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 k* 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 k*. We prove that for any p, q, the analogous vector field for the SU (p, q ) case exists on an open subset of k*. (2) Each generic dressing orbit ψ(λ) 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 π(λ) on ψ(λ) 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.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/193755 |
| Date | January 2009 |
| Creators | Lamb, McKenzie Russell |
| Contributors | Foth, Philip A., Foth, Philip A., Foth, Philip A., Pickrell, Doug, Flaschka, Hermann, Glickenstein, David |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | English |
| Detected Language | English |
| Type | text, Electronic Dissertation |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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