# Connecting Lie Algebra to modules

I am studying the section of Complete Reducibility of representations(Chapter 2) from Lie Algebras book by John Humphreys.

I got some remark which says: (Assuming $$L$$ is Lie algebra and $$V$$ is $$L$$-Module)

"Given a representation $$\phi: L \rightarrow gl(V)$$, 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."

I can't understand what "associative algebra with 1" they are referring to? What is 1?.

How all usual results for modules over associative rings hold for $$L$$?

If someone can explain, it will be a great help!

Here, $$1$$ is the identity map $$\operatorname{Id}\colon V\longrightarrow V$$. Now, consider the set $$\operatorname{End}V$$. It has a natural structure of an associative and unitary algebra (the unit being $$\operatorname{Id}$$). So, consider the subalgebra of $$\operatorname{End}V$$ spanned by $$\operatorname{Id}$$ and by the set $$\{\phi(X)\mid X\in L\}$$. That's the “associative algebra with $$1$$” that Humphreys has in mind.
• In this context, the Jordan-Hölder therem states that, for every finitely generated module $M$, there is an increasing sequence of sub-modules$$0=J_0\varsubsetneq J_1\varsubsetneq J_2\varsubsetneq \cdots\varsubsetneq J_n=M$$such that every quotient $J_k/J_{k-1}$ ($k\in\{1,2,\ldots,n\}$ is a simple module. Since this holds for modules over associative and unitary algebras, it also holds for Lie algebras, for the reason described by Humphreys. Jul 11, 2021 at 11:27
• So, we are considering an arbitrary module over associative ring (in this case, set generated by $\phi(L)$) and then we are somehow replacing our $\phi(L)$ by $L$ since $\phi(L)$ leaves the same invariant subspaces as $L$. From this, we are getting all usual results of modules to be true if module is over a lie algebra. Am I right? Jul 11, 2021 at 11:50