Batalin-Vilkovisky algebra structure on Poisson manifolds with diagonalizable modular symmetry

TL;DR

This paper establishes a Batalin-Vilkovisky algebra structure on Poisson cohomology for manifolds with diagonalizable modular vector fields, extending previous results and demonstrating its invariance under deformation quantization and Koszul duality.

Contribution

It introduces a new BV algebra structure on Poisson cohomology under diagonalizable modular symmetry, generalizing prior unimodular cases and showing its stability under key algebraic transformations.

Findings

BV algebra structure exists under diagonalizable modular vector fields

Structure is preserved under Kontsevich's deformation quantization

BV algebra is also preserved by Koszul duality in polynomial cases

Abstract

We study the ``twisted" Poincar\'e duality of smooth Poisson manifolds, and show that, if the modular vector field is diagonalizable, then there is a mixed complex associated to the Poisson complex, which, combining with the twisted Poincar\'e duality, gives a Batalin-Vilkovisky algebra structure on the Poisson cohomology. This generalizes the previous results obtained by Xu for unimodular Poisson manifolds. We also show that the Batalin-Vilkovisky algebra structure is preserved under Kontsevich's deformation quantization, and in the case of polynomial algebras it is also preserved by Koszul duality.