Riemannian Geometry on Hom-ρ-commutative Algebras

Abstract

Recently, some concepts such as Hom-algebras, Hom-Lie algebras, Hom-Lie admissible algebras, Hom-coalgebras are studied and some classical properties of algebras and some geometric objects are extended on them. In this paper by recalling the concept of Hom-ρ-commutative algebras, the authurs intend to develop some of the most classical results in Riemannian geometry such as metric, connection, torsion tensor, curvature tensor on it and also they discuss about differential operators and get some results of differential calculus by using them. The notions of symplectic structures and Poisson structures are included and an example of ρ-Poisson bracket is given.

Keywords

Extended hyper-plan / Hom-ρ-commutative algebra / ρ-Poisson bracket

References

[1] Abdaoui K, Ammar F, Makhlouf A. Construction and cohomology of color Hom-Lie algebras. Commu. Alg., 2015, 43(11): 4581-4612

[2] Ammar F, Makhlouf A. Hom-Lie superalgebras and Hom-Lie admissible superalgebras. J. Alg., 2010, 324(7): 1513-1528

[3] Armakan A R, Silvestrov S, Farhangdoost M R. Extensions of hom-Lie color algebras. Georgian Math. J., 2021, 28(1): 15-27

[4] Bakayoko, I., Modules over color Hom-Poisson algebras, J. Gen. Lie Theo. Appl., 8 (1), 2014, 6 pages.

[5] Bakayoko, I., Hom-Novikov color algebras, 2016, arXiv:1609.07813vl, 16 pages.

[6] Bongaarts P J, Pijls H G J. Almost commutative algebra and differential calculus on the quantum hyperplane. J. Math. Phys., 1994, 35(2): 959-970

[7] Ciupala C. Linear connections on almost commutative algebras. Acta. Math. Univ. Comenianiae, 2003, 72(2): 197-207

[8] Ciupala C. Differential calculus on almost commutative algebras and application to the quantum hyper-plane. Arch. Math. (BRNO) Tomus, 2005, 41: 359-377

[9] Ciupala C. 2-ρ-Derivations on a ρ-algebra and applications to the quaternionic algebra. Int. J. Geom. Meth. Mod. Phys., 2007, 4(3): 457-469

[10] Darling R W R. Differential Forms and Connections, 1994, Cambridge: Cambridge University Press

[11] Dubois-Violette M. Dérivations et calcul différentiel non commutatif. C. R. Acad. Sci. Paris, Série I, 1988, 307: 403-408

[12] Fregier, Y. and Gohr, A., On Hom-type algebras, J. Gen. Lie Theo. Appl., 4, 2010, 16 pages.

[13] Hartwig J T, Larsson D, Silvestrov S D. Deformations of Lie algebras using σ-derivations. J. Alg., 2006, 295: 314-361

[14] Larsson D, Silvestrov S D. Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities. J. Alg., 2005, 288: 321-344

[15] Larsson D, Silvestrov S D. Quasi-Lie algebras. Noncommutative Geometry and Representation Theory in Mathematical Physics, 2005, Providence, RI: Amer. Math. Soc. 241-248

[16] Majid S. Riemannian geometry of quantum groups and finite groups with nonuniversal differentials. Commun. Math. Phys., 2002, 225: 131-170

[17] Makhlouf A. Hom-alternative algebras and Hom-Jordan algebras. Int. Elect. J. Alg., 2010, 8: 177-190

[18] Makhlouf A, Silvestrov S. Hom-algebra structures. J. Gen. Lie Theo. Appl., 2008, 2: 51-64

[19] Makhlouf A, Silvestrov S. Hom-algebras and Hom-coalgebras. J. Alg. Appl., 2010, 9(4): 553-589

[20] Makhlouf A, Silvestrov S. Notes on formal deformations of Hom-associative algebras and Hom-Lie algebras. Forum Math., 2010, 22(4): 715-739

[21] Ngakeu F. Levi-Civita connection on almost commutative algebras. Int. J. Geom. Meth. Mod. Phys., 2007, 4(7): 1075-1085

[22] Ngakeu F. Graded Poisson structures and Schouten-Nijenhuis bracket on almost commutative algebras. Int. J. Geom. Meth. Mod. Phys., 2012, 9(5): 12500421-125004220

[23] Ngakeu F, Majid S, Lambert D. Noncommutative Riemannian geometry of the alternating group A ₄. J. Geom. Phys., 2002, 42: 259-282

[24] Sheng Y. Representations of Hom-Lie algebras. Alg. Rep. Theo., 2012, 15: 1081-1098

[25] Sheng Y, Chen D. Hom-Lie 2 algebras. J. Alg., 2013, 376: 174-195

[26] Sternberg S. Lectures on Differential Geometry, 1964, Englewood Cliffs, New Jersey: Prentice-Hall

[27] Weintraub S H. Differential Forms, A Complement to Vector Calculus, 1997, San Diego, CA: Academic Press, Inc.

[28] Yau D. Hom-Malcev, Hom-alternative and Hom-Jordan algebras. Int. Elect. J. Alg., 2012, 11: 177-217

[29] Yuan L. Hom-Lie color algebra structures. Commu. Alg., 2012, 40(2): 575-592