The Low-Dimensional Algebraic Cohomology of the Virasoro Algebra

TL;DR

This paper proves that the third algebraic cohomology of the Virasoro algebra with certain modules is one-dimensional, providing detailed proofs and extending known results about related algebras.

Contribution

It establishes the one-dimensionality of the third algebraic cohomology of the Virasoro algebra with the adjoint and trivial modules, filling gaps in previous proofs.

Findings

Third algebraic cohomology of Virasoro algebra is one-dimensional.

Third algebraic cohomology of Witt algebra with trivial module is one-dimensional.

Provides detailed algebraic proof independent of topology.

Abstract

The main aim of this article is to prove the one-dimensionality of the third algebraic cohomology of the Virasoro algebra with values in the adjoint module. We announced this result in a previous publication with only a sketch of the proof. The detailed proof is provided in the present article. We also show that the third algebraic cohomology of the Witt and the Virasoro algebra with values in the trivial module is one-dimensional. We consider purely algebraic cohomology, i.e. our results are independent of any topology chosen. The vanishing of the third algebraic cohomology of the Witt algebra with values in the adjoint module has already been proven by Ecker and Schlichenmaier.