Mathematics – Quantum Algebra
Scientific paper
1997-11-24
Mathematics
Quantum Algebra
Latex, 13 pages
Scientific paper
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.
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