Abstract
Let L be the sub Z[g]-module of U~ generated by a PBW basis of U~. This submodule is independent of the choice of the PBW basis. Let n': L —> L/qL be the canonical projection. Then the image B of the PBW basis is a Z-basis of L/qL and it is independent of the choice of the PBW bases. Let — : Uq^Uq be the Q-algebra involution defined by et^elf /*>->/*, q *->q~, q^q~. Then n' induces a Z-module isomorphism TT : Lr\L-*L/qL, B = (n)~(B) is a Z-basis of LC\L and Z[#]-basis of L. Moreover each element of B is fixed by —. B is called the canonical base of U~.
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