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Towers of solutions of qKZ equations and their applications to loop models. (English) Zbl 1442.81033

The authors consider quantum affine Knizhnik-Zanalodchikov (qKZ) equations associated with the \(GL_n\)-type extended affine Hecke algebra \(\mathcal{H}_n\). These Hecke algebras form a tower of algebras with respect to algebra morphisms \(\mathcal{H}_n\to\mathcal{H}_{n+1}\) that arise from the insertion morphism \(\mathcal{B}_n\to\mathcal{B}_{n+1}\) for the groups \(\mathcal{B}_n\) of affine \(n\)-braids.
The paper solves the qKZ equations taking values in modules \(V_n\) of the Hecke algebras \(\mathbb{H}_n\) such that family of solutions is compatible with the tower structure. First they consider the case where parameter of the Hecke algebra is a third root of unity and then the case of a generic parameter.
The extended affine Temperley-Lieb algebras is invariant under the inversion \(t^{\frac{1}{4}}\mapsto t^{-\frac{1}{4}}\). The last section of the paper shows how this symmetry results in a dual braid recursion for the tower of solutions obtained previously.

MSC:

81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
20C08 Hecke algebras and their representations
20F05 Generators, relations, and presentations of groups
57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
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