For an irreducible affine variety X over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on X—gauge modules and Rudakov modules, which admit a compatible action of the algebra of functions. Gauge modules are generalizations of modules of tensor densities whose construction was inspired by non-abelian gauge theory. Rudakov modules are generalizations of a family of induced modules over the Lie algebra of derivations of a polynomial ring studied by Rudakov [23]. We prove general simplicity theorems for these two types of modules and establish a pairing between them.
Israel Journal of Mathematics
School of Mathematics and Statistics

Billig, Y, Futorny, V. (Vyacheslav), & Nilsson, J. (Jonathan). (2019). Representations of Lie algebras of vector fields on affine varieties. Israel Journal of Mathematics. doi:10.1007/s11856-019-1909-z