Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups

Vitaly Tarasov; Alexander Varchenko

paper

Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups

arXiv:q-alg/9703044

Abstract

The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the quantum group $U_{q} (s l_{2})$ is a system of linear difference equations with values in a tensor product of $U_{q} (s l_{2})$ Verma modules. We solve the equation in terms of multidimensional $q$ -hypergeometric functions and define a natural isomorphism between the space of solutions and the tensor product of the corresponding evaluation Verma modules over the elliptic quantum group $E_{ρ, γ} (s l_{2})$ , where parameters $ρ$ and $γ$ are related to the parameter $q$ of the quantum group $U_{q} (s l_{2})$ and the step $p$ of the qKZ equation via $p=e^{2\piiρ}$ and $q=e^{-2\piiγ}$. We construct asymptotic solutions associated with suitable asymptotic zones and compute the transition functions between the asymptotic solutions in terms of the dynamical elliptic R-matrices. This description of the transition functions gives a connection between representation theories of the quantum loop algebra $U_{q} (g l_{2}$ and the elliptic quantum group $E_{ρ, γ} (s l_{2})$ and is analogous to the Kohno-Drinfeld theorem on the monodromy group of the differential Knizhnik-Zamolodchikov equation. In order to establish these results we construct a discrete Gauss-Manin connection, in particular, a suitable discrete local system, discrete homology and cohomology groups with coefficients in this local system, and identify an associated difference equation with the qKZ equation.

72 pages, amstex.tex (ver. 2.1) and amssym.tex are required

Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups

Abstract

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