Graduation Semester and Year

2016

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Michaela Vancliff

Abstract

A quantum $\mathbb{P}^3$ is a noncommutative analogue of a polynomial ring on four variables, and, herein, it is taken to be a regular algebra of global dimension four. It is well known that if a generic quadratic quantum $\mathbb{P}^3$ exists, then it has a point scheme consisting of exactly twenty distinct points and a one-dimensional line scheme. In this thesis, we compute the line scheme of a family of algebras whose generic member is a candidate for a generic quadratic quantum $\mathbb{P}^3$. We find that, as a closed subscheme of $\mathbb{P}^5$, the line scheme of the generic member is the union of seven curves; namely, a nonplanar elliptic curve in a $\mathbb{P}^3$, four planar elliptic curves and two nonsingular conics. Afterward, we compute the point scheme and line scheme of several (nongeneric) quadratic quantum $\mathbb{P}^3$'s related to the Lie algebra $\mathfrak{sl}(2)$. In doing so, we identify some notable features of the algebras, such as the existence of an element that plays the role of a Casimir element of the underlying Lie-type algebra.

Keywords

Algebra, Noncommutative algebra, Algebraic geometry, Regular algebra, Lie algebra, Point module, Point scheme, Line module, Line scheme

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons

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