Free Lie algebra

In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating K-bilinearity and the Jacobi identity.

One can alternatively define a free Lie algebra on a vector space V as left adjoint to the forgetful functor from Lie algebras over a field K to vector spaces over the field K – forgetting the Lie algebra structure, but remembering the vector space structure.

Ernst Witt showed that the number of basic commutators of degree k in the free Lie algebra on an m-element set is given by the necklace polynomial: where

See tensor algebra for a detailed exposition of the inter-relation between the shuffle product and comultiplication.

Subsequently, Wilhelm Magnus showed that they arise as the graded Lie algebra associated with the filtration on a free group given by the lower central series.

This correspondence was motivated by commutator identities in group theory due to Philip Hall and Witt.