Differential calculus on bigraded spaces and Koszul duality

Abstract

Let ${\mathbb{R}}^{m|n}$ be the usual superspace. The algebra of functions on it is Koszul, but its Koszul dual is not graded commutative, and in particular not the algebra of functions on ${\mathbb{R}}^{n|m}$. This contrasts with the two extreme cases, in which the polynomial and exterior algebras are Koszul dual. We remedy this discrepancy by realizing the algebra $\mathcal{O}({\mathbb{R}}^{m|n})$ as the algebra of a bigraded space, and its Koszul dual, in the bigraded sense, is then also the algebra of a bigraded space. We then show that the differential calculus structures on these two spaces are isomorphic via Koszul duality. As an application, in the second half of the paper, we study Poisson structures on these spaces and show that for quadratic bigraded Poisson structures, Koszul duality also identifies the corresponding differential calculus structures on the Poisson (co)homology groups of these two spaces. Moreover, if the Poisson structures are unimodular, then the associated Batalin-Vilkovisky algebras on the Poisson cohomology groups are also isomorphic.