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 $U_q({\widehat{sl}}_2)$. The tensor product $V({\Lambda }_i)\otimes V(-{\Lambda }_j)$ is filtered by submodules $F_n=U_q({\widehat{sl}}_2)(v_i\otimes {\bar{v}}_{n-i})$, $n\ge 0,n\equiv i-j\mathrm{mod}2$, 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 $(F_n{)}_{n\ge 0}$. The subspace of $V({\Lambda }_i)\otimes V(-{\Lambda }_j)$ of $s{l}_2$-weight $m$ is mapped to a certain space of sequences $(P_{n,l}{)}_{n\ge 0,n\equiv i-j\mathrm{mod}2,n-2l=m}$, 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 $\sqrt{-1}$, this construction settles a conjecture which arose in the study of form factors in integrable field theory.

33 pages

References in corpus (1)

• Counting minimal form factors of the restricted sine-Gordon model