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
License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.
Recommended Citation
Lim, Ian Christopher, "SOME QUADRATIC QUANTUM P³s WITH A LINEAR ONE-DIMENSIONAL LINE SCHEME" (2021). Mathematics Dissertations. 174.
https://mavmatrix.uta.edu/math_dissertations/174
Comments
Degree granted by The University of Texas at Arlington