A new family of Poisson algebras and their deformations

Cesar Lecoutre, Susan J. Sierra

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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 languageEnglish
Pages (from-to)32-86
Number of pages34
JournalNagoya mathematical journal
Early online date25 Sep 2017
Publication statusPublished - Mar 2019


  • math.RA
  • math.AG

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