Last multipliers for multivectors with applications to Poisson geometry

TL;DR

This paper extends the concept of last multipliers from vector fields to multivectors, exploring their algebraic structure and applications in Poisson geometry, including a new measure of how non-exact a Poisson structure is.

Contribution

It introduces last multipliers for multivectors, characterizes them via Witten and Marsden differentials, and applies these concepts to analyze Poisson structures and cohomology.

Findings

Last multipliers for multivectors are characterized algebraically.

Application to Poisson bivectors quantifies their deviation from exactness.

Introduction of exact Poisson cohomology for unimodular structures.

Abstract

The theory of the last multipliers as solutions of the Liouville's transport equation, previously developed for vector fields, is extended here to general multivectors. Characterizations in terms of Witten and Marsden differentials are reobtained as well as the algebraic structure of the set of multivectors with a common last multiplier, namely Gerstenhaber algebra. Applications to Poisson bivectors are presented by obtaining that last multipliers count for ''how far away'' is a Poisson structure from being exact with respect to a given volume form. The notion of exact Poisson cohomology for an unimodular Poisson structure on $IR^{n}$ is introduced.