Table Of ContentLONDON 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
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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.
