Dirac Lie groups, Dirac homogeneous spaces and the theorem of Drinfeld (Q2884653)

This paper generalizes the notions of Poisson Lie group and Poisson homogeneous spaces to their corresponding Dirac structures. Let us recall that Poisson Lie groups were introduced by Drinfeld and first studied by Lu and Weinstein, and then extended to Poisson Lie groupoids. Moreover, it is well-known that Poisson homogeneous spaces of Poisson Lie groups are in correspondence with some subspaces of the direct sum of the Lie algebra with its dual. In this work, the author proves that this correspondence (which is in fact the theorem of Drinfeld) can be generalized in Dirac category. After some recalls about Dirac manifolds and actions of Lie groups, geometric properties of Dirac Lie groups are proved and Dirac homogeneous spaces are introduced. The main theorem is then obtained. The special case of Dirac Lie groups where the characteristic subgroup is closed in the Lie group is also studied.