Infinitesimal objects associated to Dirac groupoids and their homogeneous spaces

Details Infinitesimal objects associated to Dirac groupoids and their homogeneous spaces Infinitesimal objects associated to Dirac groupoids and their homogeneous spaces

2010-09-03

arxiv.org/abs/1009.0713v4

Mathematics

Differential Geometry

Improved version of the paper (notation changed, tipos corrected), part about integrability criterion for multiplicative almos

Scientific paper

Let $(G\rr P, \mathsf D_G)$ be a Dirac groupoid. We show that there are natural Lie algebroid structures on the units $\lie A(\mathsf D_G)$ and on the core $I^\tg(\mathsf D_G)$ of the multiplicative Dirac structure. In the Poisson case, the Lie algebroid $A^*G$ is isomorphic to $\lie A(\mathsf D_G)$ and in the case of a closed 2-form, the $IM$-2-form is equivalent to the core algebroid that we find. We construct a vector bundle $\lie B(\mathsf D_G)\to P$ associated to any (almost) Dirac structure. In the Dirac case, $\lie B(\mathsf D_G)$ has the structure of a Courant algebroid that generalizes the Courant algebroid defined by the Lie bialgebroid of a Poisson groupoid. This Courant algebroid structure is induced in a natural way by the ambient Courant algebroid $TG\oplus T^*G$. The already known theorems about one-one correspondence between the homogeneous spaces of a Poisson Lie group (respectively Poisson groupoid, Dirac Lie group) and suitable Lagrangian subspaces of the Lie bialgebra or Lie bialgebroid are generalized to a classification of the Dirac homogeneous spaces of a Dirac groupoid. $\mathsf D_G$-homogeneous Dirac structures on $G/H$ are related to suitable Dirac structures in $\lie B(\mathsf D_G)$. In the case of almost Dirac structures, we find Lagrangian subspaces of $\lie B(D_G)$, that are invariant under an induced action of the bisections of $H$ on $\lie B(\mathsf D_G)$.

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