Diagram automorphisms and canonical bases for quantum affine algebras

TL;DR

This paper proves a fundamental correspondence between fixed points of canonical bases and canonical bases of subalgebras in quantum affine algebras, using elementary methods without geometric or crystal basis theories.

Contribution

It provides an elementary proof of Lusztig's bijection for finite and affine types, and explores PBW-bases via new constructions by Muthiah-Tingley.

Findings

Established the bijection between fixed points and subalgebra bases in finite and affine types.

Connected PBW-bases from different constructions, enhancing understanding of their relationships.

Abstract

Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of ${\mathfrak g}$ obtained from a diagram automorphism $\sigma$ on ${\mathfrak g}$. Let ${\mathbf B}^{\sigma}$ be the set of $\sigma$-fixed elements in the canonical basis of ${\mathbf U}_q^-$, and $\underline{\mathbf B}$ the canonical basis of $\underline{\mathbf U}_q^-$. Lusztig proved that there exists a canonical bijection ${\mathbf B}^{\sigma} \simeq \underline{\mathbf B}$ based on his geometric construction of canonical bases. In this paper, we prove (the signed bases version of) this fact, in the case where ${\mathfrak g}$ is finite or affine type, in an elementary way, in the sense that we don't appeal to the geometric theory of canonical bases nor Kashiwara's theory of crystal bases. We also discuss the correspondence between PBW-bases, by using a new type of PBW-bases of ${\mathbf U}_q^-$ obtained by Muthiah-Tingley, which is a generalization of PBW-bases constructed by Beck-Nakajima.