dorsal/arxiv
View SchemaToroidal actions on level 1 modules of U_q(\hat{sl_n})
| Authors | Y. Saito, K. Takemura, D. Uglov |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9702024 |
| URL | https://arxiv.org/abs/q-alg/9702024 |
Abstract
Recently Varagnolo and Vasserot established that the q-deformed Fock spaces due to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum toroidal algebra $U_q(sl_n,tor)$ (n > 2) with the level (0,1). In the present article we propose a more detailed proof of this fact then the one given by Varagnolo and Vasserot. The proof is based on certain non-trivial properties of Cherednik's commuting difference operators. The quantum toroidal action on the Fock space depends on a certain parameter. We find that with a specific choice of this parameter the action on the Fock spaces gives rise to the toroidal action on irreducible level-1 highest weight modules of the affine quantum algebra $U_q(\hat{sl}_n)$. Similarly, by a specific choice of the parameter, the level (1,0) vertex representation of the quantum toroidal algebra gives rise to a $U_q(sl_n,tor)$-module structure on irreducible level-1 highest weight $U_q(\hat{sl}_n)$-modules.
{
"annotation_id": "7b8cee45-5fbe-4f32-aa1d-d2dec7a34135",
"date_created": "2026-03-02T18:01:28.694000Z",
"date_modified": "2026-03-02T18:01:28.694000Z",
"file_hash": "d5c8774fa4b9ea3a9b2a9c00da324ed065501f870ca4c5a6d8b83dc586768d64",
"private": false,
"record": {
"abstract": "Recently Varagnolo and Vasserot established that the q-deformed Fock spaces\ndue to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum\ntoroidal algebra $U_q(sl_n,tor)$ (n \u003e 2) with the level (0,1). In the present\narticle we propose a more detailed proof of this fact then the one given by\nVaragnolo and Vasserot. The proof is based on certain non-trivial properties of\nCherednik\u0027s commuting difference operators. The quantum toroidal action on the\nFock space depends on a certain parameter. We find that with a specific choice\nof this parameter the action on the Fock spaces gives rise to the toroidal\naction on irreducible level-1 highest weight modules of the affine quantum\nalgebra $U_q(\\hat{sl}_n)$. Similarly, by a specific choice of the parameter,\nthe level (1,0) vertex representation of the quantum toroidal algebra gives\nrise to a $U_q(sl_n,tor)$-module structure on irreducible level-1 highest\nweight $U_q(\\hat{sl}_n)$-modules.",
"arxiv_id": "q-alg/9702024",
"authors": [
"Y. Saito",
"K. Takemura",
"D. Uglov"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Toroidal actions on level 1 modules of U_q(\\hat{sl_n})",
"url": "https://arxiv.org/abs/q-alg/9702024"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "cb0a88f7-e942-4bf2-a945-5945795c2d6b",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}