# Representations of the Lie algebra of vector fields on a sphere

## TL;DR

This paper investigates modules over the Lie algebra of vector fields on a sphere, constructing simple modules and establishing an equivalence with finite-dimensional rational L_2-modules, revealing new algebraic structures.

## Contribution

It introduces a new class of simple modules for the Lie algebra of vector fields on a sphere and proves their categorical equivalence to GL_2-modules.

## Key findings

- Constructed finitely generated simple modules over the algebra of functions on the sphere.
- Proved the monoidal category generated by these modules is equivalent to finite-dimensional rational GL_2-modules.
- Established a categorical equivalence linking vector field modules to GL_2-representations.

## Abstract

For an affine algebraic variety $X$ we study a category of modules that admit compatible actions of both the algebra of functions on $X$ and the Lie algebra of vector fields on $X$. In particular, for the case when $X$ is the sphere $\mathbb{S}^2$, we construct a set of simple modules that are finitely generated over $A$. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational $\mathrm{GL}_2$-modules.