Finite-dimensional modules for the quantum affine algebra Uq(g) and its Borel subalgebra

Let g be a Kac-Moody algebra of affine type. Let Uq( g ) be the quantized universal enveloping algebra, or quantum affine algebra, associated with g . The Borel subalgebra Uq( g )≥0 of Uq( g ) is the nonnegative part of Uq( g ) with respect to the standard triangular decomposition Uqg -⊗Uqg 0⊗Uqg +.;Let n be the number of simple roots of g and choose &egr; ∈ {-1,1}n. We prove the following: (1) Let V be a finite-dimensional irreducible Uq( g )≥0-module of type &egr;. Then the action of Uq( g )≥0 on V extends uniquely to an action of Uq( g ) on V. The resulting Uq( g )-module structure on V is irreducible and of type &egr;. (2) Let V be a finite-dimensional irreducible Uq( g )-module of type &egr;. When the Uq( g )-action is restricted to Uq( g )≥0 the resulting Uq( g )≥0-module structure on V is irreducible and of type &egr;.;Thus there is a bijection between finite-dimensional irreducible Uq( g )-modules of type &egr; and finite-dimensional irreducible Uq( g )≥0-modules of type &egr;.