Algorithm for computing canonical bases and foldings of quantum groups

Let ${\mathbf{U}}_q^{-}$ 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 ${\mathbf{U}}_q^{-}$. In the case ${\mathbf{U}}_q^{-}$ is symmetric, Antor gave a simple algorithm of computing $P$ by making use of monomial bases. By the folding theory, ${\mathbf{U}}_q^{-}$ (symmetric, with a certain automorphism) is related to a quantum group ${\underset{‾}{\mathbf{U}}}_q^{-}$ 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 ${\mathbf{U}}_q^{-}$ and for ${\underset{‾}{\mathbf{U}}}_q^{-}$.