Lyndon bases of split $\imath$quantum groups

TL;DR

This work develops Lyndon bases for split $ ext{I}$quantum groups by introducing $oldsymbol{U}^ ext{I}$-good words and Lyndon words, and constructs a concrete basis $oldsymbol{B}_{oldsymbol{ ext{L}}}$ that integrates with Xu–Yang's PBW-type bases, with a key alignment in type $A_n$. By establishing an integral condition (IC), the authors provide a pathway to canonical bases for $oldsymbol{U}^ ext{I}(rak{g})$, proving existence for type $A_n$ while highlighting limitations via examples such as type $G_2$. The results yield practical, computable bases for representation-theoretic work on split $ ext{I}$quantum groups and clarify how these canonical bases relate to, yet differ from, other $ ext{I}$canonical constructions. The work thereby advances the structural understanding of bases in the $ ext{I}$quantum setting and offers a concrete canonical basis framework in the type $A$ case, with broader implications for quantum symmetric pairs.

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 $\mathbf{U}^\imath(\mathfrak{sl}_n)$.

Theorems & Definitions (65)

• Definition 3.1.1
• Proposition 3.1.2
• proof
• Remark 3.1.3
• Proposition 3.2.1
• proof
• Definition 3.2.2
• Remark 3.2.3
• Proposition 3.2.4
• proof