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LONDON MATHEMATICAL SOCIETY MONOGRAPHS NEW SERIES Series Editors P. M. Cohn H. G. Dales LONDON MATHEMATICAL SOCIETY MONOGRAPHS NEW SERIES PreviousvolumesoftheLMSMonographswerepublishedbyAcademicPress, towhomallenquiriesshouldbeaddressed. VolumesintheNewSerieswillbe published by Oxford University Press throughout the world. NEW SERIES Diophantine inequalities R. C. Baker ‘ t N The Schur multiplier Gregory Karpilovsky ’ P Existentially closed groups Graham Higman and Elizabeth Scott P The asymptotic solution oflinear diflerential systems M. S. P. Eastham P' The restricted Burnside problem Michael Vaughan-Lee ‘ g Pluripotential theory Maciej Klimek N Free Lie algebras Christophe Reutenauer Free Lie Algebras Christophe Reutenauer Université du Québec d Montréal CLARENDON PRESS - OXFORD 1993 Oxford University Press, Walton Street, Oxford0X26DP Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Kuala Lumpur Singapore Hang Kong Tokyo Nairobi Dares Salaam Cape Town Melbourne Auckland Madrid andassociatedcompanies in Berlin Ibadan Oxfordisa trademark ofOxford University Press Published in the United States byOxford University Press Inc., New York © Christophe Reutenauer, 1993 _ Allrightsreserved. No part ofthispublicationmaybe reproduced, stored ina retrievalsystem, ortransmitted, in any form orbyanymeans, without thepriorpermission in writingofOxford University Press. Within the UK, exceptions areallowed in respectofany fairdealingfor thepurposeofresearchorprivate study,orcriticism or review, aspermitted under the Copyright, Designsand PatentsAct, 1988, or in thecaseofreprographic reproduction in accordance with the termsof licences issued by the Copyright Licensing Agency. Enquiriesconcerning reproduction outside those terms and in othercountriesshouldbesent to theRightsDepartment, Oxford UniversityPress, at theaddress above. A cataloguerecordfor this book isavailablefrom theBritishLibrary LibraryofCongress Cataloging in PublicationData Reutenauer, Christophe. Free lie algebras/Christophe Reutenauer. (London MathematicalSocietymonographs,new series, no. 7) Includesbibliographicalreferencesand index. I. Liealgebras. I. Title. II. Series. QA252.3.R48 1993 512’.55--dc20 92-27318 ISBN0-19-853679-8 TypesetbyIntegral Typesetting, Great Yarmouth, Norfolk Printed in Great BritainbySt EdmundsburyPress, BurySt Edmunds, Suflalk Ce livre est dédié d Arthur, Victor, Emile, Eva Preface Lie polynomials first appeared at the turn of the century in the work of Campbell, Baker, and Hausdorff on exponential mapping in a Lie group, which led to a result known as the Campbell—Baker—Hausdorfl‘ formula. About thirtyyearslater,Wittshowed thatthe LiealgebraofLiepolynomials is actually the free Lie algebra, and that its enveloping algebra is the (associative) algebra of noncommutative polynomials; he answered a ques- tion of Magnus—who had himself arrived at the solution—on the lower centralseries ofthefree group. Someyearsearlier,in 1933, P. Hallhadbegun commutator calculus in the free group, which led M. Hall to construct his basis of the free Lie algebra; the link between the latter and the free group is given by the work ofWitt and the Magnus transformation. In 1942 and 1944, Thrall and Brandt studied the free Lie algebra from the point of view of representation theory of the linear group, and Brandt computed the character—a formula closely related to the Witt formula. At the end of the forties, Dynkin, Specht, and Wever simultaneously discovered the characterization ofLie polynomials through the ‘left to right bracketing mapping’. Some years later, Friedrichs gave his characterization for Lie polynomials; his criterion fascinated many mathematicians, who actually proved it. After that, the subject was studied by many people, often independently. Recently, it has had a new impulse, from the point ofview ofrepresentation theory ofthe symmetric group. As far as we know, no book exists that exclusively treats free Lie algebras. Bahturin,inhisrecentbookonvarietiesofLiealgebras,devotestwochapters to free Lie algebras; Bourbaki, Jacobson, and others, give an introduction to the subject. It seems to us that the theory has become so extensive with existing results so widely scattered, to justify the publication of a book on the subject. The book is partly written in the spirit of Lothaire’s Combinatorics on words, with emphasis on the algebraic point ofview; it can be considered as aseriesofvariations on Lyndonwords;thepresentation ofthelatteris rather indirect, so the interested reader could begin by reading the corresponding section by Lothaire. In Chapter0, we give without proofthe Poincaré—Birkhoff—Witt theorem, viii Preface which enables us to prove that the Lie algebra ofLie polynomials