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}^)

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dc.creatorFeigin, B.
dc.creatorJimbo, M.
dc.creatorKashiwara, M.
dc.creatorMiwa, T.
dc.creatorMukhin, E.
dc.creatorTakeyama, Y.
dc.date2003-10-18
dc.date.accessioned2026-07-07T05:02:03Z
dc.date.available2026-07-07T05:02:03Z
dc.descriptionLet $V(Λ_i)$ (resp., $V(-Λ_j)$) be a fundamental integrable highest (resp., lowest) weight module of $U_q(\hat{sl}_{2})$. The tensor product $V(Λ_i)\otimes V(-Λ_j)$ 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 $v_i\in V(Λ_i)$ is the highest vector and $\bar{v}_{n-i}\in V(-Λ_j)$ 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 $V(Λ_i)\otimes V(-Λ_j)$ by the filtration $(F_n)_{n\geq0}$. The subspace of $V(Λ_i)\otimes V(-Λ_j)$ 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.
dc.description33 pages
dc.identifierhttps://arxiv.org/abs/math/0310284
dc.identifierhttp://arxiv.org/abs/math/0310284
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/68903
dc.subjectQuantum Algebra
dc.subjectRepresentation Theory
dc.titleA functional model for the tensor product of level 1 highest and level -1 lowest modules for the quantum affine algebra U_q(sl_{2}^)
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