Subcoalgebras and endomorphisms of free Hopf algebras

TL;DR

This paper characterizes subcoalgebras and endomorphisms of various free Hopf algebras generated by matrix coalgebras, revealing structural insights and automorphism groups relevant to Hopf algebra theory.

Contribution

It explicitly determines small subcoalgebras and endomorphisms of free Hopf algebras with specific antipode properties, advancing understanding of their internal structure.

Findings

Classification of small subcoalgebras in free Hopf algebras

Description of endomorphism structures

Identification of centers of related Hopf algebra categories

Abstract

For a matrix coalgebra $C$ over some field, we determine all small subcoalgebras of the free Hopf algebra on $C$, the free Hopf algebra with a bjective antipode on $C$, and the free Hopf algebra with antipode $S$ satisfying $S^{2d}={\rm id}$ on $C$ for some fixed $d$. We use this information to find the endomorphisms of these free Hopf algebras, and to determine the centers of the categories of Hopf algebras, Hopf algebras with bijective antipode, and Hopf algebras with antipode of order dividing 2d.