Actions of some pointed Hopf algebras on path algebras of quivers

Abstract

We classify Hopf actions of Taft algebras $T\left(n\right)$ on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful ${\mathbb{Z}}_n$-action (by directed graph automorphisms) also admits inner faithful actions of a Taft algebra. Several examples for actions of the Sweedler algebra $T\left(2\right)$ and for actions of $T\left(3\right)$ are presented in detail. We then extend the results on Taft algebra actions on path algebras to actions of the Frobenius–Lusztig kernel $u_q\left({\mathfrak{s}\mathfrak{l}}_2\right)$, and to actions of the Drinfeld double of $T\left(n\right)$.