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Noncommutative Riemann Surfaces by Embeddings in R-3
Inst Hautes Etud Sci.
Univ Haute Alsace, Lab MIA.
Univ Luxembourg.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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2009 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 288, no 2, 403-429 p.Article in journal (Refereed) Published
Abstract [en]

We introduce C-Algebras of compact Riemann surfaces ∑ as non-commutative analogues of the Poisson algebra of smooth functions on ∑. Representations of these algebrasgive rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.

Place, publisher, year, edition, pages
2009. Vol. 288, no 2, 403-429 p.
Keyword [en]
National Category
Engineering and Technology
URN: urn:nbn:se:kth:diva-7901DOI: 10.1007/s00220-009-0766-8ISI: 000265388800001ScopusID: 2-s2.0-67349214377OAI: diva2:13068
QC 20100630. Uppdaterad från submitted till published (20100630). Tidigare titel: Noncommutative Riemann Surfaces.Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2012-04-14Bibliographically 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.
Trita-MAT. MA, ISSN 1401-2278 ; 08-MA-02
National Category
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
QC 20100630Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2010-07-01Bibliographically approved

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Arnlind, JoakimHoppe, Jens
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