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

الأصل · EN

The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the quantum group Uq(sl₂) is a system of linear difference equations with values in a tensor product of Uq(sl₂) 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 Eᵨ,ᵧ(sl₂), where parameters ρ and γ are related to the parameter q of the quantum group Uq(sl₂) and the step p of the qKZ equation via p=e2ρ and q=e-2γ. 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 Uq(gl₂ and the elliptic quantum group Eᵨ,ᵧ(sl₂) 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.