Mazorchuk V. Representation of graded Lie algebras and their generalizations

The structure of graded modules over graded Lie algebras is studied. A criterion for existence of non-trivial homomorphisms between generalized Verma modules and a criterion for simplicity of generalized Verma modules are obtained. The determinant of the generalized Shapovalov form is calculated. BGG-resolution for simple modules without highest weight is constructed. Twisted generalized Weyl algebras and orthogonal Gelfand-Zetlin algebras and their weight modules are defined and studied. The supports of simple weight modules over generalized Virasoro, Witt, Kaplansky and toroidal algebras are described.