Dirac actions and Lu's Lie algebroid

TL;DR

This paper generalizes the geometric structure of Poisson actions to Dirac actions of Dirac Lie groups, extending key results and classifying actions on homogeneous spaces, thus broadening the understanding of Lie algebroid interactions.

Contribution

It extends Jiang-Hua-Lu's result to Dirac actions and classifies Dirac actions on homogeneous spaces, generalizing Drinfeld's classification for Poisson Lie groups.

Findings

Cotangent and action algebroids form a matched pair for Dirac actions.

Complete classification of Dirac actions on homogeneous spaces $H/K$.

Generalization of Drinfeld's classification to Dirac Lie groups.

Abstract

Poisson actions of Poisson Lie groups have an interesting and rich geometric structure. We will generalize some of this structure to Dirac actions of Dirac Lie groups. Among other things, we extend a result of Jiang-Hua-Lu, which states that the cotangent Lie algebroid and the action algebroid for a Poisson action form a matched pair. We also give a full classification of Dirac actions for which the base manifold is a homogeneous space $H/K$, obtaining a generalization of Drinfeld's classification for the Poisson Lie group case.