Vol. 10, No. 1, 2016

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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(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 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 uq(sl2), 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