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Dissertations / Theses on the topic 'Coadjoint orbits'
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1
Mihov, Diko. "Quantization of nilpotent coadjoint orbits." Thesis, Massachusetts Institute of Technology, 1996. http://hdl.handle.net/1721.1/38410.
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Li, Zongyi. "Coadjoint orbits and induced representations." Thesis, Massachusetts Institute of Technology, 1993. http://hdl.handle.net/1721.1/43270.
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Astashkevich, Alexander. "Fedosov's quantization of semisimple coadjoint orbits." Thesis, Massachusetts Institute of Technology, 1995. http://hdl.handle.net/1721.1/38396.
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Dai, Jialing. "Conjugacy classes, characters and coadjoint orbits of Diff⁺S¹." Diss., The University of Arizona, 2000. http://hdl.handle.net/10150/284342.
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The principal motivation of this dissertation is to understand the unitary irreducible representations and characters of Dif f⁺S¹-the group of all orientation-preserving diffeomorphisms of S¹ by studying conjugacy classes of Dif f⁺S¹ and its coadjoint orbits. For this purpose, we mainly focus on the following two topics. The first is to study the relation between a real Lie group G and its associated complex semigroup S(G), which was initiated by Oshansky. We consider two particular examples: (1) PSU(1.1) and PSL(2, C)⁺ (Chapter 1); (2) D and A (Chapter 2). We have shown that (a) The equivalence classes determined by the function q on PSL(2, C)⁺ (resp. A) are the same as the conjugacy classes in PSL(2, C)⁺ (resp. A ). In fact the restriction of q to PSL(2, C)⁺ equals the square of the "smaller" of the two eigenvalues of an element in PSL(2, C)+. (b) The fact that the representation of PSL(2, C)⁺ is of trace class makes the character of PSL(2, C)⁺ well-defined. Moreover the character of PSL(2, C)⁺ has analytic continuation onto PSU(1,1) except on a set of measure zero. Surprisingly, the extended characters are exactly Harish-Chandra global characters Xᵐ(PSU)₍₁.₁₎ =Θm. Secondly, we investigate the coadjoint orbits of Virasoro group-the central extension of Dif f⁺S¹ (Chapter 3), which has been considered before by Segal, Kirillov and (later) Witten. We improved Segal's result by parameterizing coadjoint orbits precisely in terms of following conjugacy classes in P͂S͂U͂(1,1): Par⁺₀,{n,n ∈ N},{Elln, n∈N},{Par⁽⁺/⁻⁾(n),n > 0}, {Hyp(n),n ≥ }0. We also completed Kirillov's list of representatives of coadjoint orbits, and we fleshed out the connection between Segal's and Kirillov's and Witten's work by giving the correspondence between conjugacy classes in P͂S͂U͂ (1,1) and the representatives of coadjoint orbits and stabilizers.
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André, Carlos Alberto Martins. "Irreducible characters of the unitriangular group and coadjoint orbits." Thesis, University of Warwick, 1992. http://wrap.warwick.ac.uk/110600/.
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The method of coadjoint orbits was introduced by Kirillov to study the unitary irreducible representations of a nilpotent Lie groups. Afterwards Kazhdan adapted this method to determine the irreducible complex characters of a finite unipotent group. We use this method to study the irreducible complex characters of any finite unitriangular group. In chapters 2 and 5 we established an orthogonal decomposition of the regular character of any finite unitriangular group. Chapters 3 and 4 are concerned with coadjoint orbits of any unitriangular group defined over an algebraically closed field. Chapter 3 is essentially the orbit version of chapter 2. In fact we obtain a decomposition of the dual space of the niltriangular Lie algebra as a disjoint union of invariant subvarieties. In a certain sense this decomposition corresponds to the one obtained in chapter 2 and 5.
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Nevins, Monica 1973. "Admissible nilpotent coadjoint orbits of p-adic reductive Lie groups." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/47467.
