Cohomology Of Quadratic Possion Structures In Dimension Two

First we recall the notions of Poisson algebras and Poisson(co)homology.Second we introduce the notion of differential KS structure and show that Poisson cohomology complex together with Poisson homology complex form a differential KS structure.Last based on the work of N.Nakanishi,we determine the Gerstenhaber algebra structure over the Poisson cohomology of plane quadratic Poisson structures,and as a consequence,the Poisson coho-mology of the elliptic and hyperbolic structures are isomorphic as Gerstenhaber algebras.