No abstract

We reformulate notions from the theory of quasi-Poisson g-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups and prove that quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids.We then interpret this result within the theory of Dirac morphisms and multiplicative Manin pairs, to connect our work with more traditional approaches, and also to put it into a wider context suggesting possible generalizations.

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.

Abstract. We construct a generalization of Courant algebroids that are classified by the third cohomology group H 3 (A, V ), where A is a Lie Algebroid, and V is an A-module. We see that both Courant algebroids and E 1 (M) structures are examples of them. Finally we introduce generalized CR structures on a manifold, which are a generalization of generalized complex structures, and show that every CR structure and contact structure is an example of a generalized CR structure.

The Van Est homomorphism for a Lie groupoid G ⇒ M , as introduced by Weinstein-Xu, is a cochain map from the complex C ∞ (BG) of groupoid cochains to the Chevalley-Eilenberg complex C(A) of the Lie algebroid A of G. It was generalized by Weinstein, Mehta, and Abad-Crainic to a morphism from the Bott-Shulman-Stasheff complex Ω(BG) to a (suitably defined) Weil algebra W(A). In this paper, we will give an approach to the Van Est map in terms of the Perturbation Lemma of homological algebra. This approach is used to establish the basic properties of the Van Est map. In particular, we show that on the normalized subcomplex, the Van Est map restricts to an algebra morphism.