Diagram automorphisms and canonical bases for quantum affine algebras, II

TL;DR

This paper extends the understanding of canonical bases in quantum affine algebras by establishing a bijection between bases under diagram automorphisms, even when the automorphism is not admissible, broadening previous results.

Contribution

It proves the existence of a natural bijection between canonical bases for quantum affine algebras under diagram automorphisms without the admissibility restriction.

Findings

Bijection exists even if automorphism is not admissible

Results hold for simply-laced finite or affine types

Extends previous elementary proofs to more general automorphisms

Abstract

Let ${\mathbf U}_q^-$ be the negative part of the quantum enveloping algebra, and $\sigma$ the algebra automorphism on ${\mathbf U}_q^-$ induced from a diagram automorphism. Let $\underline{\mathbf U}_q^-$ be the quantum algebra obtained from $\sigma$, and $\widetilde{\mathbf B}$ (resp. $\widetilde{\underline{\mathbf B}}$) the canonical signed basis of ${\mathbf U}_q^-$ (resp. $\underline{\mathbf U}_q^-$). Assume that ${\mathbf U}_q^-$ is simply-laced of finite or affine type. In our previous papers [SZ1, 2], we have proved by an elementary method, that there exists a natural bijection $\widetilde{\mathbf B}^{\sigma} \simeq \widetilde{\underline{\mathbf B}}$ in the case where $\sigma$ is admissible. In this paper, we show that such a bijection exists even if $\sigma$ is not admissible, possibly except some small rank cases.