Actions of some pointed Hopf algebras on path algebras of quivers

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).Comment: 29 pages. v3: Title changed, and Section 8 is ne

Discriminants of Taft Algebra Smash Products and Applications

“…Since the action of H n (λ) is inner faithful, the matrix x is nonzero [13,Lemma 2.5]. Additionally,…”

“…The goal of this paper is to consider this problem for certain smash products by H n (λ), the nth Taft algebra. Such actions have been studied previously [3,4,13].…”

Actions of quantum linear spaces on quantum algebras

“…Semisimple Hopf actions on quantum planes and quantum Weyl algebras are well-understood [8,9]. Our goal is to better understand non-semisimple Hopf actions, specifically actions by pointed Hopf algebras, which themselves have attracted much recent interest [11,17].…”

On actions of Drinfel'd doubles on finite dimensional algebras

“…Chen and Zhang classified all D(T 2 (−1))-module algebras of dimension 4 up to isomorphism as D(T 2 (−1))-modules in [8], in particular giving all D(T 2 (−1))-module algebra structures on M 2 (k). In [15], Kinser and Walton examine actions of Taft algebras on path algebras of quivers, and extend such actions to D(T n (q)).…”