm{t))

for all t it follows F ( J C I , . . . , xm) — H(x\,..., xm). Since Lj, j = 1, 2, is isomorphic to a factor loop L/Zj where Zj is a discrete central subgroup of L we have the assertion. • Lemma 1.34. Let L be a connected, locally simply connected topological loop and let G be the group topologically generated by the left translations λα (a G L) of L. We assume that the group G is locally simply connected. Let L be the universal covering loop of L and let Ζ C L be the fundamental group o£L. Then the group G topologically generated by the left translations ku (u e L) of L is the covering group of G such that the kernel of the covering map φ : G —> G is Ζ* — {λΜ ; u £ Z] (Ζ* ΞΞ Ζ). If we identify L and L with the homogeneous space G/H and G/ Η, where Η or Η is the stabilizer of 1 in L or in L, respectively, then φ(Η) = Η, Η Π Ζ* = {1} and Η is isomorphic to Η. Proof. According toTheorem A 2.14. in [51], p. 678 there exists a universal covering of L as well as that of G. The universal covering loop L of L contains the group Ζ as a central subgroup (cf. [51], p. 216). Every element of Ζ associates and commutes with any element of L. Therefore the group Z* is a central discrete subgroup of the connected group G. Since the left translations of L have the form λφ(α) = λ Μ Ζ (u e L) we have G = G/Z* and G, where Η — τ

(//), satisfying

e G. The mapping

gH η* τ(g)H

: G/H

G/H

defines an isomorphism of the loop corresponding G/H isomorphic

1

to L. The set σ(G/H)

to the section σ onto the loop on

generates G and the kernel of the action of

G on G/H is the group r _ 1 ( l ) C H. Proof

Since G/H is homeomorphic to L the s£ace G/H is simply connected. There-

fore there exists a unique map σ : G/H

—G

theorem in [54], p. 89). The mapping σ : G/H is a sharply transitive section on G/H. there exists a unique element xH τ~][σ(χΗ)]^Η

=

The space G/H the space G/H. position

such that τ ο σ = σ (cf. The lifting -> G defined by gH ι-* σ ο

Indeed, for the elements τ(ξ\ )Η and

such that σ(χΗ)τ(]>\)Η

x(g2)H

and hence

x(g2)H

g2H. is homeomorphic to ( G / τ - 1 ( l ) ) / ( / / / r _ 1 ( 1 ) ) and therefore to

The mapping ( σ | a ( G / H ) ) ~ ] &(G/H)

(σ\ff(G/H))~l

since σ : G/H

=

τ(g)H

It follows that σ : G/H

—• G/H is equal to the com-

- > a(G/H)

is a homeomorphism

is a homeomorphism (Lemma 1.5). Therefore in the

σ(G/H)

set r _ 1 [ a ( j c H ) ] the element σ ( χ Η ) is the unique element from σ ( ό / Η ) such that a(xH)giH

=

g2H.

In a connected locally compact group any open neighbourhood of the identity generates the wholej^roup and r : G -»· a(G/H)

G is a local isomorphism, hence the set

generates G.

The mapping φ : gH (->· T(g)H

is an isomorphism from the loop defined on G/H

by the section σ onto the loop defined by on G/H by the section σ . Indeed L/N is a continuous open mapping if we provide L/N with the quotient topology, i.e. the set W is open in L/N if and only if its preimage φ~χ{Ψ) is open in L. The factor loop L/N is a topological loop and the space of L/N is regular. If L is locally compact then L/N is locally compact. Proof. Since L/N has the quotient topology φ : L —>· L/N is a continuous open mapping. The subspaces {xN\ χ e L} define a congruence relation Θ with respect the operations of L. The congruence classes of Θ are closed subspaces of L since the left translation of L are homeomorphisms. The topological space L/N coincides with the space L / Θ of congruence classes of Θ. Since the quotient map Θ : L -> L / Θ is an open map the mapping 0 x 9 : L x L - > L / 0 x L / Θ is also open. From this one can readily deduce that the loop operations of L/N are continuous. Hence L/N is a topological loop. Since the points of L/N — L / Θ are closed the topological space L/N is regular (cf. [51], p. 213). A continuous open mapping maps compact neighbourhoods onto compact neighbourhoods. Therefore with L also L/N is locally compact. • Proposition 1.38. If φ : L —» L' is a continuous epimorphism of a topological loop L onto a topological loop L' then the preimage φ~χ(\') of the unit V e L' is a closed normal subloop Ν of L if the point V is closed in L'. If φ is an open map then the quotient loop L/N is homeomorphic to L'. Proof. According to [18], pp. 60-61 the preimage φ~{(\') is a normal subloop Ν of L. It is closed in L if 1' is closed in L'. Because of the previous proposition the mapping L —» L/N is continuous and open. Since φ is continuous the mapping φ : xN ι-» L' is continuous and bijective. If ^ i s open then L' be a continuous epimorphisms from a locally compact topological loop L onto a locally compact topological loop L'. If L is a countable union of compact sets then φ is an open map. A proof of this assertion may be obtained by a slight modification of the proof of the corresponding statement for locally compact groups (e.g. [44], (5.29) Theorem, pp. 42-43). Now let L be a connected differentiable loop and TV be a closed differentiable normal subloop. Let T\ Ν be the tangent space of TV at 1 e L which is contained in T\L. Since any left translation λχ (JC e L) isadiffeomorphismevery co$QikxN = xN is a differentiable submanifold of L. For the induced tangent map (λ χ ) * on the tangent bundle TL of L we have (*)

(λχ)ΛΆΝ)

=

τχ(χλ0·

Hence the mapping χ ΜΝ) is a differentiable distribution in the tangent bundle TL. Because of the relation (*) any coset χ Ν is an integral submanifold of the distribution (λχ)*(Τ\Ν). Now it follows from [15], Proposition 11.3.3 that the distribution (λ Λ )*(ΓιΛ0 is integrable. This means that each point of L is contained in the domain D of a coordinate system JC1, . . . , xn, where η = dim L, such that the connected components of the submanifolds D Π (χΝ) are given by the equations n JC^+I = c o n s t , . . . , x — const, where q = dim Ν (cf. [15], p. 194). Therefore the set of cosets χ Ν (χ ε L) forms a (#2) x

Hi, H2)

1-+ g2H2 :

L ->>

is a loop epimoφhism. The element a{(g\, g2){(p(H2) χ Μ ι χ A/2 is determined by the following system of equations:

W

L2

H\, H2)) = {x,y) in

g\=x· K2 is a homomorphism and L is isomorphic to a semidirect product G i χι K\. Proof Since the group Η = (ρ(// 2 ), Η2) does not contain any non-trivial normal subgroup of G = G1 χ G 2 the group G1 χ G2 acts effectively on L. The section · L defined by x * y = α[(α\χ)

ο (a\y)]

yields the loop (L, *) isomorphic to (L, o) such that χ * y = (x/b) • (a\y).

•

According to Lemma 1.10, the left, middle and right nuclei of isotopic loops L are isomorphic. Hence they can be recognized within the collineation group of the 3-net N ( L ) associated to the isotopism class of L. As examples for this we give the following two lemmas: L e m m a 3.2. Let L be a loop and N ( L ) be the associated 3-net. Then the group Θ consisting of the collineations of N ( L ) which preserve all the 3 pencils of lines and which leave every vertical line invariant is the set of mappings of the form (jt, y) μ* (jc, yQa) '· L χ L —» L χ L, where a is in the right nucleus Nr of L and Qa : y i—> ya. Proof If θ e Θ then Θ leaves every vertical line invariant and maps horizontal lines onto horizontal lines; hence it can be written as θ : (jc, j ) Μ* (χ, > ,α ).

3 Nets associated with loops

55

The transversal line through a point , y) contains the point (xy, l ) , t o o . It follows xy • 1 α = χ • ya. We denote a = \a. If we put jc = 1 we obtain ya = ya or a = ρα. It follows xy • a — χ • ya and a e Nr. Clearly, this condition is sufficient too. • L e m m a 3.3. Let Ή be a 3-net and Γ be a group of collineations ο/Ή. Let Nh, Nv and Nt, respectively, denote the normal subgroup of Γ which leaves invariant every horizontal, vertical and transversal line, respectively. Then Nh Π Nv = Nv η Nt = N,f)Nh = {1}. Proof. Any element of the intersection of two of these normal subgroups fixes every point of N. • Now we give an incidence geometric description of 3-nets associated with (left) Bol loops. Let V be the pencil of vertical lines and let y,· (i = 1, 2) be the other two pencils in a 3-net N. Let Xj ( j = 1, 2) be lines from V and let x\ and yi be points of X[. Let A\ e and A2 e y2 be the lines through jci and Β ι e and Bi G V2 be the lines through yi. We denote by C\ and D\ the line from which is incident with A\ Γ) X2 respectively with B\ Π X^, by C2 and Z>2 we denote the line from which is incident with A2 Π Xj respectively with B2 Π X2. We call a configuration consisting of the lines Χ ι , X2, A\, A2, B\, B2, C\, C2, D\, D2 and of the points xi,yi,A\ Π X2, B\ Π X2, A2 Π X2, B2 Π X2, C\ Π C2 and D{ Π D2 a non-closed Bol configuration and say that the 3-net Κ satisfies the Bol condition with respect to the pencil of vertical lines (or that the non-closed Bol configuration can be closed with respect to the pencil of vertical lines) if C\ Π C2 and D\C\ D2 belong to the same vertical line X3. If a 3-net satisfies this condition we call it a Bol net. It is well known that the coordinate loops of a 3-net Κ are left Bol loops if and only if Ν is a Bol net (cf. [107], pp. 50-51). The coordinate loops of a 3-net X are Moufang loops if and only if Ν satisfies the Bol conditions for two (and then for any) pencils of lines (cf. [1]). In this case X is called a Moufang net. A Bol net which is not a Moufang net is called a proper Bol net. Let Ν be a Bol net. Then to every vertical line G there exists an involutory collineation tq of X fixing G pointwise. If p is a point of N t h e n pre is the intersection of the two non-vertical lines meeting the line G in the same points as the two nonvertical lines through p. The involutory collineation τq is called the Bol reflection on G. Conversely, if Ν is a net and to every vertical line G there is an involutory collineation tq of Ν fixing G pointwise then Κ is a Bol net (cf. e.g. [33]). Let Ν be a Bol net. Since the collineations tq (G € V) preserve the pencil V of vertical lines we can consider the mapping ? c which is induced by the collineation tq on the pencil V. L e m m a 3.4. For the map a : tq —> tg there holds a ( r c , 0 t c 2 ) = «(tc, ) · α(το2)· Hence the group Γ generated by the mappings tq is a homomorphic image of the

56

3 Nets associated with loops

collineation

group Γ generated by the collineations

phism a is isomorphic

TQ. The kernel of this

homomor-

to a subgroup of the right nucleus Nr of a coordinate

loop L

of the Bol net Ή. The image of any stabilizer of a point ρ of Ή in Γ under a : Γ —» Γ is contained

in the stabilizer

subgroup of Γ consisting

of the vertical

of products

line through ρ in Γ.

of an even number of reflections

stabilizer of a point ρ e Ν in Γο is contained in the intersection Γο of the horizontal Proof

If Γο denotes the

and of the vertical line through

τG then the

of the stabilizers in

p.

The multiplicative property o f the mapping a : Γ —»· Γ is a consequence of

the fact that the collineations tg

( G e V) preserve the pencil of vertical lines. From

the Lemma 3.2 follows that the kernel of a is isomorphic to a subgroup of Nr.

Since

Γ preserves the pencil of vertical lines and Γο leaves each of the 3 pencils of lines invariant, Lemma 3.4 is proved.

•

N o w we show how a Bol reflection τQ may be expressed in terms of coordinates with respect to a coordinate loop L. Let G be the vertical line {(w, υ); ν e L } , u e L. If we denote by (z, t ) the image το (jc, y) of the point ( x , y) then (z, t) is uniquely determined by the equations zt = t —

u~x(xy).

uy and ut = xy.

This is true since ut = u zt =

• u~]

(u · X _ 1 M ) ( W _ 1 · x y )

(xy)

W e claim ζ =

w ( x _ 1 w ) and

— x y and

- M ( X _ 1 · W ( W _ 1 · xy))

=

uy

because of the Bol identity. Consequently if Κ denotes the vertical line incident with ( 1 , 1 ) then we have (*)

*GTK(x,y)

= rc(x~l,xy)

= (u •xu,u~xy)

.

Proposition 3.5. The group Γο consisting of products of an even number of reflections τg is generated by the set of mappings

(x, y )

( λ „ ρ Μ χ , λ~ιγ),

u e L.

There is

Φ of the group Γο onto the group G generated by the left

a canonical

epimorphism

translations

{ λ „ , u € L) of L such that the kernel of Φ is isomorphic

to a subgroup

of the left nucleus Ν ι of L. Proof.

Using the expression ( * ) w e obtain that the group G is isomorphic to the

group G* of permutations induced by Γο on the set of horizontal lines. The canonical epimorphism Φ * : Γο

G * is given by the projection of the map (a, β ) € Γο with

(α, β) : (χ, y) m>- ( a ( x ) , ß(y))

onto the map β \ y \-> ß(y)

: L

L. The kernel of

Φ * consists of all mappings (α, 1) of Γο which leave every horizontal line invariant. The transversal line through the point ( x , y) contains the point (1, x y ) , too. It follows that or(1) · xy = a ( x ) · y. W e denote a = a ( l ) , putting y = 1 we obtain ax =

a(x)

or a = λ α . It follows that a • xy = ax • y. It means a e N[.

•

Lemma 3.6. Let L be a Bol loop, let Γ be the group generated by the Bol

reflections

and let Γο be the group generated by products of an even number of Bol

reflections.

Then Γο has in Γ index 2.

3 Nets associated with loops

57

Proof. Since the reflection tq (G e V) interchanges the sets of horizontal and transversal lines we have tq Γο, but Γ = Γο(το). • Lemma 3.7. Let L be a Bol loop having trivial right nucleus Nr, let Γ be the group generated by the Bol reflections in the 3-net 'N(L) and let Γο be the subgroup of Γ generated by even products of Bol reflections. If the group Σ induced by Γ on the set V of vertical lines coincides with the group Σο induced by Γο then Γ is the direct product Γ = Γο x Μ where Μ is a group of order 2. Proof. According to Lemma 3.2 Γο is isomorphic to Σο· There is a normal subgroup Μ C Γ leaving every vertical line invariant such that Γ / Μ = Σ . Because of Lemma 3.2 the group Γο Π Μ = {1}. Since Γο = Σο = Σ one has Γ/Μ = Γο and hence Μ φ\ 1} (cf. Lemma 3.6). Therefore Γ = Μ χ Γ 0 . • Lemma 3.8. Let L be a loop and 'N(L) = Ή be the 3-net corresponding to coordinate loop L. If the group % of collineations of Ν operates transitively on pencil of horizontal lines then any loop L' isotopic to L is isomorphic to a loop L" multiplication of which, expressed by operations of L, is given by (x, y) ι-> χ • for some fixed a e L.

the the the a\y

Proof. Let κ be a collineation and ρ € Ν be a point. Then the coordinate loops with origin ρ and κ(ρ), respectively, are isomorphic (cf. [8], p. 46). Since % operates transitively on horizontal lines, any coordinate loop of IN" is isomorphic to a coordinate loop with the origin on the horizontal line L χ {1}, 1 e L. Any loop L' isotopic to L is isomorphic to a coordinate loop of X and the multiplication of any coordinate loop with origin on L χ {1} can be expressed by (jc, y) χ · a\y (cf. [107], p. 42). Hence the assertion is proved. • Corollary 3.9. Let L be a Bol loop. If L' is a loop isotopic to L then there is a loop L" isomorphic to L which is principal right isotopic to L'. Proof. The assertion follows from Proposition 3.5, where it is shown that the collineation group of the 3-net N(L) is transitive on the pencil of horizontal lines. • Corollary 3.10. A loop L' is isotopic to the Bol loop L if and only ifL' is right isotopic to L. Proof Every isotopism is a product of a principal isotopism and an isomorphism. The assertion follows now from the previous corollary. • Corollary 3.11. The full collineation group Θ of a proper Bol net leaves the pencil V of vertical lines invariant. Proof Assume that there is a collineation a e Θ such that a(V) φ V. Then a - 1 t e a , G G V, are involutory collineations fixing pointwise a(G) e a(V). Hence

3 Nets associated with loops

58

the Bol condition is satisfied also with respect to the pencil a(V) φ V and Ν is a Moufang net. • Lemma 3.12. Let L be a proper Bol loop and 3Vf(L) be the 3-net having L as a coordinate loop. We denote by Γ the group generated by the involutory collineations το where G is a vertical line. Let Ty be the stabilizer of the vertical line V in Γ and Θ be the full collineation group of N(L). Then Γ is a normal subgroup of Θ and Ty is a normal subgroup of the stabilizer Θ ν of V in Θ. Moreover, two coordinate loops with origins ο and ο', respectively, on V are isomorphic if and only if ο and o' are contained in the same orbit of the stabilizer Θ ν. Proof Two coordinate loops of N(L) with origin ο and o', respectively, are isomorphic if and only if ο and o' belong to the same orbit of the full collineation group Θ of N(L). If ο and o' are two points on the same vertical line V then any collineation jc e Θ with x(o) — o' leaves V invariant since L is a proper Bol loop (Lemma 3.11). Hence two loops having origins on V are isomorphic if and only if these origins belong to the same orbit of the stabilizer ©y of V in Θ. Since the set of collineations {tc, G vertical} is invariant under the conjugations with elements of Θ (Lemma 3.11) the group Γ is a normal subgroup of Θ. Hence every element of 0 y normalizes Ty and Ty is a normal subgroup of ©y. • Proposition 3.13. Let L be a loop and !N(L) = "H be the 3-net corresponding to L. If the group % of collineations of Ή acts transitively on the pencil of vertical lines then any loop L' isotopic to L is isomorphic to loop L" which is principal left isotopic to L. Proof. Since X acts transitively on the pencil of vertical lines any loop isotopic to L is isomorphic to a coordinate loop the origin of which is on the vertical line {1} χ L. The assertion follows now from [107], p. 42. • Let C be the category of topological spaces, C°°-differentiable manifolds or analytical manifolds. A 3-net [NT is called a C-net if the point set of X and any pencil of lines are objects in C and the following mappings Λ:

HX : Ή

IK,

JC

i-> V* : Ν -> V

and

JC

TX

:

>Γ

Τ

assigning to a point JC the horizontal, vertical, respectively transversal line incident with χ as well as the mappings (Τ, Η) ^

Τ ΠΗ :Τ χ %

Κ

(V, Γ) Η» V n r : V x T - > K are C-morphisms. The restrictions {G} x V ^ G, G e Oi, {G} χ IK —• G , G e T, {G} χ Τ —• G, G e V, of the last three morphisms are injective and the inverse mappings are also C-morphisms. This shows that any line of Ν is an object of C.

3 Nets associated with loops

59

If Gi and G2 are lines of the 3-net Ν and D is a pencil of lines not containing Gi and G2 then the bijection G\ — G i which assigns to each point χ e G\ the point Dx Π G2 where Dx is the line of T> incident with χ is called the perspectivity along T> from G\ to G2. We denote this mapping by [Gi, T>, G2]. Compositions of perspectivities are called projectivities between two lines. If Κ is an object of the category C then projectivities are C-morphisms. Conversely, if L is a C-loop then the associated 3-net N(L) belongs to the category C, too. The point set of Ν carries the product topology or the product differentiable structure, the lines of [NT are embedded C-subspaces of N, perspectivities and projectivities between lines are C-morphisms. The topology or differentiable structure of a pencil of lines is defined so that the intersection map from the pencil onto a line not belonging to this pencil is a C-morphism and the inverse map is also a C-morphisms. If L is a differentiable loop of dimension η then the associated 3-net is of dimension 2η and its lines are «-dimensional embedded submanifolds. The pencils of lines are also «-dimensional manifolds (cf. [51], 9.4).

Section 4

Local 3-nets

Differentiable local 3-nets on a 2n-dimensional manifold Μ are geometries associated with isotopism classes of local differentiable loops. They are a useful tool to show that any almost differentiable global loop L is differentiable if and only if the right translations of L are diffeomorphisms. Moreover, they are local models of triples of foliations of codimension n, the leaves of which intersect each other transversally. In the terminology of differential geometry such structures are called 3-webs (cf. [11], [3]). Let G be the category of topological spaces, C 00 -differentiable manifolds, or analytic manifolds. Let Ν be an object in the category C and let !K, V and Τ be 3 disjoint families of subobjects of X; the elements of !K, V or 7 we call horizontal, vertical or transversal lines, respectively. The object X is called a local 3-net in the category C if the following conditions are satisfied: (i) Every point χ of Ν is incident with precisely one horizontal line Hx e !K, one vertical line Vx e V, and one transversal line Tx e 7. The injective mappings Η : χ Η* Hx : X Ή, V : χ Η* V* : X V, and Τ : χ Η* Tx : X 7 are C-morphisms. (ii) Let G be a line and X e {iK, V, T} be a family such that G does not belong to X. Then for the injective mapping X\c : g η :G Ϊ assigning to any point g e G the line from the family X which is incident with g the image is open in X and the inverse map Xl^ 1 : X(G) —> G is a C-morphism. (iii) Any two lines G e. Χ, Η e y, where X, y e {IK, V, T} and Χ φ y, have at most one point in common. If X — y then the lines G and Η have no point in common or G = Η. (iv) Let Χι, X2, X3 e {3ΐ, V, Τ} such let ρ = Χι Π Χ2 Π Χ3 where Χ, different indices (/', j, k) (i, j,k e Xi in Χ;, Uj of Xj in Xy and £4

that Χ; φ if i ψ j (/, j = 1, 2, 3) and is a line in the family X, . For all triples of {1,2, 3}) there exist neighbourhoods t/,· of of Xk in X* such that for any e £/,· and

61

4 Local 3-nets

€ Uj the intersection £,· Π is a point q and the line in X^ which is incident with q is contained in the neighbourhood Uk. Moreover, the mapping (£,·, (-> : Uj χ Uj —> Uk is a C-morphism. R e m a r k 4.1. To each local loop L we can associate a local 3-net Ν in the same category C such that the points of Ν are the pairs (x,y) e L χ L and the families are given by

X= {{(jc.y); xeL], V = { { ( * , > ) ; yeL}, Τ — {{(*.

yeL], x e L ] ,

j ) ; χ • y - ζ, x, y e L},

z e i j .

We can reconstruct the multiplication of the local loop L in a geometric way from the local 3-net Ή: Consider on V(i i) the points ( I , * ) , ( l , y ) and ( 1 , x y ) . Then ( l , x ; y ) = T{x,y) Π V(i,i) and (x, y) = H { η ν{χ Si and s Hs : S2 —> S2 are C-isomoφhisms and for any t e .Si and s e S2 the transversal line ψ(ντ,ηΗ ρ , Hs) belongs to Wm = C/3 Π W 3 . Now for all points x, y e Sm = Si Π 52 we can define the multiplication by the

map (λ, y) h-> χ * y = ψ(ΥτχΓ\Ηρ, Hy) Π VSpm: (ρ)

χ

Sm

Changing the role of the families !K and Τ or Τ and V we can find in an analogous way as before on Vg neighbourhoods 5/ and Wi or Sr and Wr in which the C-morphisms (x, y) ι-» χ \ y : 5/ χ 5/ —» Wi or (χ, y) χ / y : Sr χ Sr —Wr are well defined. (P)

On the

ip)

intersection 5 — Sm Π 5/ Π Sr all the three operations * , \ and / are

_

(Ρ) (ρ)

(ρ)

mappings 5 x 5 ^ · W = Wm U Wi U Wr satisfying (jc(Ρ) / y )W * y = x, y(ρ) * (y(Ρ) \ λ:) = χ. Hence we obtained a local loop in the category C with identity p. Such a local loop is called a coordinate local loop of N.

