Research Article
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Year 2023, Volume: 52 Issue: 2, 340 - 355, 31.03.2023
https://doi.org/10.15672/hujms.1105421

Abstract

References

  • [1] S. Amari and H. Nagaoka, Methods of information geometry, in: Transl. Math. Monogr., Amer. Math. Soc. 191, 2000.
  • [2] Z. Bagheri and E. Peyghan, (Para-) Kähler structures on $\rho$-commutative algebras, Adv. Appl. Clifford Algebras 28 (5), 95, 2018.
  • [3] P. J. Bongaarts and H. G. J. Pijls, Almost commutative algebra and differential calculus on the quantum hyperplane, J. Math. Phys. 35 (2), 959–970, 1994.
  • [4] S.Y. Cheng and S. T. Yau, The real Monge-Amp‘ere equation and affine flat structures. In: Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. I, Science Press, New York, 339-370, 1982.
  • [5] C. Ciupala, Linear connections on almost commutative algebras, Acta. Th. Univ. Comenianiae 72 (2), 197207, 2003.
  • [6] C. Ciupala, Connections and distributions on quantum hyperplane, Czech. J. Phys. 54 (8), 921–932, 2004.
  • [7] C. Ciupala, 2-$\rho$-derivation on a $\rho$-algebra and application to the quaternionic algebra, Int. J. Geom. Meth. Mod. Phys. 4 (3), 457–469, 2007.
  • [8] M. Dubois-Violette, Dérivations et calcul différentiel non commutatif, C.R. Acad. Sci. Paris, série I. 307, 403–408, 1988.
  • [9] T. Fei and J. Zhang, Interaction of Codazzi Couplings with (Para-)Kähler Geometry, Results Math. 72 (2), 2017. DOI 10.1007/s00025-017-0711-7.
  • [10] H. Furuhata and I. Hasegawa, Submanifold theory in holomorphic statistical manifolds, in: S. Dragomir, M.H. Shahid, F.R. Al-Solamy (Eds.), Geometry of Cauchy- Riemann Submanifolds, Springer, Singapore, 179–215, 2016.
  • [11] E. Kähler, Über eine bemerkenswerte Hermtesche metrik, Abn. Sem. Unv. Hamburg 9, 173–186, 1933.
  • [12] S. L. Lauritzen, Statistical manifolds, In: Differential Geometry in Statistical Inferences, IMS Lecture Notes Monogr. Ser., 10, Inst. Math. Statist., Hayward California, 96–163, 1987.
  • [13] S. Majid, Classification of bicovariant differential calculi, J. Geom. Phys. 25, 119–140, 1998.
  • [14] S. Majid, Riemannian geometry of quantum groups and finite groups with nonuniversal differentials, Commun. Math. Phys. 225, 131–170, 2002.
  • [15] F. Ngakeu, Levi-Civita connection on almost commutative algebras, Int. J. Geom. Meth. Mod Phys. 4 (7), 1075–1085, 2007.
  • [16] F. Ngakeu, S. Majid and D. Lambert, Noncommutative Riemannian geometry of the alternating group A4, J. Geom. Phys. 42, 259–282, 2002.
  • [17] K. Nomizu and T. Sasaki,Affine Differential Geometry: Geometry of Affine Immersions, Volume 111 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1994.
  • [18] H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math. 13 (2), 213-229, 1976.
  • [19] U. Simon, Affine differential geometry. In: Dillen, F., Verstraelen, L. (eds.) Handbook of Differential Geometry, North-Holland 1, 905–961, 2000.
  • [20] K.Takano, Statistical manifolds with almost complex structures and its statistical submersions, Tensor N.S. 65, 128–142, 2004.
  • [21] K.Takano, Statistical manifolds with almost contact structures and its statistical submersions, J. Geom. 85, 171–187, 2006.

Statistical $\rho$-commutative algebras

Year 2023, Volume: 52 Issue: 2, 340 - 355, 31.03.2023
https://doi.org/10.15672/hujms.1105421

Abstract

In this article, we study Codazzi-couples of an arbitrary connection $\nabla$ with a nondegenerate 2-form $\omega$, an isomorphism $L$ on the space of derivation of $\rho$-commutative algebra $A$, which the important examples of isomorphism $L$ are almost complex and almost para-complex structures, a metric $g$ that $(g, \omega,L)$ form a compatible triple. We study a statistical structure on $\rho$-commutative algebras by the classical manner on Riemannian manifolds. Then by recalling the notions of almost (para-)Kähler $\rho$-commutative algebras, we generalized the notion of Codazzi-(para-)Kähler $\rho$-commutative algebra as a (para-)Kähler (or Fedosov) $\rho$-commutative algebra which is at the same time statistical and moreover define the holomorphic $\rho$-commutative algebras.

