Quantum Groups meet Free Probability

Moritz Weber (Universitaet des Saarlandes)

Wednesday 4th March, 2015 13:00-14:00 Maths 417

Abstract

It is a fundamental concept in operator algebras to pass from a (topological/measurable/...) space to the algebra of (continuous/measurable/...) functions over it and to study this algebra instead of the space itself. In a next step, the multiplication of such algebras is allowed to be noncommutative yielding a kind of ``noncommutative topology'' ($C^*$-algebras) or ``noncommutative measure theory'' (von Neumann algebras) etc. 
Sending compact groups through this machinery, we obtain ($C^*$-algebraic) compact quantum groups, as defined by Woronowicz in the 1980's. Starting with probability theory, we end up with free probability as developed by Voiculescu around the same time.

We will give brief introductions into compact quantum groups and free probability. The meeting point between these two theories are the so called easy quantum groups as introduced by Banica and Speicher in 2009. These objects are governed by the combinatorics of set theoretical partitions and it is amazing how several operator algebraic properties can be studied by purely combinatorial means. We will report on this particular class of quantum groups and their interplay with free probability.

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