Hochschild cohomology and deformation quantization of affine toric varieties

For an affine toric variety Spec(A), we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Under certain assumptions we compute the dimensions of the Hodge summands T(i)1(A), generalizing the existing results about the André-Quillen cohomology group T(1)1(A). We prove that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization.