Holonomy transformations for Lie subalgebroids

TL;DR

This paper introduces a new concept of holonomy transformations for Lie subalgebroids, extending classical ideas from foliation theory to a broader algebraic context, including singular cases.

Contribution

It defines an effective action of the minimal integration of Lie subalgebroids and describes it explicitly via conjugation by bisections, extending to singular subalgebroids.

Findings

Provides an explicit description of holonomy transformations for Lie subalgebroids.

Extends the notion to singular subalgebroids, broadening applicability.

Connects holonomy with conjugation by bisections in the Lie groupoid setting.

Abstract

Given a foliation, there is a well-known notion of holonomy, which can be understood as an action that differentiates to the Bott connection on the normal bundle. We present an analogous notion for Lie subalgebroids, consisting of an effective action of the minimal integration of the Lie subalgebroid, and provide an explicit description in terms of conjugation by bisections. The construction is done in such a way that it easily extends to singular subalgebroids, which provide our main motivation.