Graduation Year

2023

Document Type

Campus Only Senior Thesis

Degree Name

Bachelor of Arts

Department

Mathematics

Reader 1

Sam Nelson

Reader 2

Christopher Towse

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Abstract

This mathematical paper embarks on a journey through the rich landscape of algebraic structures, ranging from fundamental concepts of groups and fields to the intricate world of Hopf Algebras. The paper starts in Section 1.1 with a solid foundation by introducing groups and fields. We will introduce the cyclic group of order 3, and the field of the integers modulo 3, which will be used as an example to explain various ideas throughout the text. Section 1.2 offers a review of Vector Spaces and Tensor Products, and a description of the construction of the vector space C, which is defined as the elements of 𝐶3 as a vector space over the field ℤ3. Moving into Section 2, we delve into the realm of more complex algebra structures. Section 2.1 explores algebras and tensor algebras, showcasing how these structures arise naturally from vector spaces and tensor products. We also provide two examples of how the vector space C can be equipped with an algebra structure using tensor products and polynomial multiplication respectively. Section 2.2, transitions to the study of co-algebras and the process of defining a Hopf algebra. We explore co-algebraic structures as an essential component of Hopf algebras. Section 2.3 focuses on the antipode of a Hopf algebra, and Section 2.4, explores properties of algebras that might cause interest for further study. Finally, in Section 3, we provide a glimpse into the broader landscape of algebraic structures and suggest potential avenues for future research and exploration. This paper serves as a resource for mathematicians and students alike, providing an overview of some fundamental algebraic structures and a bridge to the captivating world of Hopf algebras.

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