I'm studying Humphreys Introduction to Lie Algebras and Representation Theory. I do not understand the second paragraph in page 26.
Given a representation $\phi : L \to \mathfrak gl (V)$, the associative algebra (with 1) generated by $\phi(L)$ in End$V$ leaves invariant precisely the same subspaces as $L$. Therefore, all the usual results (e.g. Jordan-Holder Theorem) for modules over associative rings hold for $L$ as well.
My Question is,
- How the associative algebra generated by $\phi(L)$ leaves L invariant?
- From 1, how can we deduce that all the results for modules over associative rings hold for L as well?
Any help would be appreciated. Thanks.