Finite index quantum subgroups of discrete quantum groups
Mao Hoshino
TL;DR
This paper establishes that finite-index quantum subgroups of a discrete quantum group arise from its unimodularization, via a precise correspondence with finite-index subhypergroups of a stable kernel–unimodular quotient hypergroup. It develops foundational equivalences for finite-index right coideals on compact quantum groups and derives imprimitivity-type results that connect algebraic, categorical, and hypergroup data. The framework yields concrete classifications, notably for free products of duals of connected simply-connected compact Lie groups, via weight/root lattice quotients $P/Q$, and extends to explicit descriptions in examples such as free unitary and $SU(2)$–based constructions. The approach provides a unified strategy to classify finite-index discrete quantum subgroups by passing to unimodular quotients and analyzing corresponding hypergroups, with broad implications for representation theory and categorical dimensions in quantum group theory.
Abstract
We show that finite index quantum subgroups of a discrete quantum group are induced from finite index quantum subgroups of the unimodularization. As an application, we classify all finite index quantum subgroups of free products of the duals of connected simply-connected compact Lie groups. We also put proofs for some fundamental facts on finite index right coideals of compact quantum groups.