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

TL;DR

The paper introduces a Koszul duality framework for bi-graded spaces, realizing the algebra of functions on $k^{m||n}$ and its dual on $k^{n\wedge m}$ as Koszul pairs. It shows that quadratic Poisson structures are preserved under this duality and develops a corresponding differential calculus for both spaces, with isomorphisms between their Poisson cohomology and mixed Poisson homology. Unimodularity is shown to be preserved under duality, yielding isomorphisms of Batalin–Vilkovisky algebras on the respective Poisson cohomologies. The results generalize known dualities for purely graded or purely super cases to the bi-graded setting and pave the way for deformation quantization in the Koszul dual context.

Abstract

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.

Theorems & Definitions (73)

• Theorem 1.1: Theorem \ref{['newmainthm1']}
• Theorem 1.2: Theorem \ref{['mainthm22']}
• Theorem 1.3: Theorem \ref{['mainthm33']}
• Remark 1.4: Novelty of the paper
• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6: Theorem \ref{['mainthm1']}