Representation theory of quantised function algebras at roots of unity

Gordon, Iain (1998) Representation theory of quantised function algebras at roots of unity. PhD thesis, University of Glasgow.

Abstract

The subject of this thesis is quantum group theory. More precisely this thesis is concerned with the representation theory of quantised function algebras of semisimple algebraic groups at roots of unity and of quantised enveloping algebras of Borel subalgebras of semisimple Lie algebras at roots of unity.
Chapter 1 is an introduction to the theory of quantum groups. We describe generic quantum groups and their specialisations to roots of unity over C, following [23] and [25]. We discuss various ring theoretic features enjoyed by these algebras which allow us define a family of finite dimensional C-algebras parametrised by an algebraic group. We end this chapter by discussing Drinfel’d duality and recalling a number of the features of the representation theory of these factors, proved in [24] and [27].
In Chapter 2 we consider some cohomological aspects of augmented algebras and Hopf algebras. We introduce two spectral sequences for augmented algebras: firstly, following [7], an analogue of the Lyndon-Hochschild-Serre spectral sequence of group theory and Lie algebras; secondly a spectral sequence which relates, for a class of filtered rings, the cohomology of the associated graded ring with the cohomology of the original ring. We discuss various aspects of these spectral sequences including their relations with inflation and restriction of cohomology. We then pursue links between the cohomology of a Hopf algebra and its representation theory. Following the well-trodden path of [34], [35], [31] and [74] we acquaint ourselves with the complexity of a finite dimensional algebra, support varieties for Hopf algebras and how these relate to the representation type of a finite dimensional algebra. One necessary condition on the success of this theory is that the Hopf algebras we consider must have finitely generated cohomology rings.
Chapter 3 is dedicated to proving that quantised function algebras at roots of unity have finitely generated cohomology, allowing the results of Chapter 2 to be applied. In fact, under a restriction on the order of the root of unity, we explicitly describe the structure of the cohomology ring - it is simply a polynomial ring concentrated in even degrees. There is already an analogue of this result in the literature for quantised enveloping algebras (of Borel subalgebras) at roots of unity, [39]. We also discuss the finite generation of cohomology for much less restrictive conditions on the order of the root of unity. In these situations, however, we are unable to prove explicit structure theorems. Much of this chapter will be published in [40].
Following Chapter 3, we compute the complexities of the algebras occuring in the family of factors of quantum groups. We first do our calculations for factors of the quantised enveloping algebra of Borel subalgebras. This has a simple description in terms of the length function on the Weyl group associated to the Borel subalgebra (or more precisely its associated semisimple Lie algebra). To prove this we have to invoke a number of results from Chapters 1 and 2. Having done this we deduce the result for factors of quantised function algebras by using Drinfel’d duality. Again this has a description in terms of the length function on the Weyl group. The results in this chapter comprise the core of [42].
In Chapter 5 we discuss the ramifications of the results of Chapter 4. In particular we completely describe the factors (for both quantised function algebras and quantised enveloping algebras of Borel subalgebras) which have finite representation type. It turns out that this question is more easily answered in the function algebra case - for quantised enveloping algebras we get a result which is perhaps indicative of something more general. We also show that there are no tame factors. To prove this we have to discuss the Azumaya locus of the quantised function algebras. It turns out that almost all the algebras we need lie on this locus and, thanks to a result of [13] and a description of the centre of the quantised function algebra in [30], we can completely describe the structure of these algebras. Having done this it’s easy to see that these are wild whenever they don’t have finite representation type.
Finally, in Chapter 6, we discuss the most degenerate case of the above factors, namely the analogues of the restricted enveloping algebra of a restricted Lie algebra. Since these factors are basic algebras they can be described as path algebras with relations. We set forth this description in the widest setting possible, generalising the results of [21] for the case of quantised enveloping algebras. The work in this chapter constitutes [41].

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering
• Funder's Name: Engineering and Physical Sciences Research Council (EPSRC)
• Supervisor's Name: Brown, Professor K.A.
• Date of Award: 1998
• Depositing User: Theses Team
• Unique ID: glathesis:1998-86189
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 17 Aug 2026 10:12
• Last Modified: 17 Aug 2026 10:13
• Thesis DOI: 10.5525/gla.thesis.86189
• URI: https://theses.gla.ac.uk/id/eprint/86189