Matched differential calculus on the quantum groups $GL_q(2,C),SL_q(2,C),C_q(2|0)$

Details Matched differential calculus on the quantum groups $GL_q(2,C),SL_q(2,C),C_q(2|0)$ Matched differential calculus on the quantum groups $GL_q(2,C),SL_q(2,C),C_q(2|0)$

1995-09-29

arxiv.org/abs/q-alg/9509030v1

Mathematics

Quantum Algebra

LaTeX, 22 pages

Scientific paper

We proposed the construction of the differential calculus on the quantum group and its subgroup with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain the differential calculus on the quantum subgroup $SL_q(2,C)$ and quantum plane $C_q(2|0)$ (''quantum matrjoshka''). We found, that there are two differential calculi, associated to the left differential Maurer--Cartan 1-forms and to the right differential 1-forms. Matched reduction take the degeneracy between the left and right differentials. The classical limit ($q\to 1$) of the ''left'' differential calculus and of the ''right'' differential calculus is undeformed differential calculus. The condition ${\cal D}_qG=1$ gives the differential calculus on $SL_q(2,C)$, which contains the differential calculus on the quantum plane $C_q(2|0)$.

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