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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,

  1. How the associative algebra generated by $\phi(L)$ leaves L invariant?
  2. 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.

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  1. Let $A$ be that associative algebra. It is, as a set, the set of all linear combinations of $\operatorname{Id}$ (which leave invariant every subspace of $V$) and endomorphisms of $V$ of the type $\phi(X)$, with $X\in L$. So, if $W\subset V$ is a subspace which is invariant with respect to the action of $L$ on $V$, then it is invariant with respect to the action of any element of $A$.
  2. Since $A$ is an associative algebra, every theorem about representations of associative algebras applies. And the invariant subspaces are the same in both cases. So, studying theorems about invariant spaces with respect to the action of $L$ is the same thing as studying invariant subspaces with respect to the action of $A$, to which you can apply those theorems.
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  • $\begingroup$ Thanks for your answer. But I still have trouble understanding 2. The book says we can apply theorems for modules over rings to modules over Lie algebras. Could you give me more hints to relate this statement to your answer? $\endgroup$ – user Mar 4 at 5:44

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