Ginzburg -Weinstein isomorphisms for pseudo-unitary groups
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.