Free Lie algebra
In mathematics, a free Lie algebra, over a given field K, is a Lie algebra generated by a set X, without any imposed relations.
Given a set X, one can show that there exists a unique free Lie algebra L(X) generated by X.
In the language of category theory, the functor sending a set X to the Lie algebra generated by X is the free functor from the category of sets 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.
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.
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