### 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

- 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.
- M. Enock and J.-M. Schwartz,
*Kac Algebras and Duality of Locally Compact Groups*. Springer Verlag, Berlin-Heidelberg-New York, 1992.
- J. Kustermans and S. Vaes, Locally compact quantum groups.
*Ann. Scient. École Norm. Sup.* **33** (2000), 837-934.
- J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting.
*Math. Scand.* **92** (2003), 68-92.
- T. Timmermann,
*An Invitation to Quantum Groups and Duality. From Hopf Algebras to Multiplicative Unitaries and beyond*. European Mathematical Society, 2008.
- A. van Daele, Locally compact quantum groups. A von Neumann algebra approach. ArXiv: math.OA/0602212.
- 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.