Modular properties of matrix coefficients of corepresentations of a locally compact quantum group

Tuesday, May 11th 2010, 15.45, Seminar on Quantization, noncommutative geometry and symmetry (Nijmegen).
Friday, May 21th 2010, 13.30, Seminar on Operator Algebras (KU Leuven).

Abstract

Locally compact (l.c.) quantum groups have been introduced in a von Neumann algebraic setting by Stefaan Vaes and Johan Kustermans in 2000. One of their main motivations was a generalization of the Pontrjagin duality theorem for abelian l.c. groups. So to every l.c. quantum group one can associate a dual l.c. quantum group, such that the double dual is the l.c. quantum group itself. The definition of a l.c. quantum group includes a von Neumann algebra together with two normal, semi-finite, faithful weights.

In this talk we will see how the modular automorphism group of the weight can be expressed in terms of matrix coefficients of corepresentations. Furthermore, there exist (Schur) orthogonality relations between matrix coefficients. As a consequence there are relations between the weights of a quantum group and the dual weights. Also it gives a tool to determine the quantum group version of the Plancherel measure as was proved by Pieter Desmedt in 2003.