Gluing Compact Matrix Quantum Groups
- PDF / 673,332 Bytes
- 36 Pages / 439.642 x 666.49 pts Page_size
- 92 Downloads / 197 Views
Gluing Compact Matrix Quantum Groups Daniel Gromada1 Received: 27 August 2020 / Accepted: 3 November 2020 / © The Author(s) 2020
Abstract We study glued tensor and free products of compact matrix quantum groups with cyclic groups – so-called tensor and free complexifications. We characterize them by studying their representation categories and algebraic relations. In addition, we generalize the concepts of global colourization and alternating colourings from easy quantum groups to arbitrary compact matrix quantum groups. Those concepts are closely related to tensor and free complexification procedures. Finally, we also study a more general procedure of gluing and ungluing. Keywords Compact matrix quantum group · Representation category · Complexification · Gluing Mathematics Subject Classification (2010) 20G42 (Primary) · 18D10 · 46L65 (Secondary)
1 Introduction The subject of this article are compact matrix quantum groups as defined by Woronowicz in [28]. A lot of attention has recently been devoted to quantum groups possessing a combinatorial description by categories of partitions. Those were originally defined in [7]. Since then, their full classification was obtained [20] and many generalizations were introduced [4, 10, 11, 15, 22]. Studying and classifying categories of partitions or generalizations thereof is useful for the theory of compact quantum groups for several reasons. The primary motivation is finding new examples of quantum groups since every category of partitions induces a compact matrix quantum group. Those are then called easy quantum groups. In addition, since the categories of partitions are supposed to model the representation categories of quantum groups, we immediately have a lot of information about the representation theory of such quantum groups (see also [12]).
Presented by: Alistair Savage Daniel Gromada
[email protected] 1
Fachbereich Mathematik, Saarland University, Postfach 151150, 66041, Saarbr¨ucken, Germany
D. Gromada
Categories of partitions provide a particularly nice way to describe the representation categories of quantum groups. Understanding the structure of partition categories and obtaining some classification results, we may obtain analogous statements also for the associated quantum groups. Such results can then be generalized and go beyond categories of partitions and easy quantum groups. Let us illustrate this on a few examples. The classification of ordinary non-coloured categories of partitions involves a special class of so-called group-theoretical categories. Those induce group-theoretical quantum groups described by some normal subgroups A ZN 2 . However, the latter definition turns out to be more general – not every group-theoretical quantum group can be described by a category of partitions [19]. Another example are glued products, which were defined in [22] in order to interpret some classification result on two-coloured categories of partitions. The definition of glued products was inspired by partitions, but it is independent of the partitio
Data Loading...
