Persistent Link:
http://hdl.handle.net/10150/195363
Title:
Poisson Structures on U/K and Applications
Author:
Caine, John Arlo
Issue Date:
2007
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let g denote the complexification of the Lie algebra of U, g=u^C. Each u-compatible triangular decomposition g= n_- + h + n_+ determines a Poisson Lie group structure pi_U on U. The Evens-Lu construction produces a (U, pi_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that these actions are Hamiltonian and the momentum maps are computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.
Type:
text; Electronic Dissertation
Keywords:
Poisson Geometry; Triangular Factorization; Symmetric Spaces
Degree Name:
PhD
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Pickrell, Douglas M.
Committee Chair:
Pickrell, Douglas M.

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titlePoisson Structures on U/K and Applicationsen_US
dc.creatorCaine, John Arloen_US
dc.contributor.authorCaine, John Arloen_US
dc.date.issued2007en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractLet X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let g denote the complexification of the Lie algebra of U, g=u^C. Each u-compatible triangular decomposition g= n_- + h + n_+ determines a Poisson Lie group structure pi_U on U. The Evens-Lu construction produces a (U, pi_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that these actions are Hamiltonian and the momentum maps are computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectPoisson Geometryen_US
dc.subjectTriangular Factorizationen_US
dc.subjectSymmetric Spacesen_US
thesis.degree.namePhDen_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorPickrell, Douglas M.en_US
dc.contributor.chairPickrell, Douglas M.en_US
dc.contributor.committeememberFlaschka, Hermannen_US
dc.contributor.committeememberFoth, Philipen_US
dc.identifier.proquest2033en_US
dc.identifier.oclc659747111en_US
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