On the third cohomology of the Lie algebra of vector fields on weighted densities on R

TL;DR

This paper computes the third cohomology of the Lie algebra of vector fields on the real line with coefficients in weighted densities, providing explicit cocycles and advancing understanding of its algebraic structure.

Contribution

It explicitly calculates the third cohomology groups of Vect(ℝ) with density coefficients and provides explicit cocycles, a novel contribution to the field.

Findings

Explicit third cohomology groups computed

Provides explicit cocycles spanning these cohomology spaces

Enhances understanding of Lie algebra of vector fields on ℝ

Abstract

Let Vect($\mathbb{R}$) be the Lie algebra of smooth vector fields on $\mathbb{R}$ and $\mathbb{F}_{\lambda}$ be the space of $\lambda$-densities on $\mathbb{R}$. Vect($\mathbb{R}$) acts on $\mathbb{F}_{\lambda}$ by Lie derivative. In this paper, we compute the third differential cohomology of the Lie algebra Vect($\mathbb{R}$) with coeffcients in the space $\mathbb{F}_{\lambda}.$ Explicit cocycles spanning these cohomology spaces are given.