Length two extensions of modules for the Witt Algebra

TL;DR

This paper classifies length two extensions of tensor modules for the Witt algebra by computing cohomology, extending modules to Laurent polynomial actions, and explicitly determining 1-cocycles.

Contribution

It provides an explicit classification of module extensions for the Witt algebra using cohomological methods and module extension techniques.

Findings

Complete classification of length two extensions obtained.

Explicit computation of all possible 1-cocycles.

Extended modules to Laurent polynomial actions for analysis.

Abstract

In this paper, we establish an explicit classification of length two extensions of tensor modules for the Witt algebra using the cohomology of the Witt algebra with coefficients in the module of the space of homomorphisms between the two modules of interest. To do this we extended our module to a module that has a compatible action of the commutative algebra of Laurent polynomials in one variable. In this setting, we are be able to directly compute all possible 1-cocycles.