Abstract/Details

Dirac maps and generalized complex geometry on homogeneous spaces


2008 2008

Other formats: Order a copy

Abstract (summary)

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.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences, Dirac maps, Generalized complex, Homogeneous spaces
Title
Dirac maps and generalized complex geometry on homogeneous spaces
Author
Milburn, Brett
Number of pages
118
Publication year
2008
Degree date
2008
School code
0118
Source
DAI-B 69/12, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
9780549915690
Advisor
Mirkovic, Ivan
Committee member
Cattani, Eduardo; Kastor, David; Markman, Eyal
University/institution
University of Massachusetts Amherst
Department
Mathematics
University location
United States -- Massachusetts
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
3336983
ProQuest document ID
304566435
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/304566435
Access the complete full text

You can get the full text of this document if it is part of your institution's ProQuest subscription.

Try one of the following:

  • Connect to ProQuest through your library network and search for the document from there.
  • Request the document from your library.
  • Go to the ProQuest login page and enter a ProQuest or My Research username / password.