Jackson-type integrals, Bethe vectors, and solutions to a difference analog of the Knizhnik-Zamolodchikov system (Q1202954)

Progress in the theory of intertwining operators for quantized affine Lie algebras leads to a construction of difference systems which can be regarded as difference analogue of systems of Knizhnik-Zamolodchikov systems of differential equations usually called quantum Knizhnik-Zamolodchikov systems. These systems are of great interest since their solutions cover all known functions of hypergeometric type. The system of difference equations \[ f(x_1,\ldots,x_ j+k,\dots,x_n) = A_j(x_1,\ldots,x_n)\, f(x_1,\ldots,x_n) \] is connected with quantum Knizhnik-Zamolodchikov systems, where \(A_j\) are operators expressed in terms of Yang-Baxter operators \(R^{lm}(x)\). The author gives a construction for solutions to this system for a certain class of operators \(R^{lm}(x)\). This class of operators includes those which can be obtained from the affine quantum algebras \(U_ q(\widehat{\mathfrak{sl}}_2)\) and its Yangian-type counterpart. Formulas for solutions are difference analogues of integral formulas for solutions to the usual Knizhnik-Zamolodchikov system.