A family of trivial ozone algebras

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.