Statistical $\rho$-commutative algebras

Yıl 2023, Cilt: 52 Sayı: 2, 340 - 355, 31.03.2023

Zahra Bagheri Esmaeil Peyghan

https://doi.org/10.15672/hujms.1105421

Öz

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.

Anahtar Kelimeler

statistical structure, Codazzi-couple, Kähle structure, Para-Kähler structure, connection, holomorphic statistical structure

Kaynakça

• [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.