More Lie algebra constructions

The ultimate aim in the series on Lie algebras I am posting here is to cover the representation theory of semisimple Lie algebras. To get there, we first need to discuss some technical tools—for instance, invariant bilinear forms.

Generalities on representations

Fix a Lie algebra . Given representations , we clearly have a representation ; given a morphism of representations , i.e. one which respects the action of , the kernel and image are themselves representations.

Proposition 1 The category of finite-dimensional representations of is an abelian category.