References

  • [1] S. Amari and H. Nagaoka, Methods of information geometry, in: Transl. Math. Monogr., Amer. Math. Soc. 191, 2000.
  • [2] Z. Bagheri and E. Peyghan, (Para-) Kähler structures on $\rho$-commutative algebras, Adv. Appl. Clifford Algebras 28 (5), 95, 2018.
  • [3] P. J. Bongaarts and H. G. J. Pijls, Almost commutative algebra and differential calculus on the quantum hyperplane, J. Math. Phys. 35 (2), 959–970, 1994.
  • [4] S.Y. Cheng and S. T. Yau, The real Monge-Amp‘ere equation and affine flat structures. In: Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. I, Science Press, New York, 339-370, 1982.
  • [5] C. Ciupala, Linear connections on almost commutative algebras, Acta. Th. Univ. Comenianiae 72 (2), 197207, 2003.
  • [6] C. Ciupala, Connections and distributions on quantum hyperplane, Czech. J. Phys. 54 (8), 921–932, 2004.
  • [7] C. Ciupala, 2-$\rho$-derivation on a $\rho$-algebra and application to the quaternionic algebra, Int. J. Geom. Meth. Mod. Phys. 4 (3), 457–469, 2007.
  • [8] M. Dubois-Violette, Dérivations et calcul différentiel non commutatif, C.R. Acad. Sci. Paris, série I. 307, 403–408, 1988.
  • [9] T. Fei and J. Zhang, Interaction of Codazzi Couplings with (Para-)Kähler Geometry, Results Math. 72 (2), 2017. DOI 10.1007/s00025-017-0711-7.
  • [10] H. Furuhata and I. Hasegawa, Submanifold theory in holomorphic statistical manifolds, in: S. Dragomir, M.H. Shahid, F.R. Al-Solamy (Eds.), Geometry of Cauchy- Riemann Submanifolds, Springer, Singapore, 179–215, 2016.
  • [11] E. Kähler, Über eine bemerkenswerte Hermtesche metrik, Abn. Sem. Unv. Hamburg 9, 173–186, 1933.
  • [12] S. L. Lauritzen, Statistical manifolds, In: Differential Geometry in Statistical Inferences, IMS Lecture Notes Monogr. Ser., 10, Inst. Math. Statist., Hayward California, 96–163, 1987.
  • [13] S. Majid, Classification of bicovariant differential calculi, J. Geom. Phys. 25, 119–140, 1998.
  • [14] S. Majid, Riemannian geometry of quantum groups and finite groups with nonuniversal differentials, Commun. Math. Phys. 225, 131–170, 2002.
  • [15] F. Ngakeu, Levi-Civita connection on almost commutative algebras, Int. J. Geom. Meth. Mod Phys. 4 (7), 1075–1085, 2007.
  • [16] F. Ngakeu, S. Majid and D. Lambert, Noncommutative Riemannian geometry of the alternating group A4, J. Geom. Phys. 42, 259–282, 2002.
  • [17] K. Nomizu and T. Sasaki,Affine Differential Geometry: Geometry of Affine Immersions, Volume 111 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1994.
  • [18] H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math. 13 (2), 213-229, 1976.
  • [19] U. Simon, Affine differential geometry. In: Dillen, F., Verstraelen, L. (eds.) Handbook of Differential Geometry, North-Holland 1, 905–961, 2000.
  • [20] K.Takano, Statistical manifolds with almost complex structures and its statistical submersions, Tensor N.S. 65, 128–142, 2004.
  • [21] K.Takano, Statistical manifolds with almost contact structures and its statistical submersions, J. Geom. 85, 171–187, 2006.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Zahra Bagheri This is me 0000-0002-5821-8999

Esmaeil Peyghan 0000-0002-2713-6253

Publication Date March 31, 2023
Published in Issue Year 2023 Volume: 52 Issue: 2

Cite

APA Bagheri, Z., & Peyghan, E. (2023). Statistical $\rho$-commutative algebras. Hacettepe Journal of Mathematics and Statistics, 52(2), 340-355. https://doi.org/10.15672/hujms.1105421
AMA Bagheri Z, Peyghan E. Statistical $\rho$-commutative algebras. Hacettepe Journal of Mathematics and Statistics. March 2023;52(2):340-355. doi:10.15672/hujms.1105421
Chicago Bagheri, Zahra, and Esmaeil Peyghan. “Statistical $\rho$-Commutative Algebras”. Hacettepe Journal of Mathematics and Statistics 52, no. 2 (March 2023): 340-55. https://doi.org/10.15672/hujms.1105421.
EndNote Bagheri Z, Peyghan E (March 1, 2023) Statistical $\rho$-commutative algebras. Hacettepe Journal of Mathematics and Statistics 52 2 340–355.
IEEE Z. Bagheri and E. Peyghan, “Statistical $\rho$-commutative algebras”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 2, pp. 340–355, 2023, doi: 10.15672/hujms.1105421.
ISNAD Bagheri, Zahra - Peyghan, Esmaeil. “Statistical $\rho$-Commutative Algebras”. Hacettepe Journal of Mathematics and Statistics 52/2 (March 2023), 340-355. https://doi.org/10.15672/hujms.1105421.
JAMA Bagheri Z, Peyghan E. Statistical $\rho$-commutative algebras. Hacettepe Journal of Mathematics and Statistics. 2023;52:340–355.
MLA Bagheri, Zahra and Esmaeil Peyghan. “Statistical $\rho$-Commutative Algebras”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 2, 2023, pp. 340-55, doi:10.15672/hujms.1105421.
Vancouver Bagheri Z, Peyghan E. Statistical $\rho$-commutative algebras. Hacettepe Journal of Mathematics and Statistics. 2023;52(2):340-55.