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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The results exhibited in this thesis are related to Schrodinger operators in three dimensions and are subdivided in two parts based on two published papers, [15] and [14]. A variant of Turbiner's conjecture is proved in the first paper while a partial classification of quasi-exactly solvable Lie algebras of first order differential operators in dimension three is exhibited in the second paper. This classification is then used to construct new quasi-exactly solvable Schrodinger operators in three dimensions. / Turbiner's conjecture posits that, for a Lie algebraic Schrodinger operator in dimension two, the Schrodinger equation is separable if the underlying metric is locally flat. This conjecture is false in general. However, if the generating Lie algebra is imprimitive and if a certain compactness requirement holds, Rob Milson proved that in two dimensions, the Schrodinger equation separates in a Cartesian or polar coordinate system. In [15], the first paper included in this thesis, a similar theorem is proved in three variables. The imprimitivity and compactness hypotheses are still necessary and another condition, related to the underlying metric, must be imposed. In three dimensions, the separation is only partial and the separation will occur in either a spherical, cylindrical or Cartesian coordinate system. / In the second paper [14], a partial classification of quasi-exactly solvable Lie algebras of first order differential operators is performed in three dimensions. Such a classification was known in one and two dimensions but the three dimensional case was still open before the beginning of this research. These new quasi-exactly solvable Lie algebras are used to construct new quasi-exactly solvable Schrodinger operators with the property that part of their spectrum can be explicitly determined. This classification is based on a classification of Lie algebras of vector fields in three variables due to Lie and Amaldi.

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In this thesis, we firstly extend elements and periodicity properties of the theta function theory to functions that represent a wider domain of symmetries and properties, graded with different amounts of p ≥ 1, p ϵ N. Unlike theta functions, these generalised, "higher-level Appell functions" K(_p) satisfy open quasiperiodicity relations, with additive theta function terms emerging as violating terms of open quasiperiodic K(_p)’s. We evaluate the S and T modular transformations of these functions and show that the S-transform of K(_p) does not just give back K(_p), but also includes p additional 0-functions which are precisely those violating the quasiperiodicity of Appell functions. This sets a new pattern of modular group representations on functions that are not double quasiperiodic. While calculating the S-transform of K(_p), a newly arising function, namely ɸ(T, μ) will be also thoroughly analysed. As two interesting applications, we firstly study the modular group action on unitary and on an admissible class of non-unitary N = 2 characters which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. Secondly we continue this study for the admissible representation of the affine Lie superalgebra sl (2|l). We see in the final result for both cases that the functions A(T, V) are the "violating" terms of unitary calculations. We lastly confirm all our results by some sets of consistency checks including an essential residue calculation. We believe this new way of using Appell functions, could be used for any other algebraic structure whose characters can be rewritten in terms of higher-level Appell functions.

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

Techn. University, Diss., 2007--Darmstadt.