Hopf-Frobenius Algebras and a Simpler Drinfeld Double

TL;DR

This paper introduces Hopf-Frobenius algebras, a generalization of Frobenius algebras based on single Hopf algebras, simplifying the construction of the Drinfeld double.

Contribution

It defines Hopf-Frobenius algebras, establishes their properties, and constructs a simpler version of the Drinfeld double using this framework.

Findings

Every finite-dimensional Hopf algebra is a Hopf-Frobenius algebra.

Hopf-Frobenius algebras are unique up to an invertible scalar.

A simpler presentation of the Drinfeld double is achieved.

Abstract

The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in the category of finite dimensional vector spaces is a Hopf-Frobenius algebra. In addition, we show that this construction is unique up to an invertible scalar. Due to this fact, Hopf-Frobenius algebras provide two canonical notions of duality, and give us a "dual" Hopf algebra that is isomorphic to the usual dual Hopf algebra in a compact closed category. We use this isomorphism to construct a Hopf algebra isomorphic to the Drinfeld double, but has a much simpler presentation.