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CuuDuongThanCong.com LIE A L G E B R A S THEORY AND ALGORITHMS CuuDuongThanCong.com North-Holland Mathematical Library Board of Honorary Editors: M Artin, H Bass, J Eells, W Feit, E J Freyd, EW Gehring, H Halberstam, L.V H6rmander, J.H.B Kemperman, W.A.J Luxemburg, E E Peterson, I.M Singer and A.C Zaanen Board of Advisory Editors." A Bj6mer, R.H Dijkgraaf, A Dimca, A.S Dow, J.J Duistermaat, E Looijenga, J.E May, I Moerdijk, S.M Mori, J.P Palis, A Schrijver, J Sj6strand, J.H.M Steenbrink, E Takens and J van Mill V O L U M E 56 ELSEVIER Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo CuuDuongThanCong.com Lie Algebras Theory and Algorithms Willem A de Graaf University of St Andrews Scotland 2000 ELSEVIER Amsterdam - Lausanne CuuDuongThanCong.com - New York - Oxford - Shannon - Singapore - Tokyo E L S E V I E R S C I E N C E B.V Sara Burgerhartstraat 25 P.O Box 211, 1000 A E A m s t e r d a m , The Netherlands 2000 Elsevier Science B.V All rights reserved This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or 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Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made First edition 2000 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for ISBN: 444 50116 O T h e paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper) Printed in The Netherlands CuuDuongThanCong.com Preface Lie algebras arise naturally in various areas of mathematics and physics However, such a Lie algebra is often only known by a presentation such as a multiplication table, a set of generating matrices, or a set of generators and relations These presentations by themselves not reveal much of the structure of the Lie algebra Furthermore, the objects involved (e.g., a multiplication table, a set of generating matrices, an ideal in the free Lie algebra) are often large and complex and it is not easy to see what to with them The advent of the computer however, opened up a whole new range of possibilities: it made it possible to work with Lie algebras that are too big to deal with by hand In the early seventies this moved people to invent and implement algorithms for analyzing the structure of a Lie algebra (see, e.g., [7], [8]) Since then many more algorithms for this purpose have been developed and implemented The aim of the present work is two-fold Firstly it aims at giving an account of many existing algorithms for calculating with finite-dimensional Lie algebras Secondly, the book provides an introduction into the theory of finite-dimensional Lie algebras These two subject areas are intimately related First of all, the algorithmic perspective often invites a different approach to the theoretical material than the one taken in various other monographs (e.g., [42], [48], [77], [86]) Indeed, on various occasions the knowledge of certain algorithms allows us to obtain a straightforward proof of theoretical results (we mention the proof of the Poincar~-Birkhoff-Witt theorem and the proof of Iwasawa's theorem as examples) Also proofs that contain algorithmic constructions are explicitly formulated as algorithms (an example is the isomorphism theorem for semisimple Lie algebras that constructs an isomorphism in case it exists) Secondly, the algorithms can be used to arrive at a better understanding of the theory Performing the algorithms in concrete examples, calculating with the concepts involved, really brings the theory to life The book is roughly organized as follows Chapter contains a general CuuDuongThanCong.com vi introduction into the theory of Lie algebras Many definitions are given that are needed in the rest of the book Then in Chapters to we explore the structure of Lie algebras The subject of Chapter is the structure of nilpotent and solvable Lie algebras Chapter is devoted to Cartan subalgebras These are immensely powerful tools for investigating the structure of semisimple Lie algebras, which is the subject of Chapters and (which culminate in the classification of the semisimple Lie algebras) Then in Chapter we turn our attention towards universal enveloping algebras These are of paramount importance in the representation theory of Lie algebras In Chapter we deal with finite presentations of Lie algebras, which form a very concise way of presenting an often high dimensional Lie algebra Finally Chapter is devoted to the representation theory of semisimple Lie algebras Again Cartan subalgebras play a pivotal role, and help to determine the structure of a finite-dimensional module over a semisimple Lie algebra completely At the end there is an appendix on associative algebras, that contains several