A rationalization of the crystal and a diagram automorphism

Let A be a symmetrizable generalized Cartan matrix, and ω a diagram automorphism of the Dynkin diagram of A. First we define a canonical action of ω on the crystal , and then prove that its fixed point subset can be identified with the crystal associated to the generalized Cartan matrix  obtained by “folding” A according to ω. Second we introduce a natural rationalization of , and equip it with a crystal-like structure, which we call a rational crystal structure. Finally, we prove that the fixed point subset of the rational crystal under the action of ω can be identified with the rational crystal associated to Â.