4-dimensional Artin-Schelter regular quadratic $\tilde{H}_4$-algebras

TL;DR

This paper classifies four-dimensional Artin-Schelter regular quadratic algebras with Heisenberg group symmetry, showing they are twists of Sklyanin or quantum Clifford algebras, enriching the understanding of algebraic structures with group actions.

Contribution

It identifies and characterizes all such algebras with specified symmetry and representation, revealing their structure as twists of known algebraic families.

Findings

Algebras are twists of Sklyanin or quantum Clifford algebras.

Classification based on Heisenberg group action and Schrödinger representation.

Provides a complete description of these regular algebras.

Abstract

In this paper, quadratic algebras on which $\tilde{H}_4$, the Heisenberg group of order 64, acts as degree-preserving algebra automorphisms are studied. In particular, we show that if $\mathcal{A}$ is a four-dimensional Artin-Schelter regular quadratic $\tilde{H}_4$-algebra with the degree one part isomorphic to the Schr\"odinger representation of $\tilde{H}_4$, then $\mathcal{A}$ is (a twist of) a four-dimensional Sklyanin algebra or (a twist of) a quantum Clifford algebra of global dimension 4.