Superpotentials and Quiver Algebras for Semisimple Hopf Actions

TL;DR

This paper studies the action of semisimple Hopf algebras on certain regular algebras, showing how to compute homological determinants, derive quiver algebras, and analyze properties like the Auslander map, with applications to quantum Kleinian singularities.

Contribution

It introduces a method to compute homological determinants via superpotentials and explicitly determines quiver algebras for smash products, extending previous results.

Findings

Homological determinant can be computed using twisted superpotentials.

The smash product algebra is a derivation-quotient algebra and Morita equivalent to a quiver algebra.

Applications to quantum Kleinian singularities demonstrate the technique's effectiveness.

Abstract

We consider the action of a semisimple Hopf algebra $H$ on an $m$-Koszul Artin-Schelter regular algebra $A$. Such an algebra $A$ is a derivation-quotient algebra for some twisted superpotential $\mathsf{w}$, and we show that the homological determinant of the action of $H$ on $A$ can be easily calculated using $\mathsf{w}$. Using this, we show that the smash product $A\,\#\,H$ is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra $\Lambda$ to which $A\,\#\,H$ is Morita equivalent, generalising a result of Bocklandt-Schedler-Wemyss. We also show how $\Lambda$ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan-Kirkman-Walton-Zhang follow using our techniques.