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