1-cocycles of the Witt algebra with coefficients in tensor product of modules

TL;DR

This paper classifies 1-cocycles of the Witt algebra with tensor density modules and uses these to determine Lie bialgebra structures on related infinite-dimensional Lie algebras.

Contribution

It provides a complete classification of 1-cocycles for the Witt algebra with tensor density modules and applies this to identify Lie bialgebra structures.

Findings

Classified all 1-cocycles of the Witt algebra with tensor density modules.

Reproduced a known theorem in a special case.

Determined Lie bialgebra structures on certain infinite-dimensional Lie algebras.

Abstract

In this paper, we classify 1-cocycles of the Witt algebra with coefficients in the tensor product of two arbitrary tensor density modules. In a special case, we recover a theorem originally established by Ng and Taft in \cite{NT}. Furthermore, by these 1-cocycles, we determine Lie bialgebra structures over certain infinite-dimensional Lie algebras containing the Witt algebra.