Some Artin-Schelter Regular Algebras From Dual Reflection Groups and their Geometry

TL;DR

This paper explores dual reflection groups acting on Artin-Schelter regular algebras, generalizing classical theorems, and investigates their algebraic and geometric properties through specific examples.

Contribution

It introduces the concept of dual reflection groups for Artin-Schelter regular algebras and analyzes their properties with new examples and geometric insights.

Findings

Identified dual reflection groups of order 16.

Studied algebraic properties of three 4-dimensional Artin-Schelter regular algebras.

Explored geometric aspects related to these algebras.

Abstract

Let $G$ be a group coacting on an Artin-Schelter regular algebra $A$ homogeneously and inner-faithfully. When the identity component $A_e$ is also Artin-Schelter regular, providing a generalization of the Shephard-Todd-Chevalley Theorem, we say that $G$ is a dual reflection group for $A$. We give two examples of dual reflection groups of order 16, and study algebraic and geometric properties of three associated Artin-Schelter regular algebras of dimension four.