2007/12/17 by Brett Milburn, Milburn, Brett
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.DG
paper · pdf · doi:10.48550/arxiv.0712.2636
openalex publication_date 2007/12/17 · arxiv created 2010/08/07 · arxiv updated 2010/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define two categories of Dirac manifolds, i.e. manifolds with complex Dirac structures. The first notion of maps I call Dirac maps, and the category of Dirac manifolds is seen to contain the categories of Poisson and complex manifolds as full subcategories. The second notion, dual-Dirac maps, defines a dual-Dirac category which contains presymplectic and complex manifolds as full subcategories. The dual-Dirac maps are stable under B-transformations. In particular we get two structures of a category on Hitchin'sgeneralized complex manifolds, i.e., two reasonable notions of generalized complex maps. We also generalize further to get categories of Dirac manifolds for which the Dirac structures lie in arbitrary exact Courant algebroids. As an example, we consider the case of a Lie group with a complex Dirac structure and establish conditions for which multiplication is a Dirac map.