Centers, cocenters and simple quantum groups. (English) Zbl 1376.16030
Summary: We define the notion of a (linearly reductive) center for a linearly reductive quantum group, and show that the quotient of a such a quantum group by its center is simple whenever its fusion semiring is free in the sense of T. Banica and R. Vergnioux [J. Noncommut. Geom. 3, No. 3, 327–359 (2009; Zbl 1203.46048)]. We also prove that the same is true of free products of quantum groups under very mild non-degeneracy conditions. Several natural families of compact quantum groups, some with non-commutative fusion semirings and hence very “far from classical”, are thus seen to be simple. Examples include quotients of free unitary groups by their centers, recovering previous work, as well as quotients of quantum reflection groups by their centers.
MSC:
| 16T05 | Hopf algebras and their applications |
| 16T20 | Ring-theoretic aspects of quantum groups |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
Citations:
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