For non-symmetric quantum groups of finite type, the canonical-to-PBW transition matrix is computable by a closed inner-product formula, and is determined exactly by the sigma-invariant data of the folded symmetric case.
Kirillov; ``Quiver Representations and Quiver Varieties'', Graduate Studies in Math
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Algorithm for computing canonical bases and foldings of quantum groups
For non-symmetric quantum groups of finite type, the canonical-to-PBW transition matrix is computable by a closed inner-product formula, and is determined exactly by the sigma-invariant data of the folded symmetric case.