Speaker: Sven Raum (KU Leuven)

Title: A duality between easy quantum groups and reflection groups (joint work with Moritz Weber)

Date/Time: January 11 (Fri), 2013, 10:00-11:00

Room: 123 Math. Sci. Building

Abstract: Recently, Wang's free quantum groups attracted increasing interest in the operator algebras community. They are a new source of examples of von Neumann algebras, which is on the one hand amenable to detailed analysis and on the other hand it gives rise to potentially different von Neumann algebras than discrete groups, representations and ergodic theory. In 2009, Banica and Speicher introduced the definition of easy quantum groups, which gives a common combinatorial framework for amongst others all free orthogonal quantum groups. We show that there is a one-to-one correspondence between a certain class of easy quantum groups and reflection groups admitting a symmetric presentation. This establishes a triangular relationship between quantum groups, reflection groups and combinatorics. Appealing to the theory of varieties of groups, we are in particular able to find uncountably many different easy quantum groups. This shows that easy quantum groups can serve as a rich new source of examples in the theory of von Neumann algebras.