Diagram automorphisms and canonical bases for quantized enveloping algebras

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 $\underline{\tilde{\mathbf{B}}}$ of ${\underline{\mathbf{U}}}_q^{-}$ by making use of the geometric theory of quivers. He proved that there is a natural bijection ${\tilde{\mathbf{B}}}^{\sigma }\to \widetilde{\underline{\mathbf{B}}}$. 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 ${\tilde{\mathbf{B}}}^{\sigma }\to \widetilde{\underline{\mathbf{B}}}$ by an elementary method.

41 pages, the final version, to appear in J. of Algebra