Poisson and Hochschild cohomology and the semiclassical limit

TL;DR

This paper explores the relationship between quantum algebras and their semiclassical limits, establishing conditions under which Hochschild cohomology deforms into Poisson cohomology, with explicit calculations for quantum matrices.

Contribution

It provides a criterion linking Hochschild and Poisson cohomology deformations and verifies this for quantum matrices, connecting quantum and classical algebraic structures.

Findings

Hochschild cohomology of quantum algebra deforms into Poisson cohomology of its limit.

Established a criterion for deformation of cohomologies in Koszul algebras.

Computed Hochschild and Poisson cohomology for 2x2 quantum matrices.

Abstract

Let $B$ be a quantum algebra possessing a semiclassical limit $A$. We show that under certain hypotheses $B^e$ can be thought of as a deformation of the Poisson enveloping algebra of $A$, and we give a criterion for the Hochschild cohomology of $B$ to be a deformation of the Poisson cohomology of $A$ in the case that $B$ is Koszul. We verify that condition for the algebra of $2\times 2$ quantum matrices and calculate its Hochschild cohomology and the Poisson cohomology of its semiclassical limit.