Convex Bases of PBW type for Quantum Affine Algebras

Abstract

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$'$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.