Relative PBW type theorems for symmetrically braided Hopf algebras

TL;DR

This paper proves that all irreducible symmetrically braided Hopf algebras in characteristic zero possess a PBW basis, and establishes conditions for such algebras to be PBW modules over certain subalgebras.

Contribution

It introduces a relative PBW theorem for symmetrically braided Hopf algebras, extending the understanding of their structure in characteristic zero.

Findings

All irreducible symmetrically braided Hopf algebras are of PBW type in characteristic zero.

Provides criteria for braided Hopf algebras to be PBW modules over subalgebras containing the coradical.

Establishes a framework for analyzing the PBW property in braided Hopf algebra contexts.

Abstract

We show that in characteristic zero all irreducible symmetrically braided Hopf algebras are of PBW type. Consequently, we obtain conditions for a braided Hopf algebra to be of PBW type as module over a braided Hopf subalgebra containing the coradical.