On the Quantum Spaces of Some Quadratic Regular Algebras of Global Dimension Four

Author

Richard Gene Chandler

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

License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

Comments

Degree granted by The University of Texas at Arlington

Recommended Citation

Chandler, Richard Gene, "On the Quantum Spaces of Some Quadratic Regular Algebras of Global Dimension Four" (2016). Mathematics Dissertations. 228.
https://mavmatrix.uta.edu/math_dissertations/228