Actions of some pointed Hopf algebras on path algebras of quivers

TL;DR

This paper classifies how certain pointed Hopf algebras, including Taft algebras and quantum groups, act on path algebras of quivers, revealing new connections between graph automorphisms and Hopf algebra actions.

Contribution

It provides a classification of Hopf actions of Taft algebras on path algebras of quivers and extends these results to quantum groups and Drinfeld doubles.

Findings

Classified Hopf actions of Taft algebras on loopless, finite, Schurian quivers.

Showed that quivers with Z_n automorphisms admit faithful Taft algebra actions.

Presented detailed examples for T(2), T(3), and extended to u_q(sl_2) and Drinfeld doubles.

Abstract

We classify Hopf actions of Taft algebras T(n) 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(2) and for actions of T(3) 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(sl_2), and to actions of the Drinfeld double of T(n).