Examples of non-semisimple Hopf algebra actions on Artin-Schelter regular algebras

Abstract

Let k be a base field of characteristic p > 0 and let U be the restricted enveloping algebra of a 2-dimensional nonabelian restricted Lie algebra. We classify all inner-faithful U -actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three.Recall that a Hopf H-action on an algebra A is called inner-faithful if there is no nonzero Hopf ideal I ⊆ H such that IA = 0 [CKWZ1, Definition 1.5]. Our goal is to construct examples of inner-faithful and homogeneous U -actions on T where U is the non-semisimple Hopf algebra given in Definition 0.1 and T is a connected graded Artin-Schelter regular algebra. The main result consists of Proposition 3.2 and Theorem 0.5 that together classify all inner-faithful U -actions on noetherian Koszul Artin-Schelter regular algebras of global dimension at most three.Throughout let k be a base field with char k = p > 0.Definition 0.1. Let U be the k-algebra generated by u and w and subject to the relations (E0.1.1) u p = 0, w p = w, [w, u](:= wu − uw) = u.

Degree bounds for Hopf actions on Artin-Schelter regular algebras

“…We remark that the preceding result is not true if we remove the hypothesis that H is semisimple [CWZ,Observation 4.1(4)].…”

Pointed Hopf actions on quantum generalized Weyl algebras

“…Suppose that A is an inner-faithful Z-graded T -module algebra where g acts as an automorphism η γ,µ as in equation (2.3) with γ = 1, and (5) µq 1−d is an m th root of unity, (6)…”

“…The first author along with Cline considered more general actions on quantum affine spaces, quantum matrices, and quantized Weyl algebras [8]. More generally, Chen, Wang, and Zhang studied actions of nonsemisimple Hopf algebras on Artin-Schelter regular algebras in characteristic p [6].…”