#### Vol. 10, No. 1, 2016

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Actions of some pointed Hopf algebras on path algebras of quivers

### Ryan Kinser and Chelsea Walton

Vol. 10 (2016), No. 1, 117–154
##### 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 ${ℤ}_{n}$-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 $T\left(n\right)$.

##### Keywords
Hopf action, module algebra, path algebra, Schurian quiver, Taft algebra
##### Mathematical Subject Classification 2010
Primary: 16T05
Secondary: 05C20, 16S99