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

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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 GLq(2,C) has to contain the 3-dimensional differential calculi on the quantum subgroup SLq(2,C), the differential calculi on the Borel subgroups Bₗ⁽²⁾(C), Bᵤ⁽²⁾(C) of the lower and of the upper triangular matrices, on the quantum subgroups Uq(2), SUq(2), Spq(2,C), Spq(2), Tq(2,C), Bₗ(C), Bᵤ(C), Uq(1), Z₋⁽²⁾(C), Z₊⁽²⁾(C) and on the their real forms. The classical limit (q→ 1) of the left differential calculus is the nondeformed differential calculus. The differential calculi on the Borel subgroups Bₗ(C), Bᵤ(C) of the SLq(2,C) coincide with two solutions of Wess-Zumino differential calculus on the quantum plane Cq(2|0). The spontaneous breaking symmetry in the WZNW model with SLq(2,R) quantum group symmetry over two-dimensional nondeformed Minkovski space and in the σ-models with SLq(2,R)/Uₚ(1), Cq(2|0) quantum group symmetry is considered. The Lagrangian formalism over the quantum group manifolds is discussed. The variational calculus on the SLq(2,R) group manifold is obtained. The classical solution of Cq(2|0) σ-model is obtained.