Lie-Rinehart and Hochschild cohomology for algebras of differential operators

TL;DR

This paper develops a spectral sequence connecting Lie-Rinehart and Hochschild cohomology for differential operator algebras, enabling explicit computations in complex algebraic structures.

Contribution

It introduces a spectral sequence framework that links Lie-Rinehart cohomology with Hochschild cohomology, facilitating new calculations for differential operator algebras.

Findings

Spectral sequence converges to Hochschild cohomology of universal enveloping algebra.

Explicit computation of Hochschild cohomology for differential operators tangent to a line arrangement.

Provides algebraic structure descriptions aiding cohomology calculations.

Abstract

Let $(S,L)$ be a Lie-Rinehart algebra such that $L$ is $S$-projective and let $U$ be its universal enveloping algebra. In this paper we present a spectral sequence which converges to the Hochschild cohomology of $U$ with values on a $U$-bimodule $M$ and whose second page involves the Lie-Rinehart cohomology of the algebra and the Hochschild cohomology of $S$ with values on $M$. After giving a convenient description of the involved algebraic structures we use the spectral sequence to compute explicitly the Hochschild cohomology of the algebra of differential operators tangent to a central arrangement of three lines.