# Actions of cocommutative Hopf algebras

## TL;DR

This paper studies the structure of prime spectra in algebras acted upon by cocommutative Hopf algebras, revealing stratification methods and properties of stable ideals under such actions.

## Contribution

It introduces a stratification of prime spectra for algebras with integral cocommutative Hopf algebra actions and proves stability properties of semiprime ideals in characteristic zero.

## Key findings

- Prime spectrum stratification in terms of commutative subalgebras
- Largest H-stable ideal in a semiprime ideal is semiprime
- Results hold under integral action assumptions in characteristic zero

## Abstract

Let $H$ be a cocommutative Hopf algebra acting on an algebra $A$. Assuming the base field to be algebraically closed and the $H$-action on $A$ to be integral, that is, it is given by a coaction of some Hopf subalgebra of the finite dual $H^\circ$ that is an integral domain, we stratify the prime spectrum $\mbox{Spec}\, A$ in terms of the prime spectra of certain commutative algebras. For arbitrary $H$-actions in characteristic $0$, we show that the largest $H$-stable ideal of $A$ that is contained in a given semiprime ideal of $A$ is semiprime as well.