Last Multipliers for Multivectors With Applications to Poisson Geometry

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.2000 Math. Subject Classification: 58A15; 58A30; 34A26; 34C40.

Holomorphic last multipliers on complex manifolds

“…Now, according to the study of real last multipliers on smooth manifolds (see [9,10]), |α| 2 is a real last multiplier for the real vector field Z R if d(|α| 2 θ R ) = 0. By direct computation we have…”

“…The first named author of this work initiated their study on manifolds in [8] and [9], where he pointed out their relationship with the Liouville equation of transport. Since then, the last multipliers have been considered in various (non-flat) settings such as: Riemannian and Poisson geometry in [9], Lie-Poisson structures in [10], weighted manifolds in [11], Lie algebroids in [12].…”

A note on the Laplace operator for holomorphic functions on complex Lie groups

“…The computations are similar to the case of smooth manifolds, [2,3,4,5], or complex manifolds [6]. More precisely, consider a holomorphic vector field of the form…”

Last multipliers on Lie algebroids

“…Our study has been inspired by the results presented in [27], using the calculus on manifolds, especially the Lie derivative, a well-known tool for the geometry of vector fields. In [5] the previous theory is extended to general multivectors by means of the curl operator (i.e. a conjugate of the usual exterior derivative with respect to contraction of a given volume form).…”

“…The first section recalls the definition of last multipliers both for vector fields and multivectors and several previous results of [5].…”