Thesis (Aeronautical and Astronautical Engineer)--Naval Postgraduate School, December 2002. / Thesis advisor(s): I. Michael Ross, Dennis Byrnes. Includes bibliographical references (p. 119-120). Also available online.

Fan, Sin Tsun Edward. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 85-87). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- Motivation --- p.5 / Chapter 1.2 --- Main results --- p.10 / Chapter 2 --- Preliminaries --- p.14 / Chapter 2.1 --- Z-gradations of Semisimple Lie Algebras --- p.14 / Chapter 2.2 --- Basic Facts about Algebraic Groups --- p.15 / Chapter 3 --- Weight Multiplicity Free Representations and Pre- homogeneous Vector Spaces --- p.18 / Chapter 3.1 --- Weight Multiplicity Free Representations --- p.18 / Chapter 3.2 --- Prehomogeneous Vector Spaces --- p.22 / Chapter 4 --- Augmentations of Dynkin Diagrams --- p.25 / Chapter 5 --- Orbit Finiteness and Prehomogeneity --- p.32 / Chapter 6 --- Termination of Z-grading --- p.36 / Chapter 7 --- Explicit Construction of Generic Elements in Simply- laced Cases --- p.42 / Chapter 8 --- The Ambient Lie Algebras of Parabolic PVS's --- p.47 / Chapter 9 --- PVS's of Twisted Affine Type --- p.52 / Chapter 10 --- "Orbit Structure of (GL2 x SL2m+1,C2 x A2C2m+1)" --- p.55 / Chapter 11 --- Nilvarieties and Generalisation of Open Orbits --- p.59 / Chapter 11.1 --- Nilvarieties and Visible Representations --- p.59 / Chapter 11.2 --- Augmeantations of Affine Dynkin Diagrams --- p.62 / Chapter 11.3 --- Classification of Irreducible Visible Representations --- p.67 / Chapter 12 --- Real Forms of PVS of Parabolic Type --- p.70 / Chapter 12.1 --- Representations of Real Reductive Lie Algebras and Satake Diagrams --- p.70 / Chapter 12.2 --- Real Forms of PVS of Parabolic Type --- p.77 / Chapter 13 --- Tables --- p.81 / Bibliography --- p.85

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 154-157). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 154-157). Also available on the Internet.

Let G be a compact Lie group and let π be an irreducible representation of G of highest weight λ. We study the operator-valued Fourier transform of the product of the j-function and the pull-back of ?? by the exponential mapping. We show that the set of extremal points of the convex hull of the support of this distribution is the coadjoint orbit through ?? + ??. The singular support is furthermore the union of the coadjoint orbits through ?? + w??, as w runs through the Weyl group. Our methods involve the Weyl functional calculus for noncommuting operators, the Nelson algebra of operants and the geometry of the moment set for a Lie group representation. In particular, we re-obtain the Kirillov-Duflo correspondence for compact Lie groups, independently of character formulae. We also develop a "noncommutative" version of the Kirillov character formula, valid for noncentral trigonometric polynomials. This generalises work of Cazzaniga, 1992.

Let G be a compact Lie group and let π be an irreducible representation of G of highest weight λ. We study the operator-valued Fourier transform of the product of the j-function and the pull-back of ?? by the exponential mapping. We show that the set of extremal points of the convex hull of the support of this distribution is the coadjoint orbit through ?? + ??. The singular support is furthermore the union of the coadjoint orbits through ?? + w??, as w runs through the Weyl group. Our methods involve the Weyl functional calculus for noncommuting operators, the Nelson algebra of operants and the geometry of the moment set for a Lie group representation. In particular, we re-obtain the Kirillov-Duflo correspondence for compact Lie groups, independently of character formulae. We also develop a "noncommutative" version of the Kirillov character formula, valid for noncentral trigonometric polynomials. This generalises work of Cazzaniga, 1992.

