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Geometry of \(q\)-hypergeometric functions, quantum affine algebras and elliptic quantum groups. (English) Zbl 0938.17012

Astérisque. 246. Paris: Société Mathématique de France, vi, 135 p. (1997).
The trigonometric quantized Knizhnik-Zamolodchikov equation associated with the quantum group \(U_{q}( {\mathfrak {sl}}_{2}) \) is solved in terms of multidimensional \(q\)-hypergeometric functions. It is represented as a system of difference equations for a function \(\Psi ( z_{1},\ldots ,z_{n}) \) with values in a tensor product \(V_{1}\otimes \ldots \otimes V_{n}\) of \(U_{q}( {\mathfrak {sl}}_{2}) \) Verma modules. A natural isomorphism of the space of solutions and the tensor product of the corresponding Verma modules over the elliptic quantum group \(E_{\rho ,\gamma }( {\mathfrak {sl}}_{2}) \), where connection between parameters is \( q=\exp ( -2\pi i\gamma) \) and \(p=\exp ( -2\pi i\rho) \) and \(p\) is the step of the quantized Knizhnik-Zamolodchikov equation. The asymptotic solutions associated with corresponding asymptotic zones are constructed. The transition functions between the asymptotic solutions in terms of the dynamical elliptic \(R\)-matrices are computed, which gives a connection between representation theories of the quantum loop algebra \( U_{q}( \widetilde{\mathfrak {gl}}_{2}) \) and the elliptic quantum group \(E_{\rho ,\gamma }( {\mathfrak {sl}}_{2}) \)and is analogous to the Kohno-Drinfeld theorem on the monodromy group of the differential Knizhnik-Zamolodchikov equation. A discrete analogue of the Gauss-Manin connection for the discrete locally trivial bundle with a discrete local system is defined, and the corresponding difference equation is considered. To realize this idea a suitable discrete de Rham complex and its cohomology group are introduced, the homology group as the dual space to the cohomology group is defined, and a family of discrete cycles, elements of the discrete homology group is constructed.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
33-02 Research exposition (monographs, survey articles) pertaining to special functions
33D60 Basic hypergeometric integrals and functions defined by them
33D70 Other basic hypergeometric functions and integrals in several variables
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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