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!
Best Answer
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.