Z3-Graded Differential Calculus on the Quantum Space R3q

Year 2013, Volume: 42 Issue: 2, 101 - 114, 01.02.2013

Ergün Yasar Ahmet Bakkaloglu

Abstract

In this work, the Z-graded differential calculus of the extended quantum 3d space is constructed. By using this differential calculus, weobtain the algebra of Cartan-Maurer forms and the corresponding quantum Lie algebra. To give a Z-graded Cartan calculus on the extendedquantum 3d space, the noncommutative differential calculus on thisspace is extended by introducing inner derivations and Lie derivatives.

Keywords

Z -graded quantum 3d space, Lie algebra, inner derivation, Lie derivation

Z3-Graded Differential Calculus on the Quantum Space R3q

Abstract

-

Keywords

-

References

• Abramov, V. and Kerner, R., E xterior differentials of higher order and their covariant generalization, J. Math. Phys., 41, 5598–5614, 2000.
• Bazunova, N., Borowiec, A. and Kerner, R., U niversal differential calculus on ternary algebras, Letters in Math. Physics, 67 (3), 195–206, 2004.
• Celik, S. and Celik, S. A., O n the Differential Geometry of GL q (1|1), J. Phys. A 31, 9685– 9694, 1998.
• Celik, S., Z 3 -graded differential geometry of quantum plane, J. Phys. A: Math. Gen. 35, 6307–6318, 2002.
• Celik, S., C artan Calculi on The Quantum Superplane, J. Math. Phys. 47 (8), Art. No: 083501, 2006.
• Celik, S. A. and Yasar, E., D ifferential Geometry of the Quantum 3-Dimensional Space, Czech. J. Phys. 56, 229–236, 2006.
• Chryssomalakos, C., Schupp, P. and Zumino, B., I nduced extended calculus on the quantum plane, hep-th /9401141.
• Dubois-Violette, M. and Kerner, R., U niversal q-differential calculus and q-analog of homological algebra, Acta Math. Univ. Comenianae, LXV (2), 175–188, 1996.
• El Baz, M., El Hassouni, A., Hassouni, Y., Zakkari, E.H., d 3 = 0, d 2 = 0 Differential Calculi On Certain Noncommutative (Super) Spaces, J. Math. Phys. 45, 2314–2322, 2004.
• Manin, Yu I., Q uantum groups and noncommutative geometry, Montreal Univ. Preprint, 19 Kerner, R., Z 3 -graded algebras and the cubic root of the Dirac operator, J. Math. Phys., 33 (1), 403–411, 1992.
• Kerner, R., Z 3 -graded exterior differential calculus and gauge theories of higher order, Lett. in Math. Phys., 36, 441–454, 1996.
• Kerner, R., T he cubic chessboard, Class. Quantum Gravity, 14 (1A), A203–A225, 1997. Kerner, R. and Niemeyer, B., C ovariant q-differential calculus and its deformations at q N = 1, Lett. in Math. Phys., 45, 161–176, 1998.
• Schupp, P., Watts, P. and Zumino, B., D ifferential Geometry on Linear Quantum Groups, Lett. Math. Phys. 25 ,139–147, 1992.
• Schupp, P., Watts, P. and Zumino, B., C artan calculus on quantum Lie algebras, hepth/9312073.
• Schupp, P., C artan calculus: Differential geometry for quantum groups, hep-th/9408170. Vainerman, L. and Kerner, R., O n special classes of n-algebras, J. Math. Phys., 37 (5), 2553–2565, 1996.
• Wess, J. and Zumino, B., C ovariant Differential Calculus on the Quantum Hyperplane, Nucl. Phys. B 18 , 302–312, 1990.
• Woronowicz, S. L., C ompact Matrix Pseudogroups, Commun. Math. Phys. 111, 613–665, 19 Woronowicz, S. L., D ifferential Calculus on Compact Matrix Pseudogroups, Commun. Math. Phys. 122, 125–170, 1989.