Abstract/Details

Ginzburg -Weinstein isomorphisms for pseudo-unitary groups


2009 2009

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Abstract (summary)

Ginzburg and Weinstein proved in [GW92] that for a compact, semisimple Lie group K endowed with the Lu-Weinstein Poisson structure, there exists a Poisson diffeomorphism from the dual Poisson Lie group K* to the dual [special characters omitted] of the Lie algebra of K endowed with the Lie-Poisson structure. We investigate the possibility of extending this result to the pseudo-unitary groups SU(p, q), which are semisimple but not compact.

The main results presented here are the following. (1) The Ginzburg-Weinstein proof hinges on the existence of a certain vector field X on [special characters omitted]. We prove that for any p, q, the analogous vector field for the SU(p, q) case exists on an open subset of [special characters omitted]. (2) Each generic dressing orbit [special characters omitted] in the Poisson dual AN can be embedded in the complex flag manifold K/T. We show that for SU(1, 1) and SU(1, 2), the induced Poisson structure [special characters omitted] on [special characters omitted] extends smoothly to the entire flag manifold. (3) Finally, we prove the Ginzburg-Weinstein theorem for the SU(1, 1) case in two different ways: first, by constructing the vector field X in coordinates and proving that it satisfies the necessary properties, and second, by adapting the approach of [FR96] to the SU(1, 1) case.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences; Geometry; Ginzburg-Weinstein isomorphisms; Lie groups; Poisson
Title
Ginzburg -Weinstein isomorphisms for pseudo-unitary groups
Author
Lamb, McKenzie Russell
Number of pages
119
Publication year
2009
Degree date
2009
School code
0009
Source
DAI-B 70/08, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
9781109324099
Advisor
Foth, Philip A.
Committee member
Flaschka, Hermann; Glickenstein, David; Pickrell, Doug M.
University/institution
The University of Arizona
Department
Mathematics
University location
United States -- Arizona
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
3369797
ProQuest document ID
304825773
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/304825773/abstract
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