facts on associative algebras that are needed in the book Along with the theory numerous algorithms are described for calculating with the theoretical concepts First in Chapter we discuss how to present a Lie algebra on a computer Of the algorithms that are subsequently given we mention the algorithm for computing a direct sum decomposition of a Lie algebra, algorithms for calculating the nil- and solvable radicals, for calculating a Cartan subalgebra, for calculating a Levi subalgebra, for constructing the simple Lie algebras (in Chapter this is done by directly giving a multiplication table, in Chapter by giving a finite presentation), for calculating GrSbner bases in several settings (in a universal enveloping algebra, and in a free Lie algebra), for calculating a multiplication table of a finitely presented Lie algebra, and several algorithms for calculating combinatorial data concerning representations of semisimple Lie algebras In Appendix A we briefly discuss several algorithms for associative algebras Every chapter ends with a section entitled "Notes", that aims at giving references to places in the literature that are of relevance to the particular chapter This mainly concerns the algorithms described, and not so much the theoretical results, as there are standard references available for them (e.g., [42], [48], [77], [86]) I have not carried out any complexity analyses of the algorithms described in this book The complexity of an algorithm is a function giving an estimate of the number of "primitive operations" (e.g., arithmetical operations) carried out by the algorithm in terms of the size of the input Now the size of a Lie algebra given by a multiplication table is the sum of the CuuDuongThanCong.com vii sizes of its structure constants However, the number of steps performed by an algorithm that operates on a Lie algebra very often depends not only on the size of the input, but also (rather heavily) on certain structural properties of the input Lie algebra (e.g., the length of its derived series) Of course, it is possible to consider only the worst case, i.e., Lie algebras having a structure that poses most difficulties for the algorithm However, for most algorithms it is far from clear what the worst case is Secondly, from a practical viewpoint worst case analyses are not very useful since in practice one only very rarely encounters the worst case Of the algorithms discussed in this book many have been implemented inside several computer algebra systems Of the systems that support Lie algebras we mention GAP4 ([31]), LiE ([21]) and Magma ([22]) We refer to the manual of each system for an account of the functions that it contains I would like to thank everyone who, directly or indirectly, helped me write this book In particular I am grateful to Arjeh Cohen, without whose support this book never would have been written, as it was his idea to write it in the first place I am also grateful to Gs Ivanyos for his valuable remarks on the appendix Also I gratefully acknowledge the support of the Dutch Technology Foundation (STW) who financed part of my research Willem de Graaf CuuDuongThanCong.com This Page Intentionally Left Blank CuuDuongThanCong.com Contents Basic constructions 1.1 Algebras: associative and Lie 1.2 Linear Lie algebras 1.3 Structure constants 1.4 Lie algebras from p-groups 1.5 On algorithms 1.6 Centralizers and normalizers 1.7 Chains of ideals 1.8 Morphisms of Lie algebras 1.9 Derivations 1.10 (Semi)direct sums 1.11 Automorphisms of Lie algebras 1.12 Representations of Lie algebras 1.13 Restricted Lie algebras 1.14 Extension of the ground field 1.15 Finding a direct sum decomposition 1.16 Notes On 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 1 10 13 17 19 21 22 24 26 27 29 33 34 38 nilpotency and solvability 39 Engel’s theorem 39 The nilradical 42 The solvable radical 44 Lie’s theorems 47 A criterion for solvability 49 A characterization of the solvable radical 51 Finding a non-nilpotent element 54 Notes 56 CuuDuongThanCong.com 380 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subalgebra of a Lie algebra Linear Algebra Appl., 52/53:743-761, 1983 CuuDuongThanCong.com Index of Symbols 387 I n d e x of S y m b o l s (adL)*, A1 ~ A2 (isomorphic algebras), 18 ALie, C(L) (centre), 15 C ,,~ C ~(equivalence of Cartan matrices), 136 CL(S) (centralizer), 14 Ci (L) (term of the upper central series), 18 Cp (coefficient space), 205 H* (dual space of H), 95 L i (term of the lower central series), 17 L (i) (term of the derived series), 17 Lo(K) (Fitting-zero component), 54 L1 (K) (Fitting-one component), 54 L1 ~
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