Weight representations of Lie algebras

Following works of G. Benkart, D. Britten, S. Fernando, V. Futorny, A. Joseph, F. Lemire, and others, in 2000 O. Mathieu achieved a major breakthrough in representation theory by classifying the simple weight representations (with finite weight multiplicities) of semisimple Lie algebras. The next natural step in the study of weight representations is to look at indecomposable representations. In this talk we will discuss recent results related to the structure of certain indecomposable weight representations and connections to quiver theory and algebraic geometry. This is a joint work with Vera Serganova.