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. The definition of the free Lie algebra generated by a set X is as follows: Let X be a set and a morphism of sets (function) from X into a Lie algebra L. The Lie algebra L is called free on X if is the universal morphism; that is, if for any Lie algebra A with a morphism of sets , there is a unique Lie algebra morphism such that . Given a set X, one can show that there exists a unique free Lie algebra generated by X. In the language of , the functor sending a set X to the Lie algebra generated by X is the free functor from the to the category of Lie algebras. That is, it is left adjoint to the forgetful functor. The free Lie algebra on a set X is naturally graded. The 0-graded component of the free Lie algebra is just the free vector space on that set. 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. 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. 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 is the Möbius function. The graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle algebra.