Dirac Actions and Lu’s Lie Algebroid

Abstract

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. Contents

Integrability of quotients in Poisson and Dirac geometry

“…Dirac-Lie groups subsume the study of Poisson groups and of the important Cartan-Dirac structures, which are central in the theory of q-Poisson manifolds [1]. In this context we obtain Theorem 6.19, which generalizes the results of [13] to Dirac homogeneous spaces [62,53].…”

“…This subsection has two main goals: we give a general criterion for the quotient of an action Lie algebroid to be integrable based on Theorem 3.2 and the construction of [25]; then we apply this criterion to the Dirac homogeneous spaces of [62,53] and obtain a result that generalizes the integration of Poisson homogeneous spaces in [13].…”

Poisson geometry from a Dirac perspective

“…In this theorem, neither G nor K are assumed to be connected or simply connected. The statement in Drinfeld's original paper [23] is slightly less precise; the version given here can be found in [54] as well as [48].…”

“…Poisson Lie group structures on compact Lie groups, and their homogeneous spaces, were studied by Lu-Weinstein [44], and independently by Levendorskii-Soibelman [36]. The theory of Poisson Lie groups, and their homogeneous spaces, has been generalized to (suitably defined) categories of Dirac manifolds; see [32], [52], [38], [54], [48].…”

“…The Dirac-geometric approach to Drinfeld's classification theorems may be found in the paper of Liu-Weinstein-Xu [42], with a generalization to Poisson homogeneous spaces for Poisson Lie groupoids. The exposition given here incorporates ideas from [38], [54] and [48]. The fact that the cotangent Lie algebroid T * M of a Poisson homogeneous space M = G/K is a reduction (G × l)/ /K of the action Lie algebroid (where l acts on G by the dressing action) was first observed by Jiang-Hua Lu [43,Theorem 4.7].…”