Thesis
Hopf-Frobenius algebras
- Creator
- Rights statement
- Awarding institution
- University of Strathclyde
- Date of award
- 2024
- Thesis identifier
- T17128
- Person Identifier (Local)
- 201772562
- Qualification Level
- Qualification Name
- Department, School or Faculty
- Abstract
- Hopf-Frobenius 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. Hopf-Frobenius algebras are present in ZX-calculus, a model for quantum circuits, and the category of linear relations, which is used to model signal-flow 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 Hopf-Frobenius algebras. We examine the conditions under which a Hopf algebra is Hopf-Frobenius, and show that the conditions are relatively minor - every Hopf algebra in the category of Vector spaces is a Hopf-Frobenius algebra. We have provided several conditions which are all equivalent to when a Hopf algebra is Hopf-Frobenius, 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 Hopf-Frobenius 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 Hopf-Frobenius case. Hence when a Hopf algebra from a category other than vector spaces is Hopf-Frobenius, it will inherit machinery from the category of finite vector spaces. We develop the theory of Hopf-Frobenius algebras by proving that Hopf algebra isomorphisms preserve Frobenius algebra structure, and using these to construct the category of Hopf-Frobenius algebras.
- Advisor / supervisor
- Duncan, Ross
- Resource Type
- DOI
- Funder
Relations
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PDF of thesis T17128 | 2024-10-15 | Public | Download |