Poincare Duality For Smooth Poisson Algebras And BV Structure On Poisson Cohomology

TL;DR

This paper establishes a duality between Poisson homology and cohomology for smooth Poisson algebras, introduces the concept of pseudo-unimodular structures, and links Batalin-Vilkovisky operators to these structures, generalizing prior geometric results.

Contribution

It proves a twisted Poincaré duality for Poisson (co)homology and characterizes when a BV operator exists in terms of pseudo-unimodularity, extending previous geometric frameworks.

Findings

Twisted Poincaré duality between Poisson homology and cohomology.

Existence of BV operator characterized by pseudo-unimodularity.

Description of modular derivation and BV operator via dual basis of Kähler differentials.

Abstract

Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular derivation, the Poisson cochain complex with values in any Poisson module is proved to be isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincar\'{e} duality is proved between the Poisson homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson structure is defined. It is proved that the Poisson cohomology as a Gerstenhaber algebra admits a Batalin-Vilkovisky operator inherited from some one of its Poisson cochain complex if and only if the Poisson structure is pseudo-unimodular. This generalizes the geometric version due to P. Xu. The modular derivation and Batalin-Vilkovisky operator are also described by using the dual basis of the K\"{a}hler differential module.