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.
Highlights
The subject of this article are compact matrix quantum groups as defined by Woronowicz in [28]
Studying and classifying categories of partitions or generalizations thereof is useful for the theory of compact quantum groups for several reasons
G being the largest quantum group contained in H1 and H2 is equivalent to IG being the smallest ideal containing IH1 and IH2, which is equivalent to CG being the smallest category containing CH1 and CH2
Summary
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]. The primary motivation is finding new examples of quantum groups since every category of partitions induces a compact matrix quantum group. 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 [12])
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