$\mathfrak{sl}(2)$-cohomology on multidifferential operators space

Abstract

In this paper, we consider the known question of the computation of the cohomology spaces for the natural action of the Lie algebra $\mathfrak{sl}(2)$ acting on the space $\Dlm$ of multidifferential operators on the line. We first show that these cohomology spaces are entirely characterized by a linear map $f$ between two explicit finite dimensional spaces, then we prove that this map has always maximal rank, allowing to get explicit recursive expression for their dimensions.

29 pages