Modular classes of skew algebroid relations

TL;DR

This paper introduces the concept of modular classes for skew algebroids and their relations, providing a unified framework that generalizes existing notions for Lie algebroids and Poisson maps.

Contribution

It defines modular classes for skew algebroids and relations, and explores their properties and applications, unifying various concepts in the theory of algebroids and Poisson geometry.

Findings

Existence of a homogeneous invariant 1-density characterizes modular skew algebroids.

Introduction of relative modular classes and their relation to holonomy.

Unified approach to modular classes of morphisms and Poisson maps.

Abstract

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E* which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e. mod(E)=0. Further, relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and that of a Poisson map.