A functional model for the tensor product of level 1 highest and level −1 lowest modules for the quantum affine algebra [formula omitted

Abstract

Let V( Λ i ) (resp., V(− Λ j )) be a fundamental integrable highest (resp., lowest) weight module of U q( sl 2) . The tensor product V( Λ i )⊗ V(− Λ j ) is filtered by submodules F n=U q( sl 2)(v i⊗ v n−i) , n≥0, n≡i−j mod 2 , where v i ∈ V( Λ i ) is the highest vector and v n−i∈V(−Λ j) is an extremal vector. We show that F n / F n+2 is isomorphic to the level 0 extremal weight module V( n( Λ 1− Λ 0)). Using this we give a functional realization of the completion of V( Λ i )⊗ V(− Λ j ) by the filtration ( F n ) n≥0 . The subspace of V( Λ i )⊗ V(− Λ j ) of sl 2 -weight m is mapped to a certain space of sequences (P n,l) n≥0,n≡i−j mod 2,n−2l=m , whose members P n, l = P n, l ( X 1,…, X l ∣ z 1,…, z n ) are symmetric polynomials in X a and symmetric Laurent polynomials in z k , with additional constraints. When the parameter q is specialized to −1 , this construction settles a conjecture which arose in the study of form factors in integrable field theory.