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Quantized Knizhnik-Zamolodchikov equations, quantum Yang-Baxter equation, and difference equations for \(q\)-hypergeometric functions. (English) Zbl 0807.17014
New difference equations are derived for general \(q\)-hypergeometric functions. The equations have a form similar to the quantum Knizhnik- Zamolodchikov equation for quantum affine algebras introduced by Frenkel and Reshetikhin. The equations are deduced in a geometric way studying the properties of the Jackson integral and \(q\)-cycles.
A complete proof of Matsuo’s formulas for solutions to \(q\)-Knizhnik- Zamolodchikov equations for \(\widehat {sl}_ 2\) is given.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
35Q40 PDEs in connection with quantum mechanics
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