Math 617 - Topics in Functional Analysis: Non-Abelian Group Duality and Quantization (Q1), MW 13-13:50, TBA

Instructor

Volker Runde, CAB 675

Office hours

TBA

Description

If G is a locally compact abelian group, and T is the unit circle, then the dual group of G is defined to be

G^ := { χ : G → T : χ is a continuous group homomorphism }.

For instance, we have Z^ = T, T^ = Z, and R^ = R. Pontryagin's duality theorem, which is one of the fundamental results of abstract harmonic analysis, asserts that G^^ = G.

For a locally compact, but not necessarily abelian group G, the question of what G^ is supposed to be is considerably more intricate. Just using the same definition of a dual group as in the abelian case leads nowhere: for instance, there is no trivial group homomorphism from SL(2,R) into T. Still, since the 1930s, various approaches towards a non-abelian group duality have been proposed, such as Kac algebras ([2]), multiplicative unitaries ([1]), and locally compact quantum groups ([4,5]). All these approaches have in common that the dual object G^ no longer needs to be a group, but still has enough group like properties for G^^ to be defined (and to equal G again).

In this course, I plan to start with the approach by S. Baaj and G. Skandalis ([1]) via multiplicative unitaries. It requires very little background beyond some knowledge of operators on Hilbert space; in particular, no knowledge of modular theory of von Neumann algebras is required. Later, I plan to at least survey S. Woronowicz's compact quantum groups ([7]) and locally compact quantum groups in the sense of J. Kustermans and S. Vaes ([4,5]).

Prerequisites

MATH 516 is an absolute necessity; MATH 519 can be taken concurrently.

Literature

  1. S. Baaj and G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres. Ann. Sci. École Norm. Sup. 26 (1993), 425-488.
  2. M. Enock and J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups. Springer Verlag, Berlin-Heidelberg-New York, 1992.
  3. J. Kustermans and S. Vaes, Locally compact quantum groups. Ann. Scient. École Norm. Sup. 33 (2000), 837-934.
  4. J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92 (2003), 68-92.
  5. T. Timmermann, An Invitation to Quantum Groups and Duality. From Hopf Algebras to Multiplicative Unitaries and beyond. European Mathematical Society, 2008.
  6. A. van Daele, Locally compact quantum groups. A von Neumann algebra approach. ArXiv: math.OA/0602212.
  7. S. L. Woronowicz, Compact quantum groups. In: A. Connes, K. Gawedzki, and J. Zinn-Justin (eds.), Symmétries Quantiques. North-Holland, 1998.

Grading

There may be occasional homework assignments, and there will definitely be a project at the end of the semester, which may lead to graduate work with me.

Last update: 9/26/09.