Irreducible modules for the quantum affine algebra and its Borel subalgebra

Let be an affine Kac–Moody Lie algebra, and let be its quantized universal enveloping algebra. Let the Borel subalgebra of be the nonnegative part of with respect to the standard triangular decomposition. Suppose ε {−1,1}n, where n is the number of simple roots of . We construct a bijection between finite-dimensional irreducible -modules of type ε and finite-dimensional irreducible -modules of type ε. In particular: (i) Let V be a finite-dimensional irreducible -module of type ε. Then the action of on V extends uniquely to an action of on V. The resulting -module structure on V is irreducible and of type ε. (ii) Let V be a finite-dimensional irreducible -module of type ε. When the -action is restricted to , the resulting -module structure on V is irreducible and of type ε.