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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.

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vii, 146 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Statistics, 1977

Thesis (Ph.D.) -- University of Adelaide, Dept. of Statistics, 1977.

Thesis (Ph.D.)--The Johns Hopkins University, 2006. / Adviser: Gregory S. Chirikjian. Includes bibliographical references.

Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves 133-138).

Thesis (M. Phil.)--University of Hong Kong, 1998. / Includes bibliographical references (leaf 46).

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.