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Jackson integral representations for solutions of the quantized Knizhnik- Zamolodchikov equation. (Russian) Zbl 0818.33012

The quantized Knizhnik-Zamolodchikov equation is a system of difference equations which is considered as a quantization of the usual Knizhnik- Zamolodchikov equation. The authors describe solutions for the quantized Knizhnik-Zamolodchikov equation associated with the quantum group \(U_ q (\text{gl} (N+1))\) and with the Lie algebra \(\text{gl} (N+1)\). They give Jackson integral representation for these solutions. The Jackson integral is a discrete analogue of the standard integral. The solutions are represented in terms of multidimensional integrals of the hypergeometric type (that is, in terms of multidimensional basic hypergeometric functions). It is shown that the quantized Knizhnik- Zamolodchikov equations are quantized Gauss-Manin connections. The authors construct asymptotic solutions for a holonomic system of difference equations, give a new formula for Bethe-ansatz eigenvectors in a tensor product of \(U_ q (\text{gl} (N+1))\) or \(\text{gl} (N+1)\) modules and indicate connections of the Jackson integral representations with the Bethe-ansatz.
Reviewer: A.Klimyk (Kiev)

MSC:

33D60 Basic hypergeometric integrals and functions defined by them
17B37 Quantum groups (quantized enveloping algebras) and related deformations
39A10 Additive difference equations
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