Hom-Jordan and Hom-alternative bimodules
DOI:
https://doi.org/10.17398/2605-5686.35.1.69Keywords:
Bimodules, alternative algebras, Jordan algebras, Hom-alternative algebras, Hom-Jordan algebras, Hom-associative algebras.Abstract
In this paper, Hom-Jordan and Hom-alternative bimodules are introduced. It is shown that Jordan and alternative bimodules are twisted via endomorphisms into Hom-Jordan and Hom-alternative bimodules respectively. Some relations between Hom-associative bimodules, Hom-Jordan and Hom-alternative bimodules are given.
Downloads
References
H. Ataguema, A. Makhlouf, S.D. Silvestrov, Generalization of n-ary Nambu algebras and beyond, J. Math. Phys. 50 (8) (2009), 083501, 15 pp.
I. Bakayoko, B. Manga, Hom-alternative modules and Hom-Poisson comodules. arXiv:1411.7957v1
S. Eilenberg, Extensions of general algebras, Ann. Soc. Polon. Math. 21 (1948), 125 – 134.
F. Gürsey, C.-H. Tze, On The Role of Division, Jordan and Related Algebras in Particle Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
N. Huang, L.Y. Chen, Y. Wang, Hom-Jordan algebras and their αk - (a, b, c)-derivations, Comm. Algebra 46 (6) (2018), 2600 – 2614.
J.T. Hartwig, D. Larsson, S.D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 292 (2006), 314 – 361.
N. Jacobson, General representation theory of Jordan algebras, Trans. Amer. Math. Soc. 70 (1951), 509 – 530.
N. Jacobson, Structure of alternative and Jordan bimodules, Osaka Math. J. 6 (1954), 1 – 71.
P. Jordan, J. von Neumann, E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. of Math. (2) 35 (1934), 29 – 64.
D. Larsson, S.D. Silvestrov, Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra 288 (2005), 321 – 344.
D. Larsson, S.D. Silvestrov, Quasi-Lie algebras, in “Noncommutative Geometry and Representation Theory in Mathematical Physics”, Contemp. Math., 391, Amer. Math. Soc., Providence, RI, 2005, 241 – 248.
D. Larsson, S.D. Silvestrov, Quasi-deformations of sl2 (F) using twisted derivations, Comm. Algebra 35 (12) (2007), 4303 – 4318.
A. Makhlouf, S. Silvestrov, Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras, Forum Math. 22 (4) (2010), 715 – 739.
A. Makhlouf, Hom-Alternative algebras and Hom-Jordan algebras, Int. Electron. J. Algebra 8 (2010), 177 – 190.
A. Makhlouf, S.D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), 51 – 64.
S. Okubo, “Introduction to Octonion and other Non-associative Algebras in Physics”, Cambridge University Press, Cambridge, 1995.
R.D. Schafer, Representations of alternative algebras, Trans. Amer. Math. Soc. 72 (1952), 1 – 17.
Y. Sheng, Representation of hom-Lie algebras, Algebr. Represent. Theory 15 (6) (2012), 1081 – 1098.
T.A. Springer, F.D. Veldkamp, “Octonions, Jordan Algebras, and Exceptional Groups”, Springer-Verlag, Berlin, 2000.
J. Tits, R.M. Weiss, “Moufang Polygons”, Springer-Verlag, Berlin, 2002.
D. Yau, Hom-algebras and homology. arXiv:0712.3515v1
D. Yau, Module Hom-algebras. arXiv:0812.4695v1
D. Yau, Hom-Maltsev, Hom-alternative and Hom-Jordan algebras, Int. Electron. J. Algebra 11 (2012), 177 – 217.
A. Zahary, A. Makhlouf, Structure and classification of Hom-associative algebras. arXiv:1906.04969V1[math.RA]
