Free Lie algebras

Christophe Reutenauer0198536798, 9780198536796

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.

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Table of contents :
Series……Page 1
Series Titles……Page 2
Title page……Page 3
Date-line……Page 4
Dedication……Page 5
Preface……Page 7
Acknowledgements……Page 10
Contents……Page 11
Index of notation……Page 15
0.1 The Poincare-Birkhoff-Witt theorem……Page 19
0.2 Free Lie algebras……Page 22
0.3 Elimination……Page 25
0.4 Appendix……Page 30
0.5 Notes……Page 31
1.1 Words, polynomials, and series……Page 32
1.2 Lie polynomials……Page 36
1.3 Characterizations of Lie polynomials……Page 37
1.4 Shuffles……Page 41
1.5 Duality concatenation/shuffle……Page 44
1.6 Appendix……Page 51
1.7 Notes……Page 57
2.1 The weak algorithm……Page 58
2.2 Subalgebras……Page 62
2.3 Automorphisms……Page 63
2.4 Free sets of Lie polynomials……Page 67
2.5 Appendix……Page 68
2.6 Notes……Page 69
3.1 Lie series and logarithm……Page 70
3.2 The canonical projections……Page 75
3.3 Coefficients of the Hausdorff series……Page 79
3.4 Derivation and exponentiation……Page 94
3.5 Appendix……Page 98
3.6 Notes……Page 100
4.1 Hall trees and words……Page 102
4.2 Hall and Poincare-Birkhoff-Witt bases……Page 107
4.3 Hall sets and Lazard sets……Page 116
4.4 Appendix……Page 119
4.5 Notes……Page 121
5.1 Lyndon words and basis……Page 123
5.2 The dual basis……Page 126
5.3 The derived series……Page 130
5.4 Order properties of Hall sets……Page 132
5.5 Synchronous codes……Page 137
5.6 Appendix……Page 142
5.7 Notes……Page 144
6.1 The free generating set of Lyndon words……Page 145
6.2 Presentation of the shuffle algebra……Page 147
6.3 Subword functions……Page 149
6.4 The lower central series of the free group……Page 154
6.5 Appendix……Page 165
6.6 Notes……Page 170
7.1 The number of primitive necklaces……Page 172
7.2 Hall words and primitive necklaces……Page 176
7.3 Generation of Lyndon words……Page 179
7.4 Factorization into Lyndon words……Page 181
7.5 Words and multisets of primitive necklaces……Page 184
7.6 Appendix……Page 188
7.7 Notes……Page 192
8.1 Action of the symmetric group and of the linear group……Page 194
8.2 The character of the free Lie algebra……Page 198
8.3 Irreducible components……Page 203
8.4 Lie idempotents……Page 212
8.5 Representations on the canonical decomposition……Page 219
8.6 Appendix……Page 224
8.7 Notes……Page 233
9.1 The descent algebra……Page 235
9.2 Idempotents……Page 242
9.3 Homomorphisms……Page 251
9.4 Quasisymmetric functions and enumeration of permutations……Page 260
9.5 Appendix……Page 266
9.6 Notes……Page 272
References……Page 274
Index……Page 285