The Link-Indecomposable Components of Hopf Algebras and Their Products

TL;DR

This paper investigates the structure of Hopf algebras using the link relation on simple subcoalgebras, providing new conditions and formulas for their decomposition and showing how components are generated by simple subcoalgebras.

Contribution

It introduces new sufficient conditions for the link relation and derives a formula for products of link-indecomposable components in Hopf algebras with the dual Chevalley property.

Findings

Provides a formula for products of link-indecomposable components.

Shows each component is generated by a simple subcoalgebra.

Generalizes results on pointed Hopf algebras from 1995.

Abstract

The link relation on simple subcoalgebras is used for decompositions of coalgebras. In this paper, we provide more sufficient conditions for this link relation, and prove a formula on the products between link-indecomposable components of Hopf algebras with the dual Chevalley property. Furthermore, we show that each of its component is generated by a simple subcoalgebra, as a faithfully flat module (in fact, a projective generator) over a Hopf subalgebra which is the component containing the unit element. Our conclusions generalize some relevant results on pointed Hopf algebras, which were established by Montgomery in 1995.