An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module

TL;DR

This paper provides a simple, elementary proof demonstrating the vanishing of the second cohomology for the Witt and Virasoro algebras with adjoint coefficients, confirming their rigidity.

Contribution

It offers the first elementary and more elegant proof of a key cohomological property of these algebras, improving upon the original cumbersome calculation-based proof.

Findings

Second cohomology vanishes for Witt and Virasoro algebras

Confirms infinitesimal and formal rigidity of these algebras

Provides a simpler proof compared to previous work

Abstract

By elementary and direct calculations the vanishing of the (algebraic) second Lie algebra cohomology of the Witt and the Virasoro algebra with values in the adjoint module is shown. This yields infinitesimal and formal rigidity or these algebras. The first (and up to now only) proof of this important result was given 1989 by Fialowski in an unpublished note. It is based on cumbersome calculations. Compared to the original proof the presented one is quite elegant and considerably simpler.