62

4 Local 3-nets

Let N(L) be the local net associated with the local loop (L, •). Then the local loop obtained from N(L) by the above construction with respect to the point ρ — (1, 1), 1 e L, is isomorphic to L. Theorem 4.2. Let N(L) be the 3-net associated with an almost differentiate loop (L, ·). The right translations Qx, χ e L, of L are diffeomorphisms if and only if the 3-net N(L) is differentiable. Proof The differentiable manifold N(L) is diffeomorphic to the product manifold L χ L. The 3-net N(L) is differentiable if and only if for every point ρ e N(L) there exists a neighbourhood U of ρ such that the restriction of the 3-net N(L) to U is a differentiable local 3-net. This is the case if and only if the coordinate local loop Lp of U with origin ρ is differentiable. The local loop Lp is induced by the Λ

Λ

coordinate loop Lp of N(L) with origin ρ in a suitable neighbourhood of ρ e Lp. The loop Lp is isomorphic to the loop (L, *) the multiplication of which is given by χ * y — x/b · a\y with respect to the map ζ μ* λ" 1 (ζ) : L —> L, where ρ = (a, b) € L χ L (cf. Lemma 3.1). According to Proposition 1.26 the loop Lp is almost differentiable if and only if the right translation Qt, : ζ π» z-b : (L, ·) —• (L, ·) is a diffeomorphism. If the coordinate loop Lp is almost differentiable then a suitable local loop Lp of Lp is differentiable (cf. Proposition 1.25). Since ρ is an arbitrary point of N(L) it follows that the 3-net N(L) is differentiable. If the 3-net N(L) is differentiable then the loop L is differentiable and hence each right translation of L is a diffeomorphism. • Corollary 4.3. An almost differentiable loop L is differentiable if and only if all right translations of L are diffeomorphisms. Important geometric sources for differentiable local 3-nets are triples of «-dimensional foliations on a 2n-dimensional manifold M. Such a foliation J is a partition {Fa, α ε A} of Μ into connected subsets with the following property: For every point of Μ there is an open neighbourhood U and a coordinate chart (x1, . . . , xn, y1, . . . , yn) such that for each leaf Fa the connected components of U Π Fa are defined by the equations yl — c o n s t , . . . , yn = const. Such a chart is a distinguished chart (cf. [130], p. 24). Let J = { Fa; a e A} and S = {Gß \ β e ß } b e η-dimensional foliations on a 2-dimensional differentiable manifold Μ such that the tangent space TpM at an intersection point ρ of the leaves Fa and Gβ is the direct sum of the tangent spaces of these leaves, i.e. TpM = TpFa®TpGß. If ( x 1 , . . . , jc", yx,..., y") is a distinguished chart of the foliation 7 defined on a neighbourhood U of ρ in Μ then the connected components of U Π Gβ can be given by the equations g V , . . . , /

1

) ^

1

,

gn(x\...,xn,y\...,yn)

= cn,

where g1,..., gn are smooth functions and cl,... ,c" are constant. Since the leaf Gß of the foliation S in the point ρ is transversal to the leaf Fa, it follows that

4 Local 3-nets

63

the Jacobian det ( f f r j Φ 0. Hence we can consider the coordinate transformation x' = gl (χ1,..., xn, y 1 , ..., yn), yl = yl, (i = 1 , . . . , η), which gives a coordinate chart in a neighbourhood U c U of ρ such that the connected components of U Π Fa are described by y] — c o n s t , . . . ,yn = const and the connected components of U Π Gß are given by χ 1 = c o n s t , . . . , x" = const. If we have a third n-dimensional foliation % = {Κγ, γ e Γ} such that the tangent space ΤρΚγ of the leaf KY through ρ is complementary to the tangent space TpFa as well as to TpGß then there exists a neighbourhood W of ρ in which the connected components of Κγ Π W can be described by the equations / ' ( χ 1 , . . . , xn, y 1 , . . . , yn) — const, where f l , f n are smooth functions (cf. Proposition 5.1 in [98], p. 39). In the neighbourhood W we consider the foliations 7ψ, Qw and %w induced by the foliations H, S and X, respectively. Each pair {Fa, Gß} of leaves Fa e 3 V and Gß e Sw has exactly one point in common in W and each point ρ e W is incident precisely with one leaf from 3~w and one leaf from Sw- Namely, the intersection point of the leaves given by the equations y 1 = )>Q, . . . , yn = y{j and x[ = XQ, ..., χ" = χβ is the point with coordinates (jcq XQ, Remark 4.4. There exists a differentiable mapping χ Sw ^ OCw', in local co1 n 1 ordinates ( ( y , . . . , y ), (λ , . . . , χ")) of 3~w x 9w this mapping can be represented by the equations / ' ( χ 1 , . . . , xn, y 1 , . . . , y") = cl. This remark together with the previous consideration gives the following Proposition 4.5. Let Μ be a 2n-dimensional differentiable manifold and , J2, 3~3 be three n-dimensional foliations of Μ. Then for any point ρ € Μ for which the tangent spaces of the leaves Fi e J, (/ = 1, 2, 3) through ρ satisfy TpM = TpFi Θ TpFj

(ι φ j, i j Ε {1, 2, 3})

there exists a neighbourhood which is a local 3-net with respect to the induced foliations as the families of lines. Conversely if one has a differentiable local 3-net Ν then for any point ρ e N there is a neighbourhood U™ and a local loop L such that the neighbourhood U™ is diffeomorphic to L χ L and the horizontal, vertical and transversal lines intersecting Up can be described by the equations {(x, y); y = const}, {(x, _y); χ — const} and {(x, y); χ · y = const}, respectively. Clearly the horizontal and vertical families of lines give foliations of Μ with n-dimensional leaves such that in any point q e U™ the tangent spaces of the horizontal and vertical leaves are complementary subspaces of the whole tangent space TqM. We consider the map (x, y) (x, xy) : L χ L —> L χ L. Then the families of transversal, vertical and horizontal lines induced in a neighbourhood Ulp of ρ can be represented as the sets xy — const}, χ = const} and {(χ,χ)'); x\xy = y = const} from which follows that the tangent spaces of the transversal and vertical leaves are complementary subspaces

64

4 Local 3-nets

of the whole tangent space of Μ at any point of Ulp. Similarly using the mapping (x, y) h-> (xy, y) : L χ L —> L χ L we get that there is a neighbourhood Up in which for any point the tangent spaces of the horizontal and transversal leaves are complementary. Definition 4.6. Let Μ be a differentiable manifold of dimension 2η and let 3~i, 3~2 and IT3 be foliations on Μ with η-dimensional leaves such that the tangent subspaces of the three leaves through any point ρ Ε Μ are pairwise complementary subspaces of the whole tangent space TpM. The structure (M, 3~i,3~2, 3"3) is called a differentiable 3-web (cf. [3], p. 4). Theorem 4.7. (i) The differentiable local 3-nets (Μ, !K, V, T) are precisely those differentiable 3-webs with foliations 3~\ = ίΚ, 3~2 = V, 3~3 = 7, for which any two leaves of different foliations have at most one point in common. (ii) For any point ρ in a differentiable 3-web (Μ, 7\, 32, 9~3) there exists a neighbourhood Up of ρ such that the 3-web induced in Up by the foliations Up is a bijection defined by the restriction of the global construction to the neighbourhood Up. Clearly this local reflection preserves the segments of vertical lines intersecting Up.

Section 5

Loop-sections covered by 1-parameter subgroups and geodesic loops

In this section we consider differentiable loops having a Lie group as the group topologi c a l ^ generated by their left translations and corresponding to differentiable sections σ . G/H G such that the image of the tangent space T\o(GJH) at 1 e G under the exponential mapping of G is contained in a(G/H). This class of loops locally coincides with opposite loops of geodesic loops with respect to a uniquely determined affine connection with vanishing curvature, and generalizes on the one hand the geodesic loops of reductive homogeneous spaces with respect to their canonical connection and on the other hand the differentiable (local) Bol loops. We emphasize that the reductive homogeneous spaces are related to differentiable left Α-loops. The groups topologically generated by the left translations of such a loop are Lie groups. Proposition 5.1. Let L be an almost differentiable loop and let G(L) be the group topologically generated by its left translations. Let a . L G(L) : χ λχ be the corresponding section. Every element ofa(L) \ {id} is contained in precisely one 1 -parameter subgroup ofG(L) if and only if the following conditions are satisfied: (i) every element of L \ {1} is contained in precisely one l-parameter of L, (ii) one has χ {yz) = (xy)zforx, subgroup of L.

subgroup

y, ζ € L if χ and y belong to the same 1 -parameter

Proof. First we assume that the section a{L) is simply covered by l-parameter subgroups of G(L). If γ is a l-parameter subgroup contained in a(L) then the restriction of the evaluation mapping o{L) —• L : λχ M>- λ χ ( 1 ) = χ to γ is an isomorphism. Hence the l-parameter subgroups of L cover L. Since in an almost differentiable loop L the 1 -parameter subgroups are uniquely determined by their tangent vector at 1 e L (cf. Proposition 1.25) condition (i) is satisfied. Condition (ii) follows from jf(yz) - λχ(λγζ)

- (λχ ο λγ)ζ = kxyz -

if x, y are contained in the same l-parameter subgroup of L.

(xy)z

66

5 Loop-sections covered by 1-parameter subgroups and geodesic loops

If the conditions (i) and (ii) are satisfied then for every 1-parameter subgroup χ (t) of L the set { λ ^ ; t e M} is a 1-parameter subgroup of G(L). • A connected Lie group L satisfies the conditions of Proposition 5.1 if and only if L admits the partition into 1-parameter subgroups, i.e. every element χ φ 1 of L is contained in precisely one 1-parameter subgroup of L. Certainly, every Lie group, for which the exponential map is a diffeomorphism, has this property. For simply connected solvable Lie groups the existence of the partition into 1-parameter subgroups implies the bijectivity of the exponential map, since according to [22] in these groups the exponential map is surjective if and only if it is injective. But there exist Lie groups, for which the exponential map is only surjective and which admit the partition into 1-parameter subgroups: Proposition 5.2. The groups SC>2(M), S03(R), PSL2(M) and the connected component E2 of the isometry group of the euclidean plane admit the partition into 1-parameter subgroups. Proof. Every element of £ 2 \{1) is either a translation or a rotation. Every element of SC>3(IR)\{1} is a rotation in R 3 fixing a line pointwise. The group PSL2(K) is the connected component of the isometry group of the hyperbolic plane (cf. Section 22). Every element a of PSL2(R)\{1} is either a hyperbolic rotation or a hyperbolic translation along a line or leaves neither a point or a line fixed, α is a hyperbolic rotation if and only if | trace aj < 2. The element α is a hyperbolic translation if and only if | trace a | > 2. Every element a which leaves neither a point nor a line invariant satisfies | trace a \ = 2 and it is conjugate to an element of the form ± This element is contained in the 1-parameter subgroup

(ί0·

t Φ 0.

t e The normalizer of A in PSL2 (K) is the group

H o

s-i

Hence every element a e PSL2(R) with | trace a \ = 2 is contained in a 1-parameter subgroup A5 = R") with e € U, φ(ε) = 0 such that for any 1-parameter subgroup yx(t) of L we have φ(γχ(ί)) = tX with a suitable X € R" (cf- [74]). If L is a Lie group then exp is the usual exponential map from the Lie algebra of L into L. A differentiable strongly left alternative loop L is completely left alternative if and only if the map exp : TeL L is surjective. Theorem 5.4. Any connected differentiable Bol loop (L, o) is strongly left alternative. Proof Let N(L) be the 3-net associated with the Bol loop L. Since every loop isotopic to L is a Bol loop the 3-net N(L) associated with L is a Bol net. Then in particular in every coordinate loop of N(L) the identity Λ: ο (χ ο χ) = (Χ ο Χ) ο χ is satisfied. According to Theorem 5.5 in [99] there exists a neighbourhood U of e in L which is simply covered by 1-parameter subgroups of L. Hence the condition (1) of Definition 5.3 is satisfied (cf. also [52]). For the proof of the condition (2) we mention that for the Bol loop (L, o) the identity xp ο (χΐ ο y) = xP+q ο y holds for any x,y e L and p,q e Ν (cf. [107],

68

5 Loop-sections covered by 1-parameter subgroups and geodesic loops

IV.6.5 Theorem). Let {expfX; t e R} be a 1-parameter subgroup of L (X e TeL). This identity is equivalent to λ&χρρΧλ&χρηΧ = kexp(p+q)X. Since λ β χ ρ (_χ) = λ ~ ρ λ : we have kexpmxXexpnX = A ex p( m+n )x for all m, η e Ζ and X e TeL. Let q = 2s, s € Ν and r, ρ € Ζ. Then we have ΚχρϊχΚχρ^χ = (Aexp|)p(Aexp|)r = ( A e x p f ) ^ = λβχρ£±τχ. Since the set {λ 6χρ ε χ', q = 2s, s e Ν, ρ e Ζ} is dense in {kcxptx', / e l } , this set is a 1 -parameter subgroup of the group G topologically generated by the left translations of L. Hence the assertion is proved. • Proposition 5.5. Let L be an almost differentiable loop such that the group G topologically generated by the left translations of L is a Lie group. We denote by Η the stabilizer subgroup of the identity e e L in G and by σ : L G the section σ \ χ \-> λχ. The loop L = G/ Η is strongly left alternative if and only if e\p[Tia(G/H)] holds, where T\a(G/H)

C

a(G/H)

is the tangent space of the submanifold a(G/H)

at 1 e G.

Proof. The relation exp[T\a (G/H)] c a{G/H) implies that if λχ — expX for χ e L, X 6 T\a(G/H) then (exp tX)e, e e L, is the 1-parameter subgroup x' of L. Consequently we have (expiX · expsX)}* = e x p t X ( e x p s X ( y ) ) = xf(xs • y) = (exp(i + = xt+s • y for all y e L. • Remark 5.6. Let L be a differentiable local loop such that the group G topologically generated by the left translations of L is a Lie group. We denote by Η the stabilizer subgroup of e e L in G. Then we can identify L with a neighbourhood U 9 Η in G/H. The local loop L is strongly left alternative if and only if for the section σ : χ hλχ :L G there exists a neighbourhood W of Η in U C G/H such that a(W) C exp T\a(U). Let L be a differentiable manifold. An affine connection V of L is a mapping which assigns to two vector fields Χ, Y on L another vector field V^- Κ on L such that (i) V(fx+gY)Z = fVxZ + gVyZ for all vector fields Χ, Υ, Ζ and smooth real functions / and g defined on L; (ii) Vx(rY + sZ) = rVxY numbers r, s\

+ sS/xZ

for all vector fields Χ, Υ, Ζ on L and real

(iii) V x ( f Y ) = ( X f ) Y + / Vx Y for all vector fields Χ, Y and smooth real functions / defined on L.

5 Loop-sections covered by 1 -parameter subgroups and geodesic loops

The map exp x : TXL —• L is defined by the property that the curve y(t)

69

— exp x tX

is the geodesic with initial values γ (0) = χ and γ (0) = Χ. The exponential map defined in the tangent space TXL at a point χ of a differentiable manifold L with respect to an affine connection V is always denoted by exp x . The exponential map defined in the tangent space TeL at the unit element e by the 1-parameter subgroups of a strongly left alternative differentiable (local) loop L is denoted by exp. For a strongly left alternative differentiable (local) loop L equipped with an affine connection V the relation exp e = exp holds if and only if the geodesies of V through the unit e coincide with the 1-parameter subgroups of L. Definition 5.7. Let L be an almost differentiable local loop with the unit element e. We call L a geodesic local loop if there exists an affine connection V on the manifold L and an open neighbourhood U of e such that for any x, y e U there exist unique geodesies exp e tX and exp e tY with Λ: = exp e X and y = exp e Υ such that x • y — exp z ozex(Y),

where τβχ

: TeL —»· TXL is the parallel translation along the

geodesic exp e tX. A differentiable (global) loop L is called a geodesic loop with respect to an affine connection V if one can choose for the neighbourhood U the whole loop L. Remark 5.8. The geodesies exp e tX through e of a geodesic (local) loop L with unit element e are the (local) 1-parameter subgroups of L. Therefore exp e X — exp X for all X 6 TeL. Proposition 5.9. A geodesic (global) Proof.

loop L is diffeomorphic to W1 for some η > 1.

In the η-dimensional manifold L there exists a convex neighbourhood V of

the unit element e containing a sphere Sn~l

such that any geodesic through e e L

intersects this sphere in precisely 2 points. Hence L\{e} is diffeomorphic to Sn~l from which follows that L is diffeomorphic to

Rn.

χ Μ •

In particular, a Lie group G is a geodesic loop with respect to the left invariant connection in the sense of this definition if and only if the exponential map for G is bijective. A further important example for a geodesic loop is the hyperbolic plane loop defined in Section 22. Let L be a differentiable loop and ρ^ : L —> L be the right translation χ ι-> xy. If φ : L —y L is a differentiable map we denote by TV^X)L the tangent map of φ. We define an affine connection V on L by

(*)

[^(oej'j/iWO),

where the differentiable curve y(t) y(t)

= X(y(t))

is the unique solution of the differential equation

with initial value y ( 0 ) = χ for the vector fields X and Υ on L.

70

5 Loop-sections covered by 1 -parameter subgroups and geodesic loops

Indeed, the operation ( * ) is linear with respect to the vector field X and additive with respect to the vector field Y. Moreover, for a differentiable function / we have (Vx{fY)

)(*)

=

at _ d ~

dt id \dt

f=0

(lQy(t)e;Vf(Y(t))Y(Y(t)))

t=0

(f(Y(t))[Qy«)e;VY(Y(t)))

/(y(0)) f=0 if=o "

= (Xf)(x)Y(x)

/

+

Y(x)

+ f{x)j

t

|i=0

(teyioGjVWO))

f(x)(VxY)(x).

A differentiable vectorfield Y is called parallel along a curve γ ( t ) if Vj> (t)Y(y ( t ) ) — 0.

Lemma 5.10. IfYo € TXL and y(t) is a differentiable curve with y(0) = χ then a vector field Y(y(t)) ifY(Y(t))

=

such that K(x) = Yq is parallel along the curve γ it) if and only

(ρΥ(ι)ρ~ι)*Υο.

Moreover, the parallel translation r Xi V : TXL TyL with respect to V is given by rxy — (ρ^,ρ"1 it is independent of the curve connecting χ and y. The curvature o/V is 0. Proof

We compute

=

τ , t=o

_ ~ _

d

dt f=0 d dt

Y(Y(s)) Y(y(s))

=

0.

f=0

This means that Y{y(t)) = ey(t)Qx[Yo is parallel along y{t). Conversely, if Y(y(/)) is parallel along the curve γ(t) then Y(y(t))

is the unique

solution of the differential equation V ^ · ) Υ — 0 with initial value F ( 0 ) = Ko-

•

Lemma 5.11. Let L be a differentiable strongly left alternative {local) loop with unit element e. Then the geodesies of L with respect to the affine connection V defined by (*) are the curves {x' • y\ t € any x, } > e l . Moreover, we have expy[(ey)*X] = for all y e L, X e TeL, where exp^ : TyL connection V.

QyexpX L is the exponential map of the affine

Proof. Since χ' = exp tX with X = ^ |?=0Jcf we have to show for each X e TeL and y € L that the curve {(exp tX)y; t e R } is a geodesic with respect to V. This is the case if and only if

5 Loop-sections covered by 1 -parameter subgroups and geodesic loops

71

for any parameter value t = s. The tangent vector field Χ (λ (s)) of the curve k(t) — ( e x p C ? c a n be written in the form Χ(λ(*)) =

^ at

(exp(J +

= (Q y )*(eexpsx)*X •

t= 0

Since L is strongly left alternative we have also X{X{s)) =

^ (exp/X • (expsX • y)) = (ρ ε χ ρ ί Λ',y)*X · at t=ο

According to the definition (*) of V we compute the covariant derivative -l

d v

*

χ

[eexpir+.OX-yteexpsA··?) = ϊ , t=0 L d (öexps X ·>>) *(i?exp(f dt n

J*

Χ(λ(ί+ί)) (i?exp(r + i ) Υ ) * ·

Since (£ y )*(£ e x P ix)*X = (eexpsX-y)*X we obtain (QexpsX-y)*X

=

0

and the first part of the assertion is proved. The curve {y(t) = exp /X • y; t e R} is the geodesic with respect to V such that ]/(0) = y and }>(0) = (gy)*X. Hence γ(ί) — exp^ and we have γ{\) = expX • y = exp y [te y )*X]. • Proposition 5.12. A differentiable local loop (L, o) with unit element e is strongly left alternative if and only if it can be represented as the opposite loop of the geodesic local loop with unit element e with respect to the affine connection V defined by (*). The multiplication of(L, o) is given by χ O y = exp-y Tety exp^T1 JC = expy O ^ ) * Ο e x p j 1 Λ: . If L is a differentiable strongly left alternative global loop then this representation is global is and only if L is completely left alternative. Proof The multiplication (x, y) χ • y of the geodesic loop with unit element e with respect to the affine connection V is given by χ · y = expj τβ,χ exp^ 1 y in a neighbourhood U of e in L. Because of εχρ χ [(ρ χ )*Κ] = ρχ exp Υ (cf. Lemma 5.11) we have exp^. = ρχ ε χ ρ ( ρ χ ) ~ ' . Moreover, rex = (ρ*)* and exp e = exp. Hence x-y

= [ρΛ6χρ(ρ^)~1](ρΛ)*εχρ_1

y = Qxy = y ox.

•

72

5 Loop-sections covered by 1 -parameter subgroups and geodesic loops

Observation. The product χ • y is defined if and only if χ and y can be connected with e by geodesies. Indeed, τβ χ = (ρ*)* is independent of the geodesic connecting e with x. Moreover, if y = exp Y\ = exp Y2 for Y\ φ YI then we have x-y = gx exp YI = gxy = y-x for i — 1 , 2 . Corollary 5.13. A connected differentiable loop L is completely left alternative if and only if L admits the representation given in Proposition 5.12. Corollary 5.14. Any differentiable local Bol loop can be represented as the opposite loop of a geodesic local loop with respect to an affine connection with vanishing curvature. Any connected differentiable Bol loop for which the exponential map is surjective has a representation given in Proposition 5.12. Proof The assertion follows from Theorem 5.4 and Proposition 5.12.

•

Remark 5.15. Remark 9.20 will show that the geodesic local loop of a symmetric space with respect to its canonical connection can be represented also as the opposite local loop of a geodesic local loop with respect to an affine connection with vanishing curvature. Moreover, if one does not prescribe the curvature of the affine connection V the same local loop may be represented as the geodesic loop with respect to different affine connections (cf. [116], 0.8, and [115]). The classification of such connections can be reduced to the classification of mappings h : L χ L —> Diff(L) with h{ 1, b) = id, 1 ,b e L, where Diff(L) is the group of auto-diffeomorphisms of L, such that the following properties are satisfied: h(a, b)x* = (h(a, b)x)', for all a, be

h(a, a-b) = λ~1λαλ0,

h(b-a', b-au)h(b, b-af) = h(b, b au)

L and t, u e R (cf. [116], 0.13, and [115]).

Let G be a Lie group, Η a closed subgroup of G such that for the corresponding Lie algebras g and Ϊ) we have the reductive decomposition g = f) 0 m and [f], m] C m with a complementary subspace m C 0. The homogeneous space G/H is called reductive if the Lie algebra g of G has a reductive decomposition with respect to the subalgebra ί) of Η. In a reductive homogeneous space G/H there exists a unique G-invariant affine connection V satisfying the following properties: (i) the geodesies of V through the point Η are the curves {(exp tX)H\ X em, (ii) the parallel translation to,ί : Th{G/H) T^txvtx)FJ{G/H) {(exptX)Ht € R} is given by (L e x p ,x)* : TH(G/H) where Lexptx : yH m> (exp tX)yH.

/ e l } with

along the geodesic TIEXPTX)H(G/H),

5 Loop-sections covered by 1 -parameter subgroups and geodesic loops

73

This connection is called the canonical connection on G/H with respect to m (cf. [103]). Lemma 5.16. Let G be a Lie group, Η a closed subgroup containing no non-trivial normal subgroup ofG such that for the corresponding Lie algebras g and f) we have the reductive decomposition g = fy φ m. Let V be the canonical connection of G/H with respect to m and let L be a geodesic local loop in G/H with unit element Η defined by V. Then the group Κ topologically generated by the left translations of L is the subgroup of G which is generated by exp m. If W is the neighbourhood of Κ Π Η in Κ /{Κ Π Η) such that σ : W —>· Κ is the local section corresponding to L then a(W) C expm and L is strongly left alternative. Proof The relation Κ — (exp m) c G follows from the property (ii) of the canonical connection V. Since a(W) consists of left translations of L we have a(W) c expm. According to Remark 5.6 L is strongly left alternative. • Proposition 5.17. Let L be a connected almost differentiable local loop such that the group G topologically generated by the left translations is a Lie group. Let Η be the stabilizer of 1 e L in G and W be an open neighbourhood of Η in the factor space G/H, such that L has a representation on W with respect to a local section o-.W^G.Ifa satisfies a(W) C expTia(W) and [fj, T{a(W)] C Τ\σ(Ψ) where I) is the Lie algebra of Η then L is the geodesic local loop with respect to the canonical connection V of the reductive homogeneous space G/H with reductive complement M = T\G(W). The multiplication of L on W has the form (exp X)H • (exp Y)H = exp ( e x p X ) / / o T H < i e K p X ) H ο exp" 1 [(exp Y)H] with Χ, Y e m , where exp xW : TXH(G/H) G/H is the exponential map and T// (exp x)H is the parallel translation with respect to V along the unique geodesic connecting the points Η and (exp X)H. If W = G/H then L is a geodesic loop with respect to the canonical connection V if and only if the exponential map of L is a dijfeomorphism. Proof. The decomposition g = m + f) with m = T\A{W) defines on W the structure of a reductive homogeneous space and its canonical connection V. The geodesic connecting Η and (exp X)H is the orbit of a 1-parameter subgroup {exp tX\ ( e l ) with X e m and the parallel translation TH,(expX)H coincides with the translation of tangent vectors by T//(expiX) : TH(G/H) ->· T(EXPTX)H(G/H) where expiX acts on G/H by left multiplication. Since the loop multiplication on W is given by xH-yH — σ(χ H)yH and we have that σ ((exp i X ) / / ) = expiX : W ->· W, X e m, the group G acting by left multiplication on G/H consists of affine transformations with respect to the canonical invariant connection V. Hence this action of G preserves the geodesies of V on G/H and we can write σ((exp X)/f)((exp Y)H) = exp ( e x p X ) / / [7iff((exp

X)H)Y],

74

5 Loop-sections covered by 1-parameter subgroups and geodesic loops

where exp (expA - )W denotes the geodesic exponential mapping of the homogeneous space G/H with affine connection V. Since σ((εχρ X)H)((exp

Y)H) = (exp X)H · (exp Y)H

with t// (exp χ)Η = T\a((e\p X)H) the first part of the assertion is proved. It follows from the definition of a geodesic loop that its exponential map is a diffeomorphism. • Definition 5.18. A loop is called a left Α-loop if each λχ,γ = k~ykxky : L -»· L is an automorphism of L, i.e. if the inner mapping group of L is contained in the automorphism group of L. A topological local loop L is called a left Α-loop if each kXjy is a local continuous automorphism of L. The mapping kx^y is an automorphism of L if and only if the identity (xy)\{x[y(u

• υ)]} = {(*3θ\[*0> · ")]} · {(xy)\[x(y

• υ)]}

holds in L. Therefore every subloop and factor loop of a left Α-loop is again left Α-loop. Also topological loops which are covering related to topological left A-loops are left Α-loops (cf. Lemma 1.33). The identity kXiy{u • ν) = k x ^ y (u) • λ χ ^ ( ν ) can be written in the form λ x , y k

u

—

and hence we have λλ^(κ) =

.

Hence, a loop L is a left Α-loop if and only if the set Λ = {λ χ ; χ e L} of left translations is invariant under the inner mapping group 0(L). According to Lemma 1.31 this is equivalent to the property that the set A is invariant under the stabilizer Η of 1 e L in the group G generated by the left translations. Lemma 5.19. Let L be a commutative Moufang loop. Then the maps xy ^-x^-y · ^ ~^ ^ >

y ^ ^»

are automorphisms of L, i.e. L is a left A-loop. Proof In any Moufang loop L we have the identity (ab){ca) = [a(bc)]a, a,b,c e L (cf. Lemma 1.1). Since L is commutative this is equivalent to (ab){ac) = a[a(bc)]. From the Bol identity it follows that we have s(st) = s2t for all s, t e L and hence

5 Loop-sections covered by 1 -parameter subgroups and geodesic loops

75

(.ab){ac) = a2(bc). Using these two relations we obtain

= (x;y) _2 |;t 2 [;y 2 (wti)] J = (*>Γ 2 {* 2 [:νϋ>(κι;))]} = (A:y)- 2 {(^)[j:(y(«i;))]} = (xy)~l =

X~JkxXy{UV)

j

.

