A new family of Poisson algebras and their deformations

Abstract

Let $\mathbb{k}$ be a field of characteristic zero. For any positive integer $n$ and any scalar $a\in \mathbb{k}$, we construct a family of Artin-Schelter regular algebras $R(n,a)$, which are quantisations of Poisson structures on $\mathbb{k}[x_0,\dots ,x_n]$. This generalises an example given by Pym when $n=3$. For a particular choice of the parameter $a$ we obtain new examples of Calabi-Yau algebras when $n\ge 4$. We also study the ring theoretic properties of the algebras $R(n,a)$. We show that the point modules of $R(n,a)$ are parameterised by a bouquet of rational normal curves in ${\mathbb{P}}^n$, and that the prime spectrum of $R(n,a)$ is homeomorphic to the Poisson spectrum of its semiclassical limit. Moreover, we explicitly describe $\mathrm{Spec}R(n,a)$ as a union of commutative strata.

34 pages; To appear in Nagoya Mathematical Journal