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

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