Duality on the quantum space(3) with two parameters
Tóm tắt
In this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.
Tài liệu tham khảo
V. G. Drinfeld, In: A. M. Gleason (Ed.), Proceedings International Congress of Mathematicians, 03–11 August 1986, Berkeley, California, USA (Amer. Math.Soc., 1988) 798
M. Jimbo, Lett. Math. Phys. 11, 247 (1986)
L. A. Takhtajan, Adv. Stud. Pure Math. 19, 435 (1989)
Y. I. Manin, Quantum Groups and Noncommutative Geometry (Centre de Reserches Mathematiques, Montreal, 1988)
Y. I. Manin, Commun. Math. Phys. 123, 163 (1989)
S. L. Woronowicz, Commun. Math. Phys. 111, 613 (1987)
L. Faddeev, N. Reshetikin, L. Takhtajan, Alg. Anal. 1, 129 (1988)
L. Alvarez-Gaume, C. Gomez, G. Sierra, Nucl. Phys. B 330, 347 (1990)
L. Alvarez-Gaume, C. Gomez, G. Sierra, Phys. Lett. B 220, 142 (1989)
G. Moore, N. Reshetikhin, Nucl. Phys. B 328, 557 (1989)
R. B. Zhang, M. D. Gould, A. J. Bracken, Commun. Math. Phys. 137, 13 (1989).
S. Majid, Fondation of Quantum Group Theory, (Cambridge University Press, Cambridge, 1995)
J. Mourad, Classical Quant. Grav. 12, 965 (1995)
J. Madore, An Introduction to Noncommutative Differential Geometry and Its Applications, 2nd edition (Cambridge University Press, Cambridge, 2000)
P. G. Castro, B. Chakraborty, Z. Kuznetsova, F. Toppan, Cent. Eur. J. Phys. 9, 841 (2011)
A. A. Altıntaş, M. Arık, A. S. Arıkan, Cent. Eur. J. Phys. 8, 819 (2010)
R. M. Ubriaco, J. Phys. A Math. Gen. 25, 169 (1992)
S. L. Woronowicz, Commun. Math. Phys. 122, 125 (1989)
J. Wess, B. Zumino, Nucl. Phys. 18, 302 (1990)
T. Brzezinski, Lett. Math. Phys. 27, 287 (1993)
A. Sudbery, Phys. Lett. B 284, 61 (1992)
T. Kobayashi, T. Uematsu, Z. Phys. 56, 193 (1992)
A. El Hassouni, Y. Hassouni, E. H. Tahri, Int. J. Theor. Phys. 35, 2517 (1996)
P. Bouwknegt, J. McCarthy, P. Nieuwenhuizen, Phys. Lett. B 394, 82 (1997)
N. Aizawa, R. Chakrabarti, J. Math. Phys. 45, 1623 (2004)
E. M. Falaki, E. H. Tahri, J. Phys. A Math. Gen. 34, 3403 (2001)
R. Coquereaux, A. O. Garcia, R. Trinchero, Rev. Math. Phys. 12, 227 (2000)
E. Baz, Mod. Phys. Lett. A 21, 2323 (2006)
Z. Bentalha, M. Tahiri, Int. J. Geom. Methods M. 4, 1087 (2007)
A. Sudbery, In: T. Curtright, D. Fairlie, and C. Zachos (Eds), Proceedings of the Workshop on Quantum Groups, 1990, Chicago, USA (World Scientific, Singapore, 1991) 697
V. K. Dobrev, J. Math. Phys. 33, 3419 (1992)
S. A. Celik, E. Yasar, Czech. J. Phys. 56, 229 (2006)
P. N. Watts, PhD Thesis, University of California (Berkeley, USA, 1994)
S. Celik, E. M. Özkan, E. Yasar, Turk. J. Math. 33, 75 (2009)
M. Ozavşar, G. Yesilot, Int. J. Geom. Methods M. 8, 1667 (2011)
M. Özavşar, G. Yesilot, Eur. J. Pure Appl. Math. 5, 297 (2012)