Commutative Poisson algebras from deformations of noncommutative algebras

Abstract

It is well-known that a formal deformation of a commutative algebra ${\mathcal A}$ leads to a Poisson bracket on ${\mathcal A}$ and that the classical limit of a derivation on the deformation leads to a derivation on ${\mathcal A}$, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalisation of it for formal deformations of an arbitrary noncommutative algebra ${\mathcal A}$. The deformation leads in this case to a Poisson algebra structure on $\Pi({\mathcal A}):=Z({\mathcal A})\times({\mathcal A}/Z({\mathcal A}))$ and to the structure of a $\Pi({\mathcal A})$-Poisson module on ${\mathcal A}$. The limiting derivations are then still derivations of ${\mathcal A}$, but with the Hamiltonian belong to $\Pi({\mathcal A})$, rather than to ${\mathcal A}$. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantised Grassmann algebra.