On the ${\mathbb Z}_3$-Graded Structures

Yıl 2023, Cilt: 5 Sayı: 2, 31 - 40, 30.12.2023

Salih Celik Sultan Çelik

Öz

After introducing some ${\mathbb Z}_3$-graded structures, we first give the definition of a ${\mathbb Z}_3$-graded quantum space and show that the algebra of functions on it, denoted by ${\cal O}(\widetilde{\mathbb C}_q^{1|1|1})$, has a ${\mathbb Z}_3$-graded Hopf algebra structure. Later, we obtain a new ${\mathbb Z}_3$-graded quantum group, denoted by $\widetilde{\rm GL}_q(1|1)$, and show that the algebra of functions on this group is a ${\mathbb Z}_3$-graded Hopf algebra. Finally, we construct two non-commutative differential calculi on the algebra ${\cal O}(\widetilde{\mathbb C}_q^{1|1})$ which are left covariant with respect to the ${\mathbb Z}_3$-graded Hopf algebra ${\cal O}(\widetilde{\rm GL}_q(1|1))$.

Anahtar Kelimeler

Z3-graded vector space, Z3-graded algebra, Z3-graded Hopf algebra, Z3-graded quantum group, Z3-graded differential calculus

Kaynakça

• Drinfeld, V. G. (1986). Quantum groups. Proceedings International Congress of Mathematicians Berkeley (p. 798-820).
• Manin, Yu I. (1988). Quantum groups and non-commutative geometry. Les publications du Centre de Recherches Mathématiques Publications CRM: Lecture notes, Univ. de Montréal.
• Connes, A. (1995). Non-commutative geometry. Academic Press, New York.
• Abe, E. (1980). Hopf Algebras. Cambridge Tracts in Mathematics vol. 74, Cambridge University Press, Cambridge.
• Faddeev, L., Reshetikhin, N., & Takhtajan, L. (1990). Quantization of Lie groups and Lie algebras. Leningrad Mathematical Journal, 1, 193-225.
• Manin, Yu I. (1989). Multiparametric quantum deformation of the general linear supergroup. Communications in Mathematical Physics, 123, 163-175.
• Chung, W. S. (1994). Quantum $Z_3$-graded space. Journal of Mathematical Physic, 35, 2497-2504.
• Çelik, S. (2017). A new $Z_3$-graded quantum group. Journal of Lie Theory, 27, 545-554.
• Woronowicz, S. L. (1989). Differential calculus on compact matrix pseudogroups. Communications in Mathematical Physics, 122, 125-170.
• Wess, J., & Zumino, B. (1991). Covariant differential calculus on the quantum hyperplane. Nuclear Physics B-Proceedings Supplements, 18(2), 302-312.
• Soni, S. K. (1991). Differential calculus on the quantum superplane. Journal of Physics A: Mathematical and General, 24(3), 619-624.
• Çelik, S. (2017). Bicovariant differential calculus on the quantum superspace ${\mathbb R}_q(1|2)$. Journal of Algebra and its Applications, 15(09), Article Number: 1650172.
• Çelik, S. (2017). Covariant differential calculi on quantum symplectic superspace $SP_q^{1|2}$ . J Journal of Mathematical Physics, 58(2), Article Number: 023508.
• Bruce, A. J. & Dublij, S. (2020). Double-graded quantum superplane. Reports on Mathematical Physics, 86(3), 383-400.
• Fakhri, H., & Laheghi, S. (2021). Left-covariant first order differential calculus on quantum Hopf supersymmetry algebra. Journal of Mathematical Physics, 62(3), Article Number: 031702.
• Schmidke, W. B., Vokos, S. P., & Zumino, B. (1990). Differential geometry of the quantum supergroup $GL_q(1|1)$. Zeitschrift für Physik C Particles and Fields, 48(2), 249-255.
• Çelik, S., & Çelik S. A. (1998). On the differential geometry of $GL_q(1|1)$. Journal of Physics A: Mathematical and General, 31(48), 9685-9694.
• Çelik, S. (2002). Differential geometry of the $Z_3$-graded quantum superplane. Journal of Physics A: Mathematical and General, 35(19), 4257-4268.
• Çelik, S. (2002). $Z_3$-graded differential geometry of the quantum plane. Journal of Physics A: Mathematical and General, 35(30), 6307-6318.
• Çelik, S. (2016). A differential calculus on $Z_3$-graded quantum superspace ${\mathbb R}_q(2|1)$. Algebras and Representation Theory, 19, 713-730.
• Çelik, S., & Çelik, S. A. (2017). Differential calculi on $Z_3$-graded Grassmann plane. Advances in Applied Clifford Algebras, 27, 2407-2427.
• Çelik, S., & Bulut, F. (2016). A differential calculus on the $Z_3$-graded quantum group $GL_q(2)$. Advances in Applied Clifford Algebras, 26, 81-96.
• Çelik, S. (2021). Left covariant differential calculi on $\widetilde{\rm GL}_q(2)$}. Journal of Mathematical Physics, 62(7), Article Number: 073504.
• Çelik, S. A. (2023). A new ${\mathbb Z}_3$-graded quantum space $\widetilde{\mathbb C}_q^3$ and its geometry. TÜBITAK 1002 Short Term R\&D Funding Program Project Number: 123F216.
• Majid, S. (1995). Foundations of quantum group theory. Cambridge University Press, Cambridge.