Weight modules over split Lie algebras

Abstract

We study the structure of weight modules $V$ with restrictions neither on the dimension nor on the base field, over split Lie algebras $L$. We show that if $L$ is perfect and $V$ satisfies $LV=V$ and $\mathcal{Z}(V)=0$, then $$L=\underset{i\in I}{⨁}{I}_{i}andV=\underset{j\in J}{⨁}{V}_j$$ with any $I_i$ an ideal of $L$ satisfying $[I_i,I_k]=0$ if $i\ne k$, and any $V_j$ a (weight) submodule of $V$ in such a way that for any $j\in J$ there exists a unique $i\in I$ such that $I_i{V}_j\ne 0,$ being $V_j$ a weight module over $I_i$. Under certain conditions, it is shown that the above decomposition of $V$ is by means of the family of its minimal submodules, each one being a simple (weight) submodule.