•

Proposition 5.20. Let L be a connected almost dijferentiable {local) leftA-loop. Then the group G topologically generated by the left translations of L is a Lie group. Let g be the Lie algebra of G and let f) be the Lie subalgebra of the stabilizer Η of\ G L in G. If σ : χ μ>· λχ : L —• G is the section corresponding to L then the tangent space m = T\a(L) of the image of the section σ at 1 e G is complementary to f) and satisfies the following two equivalent conditions: (i) Adnm c tn , (ii) m ® t ) = 0 and [f), mj c m, i.e. g is reductive with respect to (f), m). Proof Let Lzx i j h · λζ(λζ • kz 1 y) : L —L. Since L is a leftA-loop the mapping λ~ζ1λΜλζ is an automorphism of L for all u, ζ € L. Therefore, λ^λ^λ"1*

· k~ly)

= k~zl(ux) • λ~ζ (uy) .

From this follows kuLzxy = λ„λζ(λ;ιχ

• λ-'y)

= kuz[X~xz{ux)

• λ"1^)]

= L ^ y

,

or ιτ, τι ζ __ J,

uz

λ

Let Χ and Υ be vector fields on L and let y(t) be the differentiable curve on L which is the unique solution of the differential equation γ it) = X(y(t)) with initial value γ(0) = χ. We introduce (VxY)(x)

=

d dt

r=0

Κ ^ ω Γ ^ ' » ] .

which is an affine connection. This can be proved as for the affine connection defined before Lemma 5.10. From Xu L\ = Lu^xku we obtain for the tangent map

76

5 Loop-sections covered by 1 -parameter subgroups and geodesic loops

(A t t )*(L|)* = ( Ζ , ^ ) * ( λ „ ) * . With this we have (λΜ)*(νχ7)θθ =

d (λ„)*at t=ο d dt d

r=oL

dt

?=0

-1 y(i) ,-1 [(LZy(t))*^u)*Y(Y(t))]

Hence for all u e L the mapping λΜ preserves this affine connection V . This means that the group G topologically generated by the manifold {λ Μ ; u e L) is a closed subgroup of the group of all affine transformations of L with respect to V. Therefore G is a Lie group (cf. [86], Chapter I, §2-3 or [71]). L is a left Α-loop if and only if the section a(L) is invariant under conjugation by elements of Η . From this the condition (i) follows immediately. Since "T" Adexp/x^j Y = dt J t=ο we have [f), m] C m and m ® i) = g.

[X,Y] •

In Section 17 we will show that there exist differentiable loops satisfying the condition (ii) but which are not left A-loops. P r o p o s i t i o n 5.21. Let G be a Lie group, Η be a closed subgroup containing no nontrivial normal subgroup of G and let Q and t) be the corresponding Lie algebras. A ) Let L be a differentiable local loop defined in a neighbourhood U of Η and G/H by the local section σ \ U G such that the subspace m = T^(a(U)) generates g. The subspace m gives a reductive decomposition g = f) Θ m with [f), m] C m and the local loop L is a geodesic local loop of the canonical connection V of the homogeneous space G/H with the respect to m if and only if the following conditions hold: (i) L is a local left A-loop; (ii) L is strongly left alternative. B) Let L be a connected almost differentiable loop defined on G j Η by the section σ : G/H -> G such that the subspace m = 7//(CT(G/H)) generates g. The subspace m gives a reductive decomposition g = () φ m with [i), m] c m and the loop L is the geodesic loop of the canonical connection V of the homogeneous space G/H with respect to m if and only if the following conditions hold:

5 Loop-sections covered by 1-parameter subgroups and geodesic loops

77

(i) L is a left A-loop; (ii) L is strongly left alternative; (iii) The exponential map of the canonical connection V is a dijfeomorphism. Proof. First, we prove that the condition (i), (ii), respectively (i), (ii), (iii) in the cases A) and B) imply the corresponding assertions. According to Proposition 5.20 the condition (i) gives that m = 7//er (L) is a reductive complementary subspace to f) in g. As a consequence of Proposition 5.5 and Remark 5.6 we obtain from the condition (ii) that there exists a neighbourhood W of Η in G/H such that W c exp m Π σ (L). If L is a global loop then the condition (iii) gives that expm = a{L). Now the assertions in A) and B) follow from Proposition 5.17. Conversely, we suppose that the subspace m = Tfjcr(L) satisfies [f), m] C m and that L is a geodesic (local) loop of V. Therefore we have also Ad// m c m. If L is a global loop then the condition (B) (iii) is satisfied. Now we prove (ii). According to the property TH,{exp X)H — (Lcxpx)*, X € m, of the parallel translation along {(exp tX)H·, ( e M ) with respect to the canonical connection V we have in the geodesic (local) loop L the relation λ( ε χ ρ χ)// = L e x : yH m* (exp X)yH with X e m. Therefore the set of the left translations of L consists of elements e x p X , where X is any element of a suitable neighbourhood of 0 in m. If L is a geodesic global loop then this neighbourhood is equal to m. From this it follows that L is strongly left alternative. Now let e x p X (X e m) be an element of σ(Ζ,). One has Ad/, (exp X) = exp(Ad/, X) for any h e Η and hence Ad η (expm) c expm which means that L is a (local) left A-loop. • For the infinitesimal characterization of differentiable Α-loops in Proposition 5.11 the condition (ii) is essential. Namely, there exist differentiable left Α-loops which are not strongly left alternative. Examples are for instance the left conjugacy closed loops (cf. Section 17) and the loops investigated in Proposition 23.8. Theorem 5.22. Let L be a differentiable local loop having the following (i) every element ofL is contained in a unique local \-parameter (ii) one has x(yz) = (xy)zfor x,y,z 1 -parameter subgroup of L ;

properties:

subgroup;

€ L if χ and y belong to the same local

(iii) the left translations λχ (x e L) of the loop L generate topologically a local Lie group G. If we denote by g the Lie algebra of G, by f) the Lie algebra of the stabilizer Gi of 1 € L in G and by m the tangent space Te(k(L)) of the submanifold k(L) — {λ*; χ & L} in G at the identity e of G, then g is the direct sum g = ί) 0 m and m generates g.

78

5 Loop-sections covered by 1-parameter subgroups and geodesic loops

Conversely, let g be a Lie algebra, C g a subalgebra containing no non-trivial ideal of g, and let m be a complementary subspace of\) in g, i.e. g = f) Θ m. Then there exist neighbourhoods Um and U^ of 0 in m and f), respectively, such that exp : U = Um χ U^ —> G is a dijfeomorphism and every g e exp(U) can be written in a unique way as g = g m £f) where g m e exp(£/m) and gp, e exp(£/p,). Moreover the mapping (gm, km) ^ gm °km : exp(U m ) χ exp(f/ m ) exp(U m ) defined by (g m , km) ι-* gm ο km = gmkmh e exp(t/m) with a suitable h e Η is a dijferentiable local loop multiplication on L = exp(U m ) satisfying the conditions (i)-(iii). The Lie algebra of the local Lie group topologically generated by the left translations of L is g, the Lie algebra of the stabilizer of 1 e L in this group is f) and the tangent space Te(X(L)j is m. Proof. The first part of the assertion follows from the fact that the group G is locally diffeomorphic to the product A(L) Χ GJ. Conversely, let Vm and V^ be neighbourhoods of 0 in m and f), respectively, such that exp : V — Vm x V^ G is a diffeomorphism. We consider the mapping Φ : (Χ,Υ) exp X exp Y : Vm x Vf, -> G (X e V m , Y e V The tangent map 7(ο,ο)Φ of Φ at (0, 0) can be represented by the identity matrix. Hence there exist neighbourhoods Um C Vm and U^ C Vp, such that the restriction of Φ to Um x t/(j is a diffeomorphism onto exp Um exp U^. Consequently every element g e exp Um exp U^ can be written in a unique way as a product g = gmg^ withg m e exp Um andgf, € exp U^. It follows that the loop multiplication = Km(gmkm) is a well defined differentiable mapping exp Um x exp Um —» exp Um, where nm : exp Um exp U^ — e x p Um denotes the mapping assigning to / = tmt^ € exp Um exp U^ the element /m 6 exp Um. We consider the differentiable function F(x, y) = nm(x, _y) for x, y e exp Um. Clearly, we have F(x, e) = χ and F(e, y) — y for the unit e of the local Lie group exp Um exp U^ and x,y e exp Um. Moreover the tangent maps TeF(x,e) and TeF{e,y) : Te exp Um —• Te exp Um can be represented by the identity matrix. Hence in a neighbourhood W of e in exp Um there exist differentiable functions ψι and \jfr from W χ W to exp Um such that F{tyi(y, z), y) — ζ and F(x, ψν(χ, ζ)) = ζ (cf. [110], p. 59). Hence the equations χ ο km = /m and gm ο y = /m for given g m , km ,lm e W are uniquely solvable and the solutions depend differentiable on (km, lm) or (g m , /m), respectively. It means that on W a local loop L is realized. Let G be the local Lie group expg = exp Um exp U^ and Η the local Lie group exp f) = exp C/f,. Clearly we have a local section in G with respect to the stabilizer Η in the local homogeneous space G/H given by the map nm : g exp U^ ι-> πm(g). We consider the map ω : χ Η» JC exp U^ : exp Um —G/H\ we know that the composition ω ο nm gives the identity in the neighbourhood W* — {w Η \ w e W) of Η in G/H. We define on G/H the local multiplication ( g i Η ) * (g2H)

:= π m ( g l ) g 2 H

for all giH

e W*

5 Loop-sections covered by 1 -parameter subgroups and geodesic loops

79

If k\, k2 and ^ o ^ e W then {k\ ο £2) ,

hence ( G / H , *) is a local loop locally isomorphic to (L, o). Since 7r m (W*) = W is a neighbourhood of e in expm C G all local 1-parameter subgroups exp tX for X e m are contained in W. Since for a left translation with k e W we have λ\Η{χΗ) = (kH) * (xH) = kxH for all (xH) e W*, the loop (G/H, *) and hence the loop ( L , o) satisfy the conditions (i) and (ii). Because of m = Te(nm(W*)) and m generates the Lie algebra g, the loop (L, o) satisfies also the condition (iii). Moreover the Lie algebra of the local group topologically generated by the left translations of (L, o) is isomorphic to g. The isomorphism can be chosen in such a way that it induces an isomorphism of the Lie algebra of the stabilizer of 1 G L in this local group onto I) and a linear isomorphism from Te(X{Lj) onto m. •

Section 6

Bol loops and symmetric spaces

In this section we demonstrate the important role played by symmetric spaces in the theory of Bol loops. In particular to any differentiable (local) Bol loop L we can associate in a unique way a triple (g, f), m) such that 9 is the Lie algebra of the group G topologically generated by the left translations of L, the subalgebra f) of g is the Lie algebra of the stabilizer Η of 1 e L in G and m is a complementary subspace of f) in g satisfying [[m, m], m] c m and generating g. The operation [.,.] of the Lie algebra g together with the projection g m allow to equip m with a ternary and binary operation such that m becomes the structure of a Bol algebra f. As the Lie algebras for Lie groups the Bol algebras are in one-to-one correspondence to differentiable local Bol loops ([91], pp. 424-425, [90] and [116]). The proof of this fact is based on the canonical construction of a triple (g*, I)*,m*). The Lie algebra g* is constructed using the ternary operation of f in such a way that m* is the (—l)-eigenspace of an involutory automorphism of g*; the subalgebra 1)* is defined by the binary operation of f and it is a complementary subspace of m* in g*. Moreover, the Lie algebra g* has the following property: for any triple (g, f), m) associated with a local Bol loop such that the ternary algebras (m*, [[., .]*, .]*) and (m, [ [ . , . ] , . ] ) are isomorphic there exists a Lie algebra epimorphism g* —> g mapping m* onto m. We give a description of the kernel of this epimorphism. Let G be one of the following categories: topological spaces, differentiable or analytic manifolds, algebraic manifolds. Definition 6.1. Let Μ be an object in the category C and let μ : Μ χ Μ ^ Μ be a C-morphism such that the operation μ(χ, y) := χ + _y satisfies the following conditions: (i) χ + χ = x, (ii) χ + (x + y) = y, (iii) χ + (y + z) = (x + y) + (x + z), (iv) the object Μ has an open covering {Ua}aeA such that the operation (jc, y) x/y '• Ua x Ua —» Μ determined b y x / j + j = x i s well defined for each α e A.

81

6 Bol loops and symmetric spaces

We call (M, + ) a symmetric space and for any χ e Μ the mapping σχ : y χ + y the reflection at x. The condition (iv) means that for all x, y e Ua there exists a unique point ζ such that σζχ = y. Any Lie group G as well as any algebraic group G is an example of a symmetric space if we define χ + y = xy~^x for all x,y e G. Let Μ be a connected differentiate manifold equipped with an affine connection V. Let expx : TXM Μ be the exponential mapping defined by the geodesies of V through χ (cf. p. 50). A diffeomorphism φ : Μ —»· Μ is called an affine transformation if for all vector fields Χ, Y on Μ one has = ^( φ ^χ)(φ*Υ)· The affine transformations of the manifold Μ with respect to V form a Lie group and map geodesies onto geodesies (cf. [71], Chapter VI. §1.). An involutory affine transformation φ : Μ —> Μ is called a geodesic reflection at the point χ e Μ if a0ga0, g e Σο, fixes Η elementwise ([72], Chapter XI. Theorem 1.5). The automorphism σ of Σο induces an involutory Lie algebra automorphism σ* on g such that the (+l)-eigenspace of σ* is the Lie algebra f) of Η and the (—l)-eigenspace m C 0 gives a reductive decomposition 0 = f) Θ τη. If we identify Μ with G/H then the canonical connection of the reductive homogeneous space G/H coincides with the connection V given on M; this is the unique affine connection which is invariant with respect to the geodesic reflections ([72], Chapter XI. Theorem 3.1). If φ is an affine transformation φ : Μ —> Μ and σχ is the geodesic reflection at χ Ε Μ then φσχφ~ι = σφ(χ). Indeed, φ(χ) is a fixed point of φσχφ~{ and if X e Τφ(Χ)Μ then for the tangent map (φσχφ~{)* one has φ*(σχ)(φ~χ)*Χ

= φ*(°χ)Λφ*[

Χ ) = φ*(-φ*ιΧ)

=

- x .

The family {σ*; χ e Μ] of geodesic reflections allows us to define a multiplication x • y — axy on Μ satisfying

82

6 Bol loops and symmetric spaces

(1) χ • χ — x, (2) λ; · (χ • y) = y, (3) x · (y • ζ) = (x · y) · (x • z), (4) every χ has a neighbourhood U such that χ • y — y implies y = χ for all y e U. The first two properties follow immediately from the definition and property (4) expresses the fact that χ is an isolated fixed point of the geodesic reflection σχ. We obtain from φσγφ~1 = σφ(γ) the relation axayaxt = crax(y)t for all t e M, which is equivalent to χ • [y • · f)] — (x • y) -t. Putting σχζ = t we see that the property (3) is satisfied, too. A differentiable manifold Μ with a differentiable multiplication (jc, jy) ι—> χ · _y satisfying the properties (l)-(4) is a symmetric space in the sense of Loos (cf. [86], p. 63). The previous consideration shows that a symmetric space (Μ, V) in the classical sense is a symmetric space in the sense of Loos. Conversely, in a symmetric space (M, ·) in the sense of Loos there exists a unique affine connection V, called the canonical connection of symmetric space (M, ·) such that all maps σχ : y χ · y, χ ε M, are geodesic reflections with respect to V (cf. p. 83. and pp. 102-104). Hence any symmetric space (Μ, ·) in the sense of Loos is also a symmetric space (Μ, V), where V is the canonical connection of (M, ·)· Proposition 6.2. The following three categories

coincide:

(a) The category of differentiable symmetric spaces of Definition 6.1. (b) The category of symmetric spaces in the classical sense. (c) The category of symmetric spaces in the sense of Loos. Proof. We already know that the categories (b) and (c) coincide. Now we show the equivalence of (a) and (c). The first three conditions in the definitions for symmetric spaces in (a) and (c) are the same. From the condition (iv) in Definition 6.1 it follows immediately: (iv)' every jc has a neighbourhood Ux such that axy = y implies χ = y for all y in Ux, which is equivalent to the condition (4) for a symmetric space in the sense of Loos. Conversely, let (iv)' be satisfied. The manifold Μ can be equipped with a canonical connection V such that the mappings σχ, χ e Μ, locally coincide with the geodesic reflections (cf. Theorem 2.6 and its corollaries in [86], pp. 84—86). According to Theorem 8.7 in [71], p. 149, each point χ e Μ has a convex normal neighbourhood Wx with respect to the geodesies of the canonical connection V. It follows that the geodesic reflection σζ on the midpoint ζ of the unique geodesic segment of any pair of

6 Bol loops and symmetric spaces

83

points u, ν e Wx interchanges the points u and ν and hence azu = v. Consequently the equation ζ + u = ν is uniquely solvable for ζ and any given u, w e W. The equation u + ζ = ν has the unique solution ζ = u + ν because of (ii). • Definition 6.3. Let Μ be an object in the category 6. We call Μ a locally symmetric if there exists an open submanifold W of Μ χ Μ with {(λ;, λ:); Χ € Μ} C W and a C-morphism μ : W Μ such that the following conditions are fulfilled:

space

I) The partial operation μ(χ, (i) χ + χ =

y) = χ

y satisfies:

x,

(ii) χ + (x + y) =

y,

(iii) * + (y + z) = (x + y) + (x + z) for all x, y, ζ € Μ for which the expressions in (ii) and (iii) are defined. II) There is an open covering {Uy}yeA of Μ with UY χ UY C W such that the partial operation (λ:, y) ι-» x/y : UY χ UY —• Μ determined by x/y y — x is well defined for each y e A. The mapping σχ : y at χ e M.

χ + y : ( { * } χ Μ) Π W

Μ is called the local reflection

Let Μ be a connected differentiable manifold equipped with an affine connection V. The pair (Μ, V ) is called a locally symmetric space in the classical sense if for each point χ e Μ there exists neighbourhood Ux of χ on which the geodesic reflection e x p ^ X ) ι-» e x p ^ — X ) ( e x p ^ X ) e Ux, X e TXM) is (local) affine transformation. According to [86], Chapter II. §4.1 and §4.2 this definition is equivalent to that given by Loos. Remark 6.4. The category of differentiable locally symmetric spaces coincides with the category of locally symmetric spaces in the classical sense. To see this we have only to prove that in a locally symmetric space there exists an open covering {UY }YeA with the properties required in II). According to the Corollary of Theorem 4.1 in [86], p. 103, the local reflections are the local geodesic reflections. Since each point χ e Μ has a convex normal neighbourhood (cf. [71], p. 149) with respect to the geodesies of the canonical connection V of the locally symmetric space we can take these neighbourhoods for the members of the open covering {UY}Y&ALet ο be a point of the symmetric space (M, ·) and let Μ be the set {σχσ0; χ e Μ} where axy — χ • y, y e Μ. Then the set Μ generates the group Σο of displacements of Μ. If 0 denotes the Lie algebra of Σο and f) is the Lie algebra of the stabilizer Η of ο in Σο then we have a reductive decomposition g = ϊ) φ m with i) = [m, m] and m is the (—l)-eigenspace of the involutory automorphism σ of g which is given

84

6 Bol loops and symmetric spaces

by the conjugation by σ0 (cf. [86], p. 91). Moreover the set Μ is a differentiable submanifold of Σο ([86], p. 95. Proposition 3.2) and the subspace m is the tangent space T\M C g = 7"ιΣο· The canonical connection V of the symmetric space (M, ·) coincides with the canonical connection of the reductive homogeneous space Σ ο / Η with respect to the reductive decomposition g = f) Θ m (cf. [86], Theorem 2.7. p. 86). Remark 6.5. Let ( Μ , {σχ\ χ e Μ}) be a locally symmetric space and G be the local Lie group topologically generated by the displacements σχσ0, χ e Μ, for a fixed point ο e Μ. If 0 is the Lie algebra of the local group G and () the Lie algebra of the stabilizer Η of the point ο e Μ in G, then the Lie algebra 0 is the direct sum g = I) φ m of vector subspaces f) and m where Μ is diffeomorphic to exp m with respect to the map χ ι-> σχ. Moreover the local reflection σ0 induces on the Lie algebra g an involutory automorphism which has f) and m as (+1)- and (—l)-eigenspaces, respectively. The Lie algebra g is generated by m, more precisely g = m 0 [m, m]. Conversely, given a Lie algebra g, a subalgebra f) and an involutory automorphism σ : g —> 9 such that f) is the (+l)-eigenspace of g and the (—l)-eigenspace m generates g as Lie algebra, then Μ = exp m is a locally symmetric space, the local Lie group G = exp g is the local Lie group topologically generated by displacements of Μ and the mappings crg(0) = g(expG a | m e x p ~ ' ) g _ 1 with g e G are the local reflections of Μ (cf. [86], pp. 90-104). A real vector space V with a trilinear multiplication ( Χ , Υ, Ζ) ι-* (Χ, Υ, Ζ) : V x V x V — > - V i s called a Lie triple system, if the following identities are satisfied: (i) (X, X,Y)=

0;

(ii) (Χ, Υ, Ζ) + {Υ, Ζ, X) + χ + y — xy~xx : Α χ Λ - > Λ (cf. Lemma 1.15). This property means that Λ is a symmetric subspace of the symmetric space defined on G by the operation

86

6 Bol loops and symmetric spaces

y) (->· χ + y — for all x, y e G. Hence L is a Bol loop if and only if the tangent space m = T\ A is a sub-Lie triple system of the Lie triple system induced on the Lie algebra g of G by the trilinear operation (λ, y, z) [[x, yl, z] '• g χ g χ g -h>· g (cf. [86], Theorem 1.4, pp. 121-122). • The proof of this lemma gives immediately the Remark 6.8. Let G be a local Lie group topologically generated by the left translations of a differentiable local loop L. The local loop L is a local Bol loop if and only if the image of the local section corresponding to L has the form exp m, where m is a subspace of the Lie algebra g of G for which [[m, m], m ] c m holds. Let L be an analytic (local) Bol loop and G the Lie group topologically generated by its left translations. Let g be the Lie algebra of G and f) the Lie algebra of the stabilizer Η of 1 G L in G. We have g = m ® t ) where m = T\ ( a ( G / H ) ) is the tangent space at 1 G G of the image of the (local) section σ : G/H —G corresponding to the loop L. These data define on m the binary operation ( Χ , Υ) i—• [X, Y]m : m χ m —»· m where [Χ, Y] m is the projection of the Lie bracket onto m along f), and the ternary operation (Χ, Υ, Ζ) ι-» [[Χ, Υ], Ζ] : m χ m χ m —>· m since [[m, m], m] C m. Proposition 6.9. The identity

0 = [[[X, Y], z],

- [[[X, Y], w], z ] m

+ [[Z, W], [X, Y]m] - [[X, Y], [Z, W] m ] + [[X, Y]m, [Z, W]m] m holds for any Χ, Υ, Z, W G m. Proof. We have

[[[X, Yl z], w]m - [[[X, Y], w], z ] m = [[X, Y], [Z, W]]m = [[X, Y]m, [Ζ, W] m ] m + [[X, Y]m, [Z, W]n]m + [[X, Y]h, [Ζ,

WU]m,

where [U, tVJe, is the projection of [U, W] onto f) along m. Hence we obtain the assertion. • Definition 6.10. A real vector space V equipped with a trilinear operation ( . , . , . ) and with a bilinear skew-symmetric operation [[.,.]] is called a. Bol algebra if (V, (.,.,.)) is a Lie triple system and if for all Χ, Υ, Z, W G V the identity [[(Χ, Υ, Z), W]] - [[(X, y, W), Ζ]] + (Z, w, [[x, Y]]) - (X, Y, [[Z, W]]) + [[[[X, Y]], [[Z, W]]]\ = 0 holds.

6 Bol loops and symmetric spaces

87

Definition 6.11. The Bol algebra (m, ( . , . , . ) , [[·, ·]]) induced on the tangent space m = T\{a{G/H)) of the image of the section σ : G/H G corresponding to a (local) Bol loop L is called the tangent Bol algebra of L if (x, y, z) = [[*, y], ζ] and [[*, y]] = y]m> where x, y, ζ are elements of the subspace m of the Lie algebra (0, [.,.]) of the Lie group G and ζ z m : g — m is the projection of g onto m along the subalgebra of Η. Let g be a Lie algebra, Ϊ) a subalgebra of g containing no non-trivial proper ideal of g and m a vector subspace of g such that g = m ® f ) , [[m, m], m] c m and m generates g. According to Proposition 6.9 m is a Bol algebra with respect to the operations U , y, z) = [[*, y], z]

and

[[x, y]] = [x, y] m .

We consider on the vector space m 0 m Λ m, where m Λ m is the exterior product of m, the bilinear multiplication {.,.} given by Θy

a

z, u Θ ν

a

w}

= [[y, z], u] - [[υ, w], χ] Θ ^

a

u + [[y, ζ], υ]

a

w+ ν

a

[[y, ζ], w]

= (y, ζ, u) - (υ, w, χ) φ χ Λ u + (y, ζ, υ) Λ w + ν Λ (y, ζ, w) . Lemma 6.12. The subspace m a m in the algebra (m φ m A m, {., .}) is a subalgebra such that for all elements ξ, η, ζ e m A m the identity

holds. Proof. It is enough to prove this identity for ξ = A /\ Β,η = X AY, ζ = U aV with Α, Β, X, Y,U,V Ε m. We have {Α Α Β, {X A Y, U A V}} = (Λ, Β, ( X , Y, V))AV

+ (X, Y, U) Α {Α, Β, V)

+ (A, B, U) A (Χ, Υ, V) + U Α (Α, Β, (Χ, Υ, V)) and {{Α Α Β, X

A

F}, U

A

V) + {Χ Α Υ, {A

A

B, U

A

V}}

= ((Α, Β, X), Y, U) A V + U Α ((Α, Β, Χ), Υ, V) + (Χ, (Α, Β, Y),

U)AV

+ U A (Χ, (Α, Β, Υ), V) + (Χ, Υ, (A, B, U)) A V + (A, B, U) A {Χ, Υ, V) +

(X, Y, U)

A

(A, B,V) + U A (Χ, Υ, (Α, Β, V)) .

