Diagram automorphisms and canonical bases for quantized enveloping algebras (2108.06673v3)
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.