Twists of graded Poisson algebras and related properties

Document Type

Article

Publication Date

1-1-2025

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:=k[x1,…,xn] is a graded twist of a unimodular Poisson structure on A, namely, if π is a graded Poisson structure on A, then π has a decomposition [Formula presented] where E is the Euler derivation, πunim is the unimodular graded Poisson structure on A corresponding to π, and m is the modular derivation of (A,π). This result is a generalization of the same one in the quadratic setting. The rigidity of graded twisting, PH1-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 an isolated singularity.

Publication Source (Journal or Book title)

Journal of Geometry and Physics

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