Crystal Base Elements of an ExtremalWeight Module Fixed by a Diagram Automorphism II: Case of Affine Lie Algebras

Abstract

We continue the study of the fixed point subset of a crystal under the action of a Dynkin diagram automorphism, by restricting ourselves to the case of the crystal base of an extremal weight module over a quantum affine algebra. In previous works, we introduced and studied a canonical injection from the fixed point subset of the crystal base above into the crystal base of a certain extremal weight module for the associated orbit Lie algebra, in the setting of general Kac–Moody algebras. The purpose of this chapter is to prove that this (injective) map is also surjective, and hence bijective under the assumption that the Dynkin diagram automorphism fixes a distinguished vertex “0” of the (affine) Dynkin diagram.