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 {L}. Given representations {V_1, V_2}, we clearly have a representation {V_1 \oplus V_2}; given a morphism of representations {V_1 \rightarrow V_2}, i.e. one which respects the action of {L}, the kernel and image are themselves representations.

Proposition 1 The category {Rep(L)} of finite-dimensional representations of {L} is an abelian category.