Diagram automorphisms and canonical bases for quantized enveloping algebras

Abstract

Let X be a Cartan datum of symmetric type, with an admissible automorphism σ on X, and X_ the Cartan datum induced from (X,σ). Let Uq− (resp. U_q−) be the negative part of the quantized enveloping algebra associated to X (resp. X_). Lusztig constructed the canonical basis B of Uq− and the canonical signed basis B_˜ of U_q− by making use of the geometric theory of quivers. By normalizing the sign of B_˜, he obtained the canonical basis B_ of U_q−, and a natural bijection ▪. In this paper, assuming the existence of B for Uq−, we construct the canonical basis B_ of U_q−, and a bijection ▪, by an elementary method, subject to the condition that the order of σ is odd. In the case where the order is even, we obtain the corresponding result for the canonical signed basis.