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Plummer, Michael. "Stratified fibre bundles and symplectic reduction on coadjoint orbits of SU(n)." Thesis, University of Surrey, 2008. http://epubs.surrey.ac.uk/842671/.
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The problem of classifying the reduced phase spaces of the natural torus action on a generic coadjoint orbit of SU(n) is considered. The concept of a stratified fibre bundle is defined. It is proved that the orbit map of an equivariant fibre bundle is a stratified fibre bundle. This result is then used to give an iterative description of the reduced phase spaces of the torus action on a generic coadjoint orbit of SU(n). The theory is illustrated with a detailed examination of the n = 3 case, that of the two torus action on a coadjoint orbit of SU(3).
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Villa, Patrick Björn [Verfasser], Peter [Akademischer Betreuer] Heinzner, and Alan T. [Akademischer Betreuer] Huckleberry. "Kählerian structures of coadjoint orbits of semisimple Lie groups and their orbihedra / Patrick Björn Villa. Gutachter: Peter Heinzner ; Alan T. Huckleberry." Bochum : Ruhr-Universität Bochum, 2015. http://d-nb.info/1079843477/34.
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Deltour, Guillaume. "Propriétés symplectiques et hamiltoniennes des orbites coadjointes holomorphes." Phd thesis, Université Montpellier II - Sciences et Techniques du Languedoc, 2010. http://tel.archives-ouvertes.fr/tel-00552150.
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L'objet de cette thèse est l'étude de la structure symplectique des orbites coadjointes holomorphes, et de leurs projections. Une orbite coadjointe holomorphe O est une orbite coadjointe elliptique d'un groupe de Lie réel semi-simple, connexe, non compact et à centre fini, provenant d'un espace symétrique hermitien G/K, telle que O puisse être naturellement munie d'une structure kählérienne G-invariante. Ces orbites sont une généralisation de l'espace symétrique hermitien G/K. Dans cette thèse, nous prouvons que le symplectomorphisme de McDuff se généralise aux orbites coadjointes holomorphes, décrivant la structure symplectique de l'orbite O par le produit direct d'une orbite coadjointe compacte et d'un espace vectoriel symplectique. Ce symplectomorphisme est ensuite utilisé pour déterminer les équations de la projection de l'orbite O relative au sous-groupe compact maximal K de G, en faisant intervenir des résultats récents de Ressayre en Théorie Géométrique des Invariants.
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Zergane, Amel. "Séparation des représentations des groupes de Lie par des ensembles moments." Thesis, Dijon, 2011. http://www.theses.fr/2011DIJOS086/document.
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Si (π, H) est une représentation unitaire irréductible d'un groupe de Lie G, on sait lui associer son application moment Ψπ. La fermeture de l'image de Ψπ s'appelle l'ensemble moment de π. Généralement, cet ensemble est Conv(Oπ), si Oπ est l'orbite coadjointe associée à π. Mais il ne caractérise pas π : deux orbites distinctes peuvent avoir la même enveloppe convexe fermée. On peut contourner cette non séparation en considérant un surgroupe G+ de G et une application non linéaire ø de g* dans (g+)* telle que, pour les orbites générique, ø(O) est une orbite et Conv (ø(O)) caractérise O. Dans cette thèse, on montre que l'on peut choisir le couple (G+, ø), avec ø de degré ≤ 2 pour tous les groupes nilpotents de dimension ≤ 6, à une exception près, tous les groupes résolubles de dimension ≤ 4, et pour un exemple de groupe de déplacements. Ensuite, on étudie le cas des groupes G = SL(n, R). Pour ces groupes, il existe un tel couple avec ø de degré n, mais il n'en existe pas avec ø de degré 2 si n>2, il n'en existe pas avec ø de degré 3 si n=4. Enfin, on montre que l'application moment Ψπ est celle d'une action fortement hamiltonienne de G sur la variété de Fréchet symplectique PH∞. On construit un