Diagram automorphisms and canonical bases for quantized enveloping algebras

TL;DR

This paper constructs canonical bases for orbit algebras derived from symmetric Kac-Moody algebras using elementary methods, establishing a natural bijection with fixed points of the original basis.

Contribution

It provides an elementary construction of the canonical signed basis for orbit algebras, assuming the existence of the original basis, and proves a natural bijection with fixed points.

Findings

Constructed the canonical signed basis for orbit algebras.

Established a natural bijection between fixed points and the basis of the orbit algebra.

Provided an elementary method for basis construction.

Abstract

Let ${\mathbf U}^-_q$ be the negative part of the quantized enveloping algebra associated to a Kac-Moody algebra ${\mathfrak g}$ of symmetric type, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the orbit algebra ${\mathfrak g}^{\sigma}$ obtained from an admissible diagram automorphism $\sigma$ on ${\mathfrak g}$. Lusztig consructed the canonical basis ${\mathbf B}$ of ${\mathbf U}_q^-$ and the canonical signed basis $\underline{\widetilde{\mathbf B}}$ of $\underline{\mathbf U}_q^-$ by making use of the geometric theory of quivers. He proved that there is a natural bijection $\widetilde{\mathbf B}^{\sigma} \to \widetilde{\underline{\mathbf B}}$. In this paper, assuming the existence of the canonical basis ${\mathbf B}$ of ${\mathbf U}_q^-$, we construct the canonical signed basis $\widetilde{\underline{\mathbf B}}$ of $\underline{\mathbf U}_q^-$, and a natural bijection $\widetilde{\mathbf B}^{\sigma} \to \widetilde{\underline{\mathbf B}}$ by an elementary method.