Convex bases of PBW type for quantum affine algebras

This note has two purposes. First we establish that the map defined in [L, Sect. 40.2.5 (a)] is an isomorphism for certain admissible sequences. Second we show the map gives rise to a convex basis of Poincaré-Birkhoff-Witt (PBW) type for U⁺, an affine untwisted quantized enveloping algebra of Drinfel's and Jimbo. The computations in this paper are made possible by extending the braid group action by certain outer automorphisms of the algebra.