dorsal/arxiv
View SchemaExact Deformations of Quantum Groups; Applications to the Affine case
| Authors | C. Frønsdal |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9602034 |
| URL | https://arxiv.org/abs/q-alg/9602034 |
Abstract
This paper continues our investigation of a class of generalized quantum groups. The "standard" R-matrix was shown to be the unique solution of a very simple, linear recursion relation and the classical limit was obtained in the case of quantized Kac-Moody algebras of finite type. Here the standard R-matrix for generalized quantum groups is first examined in the case of quantized affine Kac-Moody algebras. The classical limit yields the standard affine r-matrices of Belavin and Drinfeld. Then, turning to the general case, we study the exact deformations of the standard R-matrix and the associated Hopf algebras. They are described as a generalized twist, $ R_\epsilon = (F^t)^{-1}RF$, where $R$ is the standard R-matrix and $F$ (a power series in the deformation parameter $\epsilon$) is the solution of a linear recursion relation of the same type as that which determines $R$. Specializing again, to the case of quantized, affine Kac-Moody algebras, and taking the classical limit of these esoteric quantum groups, one re-discovers the esoteric affine r-matrices of Belavin and Drinfeld, including the elliptic ones. The formulas obtained here are easier to use than the original ones, and the structure of the space of classical r-matrices (for simple Lie algebras) is more tranparent. In addition, the r-matrices obtained here are more general in that they are defined on the central extension of the loop groups.
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"abstract": "This paper continues our investigation of a class of generalized quantum\ngroups. The \"standard\" R-matrix was shown to be the unique solution of a very\nsimple, linear recursion relation and the classical limit was obtained in the\ncase of quantized Kac-Moody algebras of finite type. Here the standard R-matrix\nfor generalized quantum groups is first examined in the case of quantized\naffine Kac-Moody algebras. The classical limit yields the standard affine\nr-matrices of Belavin and Drinfeld. Then, turning to the general case, we study\nthe exact deformations of the standard R-matrix and the associated Hopf\nalgebras. They are described as a generalized twist, $ R_\\epsilon =\n(F^t)^{-1}RF$, where $R$ is the standard R-matrix and $F$ (a power series in\nthe deformation parameter $\\epsilon$) is the solution of a linear recursion\nrelation of the same type as that which determines $R$. Specializing again, to\nthe case of quantized, affine Kac-Moody algebras, and taking the classical\nlimit of these esoteric quantum groups, one re-discovers the esoteric affine\nr-matrices of Belavin and Drinfeld, including the elliptic ones. The formulas\nobtained here are easier to use than the original ones, and the structure of\nthe space of classical r-matrices (for simple Lie algebras) is more tranparent.\nIn addition, the r-matrices obtained here are more general in that they are\ndefined on the central extension of the loop groups.",
"arxiv_id": "q-alg/9602034",
"authors": [
"C. Fr\u00f8nsdal"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Exact Deformations of Quantum Groups; Applications to the Affine case",
"url": "https://arxiv.org/abs/q-alg/9602034"
},
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