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

We construct a differential graded Lie algebra $\text{fg}$ controlling the Poisson deformations of an affine Poisson variety. We analyse $\text{fg}$ 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.