A decomposition Theorem for pointed braided Hopf algebras

TL;DR

This paper extends a fundamental decomposition theorem for pointed braided Hopf algebras, demonstrating compatibility with comodule structures when the underlying Hopf algebra is cosemisimple.

Contribution

It generalizes the decomposition theorem to include comodule structures in the setting of cosemisimple Hopf algebras.

Findings

Decomposition is compatible with comodule structures in cosemisimple cases.

Extension of the fundamental theorem to braided Hopf algebras in Yetter-Drinfeld categories.

Provides a broader understanding of the structure of pointed braided Hopf algebras.

Abstract

A known fundamental Theorem for braided pointed Hopf algebras states that for each coideal subalgebra, that fulfils a few properties, there is an associated quotient coalgebra right module such that the braided Hopf algebra can be decomposed into a tensor product of these two. Often one considers braided Hopf algebras in a Yetter-Drinfeld category of an ordinary Hopf algebra. In this case the braided Hopf algebra is in particular a comodule, as well as many interesting coideal subalgebras. We extend the mentioned Theorem by proving that the decomposition is compatible with this comodule structure if the underlying ordinary Hopf algebra is cosemisimple.