# Journal of Noncommutative Geometry

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**Volume 9, Issue 3, 2015, pp. 665–696**

**DOI: 10.4171/JNCG/204**

Published online: 2015-10-29

Poisson and Hochschild cohomology and the semiclassical limit

Matthew Towers^{[1]}(1) University of Kent at Canterbury, Canterbury, UK

Let $\mathsf A$ be a quantum algebra possessing a semiclassical limit $A$. We show that under certain hypotheses $\mathsf A^e$ can be thought of as a deformation of the Poisson enveloping algebra of $A$, and we give a criterion for the Hochschild cohomology of $\mathsf A$ to be a deformation of the Poisson cohomology of $A$ in the case that $\mathsf A$ 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.

*Keywords: *Poisson cohomology, Hochschild cohomology, semiclassical limit, Koszul algebra, deformations

Towers Matthew: Poisson and Hochschild cohomology and the semiclassical limit. *J. Noncommut. Geom.* 9 (2015), 665-696. doi: 10.4171/JNCG/204