Normal subgroups and relative centers of linearly reductive quantum groups

We prove a number of structural and representation-theoretic results on linearly reductive quantum groups, i.e., objects dual to those of cosemisimple Hopf algebras: (a) a closed normal quantum subgroup is automatically linearly reductive if its squared antipode leaves invariant each simple subcoalgebra of the underlying Hopf algebra; (b) for a normal embedding (Formula presented.) there is a Clifford-style correspondence between two equivalence relations on irreducible (Formula presented.) - and, respectively, (Formula presented.) -representations; and (c) given an embedding (Formula presented.) of linearly reductive quantum groups, the Pontryagin dual of the relative center (Formula presented.) can be described by generators and relations, with one generator g_V for each irreducible (Formula presented.) -representation V and one relation (Formula presented.) whenever U and (Formula presented.) are not disjoint over (Formula presented.). This latter center-reconstruction result generalizes and recovers Müger’s compact-group analogue and the author’s quantum-group version of that earlier result by setting (Formula presented.).