Free Lie algebra

From Wikipedia, the free encyclopedia

In mathematics, a free Lie algebra, over a given field K, is a Lie algebra generated by a set X, without any imposed relations.

A more concrete description can be given in terms of a Hall set, which is a particular kind of subset inside the free magma on X. Elements of the free magma are binary trees, with their leaves labelled by elements of X. Hall sets are named for Philip Hall.

The universal enveloping algebra of a free Lie algebra on a set X is the free associative algebra generated by X. By the Poincaré-Birkhoff-Witt theorem it is the "same size" as the symmetric algebra of the free Lie algebra (meaning that if both sides are graded by giving elements of X degree 1 then they are isomorphic as graded vector spaces). This can be used to describe the dimension of the piece of the free Lie algebra of any given degree.

Any Lie subalgebra of a free Lie algebra is itself a free Lie algebra. This was proved by Anatoly Illarionovich Shirshov (1921-1981), a student of Kurosh, in 1953, and by Ernst Witt (1956). This result is known as Shirshov's theorem or the Shirshov-Witt theorem.