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.

11 pages