Convex bases of PBW type for quantum affine algebras

This note has two purposes. First we establish that the map defined in [L, \S 40.2.5 (a)] is an isomorphism for certain admissible sequences. Second we show the map gives rise to a convex basis of Poincar\'e--Birkhoff--Witt (PBW) type for \bup, an affine untwisted quantized enveloping algebra of Drinfel

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d and Jimbo. The computations in this paper are made possible by extending the usual braid group action by certain outer automorphisms of the algebra.

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