A new family of Poisson algebras and their deformations

TL;DR

This paper introduces a new family of Artin-Schelter regular algebras $R(n,a)$ that serve as quantisations of Poisson structures, generalising previous examples and revealing new Calabi-Yau cases with detailed geometric and ring-theoretic properties.

Contribution

The authors construct and analyze a novel family of Poisson algebra deformations, extending known examples and providing explicit descriptions of their geometric and algebraic structures.

Findings

Point modules parameterised by rational normal curves

Prime spectrum homeomorphic to Poisson spectrum

Explicit description of the spectrum as a union of strata

Abstract

Let $\Bbbk$ be a field of characteristic zero. For any positive integer $n$ and any scalar $a\in\Bbbk$, we construct a family of Artin-Schelter regular algebras $R(n,a)$, which are quantisations of Poisson structures on $\Bbbk[x_0,\dots,x_n]$. 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 $n\geq 4$. 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 ${\rm Spec}\ R(n,a)$ as a union of commutative strata.