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

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.