Reductions in Representation Theory of Lie Algebra of Vector Fields

We study representations of Lie algebras that do not have a Cartan subalgebra. The study of such representations required new techniques, one that we applied was to restrict the action of other algebraic structures that contain the Lie algebra. Our Lie algebras came from the vector fields on arbitrary varieties. We studied representations that admit the actions of the Lie algebra of vector field and the algebra of functions on the variety in a compatible way. More specifically, we studied two such classes of modules: gauge modules and Rudakov modules. We proved that gauge modules and Rudakov modules corresponding to simple glN-modules remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. We also studied the irreducibility of tensor products of Rudakov modules. Lastly, we present a complete description of tensor modules belonging to the de Rham complex as gl3-modules. We also realize these modules using GT-tableaux