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Quantum affine algebras and holonomic difference equations. (English) Zbl 0760.17006

This paper initiates the study of a \(q\)-analogue of conformal field theory (CFT). In classical CFT, a fundamental role is played by the Knizhnik-Zamolodchikov (KZ) equation, which is satisfied by the matrix elements of intertwiners between tensor products of certain representations of affine Lie algebras. The solutions of the KZ equation can be expressed in terms of hypergeometric functions and their generalizations. The authors introduce a \(q\)-analogue of the KZ equation, show its relation to representations of quantum affine algebras, and express its solutions in terms of \(q\)-hypergeometric functions.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
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