Infinitesimal objects associated to Dirac groupoids and their homogeneous spaces

Abstract

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 $A^{\ast }G$ 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 $TG\oplus T^{\ast }G$. 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 $\lie B(\mathsf D_G)$.

Improved version of the paper (notation changed, tipos corrected), part about integrability criterion for multiplicative almost Dirac structures added

References in corpus (3)

• Linear and multiplicative 2-forms
• Foliated groupoids and their infinitesimal data
• The leaf space of a multiplicative foliation

Cited by in corpus (4)

• Foliated groupoids and their infinitesimal data
• Glanon groupoids
• Dorfman connections and Courant algebroids
• The leaf space of a multiplicative foliation