Reference for category of Lie algebras?

Question

Are there any references which deal with categorical aspects of Lie algebras?

I'm looking for constructions like kernels, products, coproducts (limits and colimits in general) etc.

My goal is to get a better understanding of the category of Lie algebras (using category theory machinery).

Thanks.

Answer 1

This is the sort of thing you should work out for yourself as an exercise. Limits are computed as in vector spaces / sets, so the interesting question is how to compute colimits. This reduces to computing coproducts and coequalizers. Coproducts are given by a version of the free product (look up free Lie algebras to get a sense of how this behaves), while coequalizers are given by quotienting by a suitable ideal. It's useful to observe that taking universal enveloping algebras is a left adjoint, so preserves colimits, and also useful to think of Lie algebras as analogous to groups.