Convex polytopes and duality in the geometry of the full Kostant-Toda lattice.

Persistent Link:
http://hdl.handle.net/10150/187199
Title:
Convex polytopes and duality in the geometry of the full Kostant-Toda lattice.
Author:
Shipman, Barbara Anne.
Issue Date:
1995
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:
Our study describes the structure of the completely integrable system known as the full Kostant-Toda lattice in terms of the rich geometry of complex generalized flag manifolds and the information encoded in their momentum polytopes. The space in which the system evolves is a Poisson manifold which is essentially the dual of a Borel subalgebra of a Lie algebra, and the symplectic leaves are the coadjoint orbits. We extend the results of Ercolani, Flaschka, and Singer in (4) in which an embedding of an isospectral submanifold of the phase space into the flag manifold is used to study the geometry of the "generic" compactified level sets of a particular family of constants of motion. In a detailed analysis of the full Sl(4,C) Kostant-Toda lattice, we consider all types of level sets, in particular those which do not satisfy the genericity conditions of (4). The breakdown of these conditions is reflected in the types of nongeneric strata to which the torus orbits in a "special" level set belong. This degeneration corresponds to certain decompositions of the momentum polytopes, which we explain in terms of representation theory. We discover a fundamental two-fold symmetry intrinsic to this geometry which appears in the phase space as an involution preserving the constants of motion, and we express it in terms of duality in the flag manifold and the pairing between a representation of a Lie algebra and its dual. Several chapters are devoted to the study of a double fibration of a generic symplectic leaf by the level sets of two distinct involutive families of integrals for the full Sl(4,C) Kostant-Toda lattice. We describe the symmetries of these two fibrations and determine the monodromy around their singular fibers. Finally, we show how the configuration of the lower-dimensional symplectic leaves of the Poisson structure in this example is revealed in the geometry of the flag manifold and its momentum polytope.
Type:
text; Dissertation-Reproduction (electronic)
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Ercolani, Nicholas

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleConvex polytopes and duality in the geometry of the full Kostant-Toda lattice.en_US
dc.creatorShipman, Barbara Anne.en_US
dc.contributor.authorShipman, Barbara Anne.en_US
dc.date.issued1995en_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.abstractOur study describes the structure of the completely integrable system known as the full Kostant-Toda lattice in terms of the rich geometry of complex generalized flag manifolds and the information encoded in their momentum polytopes. The space in which the system evolves is a Poisson manifold which is essentially the dual of a Borel subalgebra of a Lie algebra, and the symplectic leaves are the coadjoint orbits. We extend the results of Ercolani, Flaschka, and Singer in (4) in which an embedding of an isospectral submanifold of the phase space into the flag manifold is used to study the geometry of the "generic" compactified level sets of a particular family of constants of motion. In a detailed analysis of the full Sl(4,C) Kostant-Toda lattice, we consider all types of level sets, in particular those which do not satisfy the genericity conditions of (4). The breakdown of these conditions is reflected in the types of nongeneric strata to which the torus orbits in a "special" level set belong. This degeneration corresponds to certain decompositions of the momentum polytopes, which we explain in terms of representation theory. We discover a fundamental two-fold symmetry intrinsic to this geometry which appears in the phase space as an involution preserving the constants of motion, and we express it in terms of duality in the flag manifold and the pairing between a representation of a Lie algebra and its dual. Several chapters are devoted to the study of a double fibration of a generic symplectic leaf by the level sets of two distinct involutive families of integrals for the full Sl(4,C) Kostant-Toda lattice. We describe the symmetries of these two fibrations and determine the monodromy around their singular fibers. Finally, we show how the configuration of the lower-dimensional symplectic leaves of the Poisson structure in this example is revealed in the geometry of the flag manifold and its momentum polytope.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairErcolani, Nicholasen_US
dc.contributor.committeememberFlaschka, Hermannen_US
dc.contributor.committeememberPickrell, Dougen_US
dc.identifier.proquest9603347en_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.