Lyndon bases of split $\imath$quantum groups

TL;DR

This paper introduces Lyndon bases for split $ extit{i}$quantum groups, explores their relationship with PBW bases, and establishes the existence of canonical bases for type A split $ extit{i}$quantum groups, advancing the understanding of their structure.

Contribution

It provides a new construction of Lyndon bases for split $ extit{i}$quantum groups and links them to PBW bases, also proving the existence of canonical bases in type A cases.

Findings

Lyndon bases are constructed for split $ extit{i}$quantum groups.

A relationship between Lyndon and PBW bases is established.

Canonical bases are proven to exist for type A split $ extit{i}$quantum groups.

Abstract

We introduce and study Lyndon bases of split $\imath$quantum groups $\mathbf{U}^\imath(\mathfrak{g})$. A relationship between the Lyndon bases and PBW-type bases was provided. As an application, we establish the existence of canonical bases for the type A split $\imath$quantum groups $\mathbf{U}^\imath(\mathfrak{sl}_n)$.