Higher Lie algebra actions on Lie algebroids

TL;DR

This paper studies strict actions of differential graded Lie algebras on Lie algebroids, showing how these actions can be integrated into group actions within the frameworks of Lie algebroids and Lie groupoids.

Contribution

It introduces a framework for integrating strict Lie algebra actions on Lie algebroids into group actions using Mackenzie's doubles.

Findings

Strict actions of DGLAs on Lie algebroids can be integrated into group actions.

The integration process utilizes Mackenzie's doubles framework.

Results connect Lie algebra actions with Lie groupoid symmetries.

Abstract

We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids. When the action is strict, we show that it integrates to group actions in the categories of Lie algebroids and Lie groupoids (i.e. actions of LA-groups and 2-groups). We perform the integration in the framework of Mackenzie's doubles.