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 $$\hbox{$L =\bigoplus\limits_{i\in I} I_{i}$ and $V = \bigoplus\limits_{j \in J} V_{j}$}$$ with any $I_{i}$ an ideal of $L$ satisfying $[I_{i},I_{k}]=0$ if $i \neq k$, 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 $I_iV_j \neq 0,$ 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.