ABSTRACT
Let G be a complex Lie (super)algebra with a Cartan subalgebra H. The problem of classification of irreducible G-modules is extremely difficult and is only known for the case of G = sl(2) ([3]). A G-module V is called H-diagonalizable (weight) if
V = ⊕λ∈H∗Vλ, where Vλ = {v ∈ V |hv = λ(h)v,∀h ∈ H}. Denote by K(G) the category of all such modules with finite-dimensional weight spaces (dimVλ < ∞,∀λ ∈ H∗). We give an overview of the classification problem of irreducible modules
in the category K(G) for the following types of G: • Reductive finite-dimensional Lie algebras. • Basic classical Lie superalgebras. • Affine Lie algebras. • Affine Lie superalgebras.
