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

Abstract

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

Polynomial Relations for q-Characters via the ODE/IM Correspondence

“…These modules are called the fundamental representations [20]. It is known [7,8] that finite-dimensional simple U q (g)-modules remain simple when restricted to U q (b). According to the classification of the former [9,10], the simple module L(Ψ) is finitedimensional if its highest -weight has the form…”