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Graph Techniques for Matrix Equations and Eigenvalue Dynamics
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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: urn:nbn:se:kth:diva-4608ISBN: 978-91-7178-845-0 (print)OAI: oai:DiVA.org:kth-4608DiVA: diva2:13070
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
List of papers
1. Eigenvalue Dynamics off the Calogero-Moser system
Open this publication in new window or tab >>Eigenvalue Dynamics off the Calogero-Moser system
2004 (English)In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 68, no 2, 121-129 p.Article in journal (Refereed) Published
Abstract [en]

By finding N(N-1)/2 suitable conserved quantities, free motions of real symmetric N x N matrices X(t), with arbitrary initial conditions, are reduced to nonlinear equations involving only the eigenvalues of X-in contrast to the rational Calogero-Moser system, for which [X(0), X(0)] has to be purely imaginary, of rank one.

Keyword
eigenvalue dynamics; symmetric matrix; integrable systems; BODY PROBLEMS
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7899 (URN)10.1023/B:MATH.0000043320.41280.76 (DOI)000225322400006 ()
Note
QC 20100630Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2017-12-14Bibliographically approved
2. Spinning Membranes
Open this publication in new window or tab >>Spinning Membranes
2004 (English)In: Physics Letters B, ISSN 0370-2693, E-ISSN 1873-2445, Vol. 599, no 1-2, 118-128 p.Article in journal (Refereed) Published
Abstract [en]

We present new solutions of the classical equations of motion of bosonic (matrix-)membranes. Those relating to minimal surfaces in spheres provide spinning membrane solutions in AdS(p) X S-q, as well as in flat space-time. Nontrivial reductions of the BMN matrix model equations are also given.

Keyword
MINIMAL-SURFACES
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7900 (URN)10.1016/j.physletb.2004.08.026 (DOI)000224053300014 ()2-s2.0-4444342869 (Scopus ID)
Note
QC 20100630 QC 20110914Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2017-12-14Bibliographically approved
3. Noncommutative Riemann Surfaces by Embeddings in R-3
Open this publication in new window or tab >>Noncommutative Riemann Surfaces by Embeddings in R-3
Show others...
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.

Keyword
QUANTIZATION; MANIFOLDS
National Category
Engineering and Technology
Identifiers
urn:nbn:se:kth:diva-7901 (URN)10.1007/s00220-009-0766-8 (DOI)000265388800001 ()2-s2.0-67349214377 (Scopus ID)
Note
QC 20100630. Uppdaterad från submitted till published (20100630). Tidigare titel: Noncommutative Riemann Surfaces.Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2017-12-14Bibliographically approved
4. Representation theory of C-algebras for a higher order class of spheres and tori
Open this publication in new window or tab >>Representation theory of C-algebras for a higher order class of spheres and tori
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.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7902 (URN)10.1063/1.2913523 (DOI)000256304400019 ()2-s2.0-44649125372 (Scopus ID)
Note
QC 20100630. Uppdaterad från submitted till published (20100630).Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2017-12-14Bibliographically approved

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