Interacting Frobenius Algebras are Hopf

TL;DR

This paper explores the deep relationship between Frobenius and Hopf algebras, showing that interacting Frobenius algebras inherently form Hopf algebras, with generalizations including phase groups and finite-dimensional effects.

Contribution

It demonstrates that interacting Frobenius algebras naturally form Hopf algebras, extending previous work by including phase groups and finite-dimensional considerations.

Findings

Interacting Frobenius algebras form Hopf algebras.

The theory includes non-trivial phase group dynamics.

The prime power dimensional case recovers previous subtheories.

Abstract

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.