SCIENTIFIC PROGRAMS AND ACTIVITIES

October 20, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
MEETING ON NONCOMMUTATIVE GEOMETRY AND QUANTUM GROUPS: COMPACT QUANTUM PRINCIPAL BUNDLES
January 3-4, 2014

Attendees will need to contact George Elliott (elliott<at>math.utoronto.ca) for building access to this meeting

Organizers:
George Elliott, Piotr M. Hajac, Jonathan Rosenberg

Friday, January 3, 2014

09:00 - 10:00 Paul F. Baum (Penn State)
The Peter-Weyl-Galois theorem for compact principal bundles

Let G be a compact Hausdorff topological group, and let X be a compact Hausdorff topological space with a given continuous action of G. The talk will prove that the action of G on X is free if and only if the canonical map resulting from viewing an appropriate algebra of functions on X as a comodule algebra over the Hopf algebra of polynomial functions on G is an isomorphism.
This is joint work with Piotr M. Hajac and Kenny De Commer.

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10:30 - 11:30 Piotr M. Hajac (Warsaw, Poland)
Free actions of compact quantum group on unital C*-algebras

Let F be a field, G a finite group, and Map(G,F) the Hopf algebra of all set-theoretic maps G - F. If E is a finite field extension of F and G is its Galois group, the extension is Galois if and only if the canonical map resulting from viewing E as a Map(G,F)-comodule is an isomorphism. Similarly, a finite covering space is regular if and only if the analogous canonical map is an isomorphism. The main result to be presented in this talk is an extension of this point of view to arbitrary actions of compact quantum groups on unital C*-algebras. I will explain that such an action is free (in the sense of Ellwood) if and only if the canonical map (obtained using the underlying Hopf algebra of the compact quantum group) is an isomorphism. In particular, we are able to express the freeness of a compact Hausdorff topological group action on a compact Hausdorff topological space in algebraic terms. Also, we can apply the main result to noncommutative join constructions and coactions of discrete groups on unital C*-algebras. (Joint work with Paul F. Baum and Kenny De Commer.)

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13:30 - 14:30 Jonathan Rosenberg (Maryland)
Levi-Civita connections for noncommutative tori

We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas.

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15:00 - 16:00 Mira Peterka (Kansas)
Stable Rank of the Theta-Deformed Spheres (slides)

We show that any theta-deformed sphere (of arbitrary dimension) has topological stable rank equal to 2 in the case that all deformation parameters are irrational. We also show that the stable rank can exceed 2 if some of the deformation parameters are irrational and others are rational. We compare these results to some related results of T. Sudo. Time permitting, we will go in a different direction and discuss some very preliminary results concerning modules over theta-deformed complex projective spaces.

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Saturday January 4, 2014

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09:00 - 10:00 Tomasz Brzezinski (Swansea, Wales)
Noncommutativity and resolution: Quantum teardrops and the noncommutative pillow.

We discuss recently studied examples of quantum or noncommutative orbifolds and argue that they provide one with an explicit illustration of noncommutative resolution of singularities. This resolution can be understood on several levels: as separation of roots in polynomial equations defining algebraic varieties; finiteness
of the projective dimension in the category of finitely generated bimodules; isomorphism of integral and differential forms; change of the C*-description. On the algebraic level, the key observation is that, when the noncommutativity is introduced, the actions of groups on manifolds by which these spaces are defined become free. In these way deformed orbifolds become bases of quantum (compact) principal bundles. Examples include quantum teardrops, weighted real projective planes and the noncommutative pillow manifold.

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11:00 - 12:00 Albert Sheu (Kansas)
Quantum lens space as a groupoid C*-algebra

Using the classification of all irreducible *-representations of the quantum lens space obtained by Brzezinski and Fairfax in their study of quantum line bundles over a quantum teardrop, we construct directly a concrete groupoid whose groupoid C*-algebra is the C*-algebraic quantum lens space. This facilitates a way to explicitly identify those quantum line bundles, found by Brzezinski and Fairfax inside the quantum lens space, among the well-classified finitely generated projective modules over the quantum teardrop.

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14:00 - 15:00 Olivier Gabriel (Goettingen, Germany)
A case study of noncommutative U(1)-principal bundles

We discuss noncommutative U(1)-principal bundles from the topological and cohomological points of view. After reviewing the C*-algebraic setting, we shift attention to Fréchet algebras. Under certain assumptions, we prove a version of the Pimsner-Voiculescu exact sequence for periodic cyclic cohomology. We discuss applications and establish that in the commutative case, our assumptions are satisfied by smooth U(1)-principal bundles.


Participant List:

Paul F. Baum (Penn State University, USA)
Tomasz Brzezinski (University of Swansea, Wales)
George Elliott (University of Toronto, Canada)
Olivier Gabriel (University of Gottingen, Germany)
Piotr M. Hajac (University of Warsaw and IMPAN, Poland)
Byung-Jay Kahng (Canisius College, USA)
Mira Peterka (University of Kansas, USA)
Jonathan Rosenberg (University of Maryland, USA)
Albert Sheu (University of Kansas, USA)
Karen Strung (University of Munster, Germany)

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