I'm reading the paper "G. Hochschild, J.-P. Serre, Cohomology of Lie algebras. Ann. of Math. (2) 57 (1953) 591–603". I want to understand the statement of the Theorem 10:
Theorem: Let $G$ a reductive Lie algebra of finite dimension over the field $F$ of characteristc $0$. Let $M$ be a finite dimensional semisimple $G-$module, such that $H^{0}(G,M)=0$. Then $H^{n}(G,M)=0$ for all $n\geq 0$.
My questions:
What is the definition of simple $G$-module?
What is the definition of semisimple $G-$module?
I think, but I'm not sure, that:
$M$ is said to be simple if and only if its only $G-$submodules are $0$ and $V$.
$M$ is said to be semisimple if and only if $M$ is the direct sum of simple $G-$modules.
Are these the definitions tha I'm looking for?
