TY  - JOUR
T1  - On the ${\mathbb Z}_3$-Graded Structures
AU  - Celik, Salih
AU  - Çelik, Sultan
PY  - 2023
DA  - December
JF  - Hagia Sophia Journal of Geometry
JO  - HSJG
PB  - Salim YÜCE
WT  - DergiPark
SN  - 2687-4261
SP  - 31
EP  - 40
VL  - 5
IS  - 2
LA  - en
AB  - 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 ${\cal O}(\widetilde{\rm GL}_q(1|1))$.
KW  - Z3-graded vector space
KW  - Z3-graded algebra
KW  - Z3-graded Hopf algebra
KW  - Z3-graded quantum group
KW  - Z3-graded differential calculus
CR  - Drinfeld, V. G. (1986). Quantum groups. Proceedings International Congress of Mathematicians Berkeley (p. 798-820).
CR  - Manin, Yu I. (1988). Quantum groups and non-commutative geometry. Les publications du Centre de Recherches Mathématiques Publications CRM: Lecture notes, Univ. de Montréal.
CR  - Connes, A. (1995). Non-commutative geometry. Academic Press, New York.
CR  - Abe, E. (1980). Hopf Algebras. Cambridge Tracts in Mathematics vol. 74, Cambridge University Press, Cambridge.
CR  - Faddeev, L., Reshetikhin, N., & Takhtajan, L. (1990). Quantization of Lie groups and Lie algebras. Leningrad Mathematical Journal, 1, 193-225.
CR  - Manin, Yu I. (1989). Multiparametric quantum deformation of the general linear supergroup. Communications in Mathematical Physics, 123, 163-175.
CR  - Chung, W. S. (1994). Quantum $Z_3$-graded space. Journal of Mathematical Physic, 35, 2497-2504.
CR  - Çelik, S. (2017). A new $Z_3$-graded quantum group. Journal of Lie Theory, 27, 545-554.
CR  - Woronowicz, S. L. (1989). Differential calculus on compact matrix pseudogroups. Communications in Mathematical Physics, 122, 125-170.
CR  - Wess, J., & Zumino, B. (1991). Covariant differential calculus on the quantum hyperplane. Nuclear Physics B-Proceedings Supplements, 18(2), 302-312.
CR  - Soni, S. K. (1991). Differential calculus on the quantum superplane. Journal of Physics A: Mathematical and General, 24(3), 619-624.
CR  - 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.
CR  - Çelik, S. (2017). Covariant differential calculi on quantum symplectic superspace $SP_q^{1|2}$ . J Journal of Mathematical Physics, 58(2), Article Number: 023508.
CR  - Bruce, A. J. & Dublij, S. (2020). Double-graded quantum superplane. Reports on Mathematical Physics, 86(3), 383-400.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - Çelik, S. (2002). Differential geometry of the $Z_3$-graded quantum superplane. Journal of Physics A: Mathematical and General, 35(19), 4257-4268.
CR  - Çelik, S. (2002). $Z_3$-graded differential geometry of the quantum plane. Journal of Physics A: Mathematical and General, 35(30), 6307-6318.
CR  - Çelik, S. (2016). A differential calculus on $Z_3$-graded quantum superspace ${\mathbb R}_q(2|1)$. Algebras and Representation Theory, 19, 713-730.
CR  - Çelik, S., & Çelik, S. A. (2017). Differential calculi on $Z_3$-graded Grassmann plane. Advances in Applied Clifford Algebras, 27, 2407-2427.
CR  - 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.
CR  - Çelik, S. (2021). Left covariant differential calculi on $\widetilde{\rm GL}_q(2)$}. Journal of Mathematical Physics, 62(7), Article Number: 073504.
CR  - 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.
CR  - Majid, S. (1995). Foundations of quantum group theory. Cambridge University Press, Cambridge.
UR  - https://dergipark.org.tr/tr/pub/hsjg/issue//1341362
L1  - https://dergipark.org.tr/tr/download/article-file/3324332
ER  -