ℤ 2 -GRADED POISSON ALGEBRAS, THEIR DEFORMATIONS AND COHOMOLOGY IN LOW DIMENSIONS

We study ℤ 2 -graded Poisson structures defined on ℤ 2 -graded commutative polynomial algebras. In small-dimensional cases, we obtain the associated Poisson ℤ 2 -graded cohomology and in some cases, deformations of these Poisson brackets and P ∞-algebra structures. We highlight differences and analogies between this ℤ 2 -graded context and the non-graded context, by studying, for example, the links between Poisson cohomology and singularities.