1997/11/24 by V. D. Gershun, Gershun, V. D.
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA #q-alg
paper · pdf · doi:10.48550/arxiv.q-alg/9711021
Latex, 13 pages
arxiv created 1997/11/24 · openalex publication_date 1997/11/24 · arxiv updated 2009/11/30 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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.