foncteur qui associe à tout G un surgroupe de Lie Fréchet G̃, de dimension infinie et, à tout π de G, une action π̃ fortement hamiltonienne, dont l'ensemble moment caractérise πTo a unitary irreducible representation (π,H) of a Lie group G, is associated a moment map Ψπ. The closure of the range of Ψπ is the moment set of π. Generally, this set is Conv(Oπ), if Oπ is the corresponding coadjoint orbit. Unfortunately, it does not characterize π : 2 distincts orbits can have the same closed convex hull. We can overpass this di culty, by considering an overgroup G+ for G and a non linear map ø from g* into (g+)* such that, for generic orbits, ø(O) is an orbit and Conv( ø(O)) characterizes O. In the present thesis, we show that we can choose the pair (G+,ø), with deg ø ≤2 for all the nilpotent groups with dimension ≤6, except one, for all solvable groups with diemnsion ≤4, and for an example of motion group. Then we study the G=SL(n,R) case. For these groups, there exists ø with deg ø =n, if n>2, there is no such ø with deg ø=2, if n=4, there is no such ø with deg ø=3. Finally, we show that the moment map Ψπ is coming from a stronly Hamiltonian G-action on the Frécht symplectic manifold PH∞. We build a functor, which associates to each G an infi nite diemnsional Fréchet-Lie overgroup G̃,and, to each π a strongly Hamiltonian action, whose moment set characterizes π
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Guieu, Laurent. "Sur la géométrie des orbites de la représentation coadjointe du groupe de Bott-Virasoro." Aix-Marseille 1, 1994. http://www.theses.fr/1994AIX11022.
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Cette thèse se compose de trois parties. Dans la première partie la structure des sous-groupes d'isotropie de la représentation coadjointe du groupe des difféomorphismes du cercle est analysée via l'utilisation du nombre de rotation. La seconde partie approfondit l'étude des liens existant entre les orbites coadjointes du groupe de Bott-Virasoro (à charge centrale non nulle) et certaines structures géométriques sur le cercle. La troisième partie est un travail effectué en collaboration avec Valentin Yu. Ovsienko et expose la construction de structures symplectiques sur certains espaces de courbes dans des variétés localement affines ou localement projectives. Cette construction rattache de manière naturelle le crochet de Gel'fand-Dikii (et l'algèbre de Virasoro) à la géométrie différentielle projective
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Heitritter, Kenneth I. J. "Mechanics of the diffeomorphism field." Thesis, University of Iowa, 2019. https://ir.uiowa.edu/etd/6761.
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Coadjoint orbits of Lie algebras come naturally imbued with a symplectic two-form allowing for the construction of dynamical actions. Consideration of the coadjoint orbit action for the Kac-Moody algebra leads to the Wess-Zumino-Witten model with a gauge-field coupling. Likewise, the same type of coadjoint orbit construction for the Virasoro algebra gives Polyakov’s 2D quantum gravity action with a coupling to a coadjoint element, D, interpreted as a component of a field named the diffeomorphism field. Gauge fields are commonly given dynamics through the Yang-Mills action and, since the diffeomorphism field appears analogously through the coadjoint orbit construction, it is interesting to pursue a dynamical action for D.
This thesis reviews the motivation for the diffeomorphism field as a dynamical field and presents results on its dynamics obtained through projective connections. Through the use of the projective connection of Thomas and Whitehead, it will be shown that the diffeomorphism field naturally gains dynamics. Results on the analysis of this dynamical theory in two-dimensional Minkowski background will be presented.
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Kemp, Graham. "Algebra and geometry of Dirac's magnetic monopole." Thesis, Loughborough University, 2013. https://dspace.lboro.ac.uk/2134/12508.
