Twists of graded Poisson algebras and related properties

TL;DR

This paper introduces a Poisson version of graded twists for polynomial rings, showing all graded Poisson structures are related to unimodular ones, and explores their properties and cohomologies.

Contribution

It generalizes the concept of graded twists to Poisson structures and characterizes all such structures on polynomial rings as twists of unimodular Poisson structures.

Findings

Every graded Poisson structure on a polynomial ring is a graded twist of a unimodular structure.

Computed Poisson cohomologies for certain quadratic Poisson structures in three variables.

Studied properties like rigidity, $PH^1$-minimality, and $H$-ozoneness of graded twists.

Abstract

We introduce a Poisson version of the graded twist of a graded associative algebra and prove that every graded Poisson structure on a connected graded polynomial ring $A:=\Bbbk[x_1,\ldots,x_n]$ is a graded twist of a unimodular Poisson structure on $A$, namely, if $\pi$ is a graded Poisson structure on $A$, then $\pi$ has a decomposition $$\pi=\pi_{unim} +\frac{1}{\sum_{i=1}^n {\rm deg} x_i} E\wedge {\mathbf m}$$ where $E$ is the Euler derivation, $\pi_{unim}$ is the unimodular graded Poisson structure on $A$ corresponding to $\pi$, and ${\mathbf m}$ is the modular derivation of $(A,\pi)$. This result is a generalization of the same result in the quadratic setting. The rigidity of graded twisting, $PH^1$-minimality, and $H$-ozoneness are studied. As an application, we compute the Poisson cohomologies of the quadratic Poisson structures on the polynomial ring of three variables when the potential is irreducible, but not necessarily having isolated singularities.