Research Article
BibTex RIS Cite

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

Year 2023, Volume: 5 Issue: 2, 31 - 40, 30.12.2023

Salih Celik Sultan Çelik

Abstract

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 ${\cal O}(\widetilde{\rm GL}_q(1|1))$.

Keywords

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

References

  • 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.

Year 2023, Volume: 5 Issue: 2, 31 - 40, 30.12.2023

Salih Celik Sultan Çelik

Abstract

References

  • 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.
There are 25 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Articles
Authors

Salih Celik 0000-0002-6590-1032

Sultan Çelik 0000-0003-3465-8209

Publication Date December 30, 2023
Published in Issue Year 2023 Volume: 5 Issue: 2

Cite

APA Celik, S., & Çelik, S. (2023). On the ${\mathbb Z}_3$-Graded Structures. Hagia Sophia Journal of Geometry, 5(2), 31-40.