Coideal subalgebras of pointed and connected Hopf algebras

TL;DR

This paper establishes a PBW basis structure for coideal subalgebras of pointed and connected Hopf algebras, generalizing previous results and introducing new combinatorial methods for a broader class of braided bialgebras.

Contribution

It proves that certain coideal subalgebras have PBW bases and are graded iterated Ore extensions, extending structure theorems to non-primitively generated braided bialgebras.

Findings

Existence of PBW bases for coideal subalgebras under mild conditions

Characterization of these subalgebras as graded iterated Ore extensions

Development of new combinatorial methods for non-primitively generated cases

Abstract

Let $H$ be a pointed Hopf algebra with abelian coradical. Let $A\supseteq B$ be left (or right) coideal subalgebras of $H$ that contain the coradical of $H$. We show that $A$ has a PBW basis over $B$, provided that $H$ satisfies certain mild conditions. In the case that $H$ is a connected graded Hopf algebra of characteristic zero and $A$ and $B$ are both homogeneous of finite Gelfand-Kirillov dimension, we show that $A$ is a graded iterated Ore extension of $B$. These results turn out to be conceptual consequences of a structure theorem for each pair $S\supseteq T$ of homogeneous coideal subalgebras of a connected graded braided bialgebra $R$ with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of $S$ over $T$. The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko for primitively generated braided Hopf algebras of diagonal type. Since in our context we don't priorilly assume $R$ to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others.