Partial compact quantum groups

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-KreÄ­n-Woronowicz reconstruction result for such partial compact quantum groups using the notion of partial fusion C⁎-categories. As examples, we consider the dynamical quantum SU(2)-groups from the point of view of partial compact quantum groups.