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A Quantum Space and Some Associated Quantum Groups

Year 2018, Volume: 22 Issue: 2, 464 - 469, 15.08.2018

Muttalip Özavşar

Abstract

In the present paper, we first introduce a quantum $n$-space on which the algebra of coordinates is $\eta$-commutative. Further, it is shown that there are  some $\sigma$-twisted derivations acting on this algebra, and the algebra of such derivations is a quantum group. Morever, we show that a bicovariant differential calculus on this space can be constructed by using $\sigma$-twisted derivations. Finally, the quantum Lie algebra is obtained by using this bicovariant differential calculus.

Keywords

Quantum spaces Quantum groups; Derivation algebras

References

  • [1] Drinfeld, VG. 1987, Quantum Groups. Amer. Math. Soc. 1987. Proceedings International Congress of Mathematicians, 03-11 August 1986, Berkeley, 798-820.
  • [2] Brzezinski, T. 1993. Remark on bicovariant differential calculi and exterior Hopf algebras. Lett Math Phys. 27 (1993), 287-300.
  • [3] Gurevich, D. Generalized Translation Operators on Lie Groups. Sov J. Cont Math Anal 18 (1983), 57-70.
  • [4] Borowiec, A., Kharchenko, V. 1995. First order optimum calculi. Bull. Soc. Sci. Lett. L 45(1995), 75-88.
  • [5] Hu, N. Quantum Divided Power Algebra, QDerivatives, and Some New Quantum Groups. J Algebra 232 (2000), 507-540.
  • [6] Madore, J. 2000. An Introduction to Noncommutative Differential Geometry and Its Applications. Cambridge, UK: Cambridge University Press.
  • [7] Majid, S. 1995. Foundation of Quantum Group Theory. Cambridge, UK: Cambridge University Press.
  • [8] Manin, Y.I. 1988. Quantum Groups and Noncommutative Geometry. Centre de Reserches Mathematiques, Montreal.
  • [9] Manin, Y.I. 1989. Multiparemetric Quantum Deformation of the General Linear Supergroup. Commu. Math. Phys. 123 (1989), 163-175.
  • [10] Sudbery, A. 1990. Non-commuting Coordinates and Differential Operators. In Proc.Workshop on Quantum Groups, Argogne, 33-51.
  • [11] Woronowicz, S.L. 1989. Differential Calculus on Compact Matrix Pseudogroups. Commun. Math. Phys. 122 (1989), 125-170.
  • [12] Ubriaco, R.M. 1992. Noncommutative Differential Calculus and q-Analysis. J. Phys. A;Math. Gen. 25(1992), 169-173.
  • [13] Watts, P. Differential Geometry on Hopf Algebras and Quantum Groups, Ph.D. Thesis, hepth/9412153v1.
  • [14] Wess, J., Zumino, B. 1990. Covariant Differential Calculus on the Quantum Hyperplane. Nucl. Phys. 18(1990), 302-312.
  • [15] Schüler, A. 1999. Differential Hopf Algebras on Quantum Groups of Type A. J. Algebra 214 (1999), 479-518.
  • [16] Scheunert, M. 1979. Generalized Lie Algebras. J Math Phys 20 (1979), 712-720 .

Year 2018, Volume: 22 Issue: 2, 464 - 469, 15.08.2018

Muttalip Özavşar

Abstract

References

  • [1] Drinfeld, VG. 1987, Quantum Groups. Amer. Math. Soc. 1987. Proceedings International Congress of Mathematicians, 03-11 August 1986, Berkeley, 798-820.
  • [2] Brzezinski, T. 1993. Remark on bicovariant differential calculi and exterior Hopf algebras. Lett Math Phys. 27 (1993), 287-300.
  • [3] Gurevich, D. Generalized Translation Operators on Lie Groups. Sov J. Cont Math Anal 18 (1983), 57-70.
  • [4] Borowiec, A., Kharchenko, V. 1995. First order optimum calculi. Bull. Soc. Sci. Lett. L 45(1995), 75-88.
  • [5] Hu, N. Quantum Divided Power Algebra, QDerivatives, and Some New Quantum Groups. J Algebra 232 (2000), 507-540.
  • [6] Madore, J. 2000. An Introduction to Noncommutative Differential Geometry and Its Applications. Cambridge, UK: Cambridge University Press.
  • [7] Majid, S. 1995. Foundation of Quantum Group Theory. Cambridge, UK: Cambridge University Press.
  • [8] Manin, Y.I. 1988. Quantum Groups and Noncommutative Geometry. Centre de Reserches Mathematiques, Montreal.
  • [9] Manin, Y.I. 1989. Multiparemetric Quantum Deformation of the General Linear Supergroup. Commu. Math. Phys. 123 (1989), 163-175.
  • [10] Sudbery, A. 1990. Non-commuting Coordinates and Differential Operators. In Proc.Workshop on Quantum Groups, Argogne, 33-51.
  • [11] Woronowicz, S.L. 1989. Differential Calculus on Compact Matrix Pseudogroups. Commun. Math. Phys. 122 (1989), 125-170.
  • [12] Ubriaco, R.M. 1992. Noncommutative Differential Calculus and q-Analysis. J. Phys. A;Math. Gen. 25(1992), 169-173.
  • [13] Watts, P. Differential Geometry on Hopf Algebras and Quantum Groups, Ph.D. Thesis, hepth/9412153v1.
  • [14] Wess, J., Zumino, B. 1990. Covariant Differential Calculus on the Quantum Hyperplane. Nucl. Phys. 18(1990), 302-312.
  • [15] Schüler, A. 1999. Differential Hopf Algebras on Quantum Groups of Type A. J. Algebra 214 (1999), 479-518.
  • [16] Scheunert, M. 1979. Generalized Lie Algebras. J Math Phys 20 (1979), 712-720 .
There are 16 citations in total.

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Journal Section Articles
Authors

Muttalip Özavşar This is me

Publication Date August 15, 2018
Published in Issue Year 2018 Volume: 22 Issue: 2

Cite

APA Özavşar, M. (2018). A Quantum Space and Some Associated Quantum Groups. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(2), 464-469.
AMA Özavşar M. A Quantum Space and Some Associated Quantum Groups. J. Nat. Appl. Sci. August 2018;22(2):464-469.
Chicago Özavşar, Muttalip. “A Quantum Space and Some Associated Quantum Groups”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22, no. 2 (August 2018): 464-69.
EndNote Özavşar M (August 1, 2018) A Quantum Space and Some Associated Quantum Groups. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22 2 464–469.
IEEE M. Özavşar, “A Quantum Space and Some Associated Quantum Groups”, J. Nat. Appl. Sci., vol. 22, no. 2, pp. 464–469, 2018.
ISNAD Özavşar, Muttalip. “A Quantum Space and Some Associated Quantum Groups”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22/2 (August 2018), 464-469.
JAMA Özavşar M. A Quantum Space and Some Associated Quantum Groups. J. Nat. Appl. Sci. 2018;22:464–469.
MLA Özavşar, Muttalip. “A Quantum Space and Some Associated Quantum Groups”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 22, no. 2, 2018, pp. 464-9.
Vancouver Özavşar M. A Quantum Space and Some Associated Quantum Groups. J. Nat. Appl. Sci. 2018;22(2):464-9.
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