An approach to Hopf algebras via Frobenius coordinates II

TL;DR

This paper explores Hopf algebras over commutative rings as Frobenius algebras, establishing key formulas and properties, including Radford's formula, antipode order, and Frobenius structures of quantum doubles.

Contribution

It extends Frobenius algebra techniques to Hopf algebras over rings, providing new proofs and generalizations of known results.

Findings

Radford's formula for the antipode's fourth power proved using Frobenius methods

Separable and coseparable Hopf algebras have antipode of order two

Quantum double is shown to be a Frobenius algebra

Abstract

We study a Hopf algebra $H$, which is finitely generated and projective over a commutative ring $k$, as a $P$-Frobenius algebra. We define modular functions in this setting, and provide a complete proof of Radford's formula for the fourth power of the antipode, using Frobenius algebraic techniques. As further applications, we extend Etingof and Gelaki's result that a separable and coseparable Hopf algebra has antipode of order two, the result of Schneider that Hopf subalgebras are twisted Frobenius extensions, and show that the quantum double is always a Frobenius algebra.