Automorphisms of Regular Algebras

TL;DR

This paper extends Manin's construction of quantum automorphism groups from quadratic to non-quadratic homogeneous algebras, specifically constructing quantum symmetry groups for certain Artin-Schelter algebras of dimension 3.

Contribution

It introduces a method to define quantum automorphism groups for non-quadratic homogeneous algebras, expanding the scope of quantum symmetry concepts beyond quadratic cases.

Findings

Constructed quantum groups for cubic Artin-Schelter algebras.

Revealed new quantum automorphism groups for cubic quantum spaces.

Connected quadratic cases to existing classifications.

Abstract

Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Talk aims to demonstrate that Manin's construction can be extended for quantum spaces which are non-quadratic homogeneous algebras. Here given a regular Artin-Schelter algebra of dimension 3 we construct the quantum group of its symmetries, i.e., the Hopf algebra of its automorphisms. For quadratic Artin-Schelter algebras these quantum groups are contained in the the classification of the GL(3) quantum matrix groups due to Ewen and Ogievetsky. For cubic Artin-Schelter algebras we obtain new quantum groups which are automorphisms of cubic quantum spaces.