Algorithm for computing canonical bases and foldings of quantum groups

Let Uq⁻ be the negative half of a quantum group of finite type. Let P be the transition matrix between the canonical basis and a PBW basis of Uq⁻. In the case Uq⁻ is symmetric, Antor gave a simple algorithm of computing P by making use of monomial bases. By the folding theory, Uq⁻ (symmetric, with a certain automorphism) is related to a quantum group Uq⁻ of non-symmetric type. In this paper, we extend the results of Antor to the non-symmetric case, and discuss the relationship between the algorithms for Uq⁻ and for Uq⁻.