## Free lie algebrasThis 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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### Contents

Lie polynomials | 14 |

Algebraic properties | 40 |

Hall bases | 84 |

Copyright | |

6 other sections not shown

### Common terms and phrases

alphabet associative algebra basis bijection coefficient commutative composition conc concatenation conjugacy class conjugate constant term convolution corresponding cycle type deduce defined definition deg(P denote derivation element enveloping algebra equal equivalent finely homogeneous finite formula free associative algebra free group free Lie algebra free monoid Hall polynomials Hall set Hall trees Hall words Hence hi+1 homogeneous Lie polynomials homogeneous polynomial idempotent identity implies induction integers isomorphism K-algebra K-module legal rise Lemma Let H letter Lie bracket Lie idempotent Lie series Lie subalgebra linear combination linear mapping linearly Lyndon word Moreover multilinear multisets nonempty word obtain partition permutation polynomials of degree primitive necklaces Proof Let proof of Theorem proper right factor prove Reutenauer Section shows shuffle algebra shuffle product standard factorization standard sequence subset subspace subword functions Suppose symmetric functions symmetric group Theorem 5.1 totally ordered unique words of length