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Description

Let $k$ be an algebraically closed field of characteristic zero. For any positive integer $n$, we construct a Calabi-Yau algebra $R(n)$, which induces a Poisson deformation of $k[x_1, \dots, x_n]$ and generalises a construction given by Pym when $n=4$. Modulo scalars, the graded automorphism group of $R(n)$ is isomorphic to $k$, and we consider not only $R(n)$ but its Zhang twist $R(a,n)$ by the automorphism corresponding to $a$. Each $R(a,n)$ induces a Poisson structure on $k[x_1, \dots, x_n]$ in the semiclassical limit, and we study this structure. We show that the Poisson spectrum of the limit is homeomorphic to $\operatorname{Spec} R(a,n)$, and explicitly describe $\operatorname{Spec} R(a,n)$ as a union of commutative strata. This is joint work with Cesar Lecoutre.