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 [Formula: see text], then [Formula: see text] with any Ii an ideal of L satisfying [Ii, Ik] = 0 if i ≠k and any Vj a (weight) submodule of V in such a way that for any j ∈J there exists a unique i ∈I such that IiVj ≠0, being Vj a weight module over Ii. 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.