Commutative-by-finite Hopf algebras and their finite dual

Couto, Miguel Angelo Marques do (2019) Commutative-by-finite Hopf algebras and their finite dual. PhD thesis, University of Glasgow.

Abstract

During the last few decades, Hopf algebras have become a topic of great interest in mathematics, especially after the discovery in the 1980s of the examples now known as quantum groups by Drinfeld. Since then Hopf algebras have been found to be quite relevant in many areas, such as mathematical physics, integrable systems, quantum computation as well as algebraic geometry, number theory and other areas of mathematics.

Throughout my PhD I have thoroughly studied Hopf algebras and the research my supervisor Professor Ken Brown and I have carried out focused mostly on a particular class of Hopf algebras we named commutative-by-finite. These are Hopf algebras H that are finitely generated modules over some Hopf subalgebra A which is both commutative and normal. Here normality is a generalization of the notion of a normal subgroup in group theory.

As the title suggests, the theme and purpose of this thesis is twofold.

First, we intend to study the properties and structure of commutative-by-finite Hopf algebras. More specifically, we investigate their homological properties, their primeness and semiprimeness, their representation theory, as well as many other structural features of these Hopf algebras. We also provide a variety of examples, among them being the well-known quantum groups whose parameter is a root of unit.

Second, we aim to understand the duals of these Hopf algebras. The dual of a Hopf algebra H is, as in the analytic sense, the set of k-linear functionals H -> k, where k is the base field. We research decompositions of the dual of commutative-by-finite Hopf algebras, breaking it down into "easier pieces" to compute. And the dual of commutative Hopf algebras, which nowadays is completely understood, motivates the study of two Hopf subalgebras of the dual of commutative-by-finite Hopf algebras.

The study of these Hopf algebras, and in particular their duals, allows one to tackle important open questions, in particular regarding their antipode and their Drinfeld double. This is left in the form of conjectures and questions at the end of the thesis for possibly future work.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Supported by funding from the Portuguese Foundation for Science and Technology.
• Keywords: Hopf algebras, dual, Drinfeld double, quantum groups, noncommutative ring theory.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Brown, Professor Kenneth
• Date of Award: 2019
• Depositing User: Mr Miguel Couto
• Unique ID: glathesis:2019-74418
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 26 Sep 2019 07:17
• Last Modified: 10 Dec 2019 10:58
• Thesis DOI: 10.5525/gla.thesis.74418
• URI: https://theses.gla.ac.uk/id/eprint/74418