Korean J. Math. Vol. 27 No. 1 (2019) pp.175-192
DOI: https://doi.org/10.11568/kjm.2019.27.1.175

Hom-Lie-Yamaguti superalgebras

Main Article Content

Donatien Gaparayi
Sylvain Attan
A. Nourou Issa


(Multiplicative) Hom-Lie-Yamaguti superalgebras are defined as a $\mathbb{Z}_2$-graded generalization of Hom-Lie Yamaguti algebras and also as a twisted generalization of Lie-Yamaguti superalgebras. Hom-Lie-Yamaguti superalgebras generalize also Hom-Lie supertriple systems (and subsequently ternary multiplicative Hom-Nambu superalgebras) and Hom-Lie superalgebras in the same way as Lie-Yamaguti superalgebras generalize Lie supertriple systems and Lie superalgebras. Hom-Lie-Yamaguti superalgebras are obtained from Lie-Yamaguti superalgebras by twisting along superalgebra even endomorphisms. We show that the category of (multiplicative) Hom-Lie-Yamaguti superalgebras is closed under twisting by self-morphisms. Constructions of some examples of Hom-Lie-Yamaguti superalgebras are given. The notion of an $nth$ derived (binary) Hom-superalgebras is extended to the one of an $nth$ derived binary-ternary Hom-superalgebras and it is shown that the category of Hom-Lie-Yamaguti superalgebras is closed under the process of taking $nth$ derived Hom-superalgebras.

Article Details

Supporting Agencies

Donatien Gaparayi


[1] K. Abdaoui, F. Ammar and A. Makhlouf, Hom-Lie Superalgebras and Hom-Lie admissible Superalgebras, Journal of algebra. 324 (7) (2010), 1513–1528. Google Scholar

[2] , K. Abdaoui, F. Ammar and A. Makhlouf, Hom-alternative, Hom-Malcev and Hom-Jordan Superalgebras, arXiv: 1304.1579v1 [math.RA] Apr 2013. Google Scholar

[3] H. Albuquerque and A. Elduque, Classification of Mal’tsev Superalgebras of small dimensions, Algebra and Logic 35 (6) (1996), 512–554. Google Scholar

[4] H. Ataguema, A. Makhlouf and S.D. Silvestrov, Generalization of n-ary Nambu algebras and beyond, J. Math. Phys. 50 (2009), 083501. Google Scholar

[5] S. Attan and A. Nourou Issa, Hom-Bol algebras, Quasigroups and related Systems 21 (2013), 131–146. Google Scholar

[6] Y. Fregier and A. Gohr, Unital algebras of Hom-associative type and surjective or injective twistings, J. of Gen. Lie Theo. and Appl. 3 (4) (2009), 285–295. Google Scholar

[7] D. Gaparayi and A. N. Issa, A twisted generalization of Lie-Yamaguti algebras, Int. J. Algebra. 6 (7) (2012), 339–352. Google Scholar

[8] D. Gaparayi and A. N. Issa, Hom-Akivis superalgebras, Journal of Algebra and Computational Applications. 6 (1) (2017), 36–51. Google Scholar

[9] J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using σ−derivations, J. Algebras 292 (2006), 314–361. Google Scholar

[10] A. N. Issa, Hom-Akivis algebras, Comment. Math. Univ. Carolin. 52 (4) (2011), 485–500. Google Scholar

[11] A. N. Issa, Supercommutator algebras of right (Hom)-alternative superalgebras, arXiv:1710.02706v1[math.RA] Google Scholar

[12] A. Koulibaly and M.F. Ouedraogo, Supersyst`emes Triples de Lie associ es aux superalg`ebres de Malcev, Africa Matematika 14 (3) (2002). Google Scholar

[13] D. Larsson and S. D. Silvestrov, Quasi-Hom-Lie algebras, central extensions and 2-cycle-like identies, J. Algebra. 288 (2005), 321–344. Google Scholar

[14] D. Larsson and S. D. Silvestrov, Quasi-Lie algebras, Comptemp.Math. 391 (2005). Google Scholar

[15] A. Makhlouf, Hom-Alternative algebras and Hom-Jordan algebras, Int. Elect. J. Alg. 8 (2010), 177–190. Google Scholar

[16] A. Makhlouf and Silvestrov S.D., Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), 51–64. Google Scholar

[17] S. Okubo, Jordan-Lie Super Algebra and Jordan-Lie Triple System, Journal of algebra 198 (1997), 388–411. Google Scholar

[18] D. Yau, Hom-algebras and homology, J. Lie Theory. 19 (2009), 409–421. Google Scholar

[19] D. Yau, Hom-Novikov algebras, J. Phys. A 44 (2011), 085202. Google Scholar

[20] D. Yau, Hom-Maltsev, Hom-alternative and Hom-Jordan algebras, Int. Elect. J. Alg. 11 (2012), 177–217. Google Scholar

[21] D. Yau , On n-ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geom. Phys. 62 (2012), 506–522. Google Scholar

[22] T. Zhang and J. Li, Representations and cohomologies of Hom-Lie-Yamaguti algebras with applications, Colloquium Mathematicum 148 (2) (2017), 131–155. Google Scholar