Poisson structure on bi-graded spaces and Koszul duality, I. The classical case

Let $\mathbb R^{m|n}$ be the usual super space. It is known that the algebraic functions on $\mathbb R^{m|n}$ is a Koszul algebra, whose Koszul dual algebra, however, is not the set of functions on $\mathbb R^{n|m}$, due to the anti-commutativity of the corresponding variables. In this paper, we show that these two algebras are isomorphic to the algebraic functions of two $\mathbb{Z}\times\mathbb{Z}$-graded spaces. We then study the Poisson structures of these two spaces, and show that the quadratic Poisson structures are preserved under Koszul duality. Based on it, we obtain two isomorphic differential calculus structures, and if furthermore the Poisson structures are unimodular, then the associated Batalin-Vilkovisky algebra structures that arise on the Poisson cohomologies of these two $\mathbb{Z}\times\mathbb{Z}$-graded spaces are isomorphic as well.