Representations and $T^{\ast}$-extensions of $\delta$-Bihom-Jordan-Lie algebras

Year 2020, , 648 - 675, 02.04.2020

Abdelkader Ben Hassine , Liangyun Chen , Juan Li

https://doi.org/10.15672/hujms.588684

Cited By: 1

Abstract

The purpose of this article is to study representations of $\delta$-Bihom-Jordan-Lie algebras. In particular, adjoint representations, trivial representations, deformations, $T^\ast$-extensions of $\delta$-Bihom-Jordan-Lie algebras are studied in detail. Derivations and central extensions of $\delta$-Bihom-Jordan-Lie algebras are also discussed as an application.

Keywords

representations, derivations, deformations, T*-extensions, $\delta$-Bihom-Jordan-Lie algebras

References

• [1] S. Benayadi and A. Makhlouf, Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms, J. Geom. Phys. 76, 38–60, 2014.
• [2] M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras, Acta Math. Univ. Comenian. (N.S.) 66 (2), 151–201, 1997.
• [3] Y. Cheng and H. Qi, Representations of Bihom-Lie algebras, arXiv:1610.04302.
• [4] G. Graziani, A. Makhlouf, C. Menini, and F. Panaite, Bihom-associative algebras, Bihom-Lie algebras and Bihom-bialgebras, SIGMA Symmetry Integrability Geom. Methods Appl. (11), Paper 086, 34 pp, 2015.
• [5] J. Hartwig, D. Larsson and S. Silvestrov, Deformations of Lie algebras using σ- derivations, J. Algebra 295 (2), 314–361, 2006.
• [6] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2), 51–64, 2008.
• [7] S. Okubo and N. Kamiya, Jordan Lie superalgebra and Jordan Lie triple system, J. Algebra 198 (2), 388–411, 1997.
• [8] L. Qian, J. Zhou, and L. Chen, Engel’s theorem for Jordan-Lie algebras and its applications, Chinese Ann. Math. Ser. A 33 (5), 517–526, 2012 (in Chinese).
• [9] S. Wang and S. Guo, Bihom-Lie superalgebra structures, arXiv:1610.02290.
• [10] J. Zhao, L. Chen, and L. Ma, Representations and $T^\ast$-extensions of Hom-Jordan-Lie Algebras, Comm. Algebra 44 (7), 2786–2812, 2016.