dorsal/arxiv
View SchemaThe consistent reduction of the differential calculus on the quantum group $GL_{q}(2,C)$ to the differential calculi on its subgroups and $\sigma$-models on the quantum group manifolds $SL_{q}(2,R)$, $SL_{q}(2,R)/U_{h}(1)$, $C{q}(2|0)$ and infinitesimal transformations
| Authors | V. D. Gershun |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9711021 |
| URL | https://arxiv.org/abs/q-alg/9711021 |
Abstract
Explicit construction of the second order left differential calculi on the quantum group and its subgroups are obtained with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain the 3-dimensional differential calculi on the quantum subgroup $SL_q(2,C)$, the differential calculi on the Borel subgroups $B_{L}^{(2)}(C)$, $B_{U}^{(2)}(C)$ of the lower and of the upper triangular matrices, on the quantum subgroups $U_{q}(2)$, $SU_{q}(2)$, $Sp_{q}(2,C)$, $Sp_{q}(2)$, $T_{q}(2,C)$, $B_{L}(C)$, $B_{U}(C)$, $U_{q}(1)$, $Z_{-}^{(2)}(C)$, $Z_{+}^{(2)}(C)$ and on the their real forms. The classical limit ($q\to 1$) of the left differential calculus is the nondeformed differential calculus. The differential calculi on the Borel subgroups $B_{L}(C)$, $B_{U}(C)$ of the $SL_{q}(2,C)$ coincide with two solutions of Wess-Zumino differential calculus on the quantum plane $C_q(2|0)$. The spontaneous breaking symmetry in the WZNW model with $SL_{q}(2,R)$ quantum group symmetry over two-dimensional nondeformed Minkovski space and in the $\sigma$-models with ${SL_{q}(2,R)/U_{p}(1)}$, $C_{q}(2|0)$ quantum group symmetry is considered. The Lagrangian formalism over the quantum group manifolds is discussed. The variational calculus on the $SL_{q}(2,R)$ group manifold is obtained. The classical solution of $C_{q}(2|0)$ {$\sigma$}-model is obtained.
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"abstract": "Explicit construction of the second order left differential calculi on the\nquantum group and its subgroups are obtained with the property of the natural\nreduction: the differential calculus on the quantum group $GL_q(2,C)$ has to\ncontain the 3-dimensional differential calculi on the quantum subgroup\n$SL_q(2,C)$, the differential calculi on the Borel subgroups $B_{L}^{(2)}(C)$,\n$B_{U}^{(2)}(C)$ of the lower and of the upper triangular matrices, on the\nquantum subgroups $U_{q}(2)$, $SU_{q}(2)$, $Sp_{q}(2,C)$, $Sp_{q}(2)$,\n$T_{q}(2,C)$, $B_{L}(C)$, $B_{U}(C)$, $U_{q}(1)$, $Z_{-}^{(2)}(C)$,\n$Z_{+}^{(2)}(C)$ and on the their real forms. The classical limit ($q\\to 1$) of\nthe left differential calculus is the nondeformed differential calculus. The\ndifferential calculi on the Borel subgroups $B_{L}(C)$, $B_{U}(C)$ of the\n$SL_{q}(2,C)$ coincide with two solutions of Wess-Zumino differential calculus\non the quantum plane $C_q(2|0)$.\n The spontaneous breaking symmetry in the WZNW model with $SL_{q}(2,R)$\nquantum group symmetry over two-dimensional nondeformed Minkovski space and in\nthe $\\sigma$-models with ${SL_{q}(2,R)/U_{p}(1)}$, $C_{q}(2|0)$ quantum group\nsymmetry is considered. The Lagrangian formalism over the quantum group\nmanifolds is discussed. The variational calculus on the $SL_{q}(2,R)$ group\nmanifold is obtained. The classical solution of $C_{q}(2|0)$ {$\\sigma$}-model\nis obtained.",
"arxiv_id": "q-alg/9711021",
"authors": [
"V. D. Gershun"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "The consistent reduction of the differential calculus on the quantum group $GL_{q}(2,C)$ to the differential calculi on its subgroups and $\\sigma$-models on the quantum group manifolds $SL_{q}(2,R)$, $SL_{q}(2,R)/U_{h}(1)$, $C{q}(2|0)$ and infinitesimal transformations",
"url": "https://arxiv.org/abs/q-alg/9711021"
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