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Representation theory of C-algebras for a higher order class of spheres and tori
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2008 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 49, no 5, 053502-1-053502-13 p.Article in journal (Refereed) Published
Abstract [en]

We construct C-algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated with a matrix, the representation theory can be understood in terms of "loop" and "string" representations, which are closely related to the dynamics of an iterated map in the plane. As a particular class of algebras, we introduce the "Henon algebras," for which the dynamical map is a generalized Henon map, and give an example where irreducible representations of all dimensions exist.

Place, publisher, year, edition, pages
2008. Vol. 49, no 5, 053502-1-053502-13 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-7902DOI: 10.1063/1.2913523ISI: 000256304400019Scopus ID: 2-s2.0-44649125372OAI: oai:DiVA.org:kth-7902DiVA: diva2:13069
Note
QC 20100630. Uppdaterad från submitted till published (20100630).Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Graph Techniques for Matrix Equations and Eigenvalue Dynamics
Open this publication in new window or tab >>Graph Techniques for Matrix Equations and Eigenvalue Dynamics
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

One way to construct noncommutative analogues of a Riemannian manifold Σ is to make use of the Toeplitz quantization procedure. In Paper III and IV, we construct C-algebras for a continuously deformable class of spheres and tori, and by introducing the directed graph of a representation, we can completely characterize the representation theory of these algebras in terms of the corresponding graphs. It turns out that the irreducible representations are indexed by the periodic orbits and N-strings of an iterated map s:(reals) 2→(reals)2 associated to the algebra. As our construction allows for transitions between spheres and tori (passing through a singular surface), one easily sees how the structure of the matrices changes as the topology changes.

In Paper II, noncommutative analogues of minimal surface and membrane equations are constructed and new solutions are presented -- some of which correspond to minimal tori embedded in S7.

Paper I is concerned with the problem of finding differential equations for the eigenvalues of a symmetric N × N matrix satisfying Xdd=0.

Namely, by finding N(N-1)/2 suitable conserved quantities, the time-evolution of X (with arbitrary initial conditions), is reduced to non-linear equations involving only the eigenvalues of Χ.

Place, publisher, year, edition, pages
Stockholm: KTH, 2008. vi, 29 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 08-MA-02
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-4608 (URN)978-91-7178-845-0 (ISBN)
Public defence
2008-02-08, F3, KTH, Lindstedtsvägen 26, Stockholm, 14:00
Opponent
Supervisors
Note
QC 20100630Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2010-07-01Bibliographically approved

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