The Hochschild cohomology of the enveloping algebra of a Lie-Rinehart pair

TL;DR

This paper introduces a spectral sequence that computes the Hochschild cohomology of the universal enveloping algebra of a Lie-Rinehart pair, linking it to Lie-Rinehart and Hochschild cohomologies.

Contribution

It presents a novel spectral sequence connecting Hochschild cohomology of the enveloping algebra with Lie-Rinehart and Hochschild cohomologies.

Findings

Spectral sequence converges to Hochschild cohomology of the enveloping algebra.

Second page involves Lie-Rinehart cohomology and Hochschild cohomology of the base algebra.

Provides a new computational tool for Hochschild cohomology in Lie-Rinehart contexts.

Abstract

Let $(S,L)$ be a Lie-Rinehart pair such that $L$ is $S$-projective and let $U$ be its universal enveloping algebra. The purpose of this paper is to present a spectral sequence which converges to the Hochschild cohomology of $U$ and whose second page involves the Lie-Rinehart cohomology of the pair and the Hochschild cohomology of $S$ with values on $U$.