2010/09/03 by Madeleine Jotz, Jotz, M. · 1 citation
Mathematics · #22A22 #53D17 #70H45 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1009.0713
openalex publication_date 2010/09/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let (G\rr P, \mathsf DG) be a Dirac groupoid. We show that there are natural Lie algebroid structures on the units \lie A(\mathsf DG) and on the core I^\tg(\mathsf DG) of the multiplicative Dirac structure. In the Poisson case, the Lie algebroid A^*G is isomorphic to \lie A(\mathsf DG) 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 DG)→ P associated to any (almost) Dirac structure. In the Dirac case, \lie B(\mathsf DG) 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⊕ 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 DG-homogeneous Dirac structures on G/H are related to suitable Dirac structures in \lie B(\mathsf DG). In the case of almost Dirac structures, we find Lagrangian subspaces of \lie B(DG), that are invariant under an induced action of the bisections of H on \lie B(\mathsf DG).