Compact Symmetric Spaces, Triangular Factorization, and Cayley Coordinates

Persistent Link:
http://hdl.handle.net/10150/195953
Title:
Compact Symmetric Spaces, Triangular Factorization, and Cayley Coordinates
Author:
Habermas, Derek
Issue Date:
2006
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. We can represent X as the homogeneous space U/K, where U is a simply connected compact Lie group, and K is the fixed point set of an involution θ of U. Let G be the complexification of U. We consider the intersections of the image of the Cartan embedding Φ : U/K → U ⊂ G : uK → uu⁻ᶿ with the strata of the Birkhoff (or triangular, or LDU) decomposition G = ⫫(w∈W) ∑(G/w), ∑(G/w) = N⁻wHN⁺ relative to a θ-stable decomposition of the Lie algebra, g = n⁻ ⊕h ⊕ n⁺. For a generic element g in this intersection, g ∈ Φ(U/K) ∩ ∑(G/1), this yields a unique triangular factorization g = ldu. Our main contribution is to produce explicit formulas for the diagonal term d in classical cases, using Cayley coordinates (this choice of coordinate is motivated by considerations beyond sheer convenience). These formulas have several applications: 1) we can compute π₀(Φ(U/K) \ ∩ ∑(G/1) ) explicitly; 2) we can compute ʃ(Φ(U/K))ᵃΦ^-iλ (where ᵃΦ is the positive part of d) using elementary techniques in rank 1 cases; 3) they are useful in explicitly calculating Evens-Lu Poisson structures on U=K (see [Caine(2006)]). Our set-up involves choosing specific representations of the various u in su(n;C) that are compatible with θ; that is, θ fixes each of the subspaces n⁻; h; and n⁺ which, in our setup, always consist of strictly lower triangular, diagonal, and strictly upper triangular matrices, respectively. The formulas contain determinants such as det(1 + X), where X is in ip, the -1-eigenspace of θ acting on the Lie algebra u. Due to the relatively sparse nature of these matrices, these determinants are often easily calculable, and we illustrate this with many examples.
Type:
text; Electronic Dissertation
Keywords:
SYMMETRIC SPACES; CAYLEY; BIRKHOFF DECOMPOSITION
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.titleCompact Symmetric Spaces, Triangular Factorization, and Cayley Coordinatesen_US
dc.creatorHabermas, Dereken_US
dc.contributor.authorHabermas, Dereken_US
dc.date.issued2006en_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. We can represent X as the homogeneous space U/K, where U is a simply connected compact Lie group, and K is the fixed point set of an involution θ of U. Let G be the complexification of U. We consider the intersections of the image of the Cartan embedding Φ : U/K → U ⊂ G : uK → uu⁻ᶿ with the strata of the Birkhoff (or triangular, or LDU) decomposition G = ⫫(w∈W) ∑(G/w), ∑(G/w) = N⁻wHN⁺ relative to a θ-stable decomposition of the Lie algebra, g = n⁻ ⊕h ⊕ n⁺. For a generic element g in this intersection, g ∈ Φ(U/K) ∩ ∑(G/1), this yields a unique triangular factorization g = ldu. Our main contribution is to produce explicit formulas for the diagonal term d in classical cases, using Cayley coordinates (this choice of coordinate is motivated by considerations beyond sheer convenience). These formulas have several applications: 1) we can compute π₀(Φ(U/K) \ ∩ ∑(G/1) ) explicitly; 2) we can compute ʃ(Φ(U/K))ᵃΦ^-iλ (where ᵃΦ is the positive part of d) using elementary techniques in rank 1 cases; 3) they are useful in explicitly calculating Evens-Lu Poisson structures on U=K (see [Caine(2006)]). Our set-up involves choosing specific representations of the various u in su(n;C) that are compatible with θ; that is, θ fixes each of the subspaces n⁻; h; and n⁺ which, in our setup, always consist of strictly lower triangular, diagonal, and strictly upper triangular matrices, respectively. The formulas contain determinants such as det(1 + X), where X is in ip, the -1-eigenspace of θ acting on the Lie algebra u. Due to the relatively sparse nature of these matrices, these determinants are often easily calculable, and we illustrate this with many examples.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectSYMMETRIC SPACESen_US
dc.subjectCAYLEYen_US
dc.subjectBIRKHOFF DECOMPOSITIONen_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.committeememberBressler, Paulen_US
dc.contributor.committeememberFoth, Philipen_US
dc.contributor.committeememberGlickenstein, Daviden_US
dc.contributor.committeememberOtto, Michaelen_US
dc.identifier.proquest1656en_US
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