Definition
- Let X be a set and i: X → L a morphism of sets from X into a Lie algebra L. The Lie algebra L is called free on X if for any Lie algebra A with a morphism of sets f: X → A, there is a unique Lie algebra morphism g: L → A such that f = g o i.
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.
As the 0-graded component of the free Lie algebra on a set X is just the free vector space on that group, 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.
Read more about this topic: Free Lie Algebra
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