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

**Time: **
Wed 2018-11-14 13.15 - 15.00

**Location: **
Room 306, House 6, Kräftriket, Department of Mathematics, Stockholm University

**Participating: **
Laura Maassen (RWTH Aachen/Saarland University)

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.