Semester

Summer

Date of Graduation

2022

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Edgar J. Fuller, Jr.

Committee Co-Chair

Harvey R. Diamond

Committee Member

Harumi Hattori

Committee Member

Cun-Quan Zhang

Committee Member

Robert A. Paysen

Abstract

The study of Lie groups has yielded a rich catalogue of mathematical spaces that, in some sense, provide a theoretical and computational framework for describing the “world in which we live.” In particular, these topological groups that represent the rigid motions of a space, the behavior of subatomic particles, and the shape of the expanding universe consist of specialized matrices. In what follows, we define a new collection of matrices with a very specific transposition relation and attempt to classify this Lie group algebraically, geometrically, and topologically. We consider fields, $\Bbb{F},$ of characteristic zero and define the group of pseudo-orthogonal matrices to be the set

$$GO(p, q; \Bbb{F}) = \{A \in GL(p + q, \Bbb{F}) | A^T \eta A = \lambda \eta for some \lambda \in \Bbb{F}^{\times}\},$$

where $\eta$ is the diagonal matrix in which the first $p$ entries are one and the remaining $q$ entries are −1. After doing so, and verifying that $GO(p, q; \Bbb{F})$ is, indeed, a Lie group, we determine a number of standard algebraic properties that $GO(p, q; \Bbb{F})$ fails to hold and the decomposition of the group as a direct product. We study the metric of $GO(p,q; \Bbb{F})$ and use this metric to determine curvature tensors of the manifold. Further, we document a number of distinct paths attempted to study the topological nature of $GO(p, q; \Bbb{F})$ and provide rationale for these failed attempts and a description of the cyclical nature of cohomology of $GO(p, q; \Bbb{F})$ and its innate cohomological equivalence with its infinite-dimensional universal covering space.

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