Rigidity of down-up algebras with respect to finite group coactions

If G is a nontrivial finite group coacting on a graded noetherian down-up algebra A inner faithfully and homogeneously, then the fixed subring AcoG is not isomorphic to A. Therefore graded noetherian down-up algebras are rigid with respect to finite group coactions, in the sense of Alev-Polo. An example is given to show that this rigidity under group coactions does not have all the same consequences as the rigidity under group actions.