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 H ⊴ G there is a Clifford-style correspondence between two equivalence relations on irreducible G - and, respectively, H -representations; and (c) given an embedding H ≤ G of linearly reductive quantum groups, the Pontryagin dual of the relative center Z (G) ∩ H can be described by generators and relations, with one generator g_Vfor each irreducible G -representation V and one relation g U = g V g W whenever U and V ⊗ W are not disjoint over H. 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 H = G . [ABSTRACT FROM AUTHOR]