Using the identity (iii) of Lie triple systems we obtain the assertion.

•

6 Bol loops and symmetric spaces

88 L e t α be the vector

subspace

o f m Λ m generated

by the elements

o f the form

{y Λ ζ, ν Λ w} 4- {υ Λ w, y α ζ] for any y, ζ, ν, u> e m. We have {r, {y

Α

ζ, ν Λ u;} + {Υ Λ w, y A Ζ } }

= {t, Ο , ζ, ν) A w + ν A (y, ζ, w) + (ν, w, y) Α ζ + y A (υ, III, z)} = -((Cy.

)>

v

> ή +

w

(y. Z, w), t) + ((ν, w, y), Z, t) + (y, (ν, w, ζ),

= - ( ( y , ζ, (υ, w, ο ) - (υ, w, (y, ζ, ο ) +

w

t))

. (y. Ζ, ο ) - (y, Ζ, (ν, w, t ) ) )

= 0

for all y, ζ, ν, w, I e m because of the identity (iii) for a Lie triple system. Since {y Λ ζ, t) = (y, z, t) = —{i, y A z] for all t,y,z e m we obtain that {f, α} = { ο , ί } = 0. Using the identity {y Λ z,

V A W]

= (y, Z,

V) A W +

V A (Y, Z,

W)

= {y A z, v] A W + V A {y A z, w}

we get {m Λ m, ν a U;} = {m Λ m, Υ} A w + ν a {m Λ m, w} for all v, w E m. Since α C m A m we obtain {α, Ν A u>} = {α, ι>} A W + Ν A {a, W} = 0. For any element w £ mAmwehave {yAz, u;} = {)>Λζ, i f } + { u ; , >Άζ} — {w, )>λζ}, where {y A z, u>} + {w, y α ζ} Ε α. If we take w e a then {w, y a z] — 0 and hence {y Λ z, w} G o. Thus ο is an ideal of m φ m A m and for the quotient algebra g* holds 0* = (m Θ m a m ) / a = m Θ (m a m)/a. P r o p o s i t i o n 6 . 1 3 . The quotient algebra respect

to the operation

lutory automorphism ofa

0 * = m φ (m A m ) / a is a Lie algebra

[., .]* which is induced

by {.,.}

σ : 0 * —> 0 * such that the subspace

and the subalgebra

with

on g * . There exists an invom is the

(-l)-eigenspace

[m, m ] * = (m A m ) / a is the (-\-\)-eigenspace

ofa.

Proof. Clearly, the bilinear operation [., .]* is skew-symmetric. As {m A M , m A M } C m A m, the vector space m A m is a subalgebra of m φ m A m. Since [y, z]* = [y Φ 0, ζ φ 0]* = {y φ 0, ζ Φ 0} + α = 0 φ Ο α Ζ + α) for any y, ζ e m we have [m, m]* = (mAm)/o, hence [m, m]* is a subalgebra of 0* satisfying m©[m, m]* = 0*. Since [[m, m], m] c m holds for the vector subspace m in the Lie algebra g we have for all x, y, ζ e m the relation [[JC + a, y + α]*, ζ + ο]* = {{JC, y}, ζ} + a. Using the isomorphism (m φ m A m ) / a = m φ (m A m ) / a we see that the mapping χ + ο χ : (m + α)/α —> m is an isomorphism of Lie triple systems and can write [[x + a,y + α]*, ζ + ο]* κ* [[χ, y], ζ]. Now we define the involutory linear map σ : 0* 0* by r ] * Φ [Λ, « r + [ly,ζ]*,

[y, ζ]*, u φ [υ, ω ] * ] * )

[υ,

«;]*]*

6 Bol loops and symmetric spaces

89

and hence σ is an automorphism. Now we compute

θ

[[χι

[y\, ζ ι Τ , X2

θ

[y2, z2]*]*,

X3

θ

[y3,

= [[*i, [^2,22]] + [ t y i . z i l , ^ ] θ

z 3 ]*]*

[χι,χιΤ

Ί* +

[iy\,ζι]*,

[yi,

Z2\*]*,

χ3 θ

= [[*1,*2],X3] + [[[yi.Zl]*.

[[xuxiT,

[ys,

£3]*]* +

23]*]

+ [[*1, [y2, Z2]], [ > 3 . « ] ]

+ [ [ [ y i , Z l ] , * 2 ] , [y3.Z3]] θ +

[ χ ,

[yi,

[[[yu

ζιΓ,

+ [ [ [ j l , Z\], Xl\, ^3] [yi,

z2]*]*,

[y 3 ,23]*]*

·

In order to prove the Jacobi identity in g* we have to consider the sum

Υ

[[*σ(1) θ [y>2 0 0

1

^

/

which is not a left translation because, in general, u\V2 + v\ φ v\U2 + V2- But a further multiplication shows that ^(xl,yUul,v])^(x2,y2,u2,v2)^(x\,yi,ui,vi) is a left translation, and hence L is a Bol loop. The group generated by the left translations is the 5-dimensional solvable Lie group of matrices

/ a 0 0 0 V 0

b 1 0 0 0

0 0 a 8 0

0 0 0 1 0

c \ d 0 0 1 /

a, b, d, g £ R and 0 < a €

The right translation Q(x,y,u,v) can be represented by the matrix

(

1 0 0 0

0 1 0 0 0

X 0 u 0 0

y ο ο u 0

0 \ y ο

/

acting on column vectors. The group generated by the right translations is the 6dimensional solvable Lie group of matrices

(

1 0 0 0 \0

0 1 0 0 0

a 0 8 0 0

\

b 0 0

a, b,c,d,h

e l

and 0 < g e

8 0

The group generated by all left and right translations is the 9-dimensional solvable Lie group of matrices

( A 0 0 0 \ 0

Β 1 0 0 0

c 0 AK Μ 0

D 0 0 Κ 0

F G 0 Η 1

B, C, D, F, G, Μ, Η e Μ; 0 < A, Κ e

)

For differentiate Moufang loops only Lie groups can occur as groups generated by all translations.

7 Bol nets

99

Theorem 7.6. Let L be a connected differentiable Moufang loop. Then the group topologically generated by all left and right translations is a connected Lie group. Before proving this theorem we show the following

Lemma 7.7. Let L be a Moufang loop. Then the group M(L) generated by all left and right translations is generated by the group of right translations and the mappings Tc : χ Μ* cjcc-1, c e L. The group M(L) is a subgroup of the semidirect product S = Ge χι Π where the group Ge is generated by the right translations of L and Π is the group of the right pseudo-automorphisms of L. Proof

The first assertion follows from the diassociativity of Moufang loops

(cf. [107], p. 96). The second assertion is the consequence of the fact that any mapping Tc (c e L) is a pseudo-automorphism (Lemma 1.2). We have S = Ge (jc · α)τ

= (χτ

χι Π since

• aTc) · c~\ where c is a companion of the pseudo-automorphism τ . α

Proof of Theorem 7.6. The group GE topologically generated by the right translations of a differentiable Moufang loop is a Lie group (Theorem 7.3) and the same holds for the group Π of continuous pseudo-automorphisms of L (cf. Theorem 2.9.(iii) in [102]). Hence the semidirect product GE χ Π is a Lie group. According to Lemma 7.7 the group M{L)

topologically generated by all left and right translations is a closed

subgroup of GQ Ά Π. It is a connected Lie group since it is generated by the connected sets GQ and {Tc\ c e L } .

•

We conclude this section by a lemma that we shall use later.

Lemma 7.8. Let L he a differentiable connected proper Bol loop and !N(L) the 3-net having L as a coordinate loop. We denote by Γ the group topologically generated by the involutory collineations TQ , where G is a vertical line, and by Θ thefull collineation group ofN(L). If Ν c Θ is the normal subgroup of Θ which leaves every vertical line invariant then Θ/Ν is an affine transformation group containing ΓΝ/Ν with respect to the canonical connection of the symmetric space induced by the set of involutions TG on the pencil of vertical lines. Proof. The symmetric space Θ on the pencil of vertical lines is given by the set of those involutions which are induced by involutory collineations J — {TG\ G vertical} on the pencil of vertical lines. Since the set J C Γ is normalized by the full collineation group Θ (cf. Lemma 3.11) the group Θ / Ν consists of automorphisms of the symmetric space 6 , and hence the elements Θ/Ν leave the canonical connection of Θ invariant-^

Section 8

Strongly topological and analytic Bol loops

In this section we prove that the category of real analytic connected Bol loops coincides with the category of connected topological Bol loops having a Lie group as the group topologically generated by their left translations. In view of the positive solution of Hilbert's fifth problem this gives a characterization of real analytic Bol loops by means of algebraic and topological properties. As consequences we obtain that any closed subloop of an analytic Bol loop is analytic and that differentiable connected Bol loops are real analytic. Definition 8.1. We call a topological Bol loop L a strongly topological Bol loop if L is locally compact, connected and locally contractible and if the group G topologically generated by the left translations of L is locally compact and has a countable basis. Lemma 8.2. Let G be a Lie group and Μ a closed topological subspace of G such that for all x, y e Μ the element xy~lx is contained in Μ. Then Μ is a real analytic submanifold of the Lie group G. Proof The symmetric space C(G) on the Lie group G defined by the reflections agk — gk~x g for all g, k e G is a real analytic symmetric space. Hence Μ is a closed symmetric subspace of C{G) and the assertion follows from Theorem 1.7 in [86]. • Theorem 8.3. A connected topological Bol loop L is strongly topological if and only if L is a real analytic Bol loop. Proof If the connected Bol loop L is real analytic then, according to Theorem 7.3 (iii), the group G topologically generated by the left translations of L is a connected Lie group and hence L is strongly topological. Let now L be a strongly topological Bol loop. Then the locally compact group G topologically generated by the left translations of L is a transitive effective transformation group. Hence it is a Lie group according to [120], 96.14 (Szenthe's theorem) or [127], We know (cf. Lemma 1.23) that the set A of left translations of L is a closed topological subspace of G and that the topological group G is the topological product Ax H, where Η is the stabilizer of 1 e L in G. The topological loop L is isomorphic to the loop on G/H corresponding to the section σ : GjΗ —> G with a(G/H) = A.

8 Strongly topological and analytic Bol loops

101

Since L is a Bol loop the closed subspace A contains the identity of G and aß~la e A for all α, β e A. Let us consider the Lie group G as the symmetric space C(G) defined on the group G. This symmetric space may be represented as the factor space G χ G/AG where AG is the diagonal subgroup {(g, g); g e G] in G (cf. [72], pp. 198-199). The projections (x, x) (x, 1) : AG —• G and C x G ^ G x { l } a r e diffeomorphisms and hence the real analytic structures of the group G and of the symmetric space C(G) are equivalent. The subset A is a closed symmetric subspace of C(G). It follows from Lemma 8.2 that A is a real analytic submanifold of G and the analytic manifold G is diffeomorphic to Α χ Η and the projection maps π\ : G A and π2 : G —• Η are real analytic. According to Proposition 1.5 the loop L is (topologically) isomorphic to the loop which is defined on A by the multiplication α ο β = π\ (aß) for all α, β e A. Clearly this multiplication ο is analytic. Since L is a Bol loop, we have α\β = a~1 ο β, where the mapping α ι-» a ~1 in the loop (A, o) is the restriction of the inverse mapping g ι-» g - 1 : G G. Therefore the operation (α, ß) h-> α\β : A χ A —»· A is real analytic. In the Bol loop (A, o) we have β/α

=

α - 1 ο (or ο (β/α))

= α - 1 ο (α ο (β/α

ο (α ο

α-1)))

.

Because of the Bol identity we obtain from this β/α

=

α " 1 ο ((« ο β) ο « " 1 ) .

It follows that also the operation (β, α) Corollary 8.4.

Every

closed

subloop

β/α : A χ A —» A is real analytic.

L' of an analytic

Bol loop

L is real

•

analytic.

Let G be the group topologically generated by the left translations of L and G' the closed subgroup of G which is topologically generated by the left translations of L'. The loop L has a real analytic representation on the real analytic manifold A = {λ^; χ e L) of its left translations. Let A' = {λ* : L —• L; χ e L'\ C A be the closed subset of the set A of left translations of L. Since λ χ λ ~ ' λ χ e A' for all λχ,λγ e A', the set A' is a closed symmetric subspace C(L') of the real analytic symmetric space C(A) defined on A and hence also of the symmetric space C(G) defined on G. C(A') is an analytic submanifold of C(A) as well as of G. Since A' is an analytic submanifold of G the assertion follows. •

Proof.

Corollary 8.5.

Any dijferentiable

connected

Bol loop

L is real

analytic.

This follows immediately from Theorem 8.3 since the group topologically generated by the left translations of L is a Lie group (Theorem 7.3 (iii)). •

Proof.

Section 9

Core of a Bol loop and Bruck loops

In this section we study the algebraic version of the symmetric space associated with a Bol loop. In the first part we investigate abstract Bol loops, in the second part we consider topological and differentiable (locally) symmetric spaces associated with Bol loops. We show that in this context the so-called strongly 2-divisible Bol loops occupy a key position. In particular, the strongly 2-divisible Bruck loops are in oneto-one correspondence to the associated symmetric spaces. Moreover, we show that differentiable Bol loops which are also left Α-loops are related to extensions of abelian groups by Bruck loops. As a by-product we prove a theorem of Levi and van der Waerden on groups of exponent 3 by methods of non-associative algebra.

9.1

Core of a Bol loop

Definition 9.1. Let L be a Bol loop. The core C(L) is defined by the binary operation (x, ;y) η-» * + y = oxy =

The core C(L) of a Bol loop L satisfies the following identities: (1) σχχ — χ + χ — χ, 2 (2) (σχ) = id, i.e. χ + (χ + y) = y, (3) σχσγσχ = a0xy, i.e. χ + (y + z) = (x + y) + (x + z). Proof We prove (2) and (3). We have χ + (χ + y) = χ • [χ • y - 1 * ] - 1 . * . Since Proposition 9.2.

(χ (3) we write

yx_1) =

- y x - 1 ) ] } = 1 we obtain (2). In order to prove

χ + (y + ζ) — χ - [y • ζ-1)']-1* = χ · · ] l = x{y~ [z(y- x)]} = (χ • y-Mtjr^zOr1*)]} = (x + y){[x~l ·ζχ~ι][χ·γ~ιχ]} = (χ + · + ?)} = (x + y) + (x + z). If the Bol loop is a centre-free group G then we can describe the group induced by the Bol reflections of the 3-net N(G) on the set V of vertical lines.

•

9.1 Core of a Bol loop

103

Proposition 9.3. Let G be a group, let N(G) be the corresponding 3-net and let Γ be the group generated by the Bol reflections τ γ (V e V). (i) If G is not an elementary abelian 2-group then the group Σο induced by the group Γο generated by products of an even number of Bol reflections on the set V of vertical lines has index 2 in the group Σ induced onV by Γ. Moreover Γο has index 2 in Γ. (ii) If G has trivial centre then Γ is isomorphic to the group Σ induced by Γ on the set V of vertical lines and hence Γ is isomorphic to the group generated by the left translations of the core C(G). (iii) If G is an abelian group, then the group Γο is isomorphic to G, and the group Σο induced by Γο on V is isomorphic to the subgroup consisting of the all squares of G. Moreover Γο = Σο if and only if the mapping χ \ x 2 : G G is surjective. Proof Let τχ and xy be the Bol reflections with respect to the vertical lines incident with the point (jc, 1) and ( j , 1), respectively. Then for the image of the point (u, v) in N(G) we have TyTx(u, v) — ry(xu~lx, x~luv) = (yx~lux_1y, y^^v) (cf. (*) in Section 3, p. 56). The group induced by Γο on the set V of vertical lines is a subgroup of the group of the mappings u yuz, u e G, for any y,z € G. If Σο = Σ then the mapping induced by the Bol reflection x\ is contained in Σο· Hence there are y, ζ e G such that u~] = yuz for all u e G. Putting u = 1 we obtain ζ = yand _1 l w = yuy~ for all u e G. Hence G is an elementary abelian 2-group. If the group Γο is not isomorphic to the group Σο induced by Γο on the set V of vertical lines then according to Lemma 3.2 there exists in Γο a collineation of the form (u, υ) m>- (u, va) with a suitable α Φ 1 in G. The group induced by Γο on the set Ή. of horizontal lines consists of the mappings ν ι—» rv : G —> G for any r € G. If G has trivial centre then the mapping υ ι-> va is not contained in the group induced by Γο on the set !K. This contradiction shows Γο = Σο and then Γ = Σ. If G is an abelian group then Γο consists of the maps (u, ν) h^ (yuy, y~lv) = (y2u,

y~lv)

and the assertion (iii) follows.

•

By Proposition 9.2 we know that the core of a Bol loop is left distributive. The next proposition characterizes Bol loops the core of which satisfies both distributivity laws. Proposition 9.4. In the core C(L) of a Bol loop L the right distributive law (x + y) + ζ = (χ + ζ) + (y + ζ)

104

9 Core of a Bol loop and Bruck loops

is satisfied if and only if in L the identity · uv)x] = u{[v · (JC ·

;T[(T>

holds for all x,u,ve

u~lx)v]u}

L.

Proof The right distributive law may be written as (JC -

·

·

= (JC -z~lx)·

In a Bol loop L we have ( χ - 1 · (X +

Y )

=

- 1

(χ · >>

· - 1

Λ:)

- 1

O + zr'Qt+z).

= 1 and hence =

Λ:

· yx~]

- 1

Applying the Bol identity and (y + z ) _ 1 = >>_1 + x{y~l[x

• (z~l(x

• y_1Jc))]l

=

x~l

+

Y

- 1

.

we obtain from the first relation

= x[z~l[x

• (CT1

· zy~l)(x

+ z))]} .

Putting u = χ + z, using ζ = x + u = χ • u~xχ and applying the Bol identity to Χ • ( Ζ ( Λ : · > ' J T ) ) we get the equivalent identity - 1

_ 1

1 [Μ

·

1 JC]

= (JC + Μ)_1{Λ: · [ y - 1 · (JC · M _1 ;c);y _1 ]w} .

Denoting υ = y~l and using Ο + u)~xt = ( Λ : - 1 + u~l)t = (Λ-1

· ux~x)t — x~l[u

· x~lt]

we have v[u · ΙΛΧ] = x~l{u

• [Υ · (Λ: ·

u~lx)v]u]

which is equivalent to x[v[u • i?jc]} = u{v · [(λ · Μ _ 1 χ)υ]«} . Using v[u · υχ] = (υ · uv)x we obtain the assertion.

•

It is well known that the loops isotopic to distributive quasigroups are isotopic to commutative Moufang loops (cf. [107], IV.5.)· But the following corollary shows that a loop L with distributive core needs not to be isotopic to a commutative Moufang loop. Corollary 9.5. The core C ( G ) of a group G is distributive if and only if every element of G commutes with each of its conjugates. A group G with this property is nilpotent of class at most 3; if it contains no element of order 3 then the nilpotency class of G is at most 2.

105

9.1 Core of a Bol loop

Proof. According to Proposition 9.4 the core C(G) is distributive if and only if for all x, u, ν € G one has xvuvxu~l = uvxu~xxv. Putting xv — ζ and uv = y we obtain z{yzy~x) = (;yz;y -1 )z which proves the first assertion. The second assertion follows from [58], Hilfsatz 6.4 and Satz 6.5, pp. 287-288. • Proposition 9.6. The core C(G) of a group is commutative if and only if the group G has exponent 3. Proof The core (C(G), + ) of the group is commutative if and only if for every x, y € G one has χ + y = xy~xx = ;y.x_1;y = y + x, or equivalently, ( j c ^ - 1 ) 3 = 1.

•

As a corollary of the previous two assertions we obtain the well known theorem of Levi and van der Warden (cf. [58], p. 290): Corollary 9.7. If G is a group of exponent 3 then every element ofG commutes all its conjugates. Proof. A commutative core is distributive.

with

•

Now we investigate the class of Bol loops L for which the core C ( L ) is a quasigroup. We call a loop L strongly 2-divisible if the mapping q : χ x1 : L —» L is bijective. Proposition 9.8. The core C(L) strongly 2-divisible.

of a Bol loop is a quasigroup

if and only if L is

Proof For given a, b e CiL) the equation a + χ = b has the unique solution χ = a + (a+x) = a + b. The equation χ + 1 = c is equivalent to x2 = c and hence it is for given c € C(L) uniquely solvable if and only if L is strongly 2-divisible. Now we assume that L is strongly 2-divisible. We consider the equation χ + a = b for given a, b e C(L). The equation u + 1 = a has the solution u = «fa. Since the mapping au : χ ι-»· u + χ is an involutory automorphism of C ( L ) we have χ + a - au(au(x)

+ 1) = au(au(b))

= b

and hence this equation is uniquely solvable if and only if ζ + 1 = cru Φ ) is uniquely solvable. But this is the case since L is strongly 2-divisible. • We denote by L+{a) — σα and / ? + ( a ) respectively the mappings χ μ·- a + χ and χ h^- λ: + a of the core (C(L), + ) of a Bol loop L.

106

9 Core of a Bol loop and Bruck loops

Proposition 9.9. Let L be a strongly 2-divisible

Bol loop.

For a given a e L the

operation xR+(a)~l

x*ay=

yL+(a)~l

+

defines on L a Bol loop La with unit element a. Fora = 1 we have x*iy where (y/x)2 = x. Any loop La has the following properties: (i) La is isotopic to the core (C(L), (ii) La has the automorphic x,y e L. (iii) La is a left

=

y/x-y^/x

+).

inverse property,

i.e. (x *a y ) - 1 = - * - 1 *a J - 1 for all

A-loop.

(iv) The mapping ou = L+(u)

is an isomorphism

from the loop La onto the loop

Lu+a(v) If L and L' are strongly 2-divisible C(L) and C(L') are isomorphic.

Bol loops which are isotopic then the cores

Proof The operation * a defines a loop on L since the equations u *a χ — ν and x u = ν are uniquely solvable for any given u, ν e L. This follows from the fact that the core of a strongly 2-divisible Bol loop is a quasigroup. The left translation : y ι-»· x *a y of the loop La is the mapping λ ^ = L+(a) ο L+(xR+(a)~l). The loop La is a Bol loop if and only if the product λ ^ λ ^ λ ^ is again a left translation. This means there exists a ζ € La such that λ^λ^λ^

(L+{a)oL+{xR+(aTx)o(L+{a)oL+(yR+{a)-x)

=

o(L+(a)oL+(xR+(arl) = L+(a)

ο L+(zR+(a)~l)

= λ< α) .

Since C(L) is left distributive the conjugate element L+(x7?+(a)_1)

ο L+(a)

ο L+(yR+(a)~l)

° L+(a)

L+(xR+(a)~l)

ο

of the left translation L+(yR+(a)~l) is again a left translation the Bol identity is satisfied in La. Now we show the property (iv). The mapping L+(u) is an isomorphism + L ( m ) : La —• Lu+a if and only if xL+(u)

*(u+a) yL+(u)

= (x *a

y)L+(u)

for any x, y e L. This identity is equivalent to (u -I- x)R+(u

+ a)~l

+ (u + y)L+(u

+ a) = [xR+(a)~l

+ yL+(a))L+

(u).

9.1 Core of a Bol loop With ν = (u + x)R+(u

+ a)~x and t = xR+(a)~l

107

we obtain

ν + [(m + a) + (u + y)] = u + [t + (a + y)]. Because of ν + (u + a) = u+x = u + (t + a) = (u + t) + (u + a) we have ν = u +1 and hence our identity is equivalent to (u + t) + [(u + a) + (u + y)] = u + [r + (a + y)] which is the consequence of the left distributive property of the core (C/L), +). Since all loops La are isomorphic we prove the automorphic inverse property for the loop La with a = 1. In this case we have JC*I

y = xR+(l)~]

+ y L + ( l ) = xR+(iy][(l

= jcÄ + (l) _ , [;y xR+(\yl]

=

+ y)_1

xR+(\)~l]

yfx-ysfx

since v(ic/? + (l) = x. The inverse of χ e L with respect to the operation *i is x~l because of χ *i x~l = yfx · sfx = 1. Hence the element ( x _ 1 *i y - 1 ) is the inverse element of χ *i y in the loop L\ if and only if it is the inverse element of (x *i y) in the loop L. But we have (x *i y) • (x~l *i y " 1 ) = ( v ^ • y V ^ H v ^ 7 · y

- 1

/^7)

and any La has the automorphic inverse property. The left Α-loop property (iii) means that each λ**^ λ ^ λ ^ : La -> La is an automorphism of the loop La. From (iv) we know that L + (w) : La —> Lu+a is an isomorphism. It follows that λ/ν)~χ{χ/νν}]]/ν

=

=

= [x/v • u~x

{u[(y/v)~xx]}]/v

x/v{(y/v)~xx/v}.

This means that the mapping ζ (->· z/v : C(L, ·)

C(L, *) is an isomorphism.

•

Since the multiplication χ *a y of a loop La in the previous proposition is given in terms of a left distributive quasigroup satisfying the identity χ + (χ + y) = y the loops La can be defined starting with any such quasigroup. Definition 9.10. Α Bruck loop is a Bol loop (L, •) satisfying the automorphic inverse property, i.e. the identity (x • y ) - 1 = λ: - 1 · y - 1 . Since the class of Bruck loops is defined by identities each subloop and factor loop of a Bruck loop is a Bruck loop. Proposition 9.11. Let (Q, +) be a left distributive quasigroup satisfying the identity χ + (x + y) = y for all x, y e Q. Then the loop (La, defined on the set Q by the multiplication x *a y = xR+{a)~x

+

yL+(a)

with unit element a has the following properties: (i) La is α Bruck loop which is isotopic to the quasigroup (Q, +). (ii) La is a left A-loop. (iii) The mapping L+(u) is an isomorphism from the loop La onto the loop

Lu+a.

(iv) The core (C(La), 0 a ) of the loop La is isomorphic to the quasigroup (Q, +) under the mapping R+(a)~x. (v) The set Aa of the left translations of the loop La is equal to the set {L+(a)L+(x);

χ e Q\.

