A functional model for the tensor product of level 1 highest and level −1 lowest modules for the quantum affine algebra Uq(sl2)

Let V(Λi) (resp., V(−Λj)) be a fundamental integrable highest (resp., lowest) weight module of Uq(sl2). The tensor product V(Λi)⊗V(−Λj) is filtered by submodules Fn=Uq(sl2)(vi⊗vn−i), n≥0, n≡i−j mod 2, where vi∈V(Λi) is the highest vector and vn−i∈V(−Λj) is an extremal vector. We show that Fn/Fn+2 is isomorphic to the level 0 extremal weight module V(n(Λ1−Λ0)). Using this we give a functional realization of the completion of V(Λi)⊗V(−Λj) by the filtration (Fn)n≥0. The subspace of V(Λi)⊗V(−Λj) of sl2-weight m is mapped to a certain space of sequences (Pn,l)n≥0,n≡i−j mod 2,n−2l=m, whose members Pn,l=Pn,l(X1,…,Xl∣z1,…,zn) are symmetric polynomials in Xa and symmetric Laurent polynomials in zk, 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.