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Partial compact quantum groups. (English) Zbl 1316.81054

Summary: Compact quantum groups of face type, as introduced by Hayashi, form a class of quantum groupoids with a classical, finite set of objects. Using the notions of weak multiplier bialgebras and weak multiplier Hopf algebras (resp. due to Böhm-Gómez-Torrecillas-López-Centella and Van Daele-Wang), we generalize Hayashi’s definition to allow for an infinite set of objects, and call the resulting objects partial compact quantum groups. We prove a Tannaka-Kreĭn-Woronowicz reconstruction result for such partial compact quantum groups using the notion of partial fusion \(C^\ast\)-categories. As examples, we consider the dynamical quantum \(\mathrm{SU}(2)\)-groups from the point of view of partial compact quantum groups.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
20G42 Quantum groups (quantized function algebras) and their representations
16T15 Coalgebras and comodules; corings
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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