A class of semisimple Hopf algebras acting on quantum polynomial algebras

TL;DR

This paper constructs new semisimple Hopf algebras of dimension 2n^2 and explores their inner faithful actions on quantum polynomial algebras, expanding the understanding of non-group-based Hopf actions.

Contribution

It introduces a class of semisimple Hopf algebras acting on quantum polynomial algebras and classifies certain actions of the Kac-Paljutkin algebra.

Findings

Constructed non-commutative, non-cocommutative semisimple Hopf algebras of dimension 2n^2.

Provided conditions for inner faithful actions on quantum polynomial algebras.

Classified inner faithful actions of the Kac-Paljutkin algebra on the quantum plane.

Abstract

We construct a class of non-commutative, non-cocommutative, semisimple Hopf algebras of dimension $2n^2$ and present conditions to define an inner faithful action of these Hopf algebras on quantum polynomial algebras, providing, in this way, more examples of semisimple Hopf actions which do not factor through group actions. Also, under certain condition, we classify the inner faithful Hopf actions of the Kac-Paljutkin Hopf algebra of dimension $8$, $H_8$, on the quantum plane.