Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups
| dc.creator | Tarasov, Vitaly | |
| dc.creator | Varchenko, Alexander | |
| dc.date | 1997-03-27 | |
| dc.date | 1999-02-03 | |
| dc.date.accessioned | 2026-07-07T09:05:00Z | |
| dc.date.available | 2026-07-07T09:05:00Z | |
| dc.description | The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the quantum group $U_q(sl_2)$ is a system of linear difference equations with values in a tensor product of $U_q(sl_2)$ 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 $E_{ρ,γ}(sl_2)$, where parameters $ρ$ and $γ$ are related to the parameter $q$ of the quantum group $U_q(sl_2)$ 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 $U_q(\widetilde{gl}_2$ and the elliptic quantum group $E_{ρ,γ}(sl_2)$ 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. | |
| dc.description | 72 pages, amstex.tex (ver. 2.1) and amssym.tex are required | |
| dc.identifier | https://arxiv.org/abs/q-alg/9703044 | |
| dc.identifier | http://arxiv.org/abs/q-alg/9703044 | |
| dc.identifier | Asterisque 246 (1997) 1--135 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/149584 | |
| dc.subject | Quantum Algebra | |
| dc.title | Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups | |
| dc.type | text |