Free Lie Algebras

This much-needed new book is the first to specifically detail free Lie algebras. Lie polynomials appeared at the turn of the century and were identified with the free Lie algebra by Magnus and Witt some thirty years later. Many recent, important developments have occurred in the field--especially from the point of view of representation theory--that have necessitated a thorough treatment of the subject. This timely book covers all aspects of the field, including characterization of Lie polynomials and Lie series, subalgebras and automorphisms, canonical projections, Hall bases, shuffles and subwords, circular words, Lie representations of the symmetric group, related symmetric functions, descent algebra, and quasisymmetric functions. With its emphasis on the algebraic and combinatorial point of view as well as representation theory, this book will be welcomed by students and researchers in mathematics and theoretical computer science.

Index of notation
Introduction	p. 1
The Poincare-Birkhoff-Witt theorem	p. 1
Free Lie algebras	p. 4
Elimination	p. 7
The polynomials	p. 14
Words, polynomials, and series	p. 14
Lie polynomials	p. 18
Characterizations of Lie polynomials	p. 19
Shuffles	p. 23
Duality concatenation/shuffle	p. 26
Algebraic properties	p. 40
The weak algorithm	p. 40
Subalgebras	p. 44
Automorphisms	p. 45
Free sets of Lie polynomials	p. 49
Logarithms and exponentials	p. 52
Lie series and logarithm	p. 52
The canonical projections	p. 57
Coefficients of the Hausdorff series	p. 61
Derivation and exponentiation	p. 76
Hail bases	p. 84
Hall trees and words	p. 84
Hall and Poincare-Birkhoff-Witt bases	p. 89
Hall sets and Lazard sets	p. 98
Applications of Hall sets	p. 105
Lyndon words and basis	p. 105
The dual basis	p. 108
The derived series	p. 112
Order properties of Hall sets	p. 114
Synchronous codes	p. 119
Shuffle algebra and subwords	p. 127
The free generating set of Lyndon words	p. 127
Presentation of the shuffle algebra	p. 129
Subword functions	p. 131
The lower central series of the free group	p. 136
Circular words	p. 154
The number of primitive necklaces	p. 154
Hall words and primitive necklaces	p. 158
Generation of Lyndon words	p. 161
Factorization into Lyndon words	p. 163
Words and multisets of primitive necklaces	p. 166
The action of the symmetric group	p. 176
Action of the symmetric group and of the linear group	p. 176
The character of the free Lie algebra	p. 180
Irreducible components	p. 185
Lie idempotents	p. 194
Representations on the canonical decomposition	p. 201
The Solomon descent algebra	p. 217
The descent algebra	p. 217
Idempotents	p. 224
Homomorphisms	p. 233
Quasisymmetric functions and enumeration of permutations	p. 242
References	p. 256
Index	p. 267
Table of Contents provided by Blackwell. All Rights Reserved.

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