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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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##### References:
 [1] [A] Aomoto, K.: Finiteness of a cohomology associated with certain Jackson integrals. Tôhoku Math. J.43, no. 1, March, 75–101 (1991) · Zbl 0769.33016 · doi:10.2748/tmj/1178227537 [2] [AK1] Aomoto, K., Kato, Y.: Aq-analog of deRham cohomology associated with Jackson integrals. Preprint [3] [AK2] Aomoto, K., Kato, Y.: Connection coefficients for A-type Jackson integrals and Yang-Baxter equation. Preprint · Zbl 0813.33010 [4] [AKM] Aomoto, A., Kato, Y., Mimachi, K.: A solution of the Yang-Baxter equation as connection coefficients of a holonomicq-difference system. Intern. Math. Research Notices, no.1, 7–15 (1992) · Zbl 0765.39002 · doi:10.1155/S1073792892000023 [5] [D] Drinfeld, V.: Quasi-Hopf algebras. Algebra and Analysis,1, no. 2, 30–46 (1987) [6] [ESV] Esnault, H., Schechtman, V., Viehweg, E.: Cohomology of local systems on the complement of hyperplanes. Invent. Math.109, 557–562 (1992) · Zbl 0788.32005 · doi:10.1007/BF01232040 [7] [FR] Frenkel, I., Reshetikhin N.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys.146, 1–60 (1992) · Zbl 0760.17006 · doi:10.1007/BF02099206 [8] [FW] Felder, G., Wieczerkowski, C.: Topological representations of the quantum groupU q s2. Commun. Math. Phys.138, 583–605 (1991) · Zbl 0722.55005 · doi:10.1007/BF02102043 [9] [K] Kohno, T.: Monodromy representations of braid groups and Yang-Baxter equations. Ann. Inst. Fourier37, 139–160 (1987) · Zbl 0634.58040 [10] [KL] Kazhdan, D., Lusztig, L.: Affine Lie algebras and quantum groups. Duke Math. J.62, 21–29 (1991) · Zbl 0726.17015 [11] [KZ] Knizhnik, V., Zamolodchikov, A.: Current algebra and Wess-Zumino models in two dimensions. Nucl. Phys. B.247, 83–103 (1984) · Zbl 0661.17020 · doi:10.1016/0550-3213(84)90374-2 [12] [M1] Matsuo, A.: Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamolodichikov equations. Preprint [13] [M2] Matsuo, A.: Quantum algebra structure of certain Jackson integrals. Preprint [14] [M3] Matsuo, A.: Free field realization ofq-deformed primary fields for $$U_q (s\hat \ell _2 )$$ . Preprint [15] [Mi] Mimachi, K.: Connection problem in holonomicq-defference system associated with a Jackson integral of Jordan-Pochhammer type. Nagoya Math. J.116, 149–161 (1989) · Zbl 0688.39002 [16] [R] Reshetikhin, N.: Jackson type integrals, Bethe vectors, and solutions to a difference analog of the Knizhnik-Zamolodchikov system. Lett. Math. Phys.26, no. 3, 153–165 (1992) · Zbl 0776.17015 · doi:10.1007/BF00420749 [17] [SV] Schechtman, V., Varchenko, A.: Arrangements of hyperplanes and Lie algebra homology. Invent Math.106, 139–194 (1991) · Zbl 0754.17024 · doi:10.1007/BF01243909 [18] [TV] Tarasov, V., Varchenko, A.: Jackson integral representations for solutions to the quantized Knizhnik-Zamolodchikov equation. Preprint, August, 1993 [19] [V] Varchenko, A.: Hypergeometric functions and the representation theory of Lie algebras and quantum groups. Preprint, 1992
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