Publications and Preprints

Publications:

Abstract: In 2010, a quantized analog of a graded Clifford algebra (GCA), called a graded skew Clifford algebra (GSCA), was proposed by Cassidy and Vancliff, and many properties of GCAs were found to have counterparts for GSCAs. In particular, a GCA is a finite module over a certain commutative subalgebra C, while a GSCA is a finite module over a (typically non-commutative) analogous subalgebra R. We consider the case that a regular GSCA is a twist of a GCA by an automorphism, and we prove, in this case, R is a skew polynomial ring and a twist of C by an automorphism.

Abstract: We prove that quadratic regular algebras of global dimension three on degree-one generators are related to graded skew Clifford algebras. In particular, we prove that almost all such algebras may be constructed as a twist of either a regular graded skew Clifford algebra or of an Ore extension of a regular graded skew Clifford algebra of global dimension two. In so doing, we classify all quadratic regular algebras of global dimension three that have point scheme either a nodal cubic curve or a cuspidal cubic curve in P^2.

Preprints:

  • Geometry of Regular Algebras of Global Dimension 4 related to Graded Skew Clifford Algebras of Global Dimension 3”, First Draft in pdf.      

Abstract: We compute point schemes of some regular algebras using (Wolfram) Mathematica. These algebras are Ore extensions of regular graded skew Clifford algebras of global dimension 3.   

  • Introduction to Noncommutative Algebraic Geometry”, First Draft in pdf.                                          

Abstract: This Lecture Notes is meant to introduce noncommutative algebraic geometry tools (which were invented by M. Artin, W. Schelter, J. Tate, and M. Van den Bergh in the late 1980s) and also graded skew Clifford algebras (which were introduced by T. Cassidy and M. Vancliff).                               

My Ph.D. thesis:

In my Ph.D. thesis, I investigated the regular graded skew Clifford algebras of low global dimension. It had three main objectives as follows:

  1. to see how many point schemes of regular graded algebras of global dimension 3 can be obtained from graded skew Clifford algebras;
  2. to see how many known examples of regular algebras of global dimension 4 can be obtained from graded skew Clifford algebras; and
  3. to determine if a certain subalgebra of a regular graded skew Clifford algebra A is a twist of the polynomial ring whenever A is a twist of a graded Clifford algebra.

To see my Ph.D. thesis, please click on the following link:

“Regular Algebras Related to Regular Graded Skew Clifford Algebras of Low Global Dimension”, Ph.D. Thesis, University of Texas at Arlington, August 2011.