Actions of some pointed Hopf algebras on path algebras of quivers

Kinser, Ryan; Walton, Chelsea

doi:10.2140/ant.2016.10.117

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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

DOI: 10.2140/ant.2016.10.117

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 $u_{q} ({s l}_{2})$ , and to actions of the Drinfeld double of $T (n)$ .

Keywords

Hopf action, module algebra, path algebra, Schurian quiver, Taft algebra

Mathematical Subject Classification 2010

Primary: 16T05

Secondary: 05C20, 16S99

Milestones

Received: 6 January 2015

Revised: 30 July 2015

Accepted: 19 September 2015

Published: 14 February 2016

Authors

Ryan Kinser
Department of Mathematics University of Iowa Iowa City, IA 52242 United States

Chelsea Walton
Department of Mathematics Temple University Philadelphia, PA 19122 United States

Vol. 10, No. 1, 2016