Length Two Extensions of Modules for the Witt Algebra

In this thesis, we analyse length two extensions of tensor modules for the Witt algebra. In 1992, a classification of these modules was found by Martin and Piard, though no explicit form of the extensions were given. In this thesis, we establish an explicit classification of these modules using a different approach. As we will show, each module extension is classified by a 1-cocycle from the cohomology of the Witt algbera with coefficients in the module of the space of homomorphisms between the two tensor modules of interest. To use this, we first extended our module to a module that has a compatible action with the commutative algebra of Laurent polynomials in one variable. In this setting, we are able to determine the possible structure of a 1-cocycle and from here, we will be able to directly compute all possible 1-cocycles.