The Bijectivity of the Antipode Revisited

TL;DR

This paper offers a concise proof of the bijectivity of the antipode in Hopf algebras with nonzero integrals, avoiding traditional assumptions and linking to classical Haar measure theory.

Contribution

It presents the first proof of antipode bijectivity without relying on the uniqueness of integrals, simplifying and strengthening foundational results.

Findings

Proves bijectivity of the antipode without assuming integral uniqueness

Derives the uniqueness of integrals as a corollary

Introduces a shorter, more direct approach to fundamental Hopf algebra results

Abstract

We provide a very short approach to several fundamental results for Hopf algebras with nonzero integrals. Besides being short, our approach is the first to prove the bijectivity of the antipode without using the uniqueness of the integrals of Hopf algebras and to obtain the uniqueness of integrals as a corollary in a way similar to the classical theory of the Haar measure on compact groups.