Translations in quantum groups

TL;DR

This paper characterizes endomorphisms of the $C^*$-algebra of functions on a quantum group as translations by classical subgroup elements, and shows these are automorphisms in the compact case, answering a specific open question.

Contribution

It provides a complete description of certain endomorphisms of quantum group function algebras and establishes their automorphism property in the compact case, resolving an open problem.

Findings

Endomorphisms respecting the regular comodule are translations by classical subgroup elements.

In compact quantum groups, these endomorphisms are automatically automorphisms.

The results hold regardless of the quantum group norm on the $C^*$-algebra.

Abstract

Let $H$ be the Hopf $C^*$-algebra of continuous functions on a (locally) compact quantum group of either reduced or full type. We show that endomorphisms of $H$ that respect its right regular comodule structure are translations by elements of largest classical subgroup of $G$. Furthermore, we show that for compact $G$ such an endomorphism is automatically an automorphism regardless of the quantum group norm on the $C^*$-algebra $H$; this answers a question of Piotr M. Hajac.