Remarks on the structure and integrability of LA-groups

TL;DR

This paper investigates the structure of LA-groups, linking their properties to VB-group representations up to homotopy, and establishes an equivalence with LA-matched pairs, with applications to integrability.

Contribution

It introduces a characterization of LA-groups via LA-matched pairs and explores their integrability, providing new insights into their structural properties.

Findings

LA-groups are equivalent to LA-matched pairs.

The Lie algebroid structure is determined by a homotopy action.

Examples of representations and actions are cataloged and analyzed for integrability.

Abstract

We study the structure of an LA-group identifying its underlying VB-group with a representation up to homotopy. We show that the Lie algebroid structure is determined by a complementary action up to homotopy of the Lie algebra of units. We identify the equations that the representation and the action need to verify in order to assemble into an LA-group, establishing an equivalence between LA-groups and LA-matched pairs. As an application, we catalog some extreme examples of representations and actions and comment on their integrability.