dorsal/arxiv
View SchemaFRT Quantization Theory for the Nonsemisimple Cayley-Klein Groups
| Authors | N. A. Gromov, I. V. Kostyakov, V. V. Kuratov |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9711024 |
| URL | https://arxiv.org/abs/q-alg/9711024 |
Abstract
The quantization theory of the simple Lie groups and algebras was developed by Faddeev-Reshetikhin-Takhtadjan (FRT). In group theory there is a remarkable set of groups, namely the motion groups of n-dimensional spaces of constant curvature or the orthogonal Cayley-Klein (CK) groups. In some sense the CK groups are in the nearest neighborhood with the simple ones. The well known groups of physical interest such as Euclidean E(n), Poincare P(n), Galileian G(n) and other nonsemisimple groups are in the set of CK groups. But many standart algebraical costructions are not suitable for the nonsemisimple groups and algebras, in particular Killing form is degenerate, Cartan matrix do not exist. Nevertheless it is possible to describe and to quantize all CK groups and algebras, as it was made for the simple ones. The principal proposal is to consider CK groups as the groups over an associative algebra $D$ with nilpotent commutative generators and the corresponding quantum CK groups as the algebra of noncommutative functions over $D$.
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"abstract": "The quantization theory of the simple Lie groups and algebras was developed\nby Faddeev-Reshetikhin-Takhtadjan (FRT). In group theory there is a remarkable\nset of groups, namely the motion groups of n-dimensional spaces of constant\ncurvature or the orthogonal Cayley-Klein (CK) groups. In some sense the CK\ngroups are in the nearest neighborhood with the simple ones. The well known\ngroups of physical interest such as Euclidean E(n), Poincare P(n), Galileian\nG(n) and other nonsemisimple groups are in the set of CK groups.\n But many standart algebraical costructions are not suitable for the\nnonsemisimple groups and algebras, in particular Killing form is degenerate,\nCartan matrix do not exist. Nevertheless it is possible to describe and to\nquantize all CK groups and algebras, as it was made for the simple ones.\n The principal proposal is to consider CK groups as the groups over an\nassociative algebra $D$ with nilpotent commutative generators and the\ncorresponding quantum CK groups as the algebra of noncommutative functions over\n$D$.",
"arxiv_id": "q-alg/9711024",
"authors": [
"N. A. Gromov",
"I. V. Kostyakov",
"V. V. Kuratov"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "FRT Quantization Theory for the Nonsemisimple Cayley-Klein Groups",
"url": "https://arxiv.org/abs/q-alg/9711024"
},
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