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

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${}^1({\mathcal{A}}_{\alpha ,\beta })\mathrm{.}$