Reutenauer C. Free Lie Algebras

Clarendon: 1993. — 285 p.The Poincaré-Birkhoff-Witt theorem.
Free Lie algebras.
Elimination.Lie polynomials.
Words, polynomials, and series.
Lie polynomials.
Characterizations of Lie polynomials.
Shuffles.
Duality concatenation/shuffle.Algebraic properties.
The weak algorithm.
Subalgebras.
Automorphisms.
Free sets of Lie polynomials.Logarithms and exponentials.
Lie series and logarithm.
The canonical projections.
Coefficients of the Hausdorff series.
Derivation and exponentiation.Hall bases.
Hall trees and words.
Hall and Poincaré-Birkhoff-Witt bases.
Hall sets and Lazard sets.Applications of Hall sets.
Lyndon's words and basis.
The dual basis.
The derived series.
Order properties of Hall set.
Synchronous codes.Shuffie algebra and subwords.
The free generating set of Lyndon words.
Presentation of the shuffle algebra.
Subword functions.
The lower central series of the free group.Circular words.
The number of primitive necklaces.
Hall words and primitive necklaces.
Generation of Lyndon's words.
Factorization into Lyndon's words.
Words and multisets of primitive necklaces.The action of the symmetric group.
The action of the symmetric group and the linear group.
The character of the free Lie algebra.
Irreducible components.
Lie idempotents.
Representations of the canonical decomposition.The Solomon descent algebra.
The descent algebra.
Idempotents.
Homomorphisms.
Quasisymmetric functions and enumeration of permutations.