The consistent reduction of the differential calculus on the quantum group $GL_{q}(2,C)$ to the differential calculi on its subgroups and $σ$-models on the quantum group manifolds $SL_{q}(2,R)$, $SL_{q}(2,R)/U_{h}(1)$, $C{q}(2|0)$ and infinitesimal transformations

V. D. Gershun

paper

The consistent reduction of the differential calculus on the quantum group $G L_{q} (2, C)$ to the differential calculi on its subgroups and $σ$ -models on the quantum group manifolds $S L_{q} (2, R)$ , $S L_{q} (2, R) / U_{h} (1)$ , $C q (2∣0)$ and infinitesimal transformations

arXiv: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 $G L_{q} (2, C)$ has to contain the 3-dimensional differential calculi on the quantum subgroup $S L_{q} (2, C)$ , 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 $U_{q} (2)$ , $S U_{q} (2)$ , $S p_{q} (2, C)$ , $S p_{q} (2)$ , $T_{q} (2, C)$ , $B_{L} (C)$ , $B_{U} (C)$ , $U_{q} (1)$ , $Z_{-}^{(2)} (C)$ , $Z_{+}^{(2)} (C)$ 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 $S L_{q} (2, C)$ 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 $S L_{q} (2, R)$ quantum group symmetry over two-dimensional nondeformed Minkovski space and in the $σ$ -models with $S L_{q} (2, R) / U_{p} (1)$ , $C_{q} (2∣0)$ quantum group symmetry is considered. The Lagrangian formalism over the quantum group manifolds is discussed. The variational calculus on the $S L_{q} (2, R)$ group manifold is obtained. The classical solution of $C_{q} (2∣0)$ { $σ$ }-model is obtained.

Latex, 13 pages

The consistent reduction of the differential calculus on the quantum group GLq​(2,C) to the differential calculi on its subgroups and σ-models on the quantum group manifolds SLq​(2,R), SLq​(2,R)/Uh​(1), Cq(2∣0) and infinitesimal transformations

Abstract

The consistent reduction of the differential calculus on the quantum group $G L_{q} (2, C)$ to the differential calculi on its subgroups and $σ$ -models on the quantum group manifolds $S L_{q} (2, R)$ , $S L_{q} (2, R) / U_{h} (1)$ , $C q (2∣0)$ and infinitesimal transformations