Equivalence of the categories of modules over Lie algebroids

TL;DR

This paper explores the relationships between Lie algebroids and their categories of modules, introducing an equivalence relation that preserves the structure of infinitesimal actions and gauge equivalences.

Contribution

It introduces a new equivalence relation for integrable Lie algebroids and demonstrates the categorical equivalence of their modules and Hamiltonian categories.

Findings

Equivalent Lie algebroids have equivalent categories of infinitesimal actions

Gauge equivalent Dirac structures have equivalent Hamiltonian categories

The new equivalence relation preserves categorical structures

Abstract

We study geometric representation theory of Lie algebroids. A new equivalence relation for integrable Lie algebroids is introduced and investigated. It is shown that two equivalent Lie algebroids have equivalent categories of infinitesimal actions of Lie algebroids. As an application, it is also shown that the Hamiltonian categories for gauge equivalent Dirac structures are equivalent as categories.