dorsal/arxiv
View SchemaSolutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups
| Authors | Pavel Etingof, Alexander Varchenko |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9708015 |
| URL | https://arxiv.org/abs/q-alg/9708015 |
| DOI | 10.1007/s002200050437 |
Abstract
The quantum dynamical Yang-Baxter (QDYB) equation is a useful generalization of the quantum Yang-Baxter (QYB) equation introduced by Gervais, Neveu, and Felder. The QDYB equation and its quasiclassical analogue (the classical dynamical Yang-Baxter equation) arise in several areas of mathematics and mathematical physics (conformal field theory, integrable systems, representation theory). The most interesting solution of the QDYB equation is the elliptic solution, discovered by Felder. In this paper, we prove the first classification results for solutions of the QDYB equation. These results are parallel to the classification of solutions of the classical dynamical Yang-Baxter equation, obtained in our previous paper q-alg/9703040. All solutions we found can be obtained from Felder's elliptic solution by a limiting process and gauge transformations. Fifteen years ago the quantum Yang-Baxter equation gave rise to the theory of quantum groups. Namely, it turned out that the language of quantum groups (Hopf algebras) is the adequate algebraic language to talk about solutions of the quantum Yang-Baxter equation. In this paper we propose a similar language, originating from Felder's ideas, which we found to be adequate for the dynamical Yang-Baxter equation. This is the language of dynamical quantum groups (or $\h$-Hopf algebroids), which is the quantum counterpart of the language of dynamical Poisson groupoids, introduced in our previous paper q-alg/9703040.
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"abstract": "The quantum dynamical Yang-Baxter (QDYB) equation is a useful generalization\nof the quantum Yang-Baxter (QYB) equation introduced by Gervais, Neveu, and\nFelder. The QDYB equation and its quasiclassical analogue (the classical\ndynamical Yang-Baxter equation) arise in several areas of mathematics and\nmathematical physics (conformal field theory, integrable systems,\nrepresentation theory). The most interesting solution of the QDYB equation is\nthe elliptic solution, discovered by Felder. In this paper, we prove the first\nclassification results for solutions of the QDYB equation. These results are\nparallel to the classification of solutions of the classical dynamical\nYang-Baxter equation, obtained in our previous paper q-alg/9703040. All\nsolutions we found can be obtained from Felder\u0027s elliptic solution by a\nlimiting process and gauge transformations. Fifteen years ago the quantum\nYang-Baxter equation gave rise to the theory of quantum groups. Namely, it\nturned out that the language of quantum groups (Hopf algebras) is the adequate\nalgebraic language to talk about solutions of the quantum Yang-Baxter equation.\nIn this paper we propose a similar language, originating from Felder\u0027s ideas,\nwhich we found to be adequate for the dynamical Yang-Baxter equation. This is\nthe language of dynamical quantum groups (or $\\h$-Hopf algebroids), which is\nthe quantum counterpart of the language of dynamical Poisson groupoids,\nintroduced in our previous paper q-alg/9703040.",
"arxiv_id": "q-alg/9708015",
"authors": [
"Pavel Etingof",
"Alexander Varchenko"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1007/s002200050437",
"title": "Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups",
"url": "https://arxiv.org/abs/q-alg/9708015"
},
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