Braid group action and canonical bases
The author shows that the action of the braid group on a quantized enveloping algebra \(U\) (possibly of infinite type) is well compatible with the canonical basis. More precisely, if \(U^+\) is the plus-part of \(U\) and we decompose \[ U^+ =\bigl(U^+ \cap T_i^{-1} (U^+) \bigr) \oplus U^+ E_i, \quad U^+ =\bigl(T_i(U^+) \cap U^+ \bigr) \oplus E_iU^+ \] then the projections of the canonical basis vectors of \(U^+\) onto the first factor which are not zero form a basis of \(U^+ \cap T_i^{-1} (U^+)\) and \(T_i(U^+) \cap U^+\) respectively, and under \(T_i\) these bases correspond. This result leads to a parametrization of a big part of the canonical basis.
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