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This thesis is concerned with the quantum Dirac magnetic monopole and two classes of its generalisations. The first of these are certain analogues of the Dirac magnetic monopole on coadjoint orbits of compact Lie groups, equipped with the normal metric. The original Dirac magnetic monopole on the unit sphere S^2 corresponds to the particular case of the coadjoint orbits of SU(2). The main idea is that the Hilbert space of the problem, which is the space of L^2-sections of a line bundle over the orbit, can be interpreted algebraically as an induced representation. The spectrum of the corresponding Schodinger operator is described explicitly using tools of representation theory, including the Frobenius reciprocity and Kostant's branching formula. In the second part some discrete versions of Dirac magnetic monopoles on S^2 are introduced and studied. The corresponding quantum Hamiltonian is a magnetic Schodinger operator on a regular polyhedral graph. The construction is based on interpreting the vertices of the graph as points of a discrete homogeneous space G/H, where G is a binary polyhedral subgroup of SU(2). The edges are constructed using a specially selected central element from the group algebra, which is used also in the definition of the magnetic Schrodinger operator together with a character of H. The spectrum is computed explicitly using representation theory by interpreting the Hilbert space as an induced representation.
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Tumpach, Barbara. "Structures kählériennes et hyperkählériennes en dimension infinie." Palaiseau, Ecole polytechnique, 2005. http://www.theses.fr/2005EPXX0014.
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Tumpach, Alice Barbara. "Varietes kaehleriennes et hyperkaeleriennes de dimension infinie." Phd thesis, Ecole Polytechnique X, 2005. http://tel.archives-ouvertes.fr/tel-00012012.
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Le premier chapitre de cette thèse est consacré, d'une part à l'étude des quotients kaehlériens et hyperkaehlériens dans le cadre banachique et, d'autre part, à la construction par quotient hyperkaehlérien (d'une variété banachique non hilbertienne par un groupe de Lie banachique) d'une variété hilbertienne qui s'identifie (en fonction de la structure complexe distinguée) soit à l'espace cotangent d'une composante connexe de la grassmannienne restreinte définie par G. Segal et G. Wilson, soit à une complexification naturelle de cette grassmannienne. Le second chapitre comprend trois parties. La première partie est consacrée à la classification des orbites coadjointes affines hermitiennes symétriques irréductibles des L*-groupes de type compact. La seconde partie est consacrée a la démonstration du théorème de Mostow pour un L*-groupe semi-simple de type compact. Dans la troisième partie, je construis une structure hyperkaehlérienne sur les orbites complexifiées des orbites coadjointes affines hermitiennes symétriques des L*-groupes semi-simples de type compact.
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Alexander, David. "Idéaux minimaux d'algèbres de groupes." Metz, 2000. http://docnum.univ-lorraine.fr/public/UPV-M/Theses/2000/Alexander.David.SMZ0041.pdf.
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Soient G un groupe de Lie nilpotent simplement connexe d'algèbre de Lie g, et l une forme linéaire sur g. La première partie de cette thèse détermine l'ensemble des idéaux bilatères fermés de A dont l'enveloppe est un caractère, où A désigne une sous-*-algèbre de Banach de L [exposant] 1(G) admettant l'espace de Schwartz S(G) comme sous-espace dense. La seconde partie établit une bijection entre certains idéaux de L [exposant]1w(G) et L[exposant]1w|N (N) où N est un sous-groupe distingué fermé d'un groupe localement compact G. La troisième partie étudie le comportement général d'un poids sur un groupe topologique. La dernière partie examine la première situation lorsque l'orbite de l est plate.
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Raffoul, Raed Wissam Mathematics & Statistics Faculty of Science UNSW. "Functional calculus and coadjoint orbits." 2007. http://handle.unsw.edu.au/1959.4/43693.
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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.
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Zoghi, Masrour. "The Gromov Width of Coadjoint Orbits of Compact Lie Groups." Thesis, 2010. http://hdl.handle.net/1807/26269.
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The first part of this thesis investigates the Gromov width of maximal dimensional
coadjoint orbits of compact simple Lie groups. An upper bound for the Gromov width
is provided for all compact simple Lie groups but only for those coadjoint orbits that satisfy a certain technical assumption, whereas the lower bound is proved only for
groups of type A, but without the technical restriction. The two bounds use very
different techniques: the proof of the upper bound uses more analytical tools, while
the proof of the lower bound is more geometric.