Proof. The assertions (i)-(iii) can be proved in the same way as in the previous proposition.

9.1 Core of a Bol loop

109

The operation θ α of the core C{La ) is given by x θα y = Χ

Cy-1 *a x) = X *a [(« + >)/?+(α)-1

= xtf+(α)_1

+ [(α + ^fl+ia)"1

+

*L+(ö)]

xL+(a)]L+(a)

+

= i Ä + ( a ) " ' + [(α + y)R+(a)-lL+(a)

+ jc]

= [ χ Λ + ^ Γ 1 + (α + y ) Ä + ( f l ) _ 1 L + ( f l ) ] + [jc/? + (Ö) _ 1 = [jc/?^)"1

+x]

y)R+(arlL+(a)]R+(a),

+ (a +

where we used y - 1 = a + y and Jc/? + (a) _1 + χ — a. 1 Indeed y *a (a + y) = yR+(a)~l + (a + y)L+(a) = yR+(a) + y = a since yR+(aΓ1 + y = yR+ia)'1 + (yR+(a)~l +a)=a. We have a + y = (a + y)R+(a)~l R+(a) = (a + y)R+(a)~l +a and hence y — (a + y)R+(a)~^ L+(a) + a

or

(a + y)R+(a)~l

L+(a) =

yR+(a)~l.

It follows (x ®a y)R+(a)~l

= XR+(a)~l

+

yR+(a)-]

and the assertion (iv) is proved. The left translations λζ by the element ζ € La is given by y

zR+(a)~l

+ yL+(a) =

yL+(a)L+(zR+(a)-1)

and hence (v) is proved.

•

Proposition 9.12. Let (L, ·) be a strongly 2-divisible Bruck loop and (C(L), + ) be the core of (L, •). The loop (La, *a) is isomorphic to (L, ·) for any a e L. If a = 1 then the mapping q : χ η» χ2 : (L, ·) (L\, *i) is an isomorphism from (L, ·) onto (L[, *i). If L and L' are isotopic strongly 2-divisible Bol loops then for any a e L and a' e L·' the Bruck loops (La, *a) and (Z/,, *a>) are isomorphic. Proof In a Bruck loop (L, ·) we have [x • (y2x))(xyΓ1

= [x • (y2x)](x~ly~l)

= xy

and also (a:>') 2 (jcj) _1 = xy. It follows χ • y2χ = (jcy) 2 for any x,y e L. This means (jty) 2 — χ · y2x

= jc 2 * ι y2

and hence the mapping q : χ κ-» χ2 is an isomorphism from (L, ·) onto (Li, *j). Since all loops (La, *a) a e L are isomoφhic the first two assertions are proved. If L and L' are strongly 2-divisible Bol loops which are isotopic then the cores C(L) and C(L') are isomorphic left distributive quasigroups (Proposition 9.9 (v)). Now it follows from Proposition 9.9 (iii) that also the loops (La, * a ) and (L',, * a ') are isomorphic. •

110

9 Core of a Bol loop and Bruck loops

Let (L, ·) be a strongly 2-divisible Bol loop and let G be the group generated by the set S = { λ χ \ χ ε L } of left translations of L. We denote by Η the stabilizer of 1 e Lin G and by π : G —>· S the projection defined by π(λχΗ) — λχ for every χ e L and h e Η. The loop (S, Δ) defined on § by λ^Δλ^ = π(λχλγ) = λ ζ is isomorphic to (L, ·) (cf. Proposition 1.5). We define on L a new multiplication by = ^/χ·(γ··>/χ) with respect to which (L, *) is a Bruck loop (Proposition 9.9). If (L, ·) is a Bruck loop then the mapping q: ζ ζ ' (L, •) —. (L, *) is an isomorphism (Proposition 9.10). Since in a Bol loop the equations λ^ λχ2 and λ^λ^λ^ = λ x-(y-x) hold we have = λ^Α(λγΑλ^) = = e S for suitable h, h' e Η. Since (L, ·) is a Bol loop we obtain h' — h~l and hence V ^ A ^ A v ^ ) = λ ^ . ( y . j t ) · Consequently S = {λ*; χ Ε L) can be equipped with the multiplication u ο ν — and the mapping χ λ χ : (L, *) -> (S, o) is an isomoφhism of Bruck loops. If the loop (L, ·) is a Bruck loop then all the loops (L, ·), (L, *), (S, Δ) and (S, o) are isomoφhic. Moreover, if L and L' are isotopic strongly 2-divisible Bol loops then (L, *) and ( L * ) are isomorphic Bruck loops (cf. Proposition 9.9). We summarize this discussion in the following Theorem 9.13. Let (L, ·) be a strongly 2-divisible Bol loop. Then the mappings of the commuting diagram (L,·)

φ \ χ l·^· χ

•

λ : χ ι-»· λχ

λ : χ ι-» λ χ

(S,o)

(δ,Δ) Φ have the following

(L,*)

:λχ^λί

properties:

(i) the mappings λ : (L, •) —> (S, Δ) and λ : (L, *) —»· (S, o) are isomorphisms of loops; (ii) the mappings φ : (L, ·) (L, *) and Φ : (S, Δ) —>· (S, o) induce a surjection from the isomorphism classes of strongly 2-divisible Bol lops onto the isomorphism classes of strongly 2-divisible Bruck loops such that the restriction of this surjection to the isomorphism classes of Bruck loops is the identity. This surjection is constant on every class of isotopic strongly 2-divisible Bol loops. We conclude this subsection by lemmas concerning collineations of the 3-net N(L) associated with a Bruck loop L. Lemma 9.14. Let L be α Bruck loop and let N(L) be the associated 3-net. Then the mapping μ : (χ, y) mv (jc, λ: - 1 y - 1 ) : N(L) —>- N(L) is an involutory collineation centralizing the group Γ generated by the Bol reflections of"N(L).

9.1 Core of a Bol loop

111

Proof. Clearly, every vertical line is fixed and the sets of horizontal and transversal lines are interchanged by μ. The mapping τζ : (χ, y) ι-»· (z • x~lz, • xy) is the Bol x reflection with respect to the vertical line through (ζ, 1). Since (z • x~ z)(z~x · xy) = z{x _ 1 [z(z _ 1 · xy)]} = zy and z(z _ 1 · xy) = xy we have μτζ(χ, y) = (ζ · x~lz, (z · x~lz)~x{z~x because of the identity (mi;) -1 = u~lv~\and proves the assertion.

• xy)" 1 ) = (z • x~Xz, (zy) _ 1 ) τζμ(χ, y) = (ζ-x~xz,

z _ 1 y _ 1 ) , which •

Lemma 9.15. Let L be α Bruck loop and let N(L) be the associated 3-net. Then the group Ν of collineations leaving every vertical line invariant is the semidirect product of the group Nr consisting of the collineations {(x, y) m>- (X, ya)\ a € Nr}, where Nr is the right nucleus of L, by the group of order two generated by the collineation μ : (χ, >>) )-> (χ, which inverts any element of Nr. Moreover the group Ν is centralized by the group Γ generated by the Bol reflections ofK(L). Proof. The assertion follows from Lemma 3.2 and Lemma 9.14. An immediate calculation yields the action of μ on Nr. According to Lemma 9.14 the collineation μ commutes with any element of Γ. For collineations of the form a : (x, y) i-> (x, ya), a e Nr,

and

τζ : (χ, y) ι-> (z · x~]z, z~X · xy)

we have α τζ(χ, y) = (ζ- x~lz, (z~l •

τζα(χ, y) = (z · x~lz, z~l(x · ya)).

But since a e Nr we can write ( z - 1 · xy)a — z~l (xy · a) — z~l(x • ya) and hence every element of Ν commutes with every Bol reflection. • At the end of this subsection we show that the left Α-loops satisfying the left inverse and the automorphic inverse property are related to abstract symmetric spaces. Proposition 9.16. Let L be a left Α-loop satisfying the left inverse property. Let G be the group generated by the left translations of L and Η be the stabilizer of 1 e L in G. If L fulfills the automorphic inverse property then there exists an involutory automorphism σ of G fixing Η elementwise and inverting any left translation λχ, χ e L, of L. Proof Every element g e G can be written in a unique way g = kah, where λα is a left translation of L and h Ε H. Let σ : G G be the bijective map defined by a(Xah) = λ" 1 /?. We show that (*)

σ {Xahkbk) = σ {λαΗ)σ (Xbk)

112

9 Core of a Bol loop and Bruck loops

for any a, b e L and h,k e H. Indeed, we have = σ (Xahkbh~l

a(XahXbk)

hk) =

σ(λαλ/,(/,)Μ)

= ty(Kh(b){^~h(b)Xakh(b))hk)

=

^äh(b)Kh(b)Xakh(b)hk

as well as a(Xah)a(Xbk)

=

=

Xa-\Xh^~\hk

= λ αΑ(« λ α -« A(A)-' λ « " ' kh{b)~' since hXbh~x

= x~ly~l

— Xh(b) and

hk

for all x, y e L. It follows that the

identity (*) is satisfied if and only if λ ~ ^ λ α λ ν = λ~_ 1 , υ _,λ ι < -ιλ ϋ -ι holds for every u, ν € L, or equivalently λαλυλυλΗ — ] Applying this map to u ~ ζ we see that this relation is equivalent to the equation λαλυ(νζ) = Xuv[(uv)(u~lζ)]. Putting ζ — we get the equivalent identity (**)

= (uv)[(uv)(ut)~l]

u[v(vt~')]

.

Since L is a left Α-loop satisfying the automorphic inverse property we have that x[y(y-lx~1)2]

= (^[λ-'λ,λ,ΟΤ1*-1)2] = (xy)(xy)~2

-

(^[λ-'λχλ^^-'χ"1)]2

= (xy)~l

holds for all x, y e L. Replacing in this identity jc by ( t u ) ~ ] and j by ί υ we obtain (fiO-^fiOKiiO-V)]2} = [(^"'(ίυ)]

-ι

or (ίυ){[(ίυ)-1(ί")]2}

(***)

Since λ ^ λ

{

λ

υ

{ =

[(tv)-\tu)2]

= (tu)[{tu){tv)~x]

.

( t v ) ~ l ( t u ) and λ ^ ' λ ^ λ ^ is an automorphism we have = [ λ ^ λ , λ ^ ν α " ) ] = k~lktkv(lrlu)2

=

2

=

[X;v^tkv(v~lu)]2 (tvrl{t[v(v~lu)2]}.

Applying this to (* * *) we obtain (tu)[(tu)(tv)~l]

= (tv)[(tv)~\tu)]2

= t[v(v~lu)2]

.

For t = 1 in (* * *) we have v(v~lu)2 = u(uv~l) which gives the relation -1 (i«)[(fM)(rtO ] = Hence the identity (**) holds and σ : G —• G is an involutory automorphism having the desired property. •

9.2 Symmetric spaces on differentiable Bol loops

9.2

113

Symmetric spaces on differentiable Bol loops

Let C be one of the following categories: topological spaces, differentiable or analytic manifolds, and algebraic varieties. If L is a (local) Bol loop in the category C then the (partial) operation (x, y)

x + >> = oxy =

xCy-1*)

which satisfies the properties (1), (2), (3) in Proposition 9.2 defines the (local) core C(L) of L. The (local) loops (L a , * ö ) treated in Proposition 9.9 are (local) Bruck loops, i.e. (local) Bol loops satisfying the automorphic inverse property. If L is a differentiable (local) Bol loop then the (local) reflections σχ satisfy the following condition: (4) every χ e C(L) has a neighbourhood U such that axy = χ + y = y implies χ == y for all y e U. The core C(L) of a differentiable (local) Bol loop is a (locally) symmetric space in the sense of Loos ([86], p. 63 and pp. 99-101). If L is a connected differentiable strongly 2-divisible Bol loop then the core (C(L), + ) is a symmetric space which is a quasigroup such that the operation " + " is differentiable. This follows from Proposition 9.8 and from the fact that the operation + may be expressed by means of the differentiable operations of L. Proposition 9.17. The core C(L) of a connected differentiable strongly Bol loop is an analytic quasigroup.

2-divisible

This proposition follows immediately from the following Theorem 9.18 ([102], Theorem 2.10). Let (L, +) be a left distributive quasigroup on a connected manifold with differentiable operation "+" satisfying the identity χ + (x + y) = y for all x, y e L. Then the quasigroup (L, +) is analytic. Proof. Since (L, + ) is a symmetric space in the sense of Definition 6.1 it follows from Corollary on p. 94 in [86] that the operation " + " is analytic. Since the equation a + χ — b is equivalent to χ = a + b the solution χ depends analytically on a and b. The symmetric space L has a representation on the space of symmetric elements of a Lie group (cf. [86], p. 73). This means the following: there exists a connected Lie group G and an involutive automorphism σ : G —> G such that L is realized on the connected component Ga of 1 e G in the set {JC e G; χσ = χ - 1 } with respect to the multiplication χ + y — xy~*x for all x, y e Ga (cf. [86], Theorem 4.6 and Proposition 4.4, pp. 182-185). Since the equation za~xz — ζ + a — b has a unique solution in Ga and 1 is contained in Ga the mapping ζ ι—^ z^ is bijective on Ga. We denote the solution of

114

9 Core of a Bol loop and Bruck loops

z + \ = b by ζ = bT-. The solution of the equation χ + a = b may be expressed by I 1 1 χ = a? + (a? + b)? since χ + a = b is equivalent to (a 5 + x )

+

(α ζ + a )

=

( a 2 + JC) +

1 =

a?

+ b

I I ι and hence a 2 + χ — (a2 + 0 ) 2 . Now we show that every element in Ga is contained in precisely one 1-parameter subgroup of Ga. Let χ be an element of Ga and (x) be the monothetic subgroup of G generated by x. Since Ga is closed in G the group (x) is contained in Ga. The group (x) is isomorphic to Ζ or it is compact (cf. [44], p. 85). If (x) is compact we denote by Κ a maximal compact subgroup of G containing (x). Since the restriction of the exponential map from the Lie algebra 0 of G to the Lie subalgebra t of Κ is surjective onto Κ ([46], p. 150, Theorem 3.2), there exists a vector X e t such that χ = expX. We have χσ = jc - 1 = exp(—X). For the automorphism σ* : g g a induced by σ we have ( t X ) * = — tX for all ( e l . But then Ga would contain a 1-dimensional torus subgroup which contradicts the fact that the mapping ζ —> ζ 2 is bijective. Hence for every jc e Ga we have (χ) = Z. Then the closure {x) 2 in Ga of the group containing (x) and with any element also its square root is isomorphic to R. Since for x, y e Ga and {jc)2 Φ (y)2 one has {x) 2 Π (y)2 = {1} and every element of Ga different from 1 is contained in precisely one 1-parameter subgroup. It follows that the mapping χ ι-» χϊ : Ga —• Ga is analytic. Hence the mapping (a, b) m> b/a = bR+(a)~l = a? + (a? + bp : L ^ L is analytic too. • Let S be a connected differentiable manifold and {σχ; χ e 5} be a set of involutory diffeomorphisms such that (5, + ) with χ + y = axy defines on S the structure of a symmetric space with a canonical connection V (cf. Section 6, Proposition 6.2). Then for any point a e S there exists a normal neighbourhood Ua of a such that the operation (x, >>) Η* Χ * y — xR+(a)~1 + yL+(a) — o*oay is well defined if we 2 (a) + + put zL (a) = a + ζ = σαζ and zR (a) = ζ + a, where | = xR+(a)~l denotes the midpoint of the geodesic segment [a, x] connecting a and x. According to [86], Theorem 2.7, p. 64, the parallel translation τα%χ along the geodesic segment [a, x] is induced by σ*σα and the local multiplication χ * y = εχρ ζ τα^χ exp" 1 y defines a 2

(a)

geodesic loop La with identity a e S. This definition gives a global loop if and only if the map exp a is a diffeomorphism. We denote by 1 another origin of the symmetric space S. Now, we consider in the normal neighbourhoods U\ and Ua, respectively, the local multiplications χ * y = Oh{ 1 x)cr\y and χ * y — σ^α· χ * (_y_1 * λ;) = σ*σ -ι σαχ = σχσ , -ι \ (*). The geodesic reflection 2 ασ 2 (a) (a) 2 · G/G'

the set of elements conjugate to a in Τ is Μα and it generates

is surjective, then

Τ.

The element a centralizes an element (zk, z - 1 ) e A' if and only if k — z~2.

Proof

Hence C — {(z, z); ζ e G,z2

e G'}(a).

We can write

z _ 1 ) = (k~lζ'1,

(zk, z~l)~la(zk,

(k']Z-1,

=

z){a(zk,

ζ~1)α)α

z ) ( z _ 1 , zk)a =

({z2k)-\z2k)a

for all k e Κ. If the mapping φ is surjective we obtain the second assertion. Proposition 10.8. Let G be a group, and let Τ — Κ (a) with a2 = generated by Μ α

=

{(ζ, ζ

mapping χ ι-> (χ, χ " 1 C(G)

_1

)α;

ζ e G} and a(z,

z_1)a

=

*-1)a

0 (y, y~l)a

— (x,

(.x,x~l)a(y,

operation jf_1)a(j,

y~l)a(x,x~l)a

= (x,

y)(x,

jc_1)a = (JC + y, (x + y ) _ 1 ) a .

=

Proposition 10.9. Let G be a group, and let Τ = Κ (a) with a2 = generated by M a = { ( ζ ,

z-1)a;

ζ £ G} and a ( z ,

generated by the left translations of the core C(G) T/Z(T),

where Z{T)

z_1)a

=

and

z2

(z_1,

is isomorphic

is the centre of Τ; the group Z(T)

(z, z), where ζ is in the centre ofG

•

1 be the group

z). Then the group to the factor

group

consists of the elements

e G'.

Since Τ is generated by Μ α the group generated by the left translations

of ( Μ α , φ ) is the factor group T/Z(T). { ( z , z ) ; ζ € G and Z(T)

Then the of the core

We get immediately

Proof

Proof.

1 be the group

(z_1,z).

: ( C ( G ) , + ) —> ( Μ α , φ ) is an isomorphism

of G onto Μα equipped with the binary

•

z2

The centralizer of α in Κ is the group

e G'} (Proposition 10.7). An element (z, z ) is contained in

if and only if ζ is in the centre of G. Using the isomorphism between Μ α and

C ( G ) the assertion follows.

•

Proposition 10.10. The equations χ + a = b for given a, b € G are solvable in the core C(G)

of a group G if and only if the map φ : xG'

x2G'

: G/G'

G/G'

is

surjective. These equations are uniquely solvable if and only if the group G is strongly 2-divisible.

124

10 Bruck loops and symmetric quasigroups over groups

Proof. The assertion follows from the previous three propositions and from Proposition 9.8. • Now we consider the special case in which L — G is a strongly 2-divisible group. Let C(G) be the core of G defined by the binary operation χ + y = axy = xy~lx on G. Theorem 10.11. Let G be a strongly 2-divisible group. Then the group G{La) generated by the left translations of α Bruck loop La isotopic to the core C(G) of G is isomorphic to the factor group K/Z(T), where Z(T) is the centre ofT — Κ (a). Moreover, ifG = G', then G(La) contains a normal subgroup which operates sharply transitively on La. Proof. The core C(G) is isomorphic to the quasigroup (Μα, θ ) (cf. Proposition 10.8). Clearly, the commutator subgroup T' of Τ is contained in K. Since we have Μ = (Μα)α = {t~[ata; t G T] C T' and Μ generates Κ we obtain Κ = Τ'. According to Proposition 10.2 the group generated by the left translations of a Bruck loop isotopic to the quasigroup (Μα, θ ) is isomorphic to the commutator subgroup ('T/Z(T))' of the group T/Z(T) (cf. Proposition 10.9). Clearly we have (T/Z(T))'

= T'/Z(T)

=

K/Z(T)

since Τ = Κ and Z(T) c K. If G = G' then because of (k~l, 1 )a(k, 1) = (Jfc"1, k)a for any k e G' = G, the normal subgroup Ν acts sharply transitively on Μα. • Theorem 10.12. Let G be a strongly 2-divisible group. Then the following conditions are equivalent: (a) the Bruck loops La isotopic to the core C(G) are commutative groups; (b) G is α nilpotent group of class at most 2. Proof. First, we prove that the condition (a) implies (b). According to the previous theorem the abelian group L\ is isomorphic to the group K/Z(T), where Κ is an extension of a group Ν by G (cf. Proposition 10.6). Since Κ has nilpotency class at most 2 this is true also for G. Vice versa, if G is a nilpotent group of class at most 2 then Κ = {(Zk, ζ" 1 ); ζ e G, k e G'} c G χ G is also a nilpotent group of class at most 2. The Bruck loop L\ is an abelian group if and only if Κ/Z(T) is an abelian group (cf. Theorem 10.11). This is the case if and

10 Bruck loops and symmetric quasigroups over groups

125

only if Z(T) contains the commutator subgroup K' of K. The group K' is generated by the products {zk,z~{){xl,x'x){k~lz-\z){rxx-\x) (χ, ζ e G ,k,l

= {zxz~xx~\ζ~λχ~λzx) G

,

G') ,

since G' is contained in the centre of G. But we have 1 = zx{x~x z~l xz)z~l x~l

= ( z x z _ 1 x _ 1 ) C x _ 1 z _ 1 * z ) = (zxz~] x_1)xzx~l

which is equivalent to x~lz~lxz = χzx~]z~l or to zxz~lx~l = z~xx~xzx Hence K' C Z(T) = {(ζ, z); ζ2 e G' and ζ is in the centre of G}.

z_1, e G'. •

Corollary 9.5, Proposition 10.5 and Theorem 10.12 show that the Bruck loops La isotopic to the core C ( G ) of a strongly 2-divisible group G are commutative proper Moufang loops only if the nilpotency class of G is 3, and they are commutative groups precisely if the nilpotent group G has the class < 2 . If a strongly 2-divisible group G is of the nilpotency class > 4 then the Bruck loops La cannot be Moufang loops. Now we characterize the strongly 2-divisible groups in different categories. This can be used for the construction of proper Bruck loops. Proposition 10.13. A finite group is strongly 2-divisible if and only if it has odd order. Proof. In a group of odd order the elements χ and x2 generate the same cyclic group. If G has even order then G contains involutions. • A connected Lie group G with Lie algebra 0 is called exponential if the exponential map exp : g —»· G is a diffeomorphism. Proposition 10.14. A connected Lie group G is strongly 2-divisible is exponential. Exponential Lie groups are solvable.

if and only if G

Proof. If we suppose that G is exponential then every element χ / 1 of G is contained in precisely one 1-parameter subgroup of G. In this 1-parameter subgroup there exists exactly one element y such that y2 = x. Since the mapping Λ: x2 leaves every 1-parameter subgroup invariant this map is bijective and it is a diffeomorphism. Now we assume that G is strongly 2-divisible. Let (u) be the monothetic subgroup of G generated by the element u e G. Since χ ι-> x2 : G —> G is a diffeomorphism, (u) is either isomorphic to Ζ or it is a finite group of odd order ([53], p. 85). If (u) = Ζ then {u) is contained in precisely one 1-parameter subgroup of G. If (u) is finite then it is contained in a maximal compact subgroup C φ {1} of G, which is connected (cf. [92], p. 188). But then G contains tori and also involutions. Hence (u) is isomorphic to Z, and it is always contained in precisely one 1-parameter subgroup and G is exponential.

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10 Bruck loops and symmetric quasigroups over groups

An exponential group G does not contain any tori different from {1}. If G were non-solvable then any simple factor of a Levi complement of the radical of G is isomorphic to the universal covering group Δ of PSL2(M) (cf. [109], p. 535). The centre Ζ of Δ is isomorphic to the infinite cyclic group. Let ζ be a generator of Z. Since in Δ there are infinitely many elements a with a2 = z, the mapping jc i-^- jc2 is on Δ not bijective, which is a contradiction. • Proposition 10.15. Every connected algebraic group G over an algebraically closed field Κ such that the map χ t-»· x2 is an algebraic automorphism of the variety G is affine and the characteristic of Κ is φ 2. Proof If the characteristic of Κ is two then, first, we assume that G contains a 1-parameter subgroup U. The group U is isomorphic either to K+ or to the multiplicative group K* of Κ. The restriction q \ y of the mapping q : χ ι-* χ2 is a bijective mapping which is not an algebraic automorphism. But the restriction of an algebraic automorphism to an invariant subvariety is an algebraic automorphism and hence G has no 1-parameter subgroup. According to [114], Corollaries 3 and 5, the group G is an abelian variety. If the characteristic of Κ is 2 then the bijective morphism φ : χ η>· Χ2 from G onto G induces a morphism φ* of the field K{G) of rational functions on the variety G into itself such that - x2 is an algebraic automorphism of the algebraic variety G if and only if G is unipotent and the characteristic of Κ is different from 2. Proof. From the previous proposition we know that the characteristic of Κ is different from 2. Then G does not contain a torus Τ Φ {1}, since otherwise it would contain involutions. Thus G is unipotent (cf. [57], p. 99). Conversely, let G be a connected unipotent group and let the characteristic ρ of Κ be different from 2. If ρ > 0, then the order of every element of G is a power of ρ and the mapping φ: χ ι-» χ2: G —» G is bijective. The mapping φ~ι is an algebraic morphism since = x 5 , where t is the order of jc 3. Now we have to treat the case that the characteristic ρ is zero. Let X = {x) be the monothetic subgroup of χ e G, i.e. the smallest closed subgroup of G containing x.

