Irreducible modules for the quantum affine algebra Uq(g) and its Borel subalgebra Uq(g)⩾0

Let g be an affine Kac–Moody Lie algebra, and let Uq(g) be its quantized universal enveloping algebra. Let the Borel subalgebra Uq(g)⩾0 of Uq(g) be the nonnegative part of Uq(g) with respect to the standard triangular decomposition. Suppose ε∈n{−1,1}, where n is the number of simple roots of g. We construct a bijection between finite-dimensional irreducible Uq(g)⩾0-modules of type ε and finite-dimensional irreducible Uq(g)-modules of type ε. In particular:(i)Let V be a finite-dimensional irreducible Uq(g)⩾0-module of type ε. 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 ε.(ii)Let V be a finite-dimensional irreducible Uq(g)-module of type ε. 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 ε.