Irreducible modules for the quantum affine algebra Uq( ̂ sl2) and its Borel subalgebra Uq( ̂ sl2) ≥0

Let Uq(ŝl2) ≥0 denote the Borel subalgebra of the quantum affine algebra Uq(ŝl2). We show that the following hold for any choice of scalars ε0, ε1 from the set {1,−1}. (i) Let V be a finite-dimensional irreducible Uq(ŝl2) -module of type (ε0, ε1). Then the action of Uq(ŝl2) ≥0 on V extends uniquely to an action of Uq(ŝl2) on V . The resulting Uq(ŝl2)-module structure on V is irreducible and of type (ε0, ε1). (ii) Let V be a finite-dimensional irreducible Uq(ŝl2)-module of type (ε0, ε1). When the Uq(ŝl2)-action is restricted to Uq(ŝl2) , the resulting Uq(ŝl2) -module structure on V is irreducible and of type (ε0, ε1). 1 The quantum affine algebra Uq(ŝl2) The affine Kac-Moody Lie algebra ŝl2 has played an essential role in diverse areas of mathematics and physics. Elements of ŝl2 can be represented as vertex operators, which are certain generating functions that appear in the dual resonance models of particle physics (see [15] and [8]). The algebra ŝl2 also features prominently in the study of Knizhnik-Zamolodchikov equations ∗Support from NSF grant #DMS–0245082 is gratefully acknowledged.