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

TL;DR

This paper proves that graded noetherian down-up algebras are rigid against finite group coactions, meaning such coactions do not produce fixed subrings isomorphic to the original algebra, highlighting a form of algebraic stability.

Contribution

It establishes the rigidity of down-up algebras under finite group coactions and provides an example showing differences from group action rigidity.

Findings

Finite group coactions do not produce isomorphic fixed subrings.

Down-up algebras are rigid with respect to finite group coactions.

An example illustrates differences from group action rigidity.

Abstract

If a nontrivial finite group coacts on a graded noetherian down-up algebra $A$ inner faithfully and homogeneously, then the fixed subring 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.