Homological unimodularity and Calabi-Yau condition for Poisson algebras

TL;DR

This paper establishes a connection between unimodularity and Calabi-Yau properties in Poisson algebras, using duality theories and bimodule structures, with implications for Poisson geometry and algebraic structures.

Contribution

It introduces a new definition of unimodular Poisson algebras via the Poisson Picard group and proves the Calabi-Yau property for enveloping algebras of unimodular Poisson structures.

Findings

Twisted Poincaré duality derived from Serre invertible bimodule.

Unimodular Poisson algebras characterized by Poisson Picard group.

Enveloping algebra is Calabi-Yau if the Poisson structure is unimodular.

Abstract

In this paper, we show that the twisted Poincar\'e duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincar\'e duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi-Yau algebra if the Poisson structure is unimodular.