A family of trivial ozone algebras

TL;DR

This paper investigates a family of Calabi--Yau algebras, including those linked to a nodal cubic, demonstrating they have trivial ozone groups and are rigid with respect to automorphisms, highlighting their unique symmetry properties.

Contribution

It introduces and analyzes a new family of Calabi--Yau algebras with trivial ozone groups and establishes their rigidity against nontrivial automorphisms.

Findings

Algebras have trivial ozone groups, with only the identity automorphism fixing the center.

Graded members are rigid; invariant rings under nontrivial automorphisms are not Artin--Schelter regular.

Includes the quadratic Artin--Schelter regular algebras associated to a nodal cubic.

Abstract

We study a family of Calabi--Yau algebras that include the quadratic Artin--Schelter regular algebras associated to a nodal cubic. It is shown that these algebras have trivial ozone group, that is, the identity is the only automorphism that fixes the center pointwise. The graded members of this family of algebras are shown to be rigid in the sense that the invaraint ring under a nontrivial group of graded automorphisms is not Artin--Schelter regular.