is the free Lie algebra; this necessitates a basis construction (in cases where the ring of scalars is not a field), which is done through Lazard elimination. The impatient reader may proceed directly to Chapter 1, where things really begin. We give several characterizations ofLie polynomials, introduce the shuflieproductandpresent Hopfalgebra-likeproperties offreeassociative algebras. Chapter2isdevotedmainlyto two results:subalgebrasoffree Liealgebras are free; automorphisms of free Lie algebras are products of elementary automorphisms. The related problem of characterizing free sets of Lie polynomials is also treated. In Chapter 3 we characterize exponentials of Lie series, and give several results on the Hausdorff series, after having connected it to the canonical projections ofthe free associative algebra. The study ofHall bases begins in Chapter 4: we construct the Hall basis ofthe free Lie algebra, and the corresponding Poincaré—BirkhoH—Witt basis ofthe free associative algebra. We show also that this basis construction is identical to the one arising from Lazard elimination. Chapter 5 gives some applications of Hall sets: the Lyndon basis, which is a particular case of a Hall basis; the calculation of the dual basis in the shuflle algebra; the construction ofa Hall basis compatible with the derived series of the free Lie algebra; and the order on the free monoid associated with a Hall set and the associated codes. In Chapter 6 we give some properties of the shuflle algebra: it is freely generated by Lyndon words, and has a remarkable presentation. Related to shuflles is the concept of subword. This leads to subword functions, a generalization of binomial coeflicients, and the Magnus transformation of the free group. Commutator calculus is presented, and connected with the Hall basis and the algebra ofsubword functions. Chapter 7 studies circular words: after giving the formulas enumerating them, we relate them to Hall sets. Two algorithms on Lyndon words are presented, and we give a natural bijection between words on an ordered alphabet and multisets ofprimitive necklaces. The Lie representation of the symmetric group (or the linear group) is consideredin Chapter8. Its characterand themultiplicities oftheirreducible representations are given. Almost all of them occur. Remarkable Lie elements, the Lie idempotents, are studied in the symmetric group algebra. Representations on the components of the canonical decomposition of the free associative algebra are also studied. Chapter9shows thecloseconnection between thefree Liealgebraand the Solomon descentalgebra ofthe symmetricgroup. Theprimitiveidempotents of the latter represent the canonical projections, and the dimension of the corresponding quasi-ideals has an interpretation in terms ofnecklaces. The action on Liemonomials ofelements ofthedescentalgebracharacterizes this Preface ix algebra, which has as a natural homomorphic image the ring of symmetric functions, and is itselfdual to the ring ofquasisymmetric functions. Each chapter ends with an appendix: each subsection can be thought of asanexercise,withhintsorreferences,andgivessomeinformationonrelated subjects; sometimes it is simply a review of related work. Montreal C.R. July 1992 Acknowledgements I first discovered the subject of this book in Gérard Lallement’s book on semigroups and in Dominique Perrin’s chapter on factorization of free monoids in Lothaire’s book; thanks to Adriano Garsia’s Combinatorics of thefree Lie algebra and the symmetric group, this subject was given a new impulse,inthedirection ofalgebraiccombinatorics. Whilewritingthis book, I had innumerable discussions with Marco Schfitzenberger, who is for me the initiator of the subject, and gave me useful advice as well as some unpublished results. During this same time, Guy Melancon wrote his Ph.D., and was of considerable help, by discussions, reading and correcting the successive versions ofthe manuscript. Paul Cohn kindly accepted this book in the London Mathematical Society Monographs series and carefully read the whole manuscript; he also communicated some unpublished results of his Ph.D. thesis. I also thank, for many discussions and correspondence, Sheila Sundaram, André Joyal, Pierre Leroux, Xavier Viennot, Hartmut Laue, Pierre Bouchard, Francois Bergeron, and Nantel Bergeron. Special thanks to Daniel Krob, who carefullyread the manuscript, and found many mistakes. I also want to thank the whole Department of Mathematics and Computer Science of the Université du Québec a Montreal, for excellent working conditions, and especially Dominique Chabot, Sonya Comtois, Lucie Lortie, Marlaine Grenier, and Diane Amatuzio for their typing. Finally, I was supported by a grant of NSERC (Canada) during the three years I wrote this book.