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

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 productm 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.

Untitled

“…The chiral space of local fields in SG-model at generic coupling constant and that in SU(2)-ITM are both isomorphic to A 2m+i [4,5,6]. In SG case, the fields are highest-weight vectors with the weight 2m + i with respect to a certain quantum group U p ( sl 2 ) for some p, and in SU(2)-ITM case, the fields are highest-weight vectors with weight 2m + i with respect to sl 2 .…”

“…In [4], Feigin, Jimbo, Kashiwara, Miwa, Mukhin, and Takeyama proved that the chiral subspace of local fields in the quantum sine-Gordon model at generic coupling constant and that in the SU(2)-invariant Thirring model (ITM) are isomorphic to the…”

Fermionic structure in the sine-Gordon model: Form factors and null-vectors

“…Actually, they considered only right chiral descendants, so, the hierarchy in question was that of the Korteweg-de-Vries (KdV) equation. Another way of counting local operators was explained from a representation theory viewpoint in [18]. Now we would like to discuss a seemingly completely different subject.…”

Composition Series of Tensor Product

“…It turns out that all of the U q (g)submodules appeared in this filtration are compatible with the canonical basis which can be proved using an important lemma due to Kashiwara and some results for Demazure modules. Motivated by the construction of the filtration in [2], we construct the composition series of V (λ) ⊗ V (µ) directly for g of any type in the same fashion. The conjecture by Lusztig is then a special case since V (µ) is also a lowest weight module for g of finite type.…”