Dirac Lie groups

A classical theorem of Drinfel'd states that the category of simply connected Poisson Lie groups H is isomorphic to the category of Manin triples (d, g, h), where h is the Lie algebra of H.In this paper, we consider Dirac Lie groups, that is, Lie groups H endowed with a multiplicative Courant algebroid A and a Dirac structure E ⊆ A for which the multiplication is a Dirac morphism.It turns out that the simply connected Dirac Lie groups are classified by so-called Dirac Manin triples.We give an explicit construction of the Dirac Lie group structure defined by a Dirac Manin triple, and develop its basic properties.Key words.Poisson Lie Groups, Multiplicative Dirac Structures, Multiplicative Courant algebroids, Lie groupoids, Lie bialgebras, Manin triples, Multiplicative Manin pairs, quasi-Poisson geometry, Group valued moment maps.AMS subject classifications.53D17 (Primary), 17B62, 53D20.0. Introduction.Dirac structures were introduced by T. Courant [6] as a common framework for closed 2-forms and Poisson structures on manifolds.He showed that the integrability condition dω = 0 for 2-forms and [π, π] = 0 for bivector fields admits a common generalization to an integrability condition on Lagrangian subbundles E ⊆ TM = T M ⊕ T * M relative to a certain bracket on Γ(TM ).Liu-Weinstein-Xu [21] generalized Courant's original set-up, replacing TM with a more general notion of a Courant algebroid A → M .The theory of Courant algebroids and Dirac structures was clarified and simplified in the work of Dorfman [7], Ševera [36, Letter no.7], Roytenberg [34], Uchino [39], and others.It has recently gained attention through the development of generalized complex geometry [11, 13], and it provides a unified setting for various types moment maps [1, 5].A Poisson Lie group is a Lie group H, equipped with a Poisson structure such that the multiplication map is Poisson.To extend this definition to Dirac geometry, it is required that the Courant algebroid A itself has a multiplicative structure.As suggested by Mehta [27] and further explored in [20], we require that A carries a VBgroupoid structure A ⇉ g over the group H ⇉ pt, in such a way that the groupoid multiplication is a Courant morphism Mult A : A×A A. (For the standard Courant algebroid A = TH this structure is automatic, with g = h * .)One then has a notion of a multiplicative Dirac structure E ⊆ A. In the case of TH these were classified in the work of Ortiz [30] and Jotz [15], independently.While [15,30] refer to multiplicative Dirac structures as Dirac Lie group structures, we will reserve this latter term for the case that the multiplication map is a Dirac morphism (i.e a morphism of Manin pairs as in [5]).For A = TH, only the Poisson Lie group structures are Dirac Lie group structures in our sense, but many more examples are obtained by considering more general Courant algebroids.These include the well known Cartan-Dirac structure (cf.[1] and references therein), and the examples in Section 5 of [16].One of the goals of this paper is to develop the theory of Dirac Lie groups in this setting.The super-geometric interpretation of Dirac Lie group structures was previously studied

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