On the Exponent of Finite-Dimensional Non-Cosemisimple Hopf Algebras

TL;DR

This paper investigates the exponent of finite-dimensional non-cosemisimple Hopf algebras, establishing conditions under which the exponent is either infinite or finite depending on the algebra's properties and characteristic.

Contribution

It proves that the exponent is infinite for certain Hopf algebras with the Chevalley property in characteristic zero, and finite for pointed Hopf algebras in positive characteristic.

Findings

Exponent is infinite for non-cosemisimple Hopf algebras with Chevalley property in characteristic 0.

Exponent is finite for non-cosemisimple pointed Hopf algebras in positive characteristic.

Abstract

In 1999, Y. Kashina introduced the exponent of a Hopf algebra. In this paper, we prove that the exponent of a finite dimensional non-cosemisimple Hopf algebra with Chevalley property in characteristic 0 is infinite, and the exponent of a finite dimensional non-cosemisimple pointed Hopf algebra in positive characteristic is finite.