dorsal/arxiv
View SchemaGeometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras
| Authors | Vitaly Tarasov, Alexander Varchenko |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9604011 |
| URL | https://arxiv.org/abs/q-alg/9604011 |
| DOI | 10.1007/s002220050151 |
| Journal | Inventiones Mathematicae, 128 (1997), 501-588 |
Abstract
The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the Lie algebra $sl_2$ is a system of linear difference equations with values in a tensor product of $sl_2$ 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 quantum group $U_q(sl_2)$ Verma modules, where the parameter $q$ is related to the step $p$ of the qKZ equation via $q=e^{pi i/p}$. We construct asymptotic solutions associated with suitable asymptotic zones and compute the transition functions between the asymptotic solutions in terms of the trigonometric $R$-matrices. This description of the transition functions gives a new connection between representation theories of Yangians and quantum loop algebras 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.
{
"annotation_id": "91c68c1f-cf66-4f01-be16-f4dc3475655c",
"date_created": "2026-03-02T18:01:28.661000Z",
"date_modified": "2026-03-02T18:01:28.661000Z",
"file_hash": "e53a4d28e29cd524d31adc655040a9d9e62bc9d3de84011249912eab69b4f99f",
"private": false,
"record": {
"abstract": "The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation)\nassociated with the Lie algebra $sl_2$ is a system of linear difference\nequations with values in a tensor product of $sl_2$ Verma modules. We solve the\nequation in terms of multidimensional $q$-hypergeometric functions and define a\nnatural isomorphism between the space of solutions and the tensor product of\nthe corresponding quantum group $U_q(sl_2)$ Verma modules, where the parameter\n$q$ is related to the step $p$ of the qKZ equation via $q=e^{pi i/p}$.\n We construct asymptotic solutions associated with suitable asymptotic zones\nand compute the transition functions between the asymptotic solutions in terms\nof the trigonometric $R$-matrices. This description of the transition functions\ngives a new connection between representation theories of Yangians and quantum\nloop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy\ngroup 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/9604011",
"authors": [
"Vitaly Tarasov",
"Alexander Varchenko"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1007/s002220050151",
"journal_ref": "Inventiones Mathematicae, 128 (1997), 501-588",
"title": "Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras",
"url": "https://arxiv.org/abs/q-alg/9604011"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "c1841056-98ec-48a3-9eb3-f2c5e9db27f0",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}