Local criteria for cocommutative Hopf algebras

TL;DR

This paper establishes a criterion for when a finite-dimensional cocommutative Hopf algebra is local, linking it to properties of its coradical filtration and primitive elements.

Contribution

It provides a new local criterion for cocommutative Hopf algebras based on their coradical filtration and primitive elements.

Findings

A finite-dimensional cocommutative Hopf algebra is local iff the subalgebra generated by its first coradical filtration is local.

For connected Hopf algebras, locality is equivalent to all primitive elements being nilpotent.

The criterion simplifies the understanding of local properties in cocommutative Hopf algebras.

Abstract

We prove that a finite-dimensional cocommutative Hopf algebra $H$ is local, if and only if the subalgebra generated by the first term of its coradical filtration $H_1$ is local. In particular if $H$ is connected, $H$ is local if and only if all the primitive elements of $H$ are nilpotent.