10 Bruck loops and symmetric quasigroups over groups

127

Let Xo be the connected component of X. We know that X/XQ is finite and XQ is isomorphic to K+. Since Xo is divisible and X is abelian, we have X = XQ Χ F with a finite cyclic group F. As a unipotent group over a field of characteristic zero has no elements of finite order different from the identity, we have X = Xo- For the restriction of φ to the vector space X we have G/N be the natural epimorphism. We denote by A the set of left translations of Q. Then the set φ(Α) is with respect to the multiplication φ(λι) • φ(λ2) =

φ(λι)φ(λ2)φ(λ\),

for all λι, λ2 e Λ, a symmetric quasigroup such that the group generated by the left translations of this quasigroup is isomorphic to the factor group 0 and σ is an involutory automorphism of G such that the restriction of the exponential map to the ( — l)-eigenspace of the automorphism σ* induced by σ on the Lie algebra of G is a diffeomorphism. The connected differentiable strongly 2-divisible Bruck loops and the connected symmetric spaces with bijective exponential map form equivalent categories. The Riemannian symmetric spaces S corresponding to differentiable strongly 2-divisible Bruck loops L are direct products of euclidean and hyperbolic symmetric spaces. If the Lie group G topologically generated by the left translations of a locally compact connected strongly 2-divisible Bruck loop is semisimple, the associated symmetric space is Riemannian and any connected semisimple Lie group with trivial centre occurs as such a group G. If the group G topologically generated by the left translations of L is solvable then G is exponential. At the end of this section we show that any connected almost differentiable strongly 2-divisible left Α-loop satisfying the left inverse and the automorphic inverse property is an analytic Bruck loop. Let (L, ·) be a connected differentiable strongly 2-divisible Bruck loop, let S = {λ*; χ 6 L] be the manifold of left translations of L and let G be the group topologically generated by S. The set S is an embedded submanifold in G (cf. Lemma 1.27). The operation ( λ χ , λγ) λχ + λ ν = λχλ~λλχ : § —• S equips S with the structure of a symmetric space (S, -f) which is isomorphic to the core C(L) of L. It follows from Proposition 9.13 that (L, ·) is isomorphic to the Bruck loop (S, o) defined on S

130

11 Topological and differentiable Bruck loops

by u ο ν — yjuvyfu.

The group of displacements

Σο =

ι · ν ι—> μ + (1 + υ) = uvu : S —S)

of the symmetric space (S, + ) coincides with the group G topologically generated by the left translations of the loop (S, o) since u u2 : S -> S is a diffeomorphism of S. Because the loop (S, o) is isomorphic to the loop (L, •) we have G = Σο = G. The set {σΗσ\\ u e S} and the set of left translations λ® of the loop (S, o) coincide, since σΗσ\ : Ν Μ* uvu and : d i-> UVU and hence ΣΑΣ\ = λ®2 (" £ S). The conjugation of Σο by the reflection σ\ induces on § the inversion u m>- « " ' since σι(σ„σι)σι = σχσα = (σ Μ σι) _ 1 . Let 0 be the Lie algebra of Σο and t the Lie algebra of the stabilizer Κ of the element 1 g S in Σο- According to [86], Theorem 3.1, the Lie algebra g has a decomposition 0 = t Θ m with [m, m] c [t, m] C m such that the conjugation by σ\ induces on m the inversion and on I the identity; the subspace m corresponds to the Lie triple system determined by the symmetric space (S, + ) . We want to show that every element of S is contained in precisely one 1-parameter subgroup of Σο and that any such subgroup is isomorphic to M. Let u be an element of S and let (u) be the monothetic subgroup of Σο generated by u. Since S is closed in Σο the subgroup (u) is contained in S. Since λχ2 = λ\ holds in a Bol loop and jc x1 : L —> L is a diffeomorphism (u) is either isomorphic to Ζ or it is a finite group of odd order ([53], p. 85). If (u) = Ζ then because u i—> u2 : S —>· S is a diffeomorphism (u) is contained in precisely one 1-parameter subgroup of Σο which lies in S. If (u) is finite then it is contained in a maximal compact subgroup C of Σο. The group C is connected (cf. [92], p. 188) and the restriction of the exponential map of the Lie algebra 0 of Σο to the Lie subalgebra s of C is surjective onto C ([46], p. 150, Theorem 3.2). Hence there exists a vector X e s such that u = exp X. We have a\ua\ = = exp(—X). For the automorphism Ad σι : 0 —> 0 induced by the conjugation by σ\ on 0 we have Ad 1 (cf. [92], p. 188.). If f) is the Lie algebra of Η then g = f) φ mo Θ mi, where mo corresponds to a euclidean and mi to a hyperbolic symmetric space (cf. [86], Theorem 1.6, p. 145.). The Lie subgroup expmo is the connected component of the centre Ζ of G. Since Ζ is homeomorphic to Mrt (Theorem 11.2) and mi —> exprrti is a diffeomorphism ([86], Theorem 2.4, p. 158.) the assertion follows. • Lemma 11.9. Let G be a Lie group topologically generated by the left translations of a locally compact connected strongly 2-divisible Bruck loop L. Then the stabilizer G ι of 1 G L in G is connected and contains a maximal compact subgroup of G. Proof Since G is connected and G/H, which is homeomorphic to L, is a euclidean space M" (Theorem 11.1) the stabilizer is connected (cf. [120], 94.4 Proposition, p. 604). Let F be a maximal compact subgroup of G\ and Κ a maximal compact subgroup of G containing F. According to [95], Theorem 3.1, the euclidean space GjG\ and the compact manifold Κ / F have the same homotopy type. Because of the isomorphism theorem of Hurewicz ([54], p. 148) all homology groups H(K/F, Z) vanish and hence the compact manifold Κ / F is a point ([23], p. 260). Consequently Κ — F. • Theorem 11.10. Let G be a Lie group topologically generated by the left translations of a locally compact connected strongly 2-divisible Bruck loop L. If the Lie algebra QofG is semisimple then L is Riemannian. Proof Let G be a connected semisimple Lie group. According to the previous lemma the stabilizer Η of 1 e L in G is connected and contains a maximal compact subgroup Κ of G. The Lie algebra g of G splits into a direct product g = gi χ · · · χ g„ of simple Lie algebras g i , . . . , g n . Then the Lie algebra t of the maximal compact subgroup Κ decomposes into t = χ · · · χ t„, where £,· is a subalgebra of g,. Because of Propositions 11.6, 11.7 and Theorem 11.2 the group G has trivial centre. Hence G is the direct product G = G\ χ • • · χ Gn, where the groups G, = expg, are algebraically simple and the groups Κ ι = exp are maximal compact subgroups

136

11 Topological and differentiable Bruck loops

of G,. The stabilizer Η contains the group Κ = K\ χ · · · χ Kn. If Η φ Κ then there exists a j e { 1 , . . . , n} such that Gj C Η. Indeed, if π, Η is the projection of Η into the group G, then π ι Η is a subgroup of Gi containing Ki. Since Η φ Κ there exists a j e { 1 , . . . , n) such that njH φ Kj. The normalizer of the group Kj in the group Gj is the group Kj itself (cf. [31], p. 269). For any element h e Η we have hKjh~] = 7tj(h)Kjnj(h)~l. Since Kj C Η the group Η contains the group KJ generated by nj(h)Kjnj{h)~x. From πjH φ Kj follows that the subgroup K* of Gj contains Kj as a proper subgroup. Since Kj is a maximal subgroup of the simple group Gj (cf. [120], p. 614) we have Gj = K* c H. This gives a contradiction because the stabilizer Η cannot contain the normal subgroup Gj of G. Hence Η = K. Using Lemma 3.3, p. 170 in [86] we obtain that the loop L is Riemannian. • We call a Lie algebra g compact if there exists a compact Lie group having g as its Lie algebra. Theorem 11.11. Every connected non-compact semisimple Lie group G with trivial centre is the group topologically generated by the left translations of a connected differentiable strongly 2-divisible Bruck loop. The connected strongly 2-divisible differentiable Bruck loops L\ and L2 having non-compact semisimple Lie groups G1 and G2 as the groups topologically generated by their left translations are isomorphic if and only if the groups G \ and G2 are isomorphic. If G \ = Gi then any two Cartan involutions σι and σι are conjugate in Aut(G) by an inner automorphism of G. Proof Let g be the Lie algebra of the non-compact semisimple group G. Then the non-compact semisimple real Lie algebra g admits a decomposition g = ϊ φ ρ , where Ms a maximal compact subalgebra and p is a complementary subspace with [£, p] C ρ, [p,p] C t holds (cf. [86], pp. 145, 154-155). To the decomposition g = t Θ ρ there exists an involutive automorphism σ of the Lie algebra g such that σ\ξ = id and σ | p = —id. There is a one-to-one correspondence between these involutive automorphisms called Cartan involutions and such decompositions. If g is the Lie algebra of a connected non-compact semisimple Lie group G with trivial centre then σ induces an automorphism of G determining a hyperbolic symmetric space associated to a strongly 2-divisible Bruck loop (Theorem 11.2). The second part of the theorem follows from the fact that in the case G\ — G2 any two Cartan involutions are conjugate under an inner automorphism of G (cf. [87], p. 155). • Lemma 11.12. Let G be the group (topologically) generated by the left translations of a strongly 2-divisible Bruck loop L and let Γ be the group generated by all reflections of the symmetric space determined by the core C(L) of L. A reflection σ ofC(L) induces on the group Γο = G of displacements ofC(L) an inner automorphism of G if and only if we have Γ = G = Tq.

11 Topological and differentiable Bruck loops

137

Proof. If σ induces the same automorphism as the conjugation by an element g e G then in the case Γ φ G the element ζ = ag φ 1 is contained in the centre of Γ. The automorphism ζ of the left distributive quasigroup C(L) leaves no element of C(L) fixed since Γ acts effectively on C(L). The reflection σ has precisely one fixed element a € C(L). But σ fixes also the element z(a) φ a. This contradiction shows Γ = G. Conversely, if a reflection σ is contained in the connected component Γο = G then clearly it induces on G an inner automorphism. • Proposition 11.13. Let G be a simple Lie group topologically generated by the left translations of a strongly 2-divisible locally compact Bruck loop L. The group G is equal to the group generated by all reflections of the core C{L) of L if and only if a maximal compact subgroup Κ of G contains an involution which is central in Κ. Proof First we assume that the connected subgroup Κ contains an involution σ which commutes with every element of Κ. Since Κ is a maximal subgroup of G (cf. [120], p. 614) the centralizer of σ in G coincides with Κ. The automorphism σ induces on the Lie algebra g of G the automorphism σ for which the (+l)-eigenspace is the Lie algebra 6 of the subgroup Κ. The decomposition g = 6 0 p, where ρ is the (—l)-eigenspace of σ , gives a Cartan decomposition. If G is equal to the group generated by all reflections of C(L) then σ is induced by an inner automorphism of G (cf. Lemma 11.12). Since σ is a Cartan involution, its centralizer in G is a maximal compact subgroup of G. • If G is a centre-free simply connected complex simple Lie group, which we consider as a real Lie group, then the Cartan involutions are not contained in G. Indeed, the maximal compact subgroups of G are algebraically simple. In contrast to this the Cartan involutions in simple centre-free Lie groups G of displacements of Hermitian symmetric spaces are contained in G, since according to [87], p. 161, the maximal compact subgroups Κ of G are not semisimple and hence Κ contains a torus subgroup. Let G be the connected component of the Lie group PSO n +i (R, / ) , where / is a symmetric bilinear form on M" +1 of index one. The maximal compact subgroups of G are isomorphic to the group SO n (R). If η is odd then SO„(R) is algebraically simple, if η is even then SO„ (R) has a central involution. Hence for even η the Cartan involutions of G are contained in G and for odd η they cannot lie in G. Proposition 11.14. Let L be a connected locally compact strongly 2-divisible Bruck loop such that the group G topologically generated by its left translations is a solvable Lie group. Then the group G* topologically generated by the reflections of the core C{L) is not connected, it contains Gas a subgroup of index 2 and G is simply connected {and hence homeomorphic to W1 for some η > 1).

11 Topological and differentiable Bruck loops

138

Proof. The stabilizer G \ of 1 e L in G contains a maximal compact subgroup Κ of G (Lemma 11.9), which is a torus. Let σ\ be the reflection on 1 in the core C ( L ) . Then σι induces an automorphism on G such that the centralizer of σι in G is G ι. The group G* is the semidirect product of G and ( σ ι ) and the commutator subgroup of G* is G (Corollary 10.3). The centre of G is homeomorphic to R " for some η > 0 (Theorem 11.2). Since any torus of G contained in the (nilpotent) commutator subgroup of G is central in G, the solvable group G is a semidirect product G = Ν χ K, where Ν is a characteristic subgroup of G homeomorphic to a euclidean space (cf. [50], 11.21 and 12.4, [109], p. 530) and Κ is a maximal torus in G. The factor group G*/Ν

is an abelian group

which is isomorphic to a direct product of Κ with a group of order 2. Since the group G* is generated by reflections of the core C ( L ) which form a conjugacy class of G*, the group G*/Ν

is generated by σ\ Ν and Κ = {1}.

•

Proposition 11.15. Let G be a centre-free solvable Lie group which is the group topologically

generated by the left translations of α Bruck loop L.

group topologically generated by the reflections of the core C(L).

Let G* be the

If Ζ φ {1} denotes

the centre of the commutator subgroup G' of G and G* is the stabilizer of the point 1 G C ( L ) in G*. then Ζ φ G] Γ) Ζ φ {1}. Proof. The group G* is the centralizer of the reflection σ\ in G*. According to [126], Satz 2.1.(χ), the group G* is equal to its normalizer in G*. Since G* contains no normal subgroup of G* we have Ζ φ G* Π Ζ . Since C(L) is a symmetric space the Lie algebra g of G has the decomposition g = m Θ [m, m], and the Lie algebra automorphism of g induced by σ\ fixes the elements of [m, m] and inverts the elements of m. Hence we have that [m, m] is the Lie algebra of the subgroup G* Π G = G\. Consequently G\ is contained in the nilpotent group G'. Hence Gι is nilpotent and Ζ centralizes G\. Assume G* Π Ζ = {1}. Then the involution σ\ centralizes no element φ 1 in Z . It follows that no reflection σχ (χ e L) can centralize an element φ 1 of Z . Hence any involution induces — id on Ζ and the product of any two involutions commutes with any element of Z . But then G as the group of displacements of C ( L ) would have Ζ as centre, which contradicts our assumption that G is centre-free. • Let G be a simply connected solvable Lie group and g be the Lie algebra of G. A Jordan-Holder

series 0 = go D gi D · · · D 0 P = 0 is a sequence of ideals of g

such that A d o acts irreducibly on g,7g/ + i, i = 0 , . . . , / ? — 1. Since G is solvable one has dim(g ( / g i + i ) < 2. If g? (/ = 0 , . . . , p) denotes the complexification of the Lie algebra g, then

has complex dimension one. A root ω : g — C of the Lie

algebra g is a linear form given by the eigenvalue of the action of g on g ^ i G { 0 , . . . , ρ — 1}. Clearly, if ω — α+ßi belongs to the same factor space linear forms a(x),

ß(x)

χ

for some

is a root then ω — a —ßi is also a root which

j. F o r * e g we have ω (χ) = a(x)+ß(x)i

with

: g —>· K, and the eigenvalues belonging to the 1-parameter

subgroups {exptx; t e R} on g-/g- + 1 are

t e

rj.

11 Topological and differentiable Bruck loops

139

Theorem 11.16. Let L be a connected locally compact strongly 2-divisible Bruck loop such that the group G topologically generated by its left translations is a solvable Lie group. Then G is exponential. Proof. Propositions 11.6 and 11.7 imply that L is a real analytic loop. According to Proposition 11.14 the connected Lie group G is simply connected. Let G* be the group topologically generated by the reflections of the core C(L). Then the factor group G*/G', where G' is the commutator subgroup of G, is generated by the cosets of oG', where σ is any reflection of C(L). Hence the group G*/G' is a semidirect product G/G' · ( a C ) , where aG' inverts any element of the abelian group G/G'. In particular the stabilizer Η of 1 e L in G is contained in G' since Η is centralized by the reflection σι on 1. The Lie algebra g of the group G has a decomposition g = m Θ [m, m], where m is the (—l)-eigenspace of the reflection (σι)* induced by σ\ and the subalgebra [m, m] is the (+l)-eigenspace of (σι)*. We have that Η = exp[m, m] is the stabilizer of 1 G L in G and exp m is the image of the section in G corresponding to the Bruck loop L. Assume that G is not exponential. According to [22], Theoreme 3, the group G is exponential if and only if for any root ω(χ) — a(x) + ß(x)i, χ e g, there exists e e l such that β = ca. This is equivalent to the relation Ker a C Ker β. Hence if G is not exponential then for a suitable root ω = a + iß there exists an χ € g such that a(;t) = 0 and β{χ) φ 0. In this case ω(x) = ß(x)i and ω(*) = —ß(x)i φ 0. Since the factor space Qci/Qci+l belonging to the roots ω and ω is the complexification of 0//0i+i and the operator adx has the different eigenvalues ß(x)i Φ —ß(x)i on 0//0Ϊ+1 we have dim(0;/0j + i) = 2 and the 1-parameter group S — (exp(ix); i e E ) induces on g,Vg, + i the compact group SC>2(K). The mappings Adg (g e G) leave 0, as well as 0, + i invariant and induce on the 2-dimensional factor space 0 ί ·/0 1+ ι similarities in the group SO2 (R) χ Μ. Let Ν be the connected component of the normal subgroup of G consisting of those elements Λ: such that Ad* is the identity on g( / g , + I . Clearly, the commutator subgroup G' is contained in N. Since Η c G' the group Ν contains H. It follows from the decomposition g = f) 0 m that G = Μ Ν where Μ = expm is the image of the section which corresponds to the Bruck loop L. Let gN be an element of G/N which induces an irrational rotation on 0;/0ι+ι· Since any element g • η (n e N) of gN induces on g, /g I + i the same rotation and gN contains elements of Μ we may assume that g e Μ. The element g is contained in precisely one 1 -parameter subgroup S of Μ which induces the rotation group S02(K) on g, /g,+i. Among the eigenvalues of Adg there are eand with irrational β e M. Since the corresponding complex eigenvectors are complex conjugate they span a 2-dimensional real subspace V c g,- C g. The subspace V is invariant under S and the group S induces on V the group SC>2(M). The involution σ\ of G* inverts S C M. Either σ\ leaves V invariant, or S • (σι) leaves the subspace V 0 σ\ (V) invariant. In the second case any element (χ, σι (*)) is fixed under σ\. In the first case the group S · {σι > acts on V in such a way

140

11 Topological and differentiable Bruck loops

that S induces on V the group of rotations and σ\ is a reflection on a 1-dimensional subspace. In both cases σ\ leaves a 1-dimensional subspace F = R · / pointwise fixed such that this subspace/7 is not invariant under S. Since e x p F = {exp tf\ t e K } C Ν we have St — ( e x p t f ) S ( t x p t f )~] c M. The 1-parameter subgroup St cannot coincide with S for all t φ 0. Indeed, if St = S for all t then exp tf normalizes S. In this case the set D, = {(exp tf)s (exp tf)~l s~l; s € 5} C S is a connected subset of the commutator subgroup G' C G. Since the set Ν Π S is discrete then for any t the set Dt consists only of the point 1. It follows that exp F and S generate a 2dimensional commutative group which is a contradiction since F is not invariant under S. Therefore there exists a to Φ 0 such that StQ φ S. Since the group S is isomorphic to R it contains a subgroup U isomorphic to Ζ such that U induces on V, respectively V φ a(V) the identity. But for this subgroup U C S we have that every element of U commutes with exp tof and we obtain St0 Π S = (exp tof)S(exp ί ο / ) - 1 Π S = U. Since for a strongly 2-divisible Bruck loop any element of Μ is contained in precisely one 1-parameter subgroup we have a contradiction. • In Section 10 we constructed to any strongly 2-divisible group G a Bruck loop L such that the group generated by the left translations of L is a factor group of G χ G (cf. Theorem 10.11). This factor group is not nilpotent if G is not nilpotent (Proposition 10.6). In particular, to any simply connected exponential Lie group G there exists an analytic Bruck loop L such that the group topologically generated by its left translations is an exponential solvable Lie group. A connected analytic Bruck loop which is not a group is at least 2-dimensional. We show that the construction given in Section 10 applied to the 2-dimensional non-commutative Lie group G leads to a proper Bruck loop. In Section 25 we prove that this Bruck loop is the only 2-dimensional Bruck loop having a solvable Lie group as the group topologically generated by its left translations. The non-commutative connected 2-dimensional Lie group G can be represented by the matrices

Since the group G has trivial centre the group topologically generated by the left translations of the Bruck loop associated to the group G is isomorphic to the subgroup Κ generated in G χ G by the set { ( ζ , ζ - 1 ) ; ζ e G}\ this subgroup Κ consists of the elements {(zk, ζ - 1 ) ; ζ e G,k e G'} where G' is the commutator subgroup of G. The group Κ consists of the pairs of matrices

An immediate computation shows that the map

11 Topological and differentiable Bruck loops

141

is an isomorphism between Κ and the solvable but not nilpotent group K* of matrices 1 0 0

b a 0

c 0 a~x

\ ; a > 0, (a, b, c) e R3

.

)

The group generated by the reflections of the core of G is / I K* by the group of order 2 given by the matrix σ\ — I 0 \ o

the semidirect product of 0 0 \ 0 1 I. 1 o )

Proposition 11.17. Let L be a strongly 2-divisible Bruck loop. The automorphism group Aut(L) of L is isomorphic to the stabilizer of a point q e C(L) in the automorphism group ofC(L). Proof The operation + on the core C(L) is given by χ + y = χ • y~lχ and hence any automorphism of L yields an automorphism of C(L) fixing the point 1 e L. Let (Q, + ) be a left distributive quasigroup isomorphic to the core C(L) of a strongly 2-divisible Bruck loop (L, o). Then the Bruck loop (Q, * ), a e Q, defined (a)

by χ * y = χR+(a)~l + yL+(a) is isomorphic to (L, o) (cf. Proposition 9.13) and (a) hence they have isomorphic automorphism groups. If a is an automorphism of (Q, + ) fixing a e Q, then a(x * y) = a(xR+(a)~l) (a) = a(x)R+(a)~l since for ζ =

+

a(y)L+(a)

+ a(y)L+(a)

= a(x) * a(y) (a)

we have a(x) = a(z + a) = a(z) + a

From this the assertion follows.

or

a(z) =

a(x)R+(a)~1. •

Corollary 11.18. Let L be a differentiable strongly 2-divisible Bruck loop. Then the group of continuous automorphisms of L is isomorphic to the stabilizer of a point in the full group of affine transformations of the symmetric space determined by the core of L. In particular any continuous automorphism of L is differentiable. Proof. Because of the previous proposition the group of continuous automorphisms of L is isomorphic to the stabilizer of a point in the group of continuous automorphisms of the core C(L) of L. The core C(L) is a differentiable symmetric space S in the sense of Loos and its group of continuous automorphisms is the full affine transformation group of C(L) with respect to the canonical connection of S. Hence it is a Lie transformation group acting differentiable on C(L). •

142

11 Topological and differentiable Bruck loops

Proposition 11.19. Let L be a connected almost differentiable left Α-loop satisfying the left inverse as well as the automorphic inverse property. Then L may be locally represented as a geodesic loop of a symmetric space with respect to its canonical connection. If L is strongly 2-divisible then L is an analytic Bruck loop. Proof Let G be the group topologically generated by the left translations of L and Η be the stabilizer of 1 e L in G. Then G is a Lie group (cf. Proposition 5.20). Let g be the Lie algebra of G and f) be the Lie algebra of Η. According to Proposition 9.16 there exists an automorphism τ of the Lie algebra g such that the (+l)-eigenspace of τ is the subalgebra f) and the (—l)-eigenspace is the tangent space m = T\ (a(G/H)) of the image of the section σ : G/Η —G corresponding to L. Hence we have ft), m] C m and [m, m] C f). It follows that the reductive homogeneous space on G/H associated with the left Α-loop L (cf. Proposition 5.20) is a symmetric space. According to Theorem 6.4 in [69] the loop L can be locally represented as the geodesic loop on a neighbourhood of Η in G/H with respect to the canonical connection of the symmetric space G/H. If L is strongly 2-divisible then the symmetric space G/H is a quasigroup (Remark 6.2 in [69]) and hence L is a strongly 2-divisible global analytic Bruck loop (cf. Proposition 9.24). •

Section 12

Bruck loops in algebraic groups

The natural notion for an algebraic quasigroup (Q, +) defined on an algebraic variety Q over an algebraically closed field Κ could be given by the requirements that the operations + , \ , / are algebraic morphisms Q χ Q —.> Q where Q χ Q carries the Zariski topology (cf. [57], p. 20). The main tool for our investigations of quasigroups and loops Q are suitable sections in the group generated by the left translations of Q. But for algebraic varieties Q over Κ it is in general not possible to endow the group of algebraic automorphisms of Q with the structure of an algebraic group. For symmetric quasigroups Q there is a possibility to avoid this difficulty. Namely if G is the group generated by the left translations of Q then the mapping χ ι-* Lx : Q —.• G determines isomorphisms from Q onto the quasigroup defined on the conjugacy class £ of the left translations of Q with respect to the multiplication La + L^ — LaLt>La. Hence we consider instead of algebraic symmetric quasigroups algebraic groups G over an algebraically closed field Κ such that G has a closed conjugacy class £ of involutions on which the operation a + β — αβα (α, β € £ ) yields a quasigroup. Since the isomorphism classes of strongly 2-divisible Bruck loops are classified by the isomorphism classes of symmetric quasigroups it is possible to obtain a classification of strongly 2-divisible Bruck loops related to algebraic groups. Let G be an algebraic group over an algebraically closed field Κ such that G is generated as an algebraic group by a class Q of conjugate involutions, where β is a connected subvariety of G. We denote by G° the connected component of G. We call G an algebraic reflection group with respect to the conjugacy class Q if G satisfies the following properties: (i) G is centre-free. (ii) The subvariety Q forms a set of representatives for the left cosets of the centralizer C c ( a ) of an involution a e Q in G. (iii) There is an element δ e Q such that the mapping χ separable algebraic morphism (cf. [57], p. 43).

χδχ \ Q ^

Q

ά

(iv) Let 7TA be the canonical projection of G onto the left cosets of CQ (A) in G. The projection na admits an algebraic section sa from the left factor space G / Cciot) to G such that sa(G/Cc(a)) = Q.

