Normal subgroups and relative centers of linearly reductive quantum groups

Abstract

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 there is a Clifford-style correspondence between two equivalence relations on irreducible - and, respectively, -representations; and (c) given an embedding of linearly reductive quantum groups, the Pontryagin dual of the relative center can be described by generators and relations, with one generator gV for each irreducible -representation V and one relation whenever U and are not disjoint over . 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 .