Partial compact quantum groups

TL;DR

This paper generalizes the concept of compact quantum groups to include infinite objects using weak multiplier structures, and establishes a reconstruction theorem with applications to dynamical quantum groups.

Contribution

It introduces partial compact quantum groups, extending Hayashi's face type quantum groups to infinite object sets, and proves a Tannaka-Kre$reve{ extrm{}}$n-Woronowicz reconstruction theorem.

Findings

Defined partial compact quantum groups using weak multiplier (bi)algebras.

Proved a reconstruction theorem for these structures.

Applied the theory to dynamical quantum SU(2) groups.

Abstract

Compact quantum groups of face type, as introduced by Hayashi, form a class of compact quantum groupoids with a classical, finite set of objects. Using the notions of a weak multiplier bialgebra and weak multiplier Hopf algebra (resp. due to B{\"o}hm--G\'{o}mez-Torrecillas--L\'{o}pez-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-Kre$\breve{\textrm{\i}}$n-Woronowicz reconstruction result for such partial compact quantum groups using the notion of a partial fusion C$^*$-category. As examples, we consider the dynamical quantum $SU(2)$-groups from the point of view of partial compact quantum groups.