12 Bruck loops in algebraic groups

144

Proposition 12.1. Let G be an algebraic reflection group with respect to the conjugacy class Q of involutions over the algebraically closed field Κ. Then the multiplication (x, y) x+y = xyx defines on Q a symmetric quasigroup (Q, +) such thatfor each a G Q the maps L+(a) : χ l·-» a + χ and R+(a) : χ ι—> χ + a are automorphisms of the algebraic variety Q. Moreover, the group G is the group algebraically generated by the left translations of(Q,+). Proof The assertion follows from the second part of Theorem 2.10 in [27], p. 262, since for any a e Q the mapping χ Μ- χ' with χ + χ' = a can be expressed by χ' = χ + a = R+(a)x. • Definition 12.2. A symmetric quasigroup (Q, + ) is called semi-algebraic if Q is an algebraic variety, and if for every a e β , the mappings R+(a)

: χ ι-»· χ + a ,

L+(a)

: χ M>- a + χ

are algebraic automorphisms of the variety Q. A strongly 2-divisible Bol loop (L, o) is called semi-algebraic, if L is an algebraic variety and, for every a 6 L, the left translation λα, the right translation ρα and the mappings χ ι-> x2 : L L as well as χ ι-> x~l : L —> L are algebraic automorphisms of the variety L.

Proposition 12.3. (i) If (Q, +) is a semi-algebraic symmetric quasigroup then for every a e Q the Bruck loop (Q, * ) defined on Q by (a)

x *

y

= xR+(a)~l

+

yL+(a)

(a) is

semi-algebraic. (ii )If(L, o) is a strongly 2-divisible semi-algebraic Bol loop then its core (C(L), + ) is a semi-algebraic symmetric quasigroup. (iii) The isomorphism classes in the category of semi-algebraic strongly 2-divisible Bruck loops are classified by the isomorphism classes in the category of semi-algebraic symmetric quasigroups. Proof

(i) One obtains immediately that the left and right translations of (Q, * ) are (a)

algebraic automorphisms of the variety Q. Putting XÄ + (Ö) _ 1 = ζ we have χ * χ — Z + (a + {z+a)) (a)

= (z + a) + [z + (z + a)] = (z + a)+a

= x+a

=

R+(a)x

12 Bruck loops in algebraic groups and hence the mapping χ

145

χ * jc is an algebraic automorphism. The inverse element (a)

χ - 1 with respect to the multiplication χ * y satisfies (a)

a

=

χ

*

=

j c / ?

+

( a )

_ 1

+

( a

+

x

-

1

)

(«)

and hence λ ; - 1 — a + [ ; t / ? + ( a ) _ 1 + a ] — a + χ = L ( a ) x . Consequently jc i-»· x ~ is an algebraic automorphism of Q. (ii) The core operation in CiL) is given by χ + y = Since the left translations, the right translations and the inversion are algebraic automorphisms of L the left translations of C(L) are algebraic automorphisms. Since the mapping jc i—^ jc2 is an algebraic automorphism of the variety L the mapping R+( 1) is an algebraic automorphism. We have +

R

+

( a ) χ

=

L

=

R

+

+

( R

+

( \ ) ~

( \ y l

a

l

+

a ) R

+

[ ( R

+

( \ ) L +

l

( \ ) ~

+

( R

a

+

l

( l ) ~

x )

l

a ) x

l} = * + ( * + ( 1 ) - 1 α + l) =

+

+

χ

a ,

and hence for any a e C(L) the right translation R+(a) is an algebraic automorphism. (iii) The core of a semi-algebraic Bruck loop L i s a semi-algebraic symmetric quasigroup (Q, + ) and the semi-algebraic Bruck loop (Q, * ) is algebraically isomorphic to L by the algebraic automorphism χ ι->· χ2 since one has (x • y)2 = x2 * y2 = χ · (y 2 ·x) (cf. proof of Proposition 9.12). The semi-algebraic Bruck loops (Q, * ) and (Q, * ) (1)

(a)

(a e Q) are isomorphic by the algebraic automorphism λ* : χ μ* a * λ: (cf. Proposition 9.9 (iv)).

•

Corollary 12.4. a n y

c

e

L

the

L e t

m a p

χ

L

b e

a

s t r o n g l y

ι—» j c · c x

i s an

2 - d i v i s i b l e a l g e b r a i c

s e m i - a l g e b r a i c

a u t o m o r p h i s m

o f

B o l

l o o p .

T h e n

f o r

L .

The core (C(L), + ) of L is a semi-algebraic symmetric quasigroup, hence any right translation R+(a) is an algebraic automorphism for any a e L. Since x + a = j c - a _ 1 x t h e assertion follows. • P r o o f

Lemma 12.5. l e t a.

b e

an

o f G

is a b e l i a n ,

L e t

e l e m e n t then

G

b e

a

g r o u p g e n e r a t e d

o f G .

A s s u m e

C

G

G ' ( a )

a n d

=

Π

b y

C g ( o c ) a

l e a v e s

the

c l a s s

=

{«}. in

G'

G

o f c o n j u g a t e

I f the no

i n v o l u t i o n s

c o m m u t a t o r

e l e m e n t

s u b g r o u p

a n d G'

fixed.

As the group G is generated by the class C of involutions the factor group G / G ' is cyclic of order two and G — G ' ( a ) . Since the group G' is commutative the set Κ — { x ~ l a x a \ χ e G'Jisagroup. W e h a v e a ( x _ 1 a x o ; ) a = — x{ax~xa) € Κ and hence a normalizes K, and the group Κ (a) is a semidirect product of AT by a . But Κ ( a ) contains any generator x ~ a x = ( x ~ a x a ) a and also any generator of the form ( a x ) ~ a { a x ) = x ~ a x , χ e G ' . If (χ -1 οτ.χαΟ α = α ( χ ~ ^ α χ α ) α = x ~ a x a ,

P r o o f .

l

x

l

l

l

146

12 Bruck loops in algebraic groups

then the element χ xax of Q would be contained in Cc(a) hypothesis G Π Cc(or) = {«}.

which contradicts the •

Lemma 12.6. Let G be an algebraic reflection group with respect to the conjugacy class Q of involutions. If a e Q, then G = Gf(a) and the commutator subgroup G' coincides with the connected component G° of G. Proof From (2.1) in [28] it follows that the commutator subgroup G' is connected and that G = G'(a). Since \G/G'\ < 2, we have G' = G° ([57], 7.3). • The next assertion follows immediately from the main theorem in [28], Theorem 2.11: Lemma 12.7. Any algebraic reflection group with respect to a conjugacy class of involutions is solvable. Theorem 12.8. If G is an algebraic reflection group with respect to a conjugacy class Q over the field Κ then the characteristic of Κ must be different from 2. Proof. We know from the previous lemmas that G = G'{a) for any a U · a is separable if and only if the mapping u (->- u2 : U U is separable. Since the characteristic of Κ is two, this mapping is a Frobenius automorphism and hence it is not an algebraic automorphism (cf. [57], p. 35), which is a contradiction. It follows from this that D is an abelian variety. Then the subgroup D(a) of G, where the mapping χ h-> xax : Q —> β for a Ε Q is separable, is an algebraic reflection group for the same reason as before the group U(a). Now, the mapping ma i-»· (ma)a(ma) = m2a : Da —> D • a is 2 separable if and only if φ : m i-> m : D —> D is separable. Since the characteristic of the field Κ is 2, the bijective morphism φ of D onto itself induces a morphism φ* of the field Κ (D) of rational functions of D into itself such that f*(K (£>)) C Κ (D)2 ([96], p. 63). Then K{D) is an inseparable extension of K(D)2 of degree 2g, where g is the dimension of D. Therefore φ* is not an automorphism of K(D) and the mapping φ is not birational ([57], p. 20) and hence not an algebraic automorphism. This contradiction proves that D = {1} and G' = L is an affine algebraic group. Such a group cannot be the connected component of an algebraic reflection group as we have seen in the first part of the proof. • Corollary 12.9. If G is an algebraic reflection group with respect to a conjugacy class Q over an algebraically closed field Κ then G is an affine algebraic group and the characteristic of Κ is different from 2. Proof From the previous theorem it follows that the characteristic of Κ is different from 2. The group G is affine if and only if its connected component G°, which is the commutator subgroup of G, is affine. If G° is a non-affine algebraic group then G° allows a factorization G° = LD, where L is a maximal normal connected affine subgroup of G° and D φ {1} is a central subgroup of G° containing involutions (cf. [114], p. 439 and Corollary 5). The group G is centre-free, hence any involution a G Q acts on D without fixed elements. But then a induces on D the mapping χ h^· jc" 1 and fixes any involution of D. This contradiction proves the assertion. • Theorem 12.10. If G is an algebraic reflection group with respect to a conjugacy class Q over an algebraically closed field Κ, then G = G'(a), a e Q, is an affine group such that the commutator subgroup G' ofG is a connected unipotent group and the field Κ has characteristic Φ 2. Proof From the Lemmas 12.6 and 12.7 together with Theorem 12.8 and Corollary 12.9 it follows that G = G'{a), a e Q, is an affine group such that the commutator

148

12 Bruck loops in algebraic groups

subgroup G' φ G is a connected solvable affine group and that the characteristic of the field Κ is different from 2. The group G' is a semidirect product G'u χ Τ, where G'u is the unipotent radical of G' and Γ is a maximal torus of G'. Since G'u is a characteristic subgroup of G' the group G'u is a normal subgroup of G. According to [28], Theorem 2.3, p. 234, the factor group G/G'u is generated by the conjugacy class Q* = {aG'u\ a € Q} of involutions, which is a closed subvariety of G / G ' u , and which forms a system of representatives for the left cosets of the centralizer CG/G'u ( a G ' u ) of ocG'u in the group G/G'u. Because of Theorem 2.4 in [28], p. 235, the factor group ( G / G ' u ) / Z , where Ζ is the centre of the group G/Gfu, has only trivial centre. But G/G'u is isomorphic to a semidirect product S = Τ χ (a*), where Γ is a maximal torus of G' and α* is an involutory automorphism of Τ induced by α e Q. The factor group S/Z(S), where Z(S) is the centre of S, is again a semidirect product Τ χ (α*), where Τ as a homomorphic image of a torus is again a torus and a * is an involutory automorphism of Τ induced by a*. Since the group Τ χ (a*) is centre-free the automorphism a * must act on Τ without fixed points. The maximal elementary abelian 2-subgroup A of Τ has the order 2r, where r = dim Τ, and it is invariant under a*. But a * cannot act on A without fixed points which contradicts the fact that the group Τ χ (α?*) is centre-free. • A sufficiently large class of examples of semi-algebraic symmetric quasigroups such that the corresponding semi-algebraic Bruck loops are not commutative can be obtained, using the results of Section 10, in the following way: Let G be a connected unipotent algebraic group over the algebraically closed field Κ of characteristic φ 2, the nilpotency class of which is greater then two. According to Proposition 10.16 G is strongly 2-divisible. Let Μ be the subvariety Μ = {(ζ, ζ - 1 ) ; ζ e G} of the direct product G χ G and let α be the involutory algebraic automorphism of G χ G defined by (x, y)a — (y, x). Clearly a inverts any element of M . Let Τ be the group algebraically generated by Μ and a . Then Μ α is a conjugacy class in Τ and the operation (.χ, x~])a

+ ( j , y - 1 ) « = (je, Jt - 1 )a(;y, )> _1 )α:(χ, x~l)a

= (jc_y_1x, jc _1 _yx _1 )ö!

defines a symmetric quasigroup on Μ α such that the group algebraically generated by the left translations of (Μα, + ) is algebraically isomorphic to the group T/Z(T), where Z(T) is the centre of Τ (cf. Propositions 10.6, 10.8, 10.9 and 10.10). Clearly, the left translations are algebraic automorphisms of the variety Μ α . It follows from Proposition 10.16 and from the proof of (ii) of Proposition 12.3 that the right translations are algebraic automorphisms. Hence (Μα, + ) is a semi-algebraic symmetric quasigroup. The algebraic group Γ is a semidirect product Γ = Κ (a) with a 2 = l a n d the group Κ consists of the elements (zk, z - 1 ) for any z e G and k e G' (cf. Propositions 10.6 and 10.8). Hence on the one hand the group Κ is as an algebraic variety the direct product of the variety Μ = {(ζ, ζ - 1 ) ; ζ e G} and of the group G'. On the other

12 Bruck loops in algebraic groups

149

hand the group Κ is as an algebraic variety also the direct product of Μ and of the centralizer F — {(.χ,χ); χ e G'} of a in K, since for any element of Κ we have (zk, z - 1 ) = ( y / z k z , Vzkz )(Vzkz · z~l, Vzkz • z - 1 ) , where vGfis the unique square root of A: in G', and(z, z _ 1 ) ( J C , X ) = {x~^z(z~xxzx), withz _ 1 xzjc e G'. Hence the algebraic variety Τ = Κ {a) has a decomposition as algebraic variety into the direct product Τ = Μ α χ F(a). It follows from [27], Theorem 2.10, p. 262, that T/Z(T) is an algebraic reflection group with respect to the conjugacy class Μ α .

S e c t i o n

1 3

Core-related Bol loops

Any Bol loop L which has the Lie group G as the group topologically generated by its left translations defines a symmetric space on the image {λ^; χ e L} of the section χ λχ : L = G/Ge —> G, where Ge is the stabilizer of the unit e e L in G. The Lie triple system of this symmetric space is determined by the tangent space of the image of the section in the Lie algebra of the group G. In this section we describe how the various groups G can differ which yield the same symmetric space and hence the same Lie triple system. This means that we investigate Lie groups topologically generated by the left translations of differentiable Bol loops L which have the same core C(L). According to Section 6 the differentiable local Bol loops L are in one-to-one correspondence with the triples (β, f), m), where β is a Lie algebra, f) is a subalgebra of β containing no ideals φ {0} of β and m is a subspace of β generating β such that β = f) θ m and for every ξ, η, ζ e m one has [[£, η], ζ] e m. Such triples (ß, f), m) we call Bol triples, since β is the Lie algebra of the local Lie group topologically generated by the left translations of the local Bol loop corresponding to the local section σ : exp β / e x p f) —> expß the image of which is expm (cf. Theorem 6.15). There is a mapping C : L C(L) assigning to every local Bol loop its local core C(L) defined on exp m. Since C(L) on exp m is a locally symmetric space and the Lie triple system of C(L) is given on m by (Χ, Υ, Ζ) — [[X, F], Z] we have the following commutative diagram: L

— C ( L ) Ψ

ψ

(β, f),m) T(C)

• v(m, L[ [ . , . ] , . ]J/)

where

g is the Lie algebra isomorphism induced by the group isomorphism a. (ii) The assertion follows from Corollary 7.5 and from Corollary 1.12 (v). • From Corollaries 6.18, 6.19 and 6.22 the following claim follows immediately. Remark 13.3. Let (g,, fy,m(·) (i = 1,2) be core-related Bol triples such that gi satisfied one of the following conditions:

152

13 Core-related Bol loops

(1) gi has trivial centre; (2) the commutator subalgebra of gi is semisimple; (3) gi = [mi, mi] φ mi and [mi, mi] intersects the centre of gi trivially. If g2 satisfies also one of these three conditions then gi = g2 is isomorphic to the Lie algebra of the group of displacements of the symmetric space corresponding to m,· (/ = 1,2). Now we want to give examples for core-related triples belonging to non-isotopic local Bol loops. Theorem 13.4. A triple (g, \), m) with non-abelian Lie algebra g of dimension η belonging to a local Bol loop is core-related to the triple (Mm, 0, R m ), where m = dim m, if and only //(g, f), m) satisfies the following conditions: (i) g is nilpotent of class 2 and m > (ii) m contains an abelian subalgebra α ofgwith dim α = n—m such that αΠ^ = {0}, where 3 is the centre of q; (iii) the subalgebra f) has the form {((*), x); χ e b}, where b is a central subalgebra of g complementary to m and φ : b —»· 0 is a linear isomorphism. Proof The triple (g, i), m) with non-abelian Lie algebra g and dimm — m is corerelated to the triple (R m , 0, Rm) only if g is a nilpotent Lie algebra of class 2 (cf. [102], Theorems 3.8 and 3.9). Since g is nilpotent of class 2 and is generated by m as a Lie algebra there exists a subalgebra b of the centre 3 of g such that g is the direct sum g = m Θ b. The subalgebra f) of the triple contains no non-trivial ideal of g; hence fj η b = {0}. Moreover f) Π 3 = {0} and therefore f) is abelian. Since g = m φ f) there exists a Lie algebra isomorphism φ : b m such that f) = {( 3) having a 2-dimensionaI centre generated by the linearly independent vectors X\,..., Xn+2 such that [Xi,Xj] = k,jXn+\ + /x,yX„+2, where not all λ ι ; and μ, 7 are zero (i,j = 1 , . . . , η), but [X(·, X„+i] = [X,, Xn+2] = 0 as well as [X„+i, X„+i] = 0. If tn = ( X i , . . . , Xn) and if we can choose in m the commuting elements Υ, Ζ not belonging to the centre of g then (g, [), m), where f) = R(Y + X n +i) + M(Z + Xn+i), is again a triple core-related to the triple (R n , 0, R"). We notice according to Theorems 3.9 and 3.10 in [102] that the local Bol loops corresponding to these examples cannot be extended to global differentiable Bol loops. Let G be a (local) group. The group of displacements of the (locally) symmetric space on the core C(G) of G is a group G* having an involutory automorphism a such that the set G has an embedding ί : G —> G* onto the set La = {*σ(*) _ 1 ; χ e G*} generating G* and having the property i(x) • i(y)~li(x) = t(jc^ _1 jc) for all x,y € G. In this case t is an isomorphism of the (local) core C{G) onto La. The mapping σ : (u, v) m>- (v, u) : G χ G —> G χ G is an involutory automorphism of (G Χ G ) / A Z , where Δ Ζ = {(Z, Z); ζ 6 Z ( G ) } and Z ( G ) is the centre of G . Then we have La = {(Μ, Μ _ 1 ) Δ Ζ ; u G G } . The group G * of displacements of the symmetric space on G is a subgroup of (G Χ G)/AZ generated by La. If G is a local Lie group then the mapping χ x2 : G —»· G is a local diffeomorphism for suitable neighbourhoods of 1 and the (geodesic) local Bruck loop Β corresponding to the symmetric space La is core-related to the local group G. The group topologically generated by the left translations of Β is G* which is not isomorphic to G if G is non-abelian. If for instance G is a quasi-simple Lie group then the group G* of displacements is isomorphic to G Χ G / Δ Ζ since the only connected non-trivial normal subgroups of G Χ G are {1} Χ G and G Χ {1}. If G is a locally compact local topological group then the same fact holds if we restrict ourselves to compact neighbourhoods of G. In case of a discrete loop we have to assume that χ ι-»· χ 2 : G —» G is a bijection in order to obtain the same result. The above examples show that there exist core-related Bol triples (g, f), m) and (g', [)', m') with non-isomorphic g and g'. This is a motivation for the following Definition 13.6. The Bol triples (g, f), m) and (g\ Ϊ)', m') are called strongly corerelated if m' = m and hence g' = g. Remark 13.7. Let (g, f), m) be a triple corresponding to a local Bol loop and let t)' be any subalgebra containing no ideal φ 0 such that g as a vector space is the direct sum g = m ® ty. Then (g, f)\ m) is a triple strongly core-related to (g, f), m). This characterization is not very effective, since it is difficult to determine all subalgebras which are complementary to m. Hence we study some examples.

154

13 Core-related Bol loops

Let ο be a Lie algebra and a an outer automorphism of the commutator subalgebra ο' = [ο, a] such that a(x) φ —χ for all 0 φ χ e a. (This is in particular the case if a has finite odd order.) Let fj and f)' be the subalgebras of α θ α given by f) = {(u>, w), w € a'} and ()' = {(u\ a(u;)), w e a'}. Let m be the subspace m = {(*, —jc), jc g a} in α Θ a. Clearly, [m, m] = f), [I), m] C m and m i l t ] = {0}. Moreover f)' Π m = {0} since α Ο ) φ —χ for all * e α \ {0}. Furthermore m φ tf c m φ f), since (x, - * ) + {w, a ( w ) ) — (x, —x) + ^ (w — a(w), —w -f a(w)) -I- ^ (w 4- a(w), w + a(w)) for any jc g α and w € a'. The linear mapping w and hence g = m ® f ) = m ® l ) ' .

e m φ f)

w + a(w) : a' —> o' is bijective

Proposition 13.8. The local Bol loops belonging to the triples (α φ a, (), m) and (ο φ a, I)', m), where I), f)' are subalgebras and m is a subspace constructed above in the Lie algebra α φ a, are not isotopic. The local Bol loop belonging to (α Φ a, I), m) is α Bruck loop, but the local Bol loop belonging to (α φ a, f)', m) is not α Bruck loop. Proof. The local Bol loops corresponding to the triples (α φ a, f), m) and ( ο φ α , f)\ m) are isomorphic if and only if there exists an automorphism β of α φ α with ß(m) — m and β(ί)) = f)'. But ß(i)) = /3([m, m]) = [ß(m), ß(m)] = [) which is a contradiction. If these local Bol loops were isotopic then it would exist an inner automorphism Ad( gN g 2 ) : α φ α -»• a © a w i t h Ad( g l i g 2 ) f) = f)' or, equivalently, for any (w, w) e i) we would have (Adg, w, Ad g 2 w) = (Ad gl w, a o A d g l w). H e n c e a = Ad g2 o ( A d g l ) _ 1 . This is a contradiction since a is an outer automorphism of a'. The local Bol loop belonging to the triple (α φ α, Ϊ), m) is a Bruck loop since [jc, _y]m = 0 for any x, y e m (cf. Proposition 9.24). If the local loop belonging to ( α φ α , f)', m) would be a Bruck loop then [x, = 0 for any x, y e m, where [x, denotes the projection of [x, y] onto m along (]'. The mapping w i-» w + a(w) of a' onto a! is bijective. Therefore any element of [m, m] can be written uniquely as (w + a{w), w+a(w)) = (a(w) — w, w — a(w)) +2(w, a(w)) with (a(w) — w, w — a (w)) e m and 2 ( w , a(w)) e Since α φ id there exists w e a' such that its projection onto m along t)' is different from 0 and the proposition is proved. • If we take in the previous proposition for the Lie algebra α the real or complex Lie algebra of type D4 (sog(R) or sos(C)) and for a an outer automorphism of order 3 (cf. [31], p. 169) of o, then we see that also for suitable semisimple Lie algebras 9 = α φ α there exist strongly core-related triples ( α φ α , f), m) and ( α φ α , f)', m) such that the corresponding local Bol loops are not isotopic. There exist also in the case of a compact simple Lie algebra g strongly core-related triples (fl, i)i, m) and (g, (72, nt) such that the corresponding local Bol loops are not isotopic.

155

13 Core-related Bol loops

Example 13.9. Let g = sos(R) be the Lie algebra of the group SOg(M). This group is the group topologically generated by the left translations of the Moufang loop L\ of Cayley numbers of norm 1 (cf. [120], Lemma 11.22) as well as the group of displacements of the symmetric space on the sphere S1. Hence SOs(M) is also the group topologically generated by the left translations of the local geodesic loop Li of S1. The local loops L ι and Lj are not isotopic since the Bruck loop L2 is not a Moufang loop. Since on the manifold S 7 there is only one symmetric space structure the local loops L\ and L2 belong to strongly core-related triples (g, f)i, m) and (g, f)2, m). Example 13.10. Let be g = so8(M) and f) = so 5 (K) θ 303(E). The subalgebra f) of g is centralized by the involution Addiag( 1,1,1,-1,-1,-1,-1,-1)» which is an outer automorphism of sog(M). In the Lie subalgebra sos(lR) there are 3 conjugacy classes of subalgebras isomorphic to f) under Adso8(K) (cf· [26], Chapter 6, Theorem 6.3). If r is an outer automorphism of sog (M) of order 3 then the subalgebras f) 1 = [), [)2 = r ([)) and f)3 = r 2 (f)> are representatives of the 3 conjugacy classes. In the projective linear group PSOg(IR) the group H2 = PSp(2) χ PSp(4) can be chosen as a representative of the group exp r(f)) (cf. [26], Chapter 1, Theorem 1.4). The group PSp(2) χ PSp(4) is locally isomorphic to the direct product of the linear group Sp(4) of the 2-dimensional quaternionic right vector space preserving the norm yV(^^j) = xx + yy and of the group Sp(2) of multiplications with scalars of norm one. Let (1, i, j, k) be the natural basis of the quaternion field H. Then the mapping

1

/

0 0

J

(

/ 1 0 0 0 \ 0 1 0 0 0 0 1 0 0 0 0 1 0 0

1

/

\

0

0

h^·

I

1 ,

^

-1 0 0 0 -1 0 0 V 0 can be extended to an isomorphism

/ φ : (x\, X2, χτ,, X4)

V

k 1—>

)

1 0 ο Ν 0 0 0 0 0 -1 0 1 0 /

0 -1 0 0 0

0 0 \ -1

0

0

0 1 0

-1 0 0

X4

Xl

X3 —X4

X4

Χι

-X2

-X3

X2

Xl

XI

X2

-X2 -X3 -X4

X3

1

0 0 0 /

^

:Η

onto the algebra EI of matrices of the form / e = V

Xl

x

2

X3

X4

-XI

Xl

-X4

X3

-X3 —X4

X4

Xl

-X2

-X3

X2

Xl

\ (X\,X2,X3,xa) /

\

e

EI

156

13 Core-related Bol loops

Then we have (x\,x2,xi,x4) (u, v) • (x, y) and c is a companion of the right pseudo-automorphism a(y). Hence the group Ρ contains the group S generated by the maps P(xj), x € L, as a normal subgroup, and consequently the group Ρ is the semidirect product of S by the group (1, G). The action of the subgroup (1, G) on S is determined by the homomorphism σ. Clearly, if a ( G ) = {1} then we obtain the direct product L χ G. If L is a differentiable connected right Bol loop and G is a connected Lie group then the group topologically generated by the right translations of any pseudo-direct product L χ σ G is a connected Lie group (cf. Theorem 7.3). There are many pseudodirect products L χσ G which are differentiable connected loops since the closure of the group Π of all continuous pseudo-automorphisms of L in the Lie group of semi-automorphisms of L is a Lie group (cf. Theorem 2.9 in [102]).

167

14.1 Pseudo-direct products

For a loop L satisfying the right inverse property a pseudo-direct product L χσ G does not necessarily have the right inverse property. Theorem 14.1. Let L be a loop with the right inverse property. A pseudo-direct product L χσ G satisfies the right inverse property if and only ifa(G) is a group of automorphisms of L. Moreover L χσ G is a right Β ol loop if and only if L ando(G) satisfy the identity [(ζχ)>]χ σ ( ί ) = z[(xy)xa(t)]

for all x,y,zeL

andt e G.

