Normal subgroups and relative centers of linearly reductive quantum groups
Abstract
We prove a number of structural and representationtheoretic results on linearly reductive quantum groups, i.e. objects dual to that 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 $\mathbb{H}\trianglelefteq \mathbb{G}$ there is a Cliffordstyle correspondence between two equivalence relations on irreducible $\mathbb{G}$ and, respectively, $\mathbb{H}$representations; and (c) given an embedding $\mathbb{H}\le \mathbb{G}$ of linearly reductive quantum groups the Pontryagin dual of the relative center $Z(\mathbb{G})\cap \mathbb{H}$ can be described by generators and relations, with one generator $g_V$ for each irreducible $\mathbb{G}$representation $V$ and one relation $g_U=g_Vg_W$ whenever $U$ and $V\otimes W$ are not disjoint over $\mathbb{H}$. This latter centerreconstruction result generalizes and recovers Müger's compactgroup analogue and the author's quantumgroup version of that earlier result by setting $\mathbb{H}=\mathbb{G}$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.08804
 Bibcode:
 2021arXiv211008804C
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Category Theory;
 Mathematics  Representation Theory;
 16T05;
 20G42;
 16T20
 EPrint:
 14 pages + references