ORCID Identifier(s)

0000-0001-7834-0994

Graduation Semester and Year

2021

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Michaela Vancliff

Abstract

It is believed that quadratic Artin-Shelter regular (AS-regular) algebras of global dimension four (sometimes called quadratic quantum P3s can be classified using a geometry similar to that developed in the 1980’s by Artin, Tate, and Van den Bergh. Their geometry involved studying a scheme (later called the point scheme) that parametrizes the point modules over a graded algebra. The notion of line scheme (which parametrizes line modules) was introduced later by Shelton and Vancliff. It is known that “generic” quadratic quantum P3s have a finite point scheme and one-dimensional line scheme. A family of algebras with these properties is presented herein where each member has a line scheme that is a union of lines. Moreover, we prove that if a quadratic quantum P3, denoted A, is an Ore extension of a quadratic quantum P2, denoted B, then the point variety of B is embedded in the line variety of A. Indeed, this result is generalized to prove that, under certain conditions, if A is a quadratic quantum P3 that contains a subalgebra isomorphic to a quadratic quantum P2, then the point variety of the subalgebra is embedded in the line variety of A.

Keywords

Noncommutative algebra, Algebraic geometry, Computational algebra, Point schemes, Line schemes, Regular algebras

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons

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