HopfFrobenius algebras are an algebraic structure present in two distinct areas of applied category theory. They consist of two Frobenius algebras and two Hopf algebras such that their structure maps overlap – i.e. a Frobenius algebra shares its monoid with one Hopf algebra, and its comonoid with the other Hopf algebra.
HopfFrobenius algebras are present in ZXcalculus, a model for quantum circuits, and the category of linear relations, which is used to model signalflow graphs and graphical linear algebra. Both of these are exemplary examples of how string diagrams can be used, and the algebras are both commutative.
This thesis focuses on the noncommutative case of HopfFrobenius algebras. We examine the conditions under which a Hopf algebra is HopfFrobenius, and show that the conditions are relatively minor  every Hopf algebra in the category of Vector spaces is a HopfFrobenius algebra. We have provided several conditions which are all equivalent to when a Hopf algebra is HopfFrobenius, which makes checking if a given Hopf algebra is Hopf Frobenius relatively straightforward. This is beneficial, as when a Hopf algebra is Hopf Frobenius, we have more morphisms and equations to work with, and the string diagrams of HopfFrobenius algebras have a pleasing topology. In addition, we demonstrate in the final section of this thesis that many theorems about Hopf algebras in finite dimensional vector spaces can be lifted to the HopfFrobenius case. Hence when a Hopf algebra from a category other than vector spaces is HopfFrobenius, it will inherit machinery from the category of finite vector spaces.
We develop the theory of HopfFrobenius algebras by proving that Hopf algebra isomorphisms preserve Frobenius algebra structure, and using these to construct the category of HopfFrobenius algebras.
Date of Award  19 Sept 2024 

Original language  English 

Awarding Institution   University Of Strathclyde


Supervisor  Neil Ghani (Supervisor) & Glynn Winskel (Supervisor) 
