Laura Maassen: The intertwiner spaces of non-easy group-theoretical quantum groups
Tid: On 2018-11-14 kl 13.15 - 15.00
Föreläsare: Laura Maassen (RWTH Aachen/Saarland University)
Plats: Room 306, House 6, Kräftriket, Department of Mathematics, Stockholm University
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 < GLn(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.