dorsal/arxiv
View SchemaGeneral Construction of Nonstandard $R_h$-matrices as Contraction Limits of $R_{q}$-matrices
| Authors | B. Abdesselam, A. Chakrabarti, R. Chakrabarti |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9706033 |
| URL | https://arxiv.org/abs/q-alg/9706033 |
| DOI | 10.1142/S021773239800084X |
Abstract
A class of transformations of $R_q$-matrices is introduced such that the $q\to 1$ limit gives explicit nonstandard $R_{h}$-matrices. The transformation matrix is singular itself at $q\to 1$ limit. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. Our method can be implemented systematically for R-matrices of all dimensions and not only for $sl(2)$ but also for algebras of higher dimensions. Explicit constructions are presented starting with ${\cal U}_q(sl(2))$ and ${\cal U}_q(sl(3))$, while choosing $R_q$ for (fund. rep.)$\otimes$(arbitrary irrep.). The treatment for the general case and various perspectives are indicated. Our method yields nonstandard deformations along with a nonlinear map of the $h$-Borel subalgebra on the corresponding classical Borel subalgebra. For ${\cal U}_h(sl(2))$ this map is extended to the whole algebra and compared with another one proposed by us previously.
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"abstract": "A class of transformations of $R_q$-matrices is introduced such that the\n$q\\to 1$ limit gives explicit nonstandard $R_{h}$-matrices. The transformation\nmatrix is singular itself at $q\\to 1$ limit. For the transformed matrix, the\nsingularities, however, cancel yielding a well-defined construction. Our method\ncan be implemented systematically for R-matrices of all dimensions and not only\nfor $sl(2)$ but also for algebras of higher dimensions. Explicit constructions\nare presented starting with ${\\cal U}_q(sl(2))$ and ${\\cal U}_q(sl(3))$, while\nchoosing $R_q$ for (fund. rep.)$\\otimes$(arbitrary irrep.). The treatment for\nthe general case and various perspectives are indicated. Our method yields\nnonstandard deformations along with a nonlinear map of the $h$-Borel subalgebra\non the corresponding classical Borel subalgebra. For ${\\cal U}_h(sl(2))$ this\nmap is extended to the whole algebra and compared with another one proposed by\nus previously.",
"arxiv_id": "q-alg/9706033",
"authors": [
"B. Abdesselam",
"A. Chakrabarti",
"R. Chakrabarti"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1142/S021773239800084X",
"title": "General Construction of Nonstandard $R_h$-matrices as Contraction Limits of $R_{q}$-matrices",
"url": "https://arxiv.org/abs/q-alg/9706033"
},
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