# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.04947