A differential graded Lie algebra controlling the Poisson deformations of an affine Poisson variety

TL;DR

This paper constructs a differential graded Lie algebra that governs Poisson deformations of affine Poisson varieties and provides explicit cohomology descriptions for certain toric cases.

Contribution

It introduces a new differential graded Lie algebra framework for controlling Poisson deformations and analyzes specific cases of affine Gorenstein toric varieties.

Findings

Construction of a dg Lie algebra controlling Poisson deformations.

Explicit descriptions of Hochschild cohomology for 3D affine Gorenstein toric varieties.

Analysis of the dg Lie algebra in the case of affine Gorenstein toric Poisson varieties.

Abstract

We construct a differential graded Lie algebra $\fg$ controlling the Poisson deformations of an affine Poisson variety. We analyse $\fg$ in the case of affine Gorenstein toric Poisson varieties. Moreover, explicit description of the second and third Hochschild cohomology groups is given for three-dimensional affine Gorenstein toric varieties.