Diagram automorphisms and canonical bases for quantum affine algebras

Abstract

Let Uq− be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra g, and U_q− the algebra corresponding to the orbit algebra of g obtained from a diagram automorphism σ on g. Let Bσ be the set of σ-fixed elements in the canonical basis of Uq−, and B_ the canonical basis of U_q−. Lusztig proved that there exists a canonical bijection Bσ≃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 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 for PBW-bases, by using a new type of PBW-bases of Uq− obtained by Muthiah-Tingley, which is a generalization of PBW-bases constructed by Beck-Nakajima.