Classify finite-dimensional Hopf algebra actions on A_N

Classify all actions of finite-dimensional Hopf algebras on the algebra A_N = ⟨x,y⟩/(yx − xy − x^N), including identifying which Hopf algebras can act and describing the actions explicitly. The goal is to determine the "quantum finite groups of automorphisms" of A_N, beyond the generalized Taft Hopf algebras considered in the paper.

Background

After classifying finite subgroups of Aut(A_N), the authors note that the next classical step would be to describe invariant subalgebras, which they find difficult. They thus turn to a broader problem: understanding quantum symmetries via Hopf algebra actions on A_N.

They explicitly state that at the general level of all finite-dimensional Hopf algebras, they do not know how to approach the problem and therefore restrict their analysis to generalized Taft Hopf algebras, proving a negative result for those. This leaves open the full classification for finite-dimensional Hopf algebras beyond this subclass.

References

We instead take a less classical direction and try to extend the result of the theorem and find all actions of finite dimensional Hopf algebras on A_N — that is, to put it in a colorful language, to find all quantum finite groups of automorphisms of A_N. Now, at that level of generality we do not know how to approach the problem, so we restrict ourselves to looking for all actions of generalized Taft Hopf algebras on A_N.

— On the derivations and automorphisms of the algebra $k\langle x, y\rangle/(yx-xy-x^N)$ (2402.11962 - Suárez-Álvarez, 19 Feb 2024) in Introduction