Laura Maassen: The intertwiner spaces of non-easy group-theoretical quantum groups

Abstract: 

With some physical motivation in mind quantum groups first appeared in the 1980's. By now the term "quantum group" is used for various mathematical objects. In this talk I will introduce the approach of Woronowicz generalising compact matrix groups in the setting of C*-algebras. Both parts of my talk will only assume a basic background in group theory and algebra.
 
Consider a compact matrix group G < GL_n(C) and C(G), the continuous complex valued functions on G. Then C(G) is a C*-algebra with a dualised group multiplication and some dualised group properties. Woronowicz defines a compact matrix quantum group as a general C*-algebra with this dual group structure. In fact, by the Gelfand duality a compact quantum group arises from a classical compact group if and only if the C*-algebra is commutative. Hence in the "quantum cases" we talk about non-commutative operator algebras.

By a Tannaka-Krein result of Woronowicz any compact matrix quantum group can be fully recovered from its intertwiner spaces. Due to this I will introduce orthogonal easy quantum groups, which are a class of compact matrix quantum groups introduced by Banica and Speicher with a nice combinatorial structure of their intertwiner spaces.
 
In the second half of my talk I will introduce group-theoretical quantum groups. They form an uncountably large class of compact matrix quantum groups and group-theoretical easy quantum groups played a key role in the classification of orthogonal easy quantum groups. As there are in general not so many well studied examples of non-easy quantum groups I studied the structure of non-easy group-theoretical quantum groups. In this talk I will show that their intertwiner spaces also have a combinatorial structure and provide this construction explicitly.