Poisson derivations of a semiclassical limit of a family of quantum second Weyl algebras

Abstract: In previous paper, we studied deformations $A_{\alpha,\beta}$ of the second Weyl algebra and computed their derivations. In the present paper, we identify the semiclassical limits $\mathcal{A}{\alpha,\beta}$ of these deformations and compute their Poisson derivations. Our results show that the first Hochschild cohomology group HH$^1(A{\alpha,\beta})$ is isomorphic to the first Poisson cohomology group HP$^1( \mathcal{A}_{\alpha,\beta}).$