Representations of Lie algebras of vector fields on affine varieties

TL;DR

This paper introduces gauge and Rudakov modules over the Lie algebra of vector fields on affine varieties, generalizing tensor densities and induced modules, with proven simplicity and a pairing between them.

Contribution

It defines new classes of modules for Lie algebras of vector fields on affine varieties, extending previous concepts and establishing their fundamental properties.

Findings

Gauge modules generalize tensor density modules.

Rudakov modules extend induced modules over derivation Lie algebras.

Simplicity theorems and a pairing between the modules are established.

Abstract

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. We prove general simplicity theorems for these two types of modules and establish a pairing between them.