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

摘要

Explicit construction of the second order left differential calculi on thequantum group and its subgroups are obtained with the property of the naturalreduction: the differential calculus on the quantum group $GL_q(2,C)$ has tocontain 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 thequantum 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 ($q\to 1$) ofthe left differential calculus is the nondeformed differential calculus. Thedifferential 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 calculuson 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 inthe $\sigma$-models with ${SL_{q}(2,R)/U_{p}(1)}$, $C_{q}(2|0)$ quantum groupsymmetry is considered. The Lagrangian formalism over the quantum groupmanifolds is discussed. The variational calculus on the $SL_{q}(2,R)$ groupmanifold is obtained. The classical solution of $C_{q}(2|0)$ {$\sigma$}-modelis obtained.