Free Lie Algebras 论文

摘要

Publisher Summary This chapter explores that the Lie algebra of Lie polynomials is the free Lie algebra. Lie polynomials appeared at the end of the 19th century and the beginning of the 20th century in the work of Campbell, Baker and Hausdorff on exponential mapping in a Lie group, which lead to the Campbell–Baker–Hausdorff formula. Around 1930, Witt showed that the Lie algebra of Lie polynomials is the free Lie algebra, and that its enveloping algebra is the associative algebra of noncommutative polynomials. The Poincare–Birkhoff–Witt theorem is proved, and shows that the free Lie algebra is related to the lower central series of a free group. The full linear group acts on Lie polynomials, and the symmetric group acts on those which are multilinear. The Lie representation of the symmetric group is induced from any faithful representation of a subgroup generated by a full cycle. Automorphism of a free Lie algebra are always tame, and are characterized by a Jacobian condition. The full linear group acts on Lie polynomials, and the symmetric group acts on those which are multilinear. The descent algebra is dual to the ring of quasi-symmetric functions which is, therefore, a free commutative algebra.