Irreducible modules for the quantum affine algebra [formula omitted] and its Borel subalgebra

Abstract

Let U q ( b ) denote the standard Borel subalgebra of the quantum affine algebra U q ( sl ˆ 2 ) . We show that the following hold for any choice of scalars ɛ 0 , ɛ 1 from the set { 1 , − 1 } : (i) Let V be a finite-dimensional irreducible U q ( b ) -module of type ( ɛ 0 , ɛ 1 ) . Then the action of U q ( b ) on V extends uniquely to an action of U q ( sl ˆ 2 ) on V. The resulting U q ( sl ˆ 2 ) -module structure on V is irreducible and of type ( ɛ 0 , ɛ 1 ) . (ii) Let V be a finite-dimensional irreducible U q ( sl ˆ 2 ) -module of type ( ɛ 0 , ɛ 1 ) . When the U q ( sl ˆ 2 ) -action is restricted to U q ( b ) , the resulting U q ( b ) -module structure on V is irreducible and of type ( ɛ 0 , ɛ 1 ) .