# A new family of Poisson algebras and their deformations

Cesar Lecoutre, Susan J. Sierra

Research output: Contribution to journalArticlepeer-review

## 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.
Original language English 32-86 34 Nagoya mathematical journal 233 25 Sep 2017 https://doi.org/10.1017/nmj.2017.29 Published - Mar 2019

• math.RA
• math.AG

## Fingerprint Dive into the research topics of 'A new family of Poisson algebras and their deformations'. Together they form a unique fingerprint.

• ### Moduli Techniques in Graded Ring Theory and Their Applications

Sierra, S.

EPSRC

1/09/1528/02/19

Project: Research