Die AdS/CFT-Dualität zwischen N=4 SYM und dem AdS_5 × S^5 Superstring zeigt Quanten-Integrabilität im planaren Limes und erlaubte die Konstruktion mächtiger Methoden, welche das Spektrale Problem zu lösen scheinen. Unser Verständnis der direkten Quantisierung des AdS_5 × S^5 Superstrings ist jedoch weiterhin unbefriedigend und besonders das Spektrum kurzer Stringzustände war bisher nur in führender Ordnung in starker ''t Hooft-Kopplung bekannt. In dieser Arbeit untersuchen wir verschiedene Methoden der perturbativen Quantisierung kurzer Strings über die führende Ordnung hinaus, wodurch wir uns auch einen besseres Verständnis der vorhandenen Quanten-Symmetrien erhoffen. Wir fokusieren auf die niedrigst angeregten Stringzustände, dual zum Konishi-Supermultiplet, und begutachten kritisch eine angeblichen Berechnung der Konishi anomalen Skalendimension im Pure-Spinor-Superstring-Formalismus. Als nächstes betrachten wir den bosonischen AdS_5 × S^5 String in statischer Eichung und konstruieren eine sog. Einzelmoden-Stringlösung, eine Veralgemeinerung des pulsierenden Strings durch unbeschränkte Nullmoden. Diese ist klassisch integrabel und quanteninvariant unter den Isometrien SO(2,4) × SO(6). Mögliche Korrekturen der vernachlässigten Supersymmetrie werden heuristisch berücksichtigt, wodurch die ersten Quantenkorrekturen der Konishi anomale Skalendimension reproduzieren werden. Wir implementieren statische Eichung für den AdS_5 × S^5 Superstring und finden elegante Ausdrücke für die Lagrangedichte und Superladungen. Unter Beschränkung auf das Superteilchen finden wir auf zwei unterschiedliche Arten kanonische Koordinaten in quadratischer Ordnung in Fermionen. Schließlich betrachten wir eine weitere Quantisierungsmethode: Da der Einzelmoden-String die SO(2,4) × SO(6)-Bahn des pulsierenden Strings ist, wenden wir Bahn-Methoden-Quantisierung auf das Teilchen und Spinning Strings in bosonischem AdS_3 × S^3 an und erhalten konsistente Ergebnisse für die Spektra. / The initial AdS/CFT duality pair, the duality between N=4 SYM and the AdS_5 × S^5 superstring, appears to enjoy quantum integrability in the planar limit, which allowed to devise powerful methods ostensibly solving the spectral problem. However, quantization of the AdS_5 × S^5 superstring from first principles is still an open question and especially the spectrum of short string states has previously been derived only at leading order in large ''t Hooft coupling. In this thesis we investigate possible routes to quantize short string states perturbatively beyond the leading order, where equally our aim is to gain better appreciation of the quantum symmetries at play. A prominent role is played by the lowest excited string states, dual to the Konishi supermultiplet, and we start by reviewing critically an asserted derivation of the Konishi anomalous dimension in the setup of pure spinor string theory. Next, we constrain ourselves to bosonic AdS_5 × S^5 String in static gauge, where we construct a so-called single-mode string solution, a generalization of the pulsating string allowing for unconstrained zero-modes. This solution shows classical integrability and invariance under the isometries SO(2,4) × SO(6) at the quantum level. Arguing heuristically about the effects of supersymmetry, we indeed recover the first non-trivial quantum correction to the Konishi anomalous dimension. We continue by implementing static gauge for the full AdS_5 × S^5 superstring and find elegant expressions for the Lagrangian density and the supercharges. We then constrain our interest to the superparticle and, using two different methods, find canonical coordinates at quadratic order in fermions. We conclude by exploring another quantization scheme: As the single-mode string is nothing but the SO(2,4) × SO(6) orbit of the pulsating string, we apply orbit method quantization to the particle and spinning string solutions in bosonic AdS_3 × S^3 yielding consistent results for the spectra.