Proof Denoting the right translation in L χσ G by P(x,y) we obtain P(x>)3 is contained in the nucleus of L. Proof The mappings T{y) : χ : L —• L, y € L, are pseudo-3 automorphisms with companion v (Lemma 1.2). A pseudo-automorphism is an automorphism if and only if its companion is contained in the nucleus. • Since the nucleus of a Moufang loop is a normal subgroup (Lemma 1.8 (a)), we obtain the following

14 Products and loops as sections in compact Lie groups

170

Observation 14.7. The crossed product L χ ( σ ) Μ for a simple Moufang loop L has the right inverse property if and only if the subloop σ ( Μ ) C L has exponent 3. Proposition 14.8. Let L be a topological Moufang loop defined on a connected topological manifold. If the image of the mapping γ : χ x3 : L ^ L is contained in the nucleus Ν of L then L is a Lie group. Proof The connected component NQ of the nucleus Ν is a normal subgroup of L; it is a connected Lie group ([92], p. 184). In the connected factor loop L* = L/NQ, which is a topological manifold, each element Φ 1 has the order 3 since Y(L) c NQ. If L* = L/NO were not the identity then the set A of left translations of L* were a connected set in the homeomorphism group of L* with respect to the compact-open topology. Each homeomorphism of L* extends uniquely to a homeomorphism of the one-point compactification L* = L* U {oo} and the compact-open topology of the homeomorphism group of L* is induced by the topology of the uniform convergence of the homeomorphism group of L* (cf. [51], p. 127). Let s(t) be a continuous curve on A with 5(0) = id e A. Since l i m ^ o ^ O = id with respect to the topology of uniform convergence in the homeomorphism group of L* the images s(t)x converge uniformly to x. Since every element of A is periodic we obtain a contradiction to Newman's theorem in the version of P. A. Smith [124], • Corollary 14.9. If a topological commutative Moufang loop L is defined on a connected topological manifold then L is an abelian Lie group. Proof. The assertion follows from the previous proposition since in a commutative Moufang loop the mapping χ x 3 is a centralizing endomorphism (cf. Lemma 1.8 (b)).D Proposition 14.10. Let L be a topological Moufang loop such that the nucleus of L is trivial. If Μ is a connected topological loop and σ : Μ —» L is a continuous homomorphism such that σ(Μ) is a topological manifold of dimension > 0 then the crossed product L Χ(σ) Μ does not have the right inverse property. Proof Assume that L Χ( σ ) Μ has the right inverse property. From Corollary 14.6 it follows that every element φ 1 of σ(Μ) has the order 3. Since σ{Μ) is a connected Moufang loop, it follows from Proposition 14.8 that σ(Μ) is a Lie group in which χ 3 = 1 for any χ e σ ( Μ ) . Hence σ ( Μ ) — {1} which is a contradiction. •

14.3

Non-classical differentiable sections in compact Lie groups

Every compact connected differentiablp Moufang loop is isomorphic to a factor loop [Κ χ (S 7 )*]/C, where Κ is a compact connected Lie group, S 7 is the multiplicative Moufang loop of Cayley numbers of norm 1 and C is a central subgroup of the direct product KxS7x---xS7 = Kx ( S 7 ) k (cf. [51], p. 226, Theorem 9.3.13). Since

14.3 Non-classical differentiable sections in compact Lie groups

171

the group topologically generated by all left and right translations of the loop S1 is the group SOg(lR) (cf. [120], Lemma 11.22, p. 16) the group topologically generated by all left and right translations of the loop Κ χ (S 7 ) k is compact. Hence for any compact connected differentiable Moufang loop L the group topologically generated by all left and right translations is a compact connected Lie group. The group topologically generated by the right translations of the pseudo-direct product L χσ G for a compact Lie group G is compact, since it is an extension of a compact Lie group by a compact Lie group. The group topologically generated by the right translations of the crossed product L Χ(σ) Μ, where L and Μ are topological loops and σ is an open continuous homomorphism, is a closed subgroup of the direct product M(L) χ Ρ (Μ), where M(L) is the group topologically generated by all right and left translations of L and P(M) is the group topologically generated by all right translations of M. These considerations give us the following Theorem 14.11. Let L and Μ be connected compact differentiable Moufang loops and let G be a connected compact subgroup of the Lie group Π of all continuous pseudo-automorphisms of L. Then the group Ρ topologically generated by the right translations of any pseudo-direct product L χσ G as well as of any crossed product L χ (σ) Μ is a compact connected Lie group. Now we determine for a compact connected differentiable Moufang loop L the connected component of the group of continuous pseudo-automorphisms and show that there are many possibilities for the construction of pseudo-direct products L x a G with a(G) φ {1} for which the group Ρ is a compact Lie group. But then also the group topologically generated by all left and right translations is a compact Lie group. Proposition 14.12. Let Lbea Moufang loop such that the 3rd powers of the elements of L generate L. Then the group Π of pseudo-automorphisms of L is the product Τ (L) · Aut(L) of the group Τ (L) generated by the maps Τ (*) : y x~lyx : L —L, χ G L, and of the automorphism group Aut(L) of L; the group T(L) is a normal subgroup of Π. Proof According to Lemma 1.2, all elements of L are companions of pseudoautomorphisms belonging to T(L). Let θ be an arbitrary pseudo-automorphism of L with a companion a e L and let φ be a pseudo-automorphism in T(L) having ( a 0 ) - 1 as a companion. It follows from [19], p. 62, that θφ is an automorphism a since it has αθ(αθ)~[ = 1 as a companion. Hence we have θ = αφ~ι. Moreover the group T(L) is normalized by the group Aut(L) since for any generator T(x) we have a~lT(x)a = T(xa). • Applying this proposition to connected differentiable Moufang loops we obtain the following

172

14 Products and loops as sections in compact Lie groups

Theorem 14.13. The group Π of pseudo-automorphisms of a connected differentiate Moufanß loop L is the product T(L) · Aut(L), and T(L) is a normal subgroup ο/Π. If Π denotes the closure of the group of continuous pseudo-automorphisms ofL in the group of homeomorphisms ofL then Π is the product T(L) · Aut(L), where T(L) is the closure of T(L) in Π and Aut(L) is the group of continuous automorphisms ofL. Proof. The subloop L 3 of L generated by the 3rd powers of the elements of L is a characteristic normal subloop of L (cf. [ 19], Theorem 2.1, p. 63). The factor loop Ljl? is a differentiable connected Moufang loop of exponent 3. It follows that L — L? (cf. Proposition 14.8). • In the Bol algebra (m, ( . , . , . ) , [., .]) of a differentiable Moufang loop one has the following identities: (i) (X, Y, Z) =

Y], Z] + [[Y, Z], X] + [[Z, X], Y]},

(ii) [[Χ, Υ], [Χ, Ζ]] = [[[Χ, Υ], Ζ], X] + [[[Y, Z], X], X] + [[[Z, X], X], Y]. Any vector space with a bilinear skew-symmetric multiplication [ . , . ] satifying (ii) is called a Malcev algebra (cf. [51], Definition IX.6.26). Theorem 14.14. The connected component Πο of the topological pseudo-automorphism group Π of a compact connected differentiable Moufang loop is a compact Lie group. Proof. The .group Π, according to the previous theorem, is the product Τ (L)Aul(L). The group T(L) is connected and hence it is a normal subgroup of Πο· As a closed subgroup of the group generatedjjy all left and right translations, T(L) is a compact Liej*roup. Clearly Πο = T(L)Aut(L)o, where Aut(L)o is the connected component of Aut(L). The connected component of the centre of L is a torus group. Since it is invariant under the automorphism group of L and has a totally disconnected automorphism group the group Aut(L)o induces the identity map on the centre. The group Aut(L)o induces on the Malcev algebra of L (cf. [51], pp. 250-251) an automorphism group Aut(L)* which is isomorphic to Aut(L)o- The Malcev algebra of L is the direct product of a compact Lie algebra g and of a Malcev algebra m which is the direct product of Malcev algebras belonging to the Moufang loop S1. The group Aut(L)* is isomorphic to a connected group A of automorphisms of the simply connected Moufang loop exp(g χ m) which is the universal covering of L (cf. [100], Theorem 3.4, p. 223). The simply connected Moufang loop exp(g χ m) is isomorphic to R m x i : x (S7)k, where Κ is a semisimple compact Lie group and M'n is the centre of exp(g χ m). The connected component of the automorphism group of R m χ Κ χ (S 7 ) k leaves R m , Κ and each factor S 7 invariant and hence it is locally isomorphic to the direct product SL m (R) χ Κ χ (Ö2) k , where G2 is the compact 14-dimensional exceptional Lie group. Since the group Aut(L)* is a group

14.4 Differentiable local Bol loops as local sections in compact Lie groups

173

of automorphisms of a compact connected Moufang loop it induces the identity map on the centre R m of exp(g χ m). Hence the group Aut(L)* is isomorphic to a factor group of ΛΓ χ (G2) k and it is a compact connected Lie group. •

14.4

Differentiable local Bol loops as local sections in compact Lie groups

Previously in this section we have constructed differentiable loops which can be realized as smooth sharply transitive sections in compact connected Lie groups. None of these examples are proper simple Bol loops. Namely we shall prove in the next section that any differentiable compact connected global simple Bol loop must be a Moufang loop (cf. Corollary 16.8). For differentiable local Bol loops the situation is completely different. There are many examples of differentiable local Bol loops having a compact Lie group as the group topologically generated by their left translations. According to Lemma 1.15, Theorem 7.3 (iii) and to Theorem 5.22 every differentiable local Bol loop is uniquely determined by a triple (g, f), m), where g is a Lie algebra, f) is a subalgebra of g containing no ideals φ {0} of g and m is a subspace of g such that g = fj Θ m and [[m, m], m] C m. Let g be a compact Lie algebra and let σ be an involutory automorphism of g. Then g = i) 0 m, where the subalgebra f) is the (-l-l)-eigenspace of σ and m is the (—l)-eigenspace of σ. If f) contains no ideal φ {0} of g then the triple (g, f), m) determines a local Bruck loop such that the group topologically generated by its left translations is the compact Lie group expg. According to Proposition 5.17 the local loop L determined by the local section exp m in the compact Lie group exp g is the local geodesic loop on the symmetric space expg/expf) with respect to its canonical connection and the compact group exp g is topologically generated by the left translations of L. Taking in particular expg = SO„(R) and exp f) = SO n _i(R) we have differentiable local Bruck loops as local geodesic loops on any (η — 1)-sphere SO„ (R)/ SO„_i (M).

Section

15

Loops on symmetric spaces of groups

In this section we give a construction principle for Bol loops in any category such that the groups generated by the left translations are far from being solvable. In particular we obtain many proper Bol loops having a direct product of simple groups as the group generated by the left translations. In contrast to this it seems that Bol loops which are not isotopic to Bruck loops and which have simple groups as the group generated by left translations are rare. The so-called core loops play an important role here. They are Bol loops which are repeated extensions of groups, considered as symmetric spaces, by groups.

15.1

Basic constructions

Let U and V be groups and let a : U —V be a non-trivial homomorphism such that the following two conditions are satisfied: (i) a(U) contains no non-trivial normal subgroup of V, (ii) the set { (Λ, λ γ - 1 ) ; χ G U } generates the direct product U χ U.

Let G be the semidirect product V xi (U χ U) given by the multiplication tel. 82, g3)(hi,h2, hi) = (g\hf, g2h2, £3^3) where h8 = ghg~l. Consequently we have (h k ) g = gkhk~lg-1 h G V and g, k e U. We denote the subgroup (a(U), 1 ,U) = {(a(u)), of G by H. Let σ : G/H

= hgk for any

,«);«€{/}

1

G be the section

σ : (α,χ, 1 )H ι-» (aa(x~l)x,

χ, x~l),

a e V, χ e U.

This section is sharply transitive since for any ( a \ , x \ , 1 )H and (a 2 ,X2, a, G V, χi G U, i = 1, 2, there exists the unique element ,

.

- K m -(X2X7])

λ = {α2α(χιχ2 ) "ι

,*2χι

-1

-K ,x\x2 )

= {α2[α(χ\χ2λ)α\ *l ]xl,x2X\X,x\*2X)

€ a{G/H)

with

15.1 Basic constructions

175

such that λ (α ι, jc ι, I) Η — {d2,X2, 1 )H. Hence the multiplication

(a, x,l)H*

(b, y, \)H = σ((α, χ, 1 )H)(b, y, 1 )H = (a a{x~x)x,x,x-{){b,y,

1 )H

= (aa(x~l)xbxa(x)xy,xy,

1 )H

= (a[a(x~l)ba(x)y]x,

xy, 1 )H

defines a loop on G/H. Because of the condition (ii) the group G is generated by the elements of a ( G / H ) . Since for any (aa(x~l)x, x, ; t _ 1 ) and ( b a ( y ~ ] ) y , y, ;y _ 1 ) belonging to a{G/H) we have

{αα(χ~λ)χ,

χ, x~])(ba(y~l)y,

y, y~l)(a

= (ca(xyx)~xyx,

χ, JC-1)

xyx,

(Λ^Λ:)-1)

with a suitable c e V the loop L is a Bol loop. The normal subgroup (V, 1, 1) of G is contained in a(G/H) since σ((α, 1, 1 )H) = {a, 1, 1), and hence (V, 1, 1 )H is a subgroup of L. The mapping π : (a, χ, 1)Η t-> χ : L U is an epimorphism onto the group U since π {{α, χ, 1 )H *(b, y, 1)//] = xy, its kernel is the normal subgroup (V, 1, 1 )H of L. L is a Moufang loop if and only if it satisfies the right inverse property. Since [(α,χ,Ι)ΗΓ1 = ( Κ be an endomorphism. We denote by Η the subgroup (ρ(Κ),

Κ) ofG and

by Σ the set {(χ, χ - 1 ) ; χ e K). The set Σ is the image of a section σ : G j Η —• G if and only if the mapping χ ι—> χρ(χ)

:Κ

Κ isbijective.

The section σ : G/H

acts sharply transitively on G/H if and only if the mapping χ ι—»· χαρ(χ)

G

. Κ ^

Κ

is bijective for any u € K. Any loop constructed in this way is a Bol loop. Proof Σ is the image of a section if and only if any element (a, b) e G = Κ χ Κ can be decomposed uniquely as a product (a, b) — (χ, χ _ 1 ) ( ρ ( 0 > t, χ e Κ which is equivalent to the unique solvability of the equation χρ(χ) — αρφ)~Κ The section σ acts sharply transitively if and only if for all (αϊ, ^ i ) , (02. ^2) e G there exist unique x,h e Κ such that (x, x~x){a\,b\) — {α2^2){ρ(Κ),Κ) holds or equivalently the λ equation χα\ρφ\)~ ρ{χ) = is uniquely solvable. The Bol property follows from Remark 1.14 (b). Proposition ρ : Κ —> Κ the image of subgroup Η

•

15.2. Let Κ be a group, G be the direct product G = Κ χ Κ and let be an endomorphism. We assume that the set Σ = {(χ, Jt - 1 ); χ 6 Κ] is a section σ : G/H -» G acting sharply transitively with respect to the = (ρ(Κ), Κ) of G. Let L be the loop corresponding to the section σ.

The subgroup (ρ (Μ), Μ) of tained in the centre Z(K) Ν = [η G Κ; ρ(η) e Z(K)} this property. If U is the group translations of L is isomorphic

Η is a normal subgroup ofG if and only ifρ(Μ) is conof Κ and the subgroup (ρ(Ν), N) defined by = ρ _ 1 [ ρ ( Ζ Ο Π Ζ(Κ)] is maximal with respect to generated by Σ then the group generated by the left to the group U/(U Π (ρ(Ν), Ν)).

The loop L is a group if and only if for any commutator j c j x - 1 ^ - 1 with x, y e Κ there exists an element u in the normal subgroup Ν of Κ such that Jtyx~1;y~1 = uρ(u). If L is a group then the commutator subgroup K' of Κ is contained in Ν and Q(K') G is associative if and only Σ is closed with respect to the multiplication of G modulo the normal subgroup (ρ(Ν), Ν). This means that for any x, y e Κ there exists ζ e Κ and u € Ν such that Jt - 1 )(>\ y - 1 ) = (ζ, ζ _ 1 )(ρ(«), u). -1 -1 This is equivalent to xy = ζρ(κ), j c } » = z~lu or xy — ιιγχρ(ιι) = uρ(u)yx x l because ρ (μ) Κ induces an

•

Let G be the product G = Κ χ Κ, the endomorphism ρ : Κ —»• Κ, the set Σ = {(jc, y - 1 ) ; χ G K] and Η = (ρ(Κ), Κ ) as in the previous proposition. If the loop L which corresponds to the set Σ is a group then the structure of L can essentially differ from the structure of Κ as the following example shows. Let C be the Lie group homeomorphic to K 3 the multiplication of which is given by (xi,x2,*3)(yuy2,

>3) = O l +yi,X2

+ y2, X3 + >3

,

(Οι, Χ2ι -£3)5 Ο ι , >2, >3) € R 3 ). Let Κ be the direct product Κ = C χ C and let G be G = Κ χ Κ. We denote by ρ : Κ —• Κ the homomorphism ρ : (Οι,

Χ2, xs),

( y i , y2,

>3))

( Ο ι , *2, *3), Ο ι , *2, -*3» : Κ

Κ.

According to Proposition 15.1 the set Σ is the image of a section corresponding to a loop if and only if for every element ((a\, 3),

-*2.«3), ( - y i ,

-y2,

-«3»;

xi, yi, " 3 , V3 e

R}.

178

15 Loops on symmetric spaces of groups

Since for the subgroup Η = (ρ(Κ), Κ) we have Η = {((*1,*2,*3), (X\,X2,*3),

(XI,X2,X3), Ol, >2, B)); Xi, )>i € R}

it follows υ η Η = {((0, 0, α), (0, 0, a), (0, 0, a), (0, 0, b))] a, be

R).

For the cosets g(U Π Η) of the normal subgroup U Π Η in the group U we may choose as a system S of representatives the elements of the form ((*i,*2.*3),

ΟΙ.^,Β),

-*2,0),

(->1,-3^2.0))

with Xi, yt e E. Now we express the multiplication in the factor group U/(U on the system S of representatives. We have {(x\,x2,x3),

(yi,y2, ys), ( - * i . -X2,0),

• ((U],U2, =

((*! ( - X I

+

us),

(V\,V2,

ui,x2

+

- Ml,

- X 2

v3),

u2,X3 - U

+ 2

(—yi, -yi,

0)) (U η Η)

( - M l , - « 2 , 0 ) , (~V\,

Μ3 + X I U 2 ) ,

, X \ U 2 ) ,

-v

Ol + v\,y2

( - y i - V\, -y2

2

, 0 ) ) (U Π

x2 - U2, 0), ( - J 1 - Ui, -y2

Η)

+ v2, y3 + V3 +

y\v2),

- v2, y\v2)) (U Π Η)

= (Ol + Ml, X2 + U2,X3 + U3), Ol + Vi, y2 + V2, ys + V3~ ( - * i - Ml,

OH)

+

y\Vl),

- v2, 0)) (U Π Η) .

Therefore the group L = U / (U Π Η) is the direct product of Μ and of the 5-dimensional nilpotent Lie group with the multiplication ( x \ , x 2 , y \ , y i , y3)("i, "2, f i , v2, v3) = Οι + Μ 1,X2+U2, yi + VI, y2 + v2, J3 + V3 - XiU2 + yiv2) , o,·, yi, Ui, Vi g E). Definition 15.3. Let A' be a group, let G be the direct product G = Κ χ Κ and let φ : Κ —> Κ be an endomorphism. If the set {Ο, χ - 1 ) ; χ e A"} is the image of a sharply transitive section σ : G/H —• G with Η = (φ(Κ), Κ) then we call the corresponding Bol loop the core loop of the group Κ with respect to φ. Clearly, if φ : Κ —> Κ is the trivial endomorphism (φ(Κ) corresponding core loop L is isomorphic to the group Κ.

— {1}) then the

Lemma 15.4. Let L be a core loop of the group Κ with respect to the endomorphism φ :Κ Κ such that Ker φ Φ {1}. Then the subgroup (1, Ker φ) induces the identity map on the loop L = (Κ χ Κ)/(φ(Κ), Κ). Proof. Since the subgroup Ker φ is normal in Κ the subgroup (1, Ker φ) contained in (φ(Κ),Κ) is a normal subgroup in A' χ • If φ : Κ — Κ is the identity map id then we have the following

179

15.1 Basic constructions

Proposition 15.5. Let Κ be a strongly 2-divisible group and let φ : Κ Κ be the 1 identity map id. Then the set {(x, x~ )·, χ e K} is the image of a sharply transitive section σ : (Κ χ Κ)/(φ(Κ), Κ) — Κ χ Κ and the core loop L corresponding to σ is α Bruck loop isotopic to the core of Κ. Proof The elements of the factor space (Κ χ Κ)/(φ(Κ), Κ) can be represented as (χ, 1 )(φ(Κ), Κ), χ e Κ. Then the loop multiplication has the form (χ, 1 Κφ(Κ),

Κ) * (y, 1)(φ(Κ), Κ) = (VI, y/Tl)(y,

= (V^y,

νΐ~*)(φ(Κ),

Κ) = (Viyyft,

\)(φ(Κ),

1 )(φ(Κ),

Κ)

Κ).

According to Proposition 9.9 this loop is a Bruck loop which is isotopic to the core of AT. • Proposition 15.6. Let L2 be a loop having G2 as the group generated by the left translations, the subgroup H-i c Gi as the stabilizer of 1 € L2 in G2 and - uxf (u) : Κ Κ is bijective for every Λ: e Κ (cf. Proposition 15.1). In this case the section σ acts sharply transitively because for any (a\, b\, c\),

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15 Loops on symmetric spaces of groups

(βι, b2, c2) € G there exist unique elements (u, u ι,σ2(ζΗ2)) k e K,h e H2 such that (ua\,u~lb\,a2(zH2)ci) Indeed, the equation a2(zH2)c\ system of equations

=

ε a{G/H)

and

(a2f(k)g(h),b2k,c2h).

— c2h is uniquely solvable for ζ and h. Moreover the

ua\ = a2f(k)g(h)

= [a2g(h)]f(k)

,

l

u~ b\ — b2k is also uniquely solvable for u and k since Σ is the image of a section which acts sharply transitively. According to Definition 2.1 the loop L is a Scheerer extension of the normal subloop L \ bytheloopL2. TheloopLi is a Bol loop (cf. Proposition 15.1). If L is a Bol loop or it has the left inverse property then the factor loop L/L\ = L2 has the same property. Conversely, if L2 has the left inverse property then we have [A2{G2/H2)}-1 = σ2(β2/Η2) and hence [A(G/H)]~L = A(G/H). If L2 is a Bol loop then xyx € CR2(G2/H2) if jc, y e O2(G2/H2). Therefore the element (u, μ - 1 , x)(v,

y)(u, u~l, x) is contained in a(G/H)

x, y € O2(G2/H2),

i. e. L is a Bol loop.

in case u,v

e

Κ and

•

Proposition 15.7. Let L be a core loop of the group Κ with respect to the endomorphism φ : Κ —• Κ. Let Κ be the direct product Κ = Α χ Β where A is an abelian and Β is a perfect group {i.e. Β is equal to its commutator subgroup). We assume that one of the following conditions is satisfied: (a) The group Β is finite or it contains no subgroup φ {1} isomorphic to a subgroup of A. (b) Κ is a connected Lie group and the homomorphism φ is continuous. Then L is the direct product of the abelian group A and of the core loop of the group Β with respect to the endomorphism ψ\β : Β —r Β induced by φ. Proof. The core loop L is given by the group G — AxBxAxB, the subgroup Η = (φ(Α χ Β), Αχ Β) and by the section σ : G/H G the image of which is the set a(G/H) = {(a, b, a~x, b~1)·, a e Λ, b e B). We denote φ{Α χ {1}) = a(A), ^({1} χ Β) = β(Β). We have φ = α χ β and a (a) — (aA(a), aB(a)),

a Β is injective. Moreover, the set {bßB{b)', b e Β] Π ctB(A) = {1} since otherwise for dßB(d) = c € A the element c φ 1 could be decomposed as c = lßB(l)c = dßB{d) which is a contradiction. If Β is a finite group then the mapping b t-> bßB(b) : Β —> Β is a bijection and hence A = {1}. Proposition 1.19 gives again the assertion. If Κ is a connected Lie group and φ is a continuous endomorphism then the mapping ω : b i-> bßB (b) : Β —> Β of the Lie group Β into Β is a continuous injective mapping. Let W be a compact neighbourhood in B. Then co(W) is homeomorphic to W since any continuous bijective map from a compact space onto a Hausdorff space is a homeomorphism. It follows that ω(Β) contains an open submanifold of B. The mappings ν

σ'((1, ν, 1, l ) ( / / n £/)) = (a, b, a~l, b~l) η* (α, b) : Β

Α χ Β

and ωχαΒ :Bx A ^ B x B are continuous. Therefore Β is homeomorphic to the topological product ω(Β) χ aB{A). Since Β and ω(Β) are connected the group

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15 Loops on symmetric spaces of groups

1. Since a Β (A) is Lie group we have the equation η = dim Β = dim ω(Β) + dim ag(A). The dimension of co(B) is η because it contains an open submanifold of B. Therefore dimafl(A) = 0 and Λ as a connected group must be trivial. Now the assertion follows from Proposition 1.19. •

15.2

A fundamental reduction

A group G is perfect if the commutator subgroup coincides with G. Definition 15.8. A group G is quasi-simple if G is perfect and if every normal subgroup of G is contained in the centre of G. A group G is strongly semisimple if G is the direct product of finitely many quasi-simple groups. An additively written abelian group A is called divisible if any equation nx = a has at least one solution for any η € Ζ and α 6 Λ. In a divisible abelian group any subgroup has a complementary subgroup. Definition 15.9. We call a group G strongly reductive if it is a direct product of a divisible abelian group and of a strongly semisimple group. In a strongly reductive group G = A χ G', where A is an abelian group and G' is the commutator subgroup of G, for all normal subgroups Ν and Μ with Ν = Β χ Ν', M = CxM',B,C G. The normal subgroups Ker φ C Ker φ2 C · · · C Ker (lk+iak+\(yk+\ . . . , lk+iak+i(Yk+i

ο