A functional model for the tensor product of level 1 highest and level -1 lowest modules for the quantum affine algebra U_q(sl_{2}^)

Abstract

Let $V(\Lambda_i)$ (resp., $V(-\Lambda_j)$) be a fundamental integrable highest (resp., lowest) weight module of $U_q(\hat{sl}_{2})$. The tensor product $V(\Lambda_i)\otimes V(-\Lambda_j)$ is filtered by submodules $F_n=U_q(\hat{sl}_{2})(v_i\otimes \bar{v}_{n-i})$, $n\ge 0, n\equiv i-j\bmod 2$, where $v_i\in V(\Lambda_i)$ is the highest vector and $\bar{v}_{n-i}\in V(-\Lambda_j)$ is an extremal vector. We show that $F_n/F_{n+2}$ is isomorphic to the level 0 extremal weight module $V(n(\Lambda_1-\Lambda_0))$. Using this we give a functional realization of the completion of $V(\Lambda_i)\otimes V(-\Lambda_j)$ by the filtration $(F_n)_{n\geq0}$. The subspace of $V(\Lambda_i)\otimes V(-\Lambda_j)$ of $sl_2$-weight $m$ is mapped to a certain space of sequences $(P_{n,l})_{n\ge 0, n\equiv i-j\bmod 2,n-2l=m}$, 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 $\sqrt{-1}$, this construction settles a conjecture which arose in the study of form factors in integrable field theory.