## More Lie algebra constructionsJuly 28, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: , , ,
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.