Note on Invariance and Finiteness for the Exponent of Hopf Algebras

TL;DR

This paper compares two notions of exponent in finite-dimensional Hopf algebras, analyzing their invariance properties and conditions under which they are finite or infinite across different characteristics.

Contribution

It clarifies the invariance of these exponents under twisting and Drinfeld double operations and characterizes their finiteness based on algebraic properties and characteristic.

Findings

Both exponents are invariant under twisting and Drinfeld double.

Exponents are infinite in characteristic 0 if certain properties hold.

Exponents are finite in positive characteristic under the same conditions.

Abstract

There are two notions of exponent of finite-dimensional Hopf algebras introduced and studied in the literature. In this note, we discuss and compare their properties including invariance and finiteness in this note. Specifically, one notion is invariant under twisting and taking the Drinfeld double, just like the other one. We also find that if the non-cosemisimplicity and dual Chevalley property hold, both exponents are infinite in characteristic $0$ but finite in positive characteristic.