Dirac Lie groups, Dirac homogeneous spaces and the Theorem of Drinfeld

The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfel$'$d (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra associated to the Poisson Lie group is proved to hold in this more general setting

Dirac groupoids and Dirac bialgebroids

We describe infinitesimally Dirac groupoids via geometric objects that we call Dirac bialgebroids. In the two well-understood special cases of Poisson and presymplectic groupoids, the Dirac bialgebroids are equivalent to the Lie bialgebroids and IM-$2$-forms, respectively. In the case of multiplicative involutive distributions on Lie groupoids, we find new properties of infinitesimal ideal systems.Comment: New expanded version; the construction of the Manin pair associated to an LA-Dirac structure has moved from arXiv:1209.6077 to here. Added background on double vector bundles, VB-algebroids and 2-term representations up to homotop

Invariant generators for generalized distributions

The existence of invariant generators for distributions satisfying a compatibility condition with the symmetry algebra is proved