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

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.