Compact coalgebras, compact quantum groups and the positive antipode

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 Nakayama automorphism– in order to enhance our understanding of the antipode itself.