The second part of the thesis is a short report on a joint project with my supervisor, which was concerned with the relationship between two different definitions of orbifolds: one using Lie groupoids and the other involving diffeologies. The results are summarized in Chapter 5 of this text.
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Hudon, Valérie. "Study of the coadjoint orbits of the Poincare group in 2 + 1 dimensions and their coherent states." Thesis, 2009. http://spectrum.library.concordia.ca/976538/1/NR63402.pdf.
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The first main objective of this thesis is to study the orbit structure of the (2 + 1)-Poincare group R2,1 [Special characters omitted.] 2,1 {604} SO (2,1) by obtaining an explicit expression for the coadjoint action. From there, we compute and classify the coadjoint orbits. We obtain a degenerate orbit, the upper and lower sheet of the two-sheet hyperboloid, the upper and lower cone and the one-sheet hyperboloid. They appear as two-dimensional coadjoint orbits and, with their cotangent planes, as four-dimensional coadjoint orbits. We also confirm a link between the four-dimensional coadjoint orbits and the orbits of the action of SO (2, 1) on the dual of [Special characters omitted.] 2,1 . The second main objective of this thesis is to use the information obtained about the structure to induce a representation and build the coherent states on two of the coadjoint orbits. We obtain coherent states on the hyperboloid for the principal section. The Galilean and the affine sections only allow us to get frames. On the cone, we obtain a family of coherent states for a generalized principal section and a frame for the basic section
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Payette, Jordan. "Les actions de groupes en géométrie symplectique et l'application moment." Thèse, 2014. http://hdl.handle.net/1866/11640.
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Ce mémoire porte sur quelques notions appropriées d'actions de groupe sur les variétés symplectiques, à savoir en ordre décroissant de généralité : les actions symplectiques, les actions faiblement hamiltoniennes et les actions hamiltoniennes. Une connaissance des actions de groupes et de la géométrie symplectique étant prérequise, deux chapitres sont consacrés à des présentations élémentaires de ces sujets. Le cas des actions hamiltoniennes est étudié en détail au quatrième chapitre : l'importante application moment y est définie et plusieurs résultats concernant les orbites de la représentation coadjointe, tels que les théorèmes de Kirillov et de Kostant-Souriau, y sont démontrés. Le dernier chapitre se concentre sur les actions hamiltoniennes des tores, l'objectif étant de démontrer le théorème de convexité d'Atiyha-Guillemin-Sternberg. Une discussion d'un théorème de classification de Delzant-Laudenbach est aussi donnée. La présentation se voulant une introduction assez exhaustive à la théorie des actions hamiltoniennes, presque tous les résultats énoncés sont accompagnés de preuves complètes. Divers exemples sont étudiés afin d'aider à bien comprendre les aspects plus subtils qui sont considérés. Plusieurs sujets connexes sont abordés, dont la préquantification géométrique et la réduction de Marsden-Weinstein.This Master thesis is concerned with some natural notions of group actions on symplectic manifolds, which are in decreasing order of generality : symplectic actions, weakly hamiltonian actions and hamiltonian actions. A knowledge of group actions and of symplectic geometry is a prerequisite ; two chapters are devoted to a coverage of the basics of these subjects. The case of hamiltonian actions is studied in detail in the fourth chapter : the important moment map is introduced and several results on the orbits of the coadjoint representation are proved, such as Kirillov's and Kostant-Souriau's theorems. The last chapter concentrates on hamiltonian actions by tori, the main result being a proof of Atiyah-Guillemin-Sternberg's convexity theorem. A classification theorem by Delzant and Laudenbach is also discussed. The presentation is intended to be a rather exhaustive introduction to the theory of hamiltonian actions, with complete proofs to almost all the results. Many examples help for a better understanding of the most tricky concepts. Several connected topics are mentioned, for instance geometric prequantization and Marsden-Weinstein reduction.