Аннотация:Let g\mathfrak {g} denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let h\mathfrak {h} denote a Cartan subalgebra of g\mathfrak {g}. In this paper we study finitely generated g\mathfrak {g}-modules that decompose into direct sums of finite dimensional h\mathfrak {h}-weight spaces. We show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules, those we call torsion free modules. We also show that if g\mathfrak {g} is a simple Lie algebra that admits a torsion free module, then g\mathfrak {g} is of type AA or CC.
