dorsal/arxiv
View SchemaGeometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups
| Authors | Vitaly Tarasov, Alexander Varchenko |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9703044 |
| URL | https://arxiv.org/abs/q-alg/9703044 |
| Journal | Asterisque 246 (1997) 1--135 |
Abstract
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 $E_{\rho,\gamma}(sl_2)$, where parameters $\rho$ and $\gamma$ are related to the parameter $q$ of the quantum group $U_q(sl_2)$ and the step $p$ of the qKZ equation via $p=e^{2\pii\rho}$ and $q=e^{-2\pii\gamma}$. 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 $E_{\rho,\gamma}(sl_2)$ 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.
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"abstract": "The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation)\nassociated with the quantum group $U_q(sl_2)$ is a system of linear difference\nequations with values in a tensor product of $U_q(sl_2)$ Verma modules. We\nsolve the equation in terms of multidimensional $q$-hypergeometric functions\nand define a natural isomorphism between the space of solutions and the tensor\nproduct of the corresponding evaluation Verma modules over the elliptic quantum\ngroup $E_{\\rho,\\gamma}(sl_2)$, where parameters $\\rho$ and $\\gamma$ are related\nto the parameter $q$ of the quantum group $U_q(sl_2)$ and the step $p$ of the\nqKZ equation via $p=e^{2\\pii\\rho}$ and $q=e^{-2\\pii\\gamma}$.\n We construct asymptotic solutions associated with suitable asymptotic zones\nand compute the transition functions between the asymptotic solutions in terms\nof the dynamical elliptic R-matrices. This description of the transition\nfunctions gives a connection between representation theories of the quantum\nloop algebra $U_q(\\widetilde{gl}_2$ and the elliptic quantum group\n$E_{\\rho,\\gamma}(sl_2)$ and is analogous to the Kohno-Drinfeld theorem on the\nmonodromy group of the differential Knizhnik-Zamolodchikov equation.\n In order to establish these results we construct a discrete Gauss-Manin\nconnection, in particular, a suitable discrete local system, discrete homology\nand cohomology groups with coefficients in this local system, and identify an\nassociated difference equation with the qKZ equation.",
"arxiv_id": "q-alg/9703044",
"authors": [
"Vitaly Tarasov",
"Alexander Varchenko"
],
"categories": [
"q-alg",
"math.QA"
],
"journal_ref": "Asterisque 246 (1997) 1--135",
"title": "Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups",
"url": "https://arxiv.org/abs/q-alg/9703044"
},
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