Date of Award

2025

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Fernando Gouvêa

Second Advisor

Jack Petok

Abstract

A familiar construction associated to any commutative ringRwith1is its group of units, traditionally denoted by Rx= {u in R | uv = 1 for some v in R}. This is but one out of many ways to get a group from a ring. To see at least one other way, we need a mild change in perspective: units may instead be characterized as elements for which the linear transformation f(r) = u ⋅ r is an isomorphism of R as a module over itself. That is to say, Rx = GL(1, R). We now have the “first” general linear group over R, so we can keep going: GL(2, R), GL(3, R), GL(4, R)and so on are naturally the 2x2, 3x3, 4x4, etc. matrix groups with invertible determinants. In general, GL(n) is an instance of an affine group. This thesis aims to present the beginnings of the theory behind such objects. To do so, we will take a top-down approach. The first section will introduce the category-theoretic underpinnings of the subject to make sense of the discussion that follows. This section is concluded with the Yoneda lemma and the corresponding embedding. The subsequent section introduces affine groups and Hopf algebras, followed by a discussion of some classic examples of both. Lastly, we will look at the tiniest bit of representation theory of affine groups, culminating with proof that every affine algebraic group over a field embeds into some GL(n).

Keywords

Affine group scheme, Hopf algebra, comultiplication, comodule, category theory, group functor

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