dorsal/arxiv
View SchemaQuasi Hopf Deformations of Quantum Groups
| Authors | Christian Frønsdal |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9611028 |
| URL | https://arxiv.org/abs/q-alg/9611028 |
| Journal | Lett.Math.Phys. 40 (1997) 117-134 |
Abstract
The search for elliptic quantum groups leads to a modified quantum Yang-Baxter relation and to a special class of quasi-triangular quasi Hopf algebras. This paper calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac-Moody algebras are more rigid than their loop algebra quotients; and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with $sl(n)$ are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang-Baxter relation and are calculated more or less explicitly. The modified classical Yang-Baxter relation is obtained, and the elliptic solutions are worked out explicitly. The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik-Zamolodchikov-Bernard equation, to throw some light on the quasi Hopf structure of conformal field theory and (perhaps) the Calogero-Moser models.
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"abstract": "The search for elliptic quantum groups leads to a modified quantum\nYang-Baxter relation and to a special class of quasi-triangular quasi Hopf\nalgebras. This paper calculates deformations of standard quantum groups (with\nor without spectral parameter) in the category of quasi-Hopf algebras. An\nearlier investigation of the deformations of quantum groups, in the category of\nHopf algebras, showed that quantum groups are generically rigid: Hopf algebra\ndeformations exist only under some restrictions on the parameters. In\nparticular, affine Kac-Moody algebras are more rigid than their loop algebra\nquotients; and only the latter (in the case of sl(n)) can be deformed to\nelliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts\nthis restriction. The full elliptic quantum groups (with central extension)\nassociated with $sl(n)$ are thus quasi-Hopf algebras. The universal R-matrices\nsatisfy a modified Yang-Baxter relation and are calculated more or less\nexplicitly. The modified classical Yang-Baxter relation is obtained, and the\nelliptic solutions are worked out explicitly. The same method is used to\nconstruct the Universal R-matrices associated with Felder\u0027s quantization of the\nKnizhnik-Zamolodchikov-Bernard equation, to throw some light on the quasi Hopf\nstructure of conformal field theory and (perhaps) the Calogero-Moser models.",
"arxiv_id": "q-alg/9611028",
"authors": [
"Christian Fr\u00f8nsdal"
],
"categories": [
"q-alg",
"math.QA"
],
"journal_ref": "Lett.Math.Phys. 40 (1997) 117-134",
"title": "Quasi Hopf Deformations of Quantum Groups",
"url": "https://arxiv.org/abs/q-alg/9611028"
},
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