The canonical central exact sequence for locally compact quantum groups

TL;DR

This paper introduces a canonical exact sequence for locally compact quantum groups, defining their center and inner automorphisms, and characterizes normal subgroups via invariance under these automorphisms.

Contribution

It establishes a natural isomorphism between the quantum group of inner automorphisms and the quotient by the center, providing a new structural understanding.

Findings

Isomorphism between inner automorphisms and quotient by the center

Characterization of normal quantum subgroups as invariant under inner automorphisms

Discussion of examples illustrating the theory

Abstract

For a locally compact quantum group $\mathbb{G}$ we define its center, $\mathscr{Z}(\mathbb{G})$, and its quantum group of inner automorphisms, $\mathrm{Inn}(\mathbb{G})$. We show that one obtains a natural isomorphism between $\mathrm{Inn}(\mathbb{G})$ and $\mathbb{G}/\!\mathscr{Z}(\mathbb{G})$, we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.