The Lie algebra over a commutative ring generated by a set subject only to bilinearity, alternation and the Jacobi identity. Every map from to the underlying set of an -Lie algebra extends uniquely to a Lie algebra homomorphism. It can be constructed from bracketed words by quotienting by those identities; elements are represented by Lie polynomials.
A finite linear combination of iterated Lie brackets in formal variables. It evaluates naturally in every Lie algebra, and every Lie algebra homomorphism preserves its evaluation. The homogeneous terms of the Baker--Campbell--Hausdorff formula are rational Lie polynomials; this permits the group law to be reconstructed from the bracket wherever the series converges.
Articles by others on the same topic
The concept of a free Lie algebra arises in the context of algebra, specifically in the study of Lie algebras. A Lie algebra is a vector space equipped with a binary operation (called the Lie bracket) that satisfies two properties: bilinearity and the Jacobi identity.