Compact coalgebras, compact quantum groups and the positive antipode

TL;DR

This paper explores the algebraic structure of compact quantum groups and o-coalgebras, providing new proofs and insights into the positive antipode, its properties, and related automorphisms, with a focus on algebraic methods.

Contribution

It offers new algebraic proofs for characterizations of compactness and the uniqueness of the compact involution in compact quantum groups, and introduces the positive antipode as a key automorphism.

Findings

Characterization of compactness via positive definite integrals

Elementary proof of the uniqueness of the compact involution

Analysis of the positive antipode and its properties

Abstract

In this article -that has also the intention to survey some known results in the theory of compact quantum groups using methods different from the standard and with a strong algebraic flavor- we consider compact o-coalgebras and Hopf algebras. In the case of a o-Hopf algebra we present a proof of the characterization of the compactness in terms of the existence of a positive definite integral, and use our methods to give an elementary proof of the uniqueness - up to conjugation by an automorphism of Hopf algebras- of the compact involution appearing in [4]. We study the basic properties of the positive square root of the antipode square that is a Hopf algebra automorphism that we call the positive antipode. We use it -as well as the unitary antipode and the Nakayama automorphism- in order to enhance our understanding of the antipode itself.