Dirac maps and generalized complex geometry on homogeneous spaces

• Main Author: Milburn, Brett
• Language: ENG
• Published: ScholarWorks@UMass Amherst 2008
• Subjects: Mathematics
• Online Access: https://scholarworks.umass.edu/dissertations/AAI3336983

Generalized complex geometry [14] and, more generally, Dirac geometry [8], [9], unify several familiar geometric structures into one uniform viewpoint. In this thesis, we consider equivariant Dirac structures on homogeneous spaces as well as a new notion of morphisms between manifolds equipped with Dirac structures. We offer a description of equivariant Dirac structures on homogeneous spaces, which is then used to construct new examples of generalized complex structures. We consider Riemannian symmetric spaces, quotients of compact groups by closed connected subgroups of maximal rank, and nilpotent orbits in [special characters omitted] For each of these cases, we completely classify equivariant Dirac structures. Additionally, we consider equivariant Dirac structures on semisimple orbits in a semisimple Lie algebra. Here equivariant Dirac structures can be described in terms of root systems or by certain data involving parabolic subagebras. We provide two new notions of Dirac maps, giving two different Dirac categories. The first generalizes holomorphic and Poisson maps, while the second dual notion generalizes both symplectic and holomorphic maps. As an application, we consider Dirac groups (i.e. groups with Dirac structure such that group multiplication is a Dirac map). We explain the conditions under which a group with Dirac structure is a Dirac group. More precisely, we explain the data and conditions for a Dirac group. Dirac groups turn out to be a generalization of Poisson groups.