Taft algebra actions on preprojective algebras

We classify actions of generalized Taft algebras on preprojective algebras of extended Dynkin quivers of type $A$. This may be viewed as an extension of the problem of classifying actions on the polynomial ring in two variables. In cases where the grouplike element acts via rotation on the underlying quiver, we compute invariants of the Taft action and, in certain cases, show that the invariant ring is isomorphic to the center of the preprojective algebra.