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

Peter Goetz, Ellen E. Kirkman, W. Frank Moore, Kent B. Vashaw

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.

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

Abstract

Let be a group coacting on an Artin-Schelter regular algebra homogeneously and inner-faithfully. When the identity component is also Artin-Schelter regular, providing a generalization of the Shephard-Todd-Chevalley Theorem, we say that is a dual reflection group for . 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.

Paper Structure

This paper contains 16 sections, 31 theorems, 79 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dual reflection groups: necessary conditions
  4. Graded algebras as modules over subalgebras
  5. The Groups M16 and SD16 are Dual Reflection Groups
  6. The modular group M16 is a dual reflection group
  7. The semidihedral group of order 16 is a dual reflection group
  8. Algebraic and homological properties of R,S and T
  9. Geometry associated to R, S and T
  10. Computing point and line schemes
  11. The point and line schemes of R
  12. The point and line schemes of S and T
  13. Remarks and Questions
  14. Dual reflection groups and their associated AS regular algebras
  15. Rulings on quadrics and fat points
  16. ...and 1 more sections

Key Result

Theorem 2.1.5

KKZ3 Let $A$ be a noetherian AS regular domain and $G$ a finite group that coacts on $A$ as a dual reflection group. Let $H=\mathbb K^G$. Then:

Theorems & Definitions (66)

  • Definition 1.0.1
  • Definition 2.1.1
  • Example 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4
  • Theorem 2.1.5
  • Example 2.1.6
  • Example 2.1.7
  • Example 2.1.8
  • Definition 2.1.9
  • ...and 56 more