Bihom-Nijienhuis operators and $T^*$-extensions of Bihom-Lie superalgebras

The purpose of this article is to study Bihom-Nijienhuis operators and $T^*$-extensions of Bihom-Lie superalgebras. We show that the deformation generated by a Bihom-Nijienhuis operator is trivial. Moreover, we introduce the definition of $T^*$-extensions of Bihom-Lie superalgebras and show that $T^*$-extensions preserve many properties such as nilpotency, solvability and decomposition in some sense. In particular, we discuss the equivalence of $T^*$-extensions.

Keywords:

Bihom-Lie superalgebras Bihom-Nijienhuis operators, deformations, $T$*-extensions,

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K. Abdaoui, F. Ammar and A. Makhlouf, Constructions and cohomology of Hom-Lie color algebras, Comm. Algebra 43 (11), 4581-4612, 2015.
F. Ammar, I. Ayadi, S. Mabrouk and A. Makhlouf, Quadratic color Hom-Lie algebras, arXiv:1204.5155.
F. Ammar Z. Ejbehi and A. Makhlouf, Cohomology and deformations of Hom- algebras, J. Lie Theory 21 (4), 813-836, 2011.
F. Ammar and A. Makhlouf, Hom-Lie superalgebras and Hom-Lie admissible super- algebras, J. Algebra 324 (7), 1513-1528, 2010.
F. Ammar, A. Makhlouf and N. Saadoui, Cohomology of Hom-Lie superalgebras and q-deforemed Witt superalgebra, Czechoslovak Math. J. 63(138) (3), 721-761, 2013.
M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras, Acta Math. Univ. Comenian. (N.S.) 66 (2), 151-201, 1997.
I. Bajo, S. Benayadi and A. Medina, Symplectic structures on quadratic Lie algebras, J. Algebra 316 (1), 174-188, 2007.
S. Benayadi and A. Makhlouf, Hom-Lie algebras with symmetric invariant nondegen- erate bilinear forms, J. Geom. Phys. 76, 38-60, 2014.
Y. Cheng and H. Qi, Representations of Bihom-Lie algebras, arXiv:1610.04302.
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(86) (1), 1-34, 2015.
J. Hartwig, D. Larsson and S. Silvestrov, Deformations of Lie algebras using - derivations, J. Algebra 295 (2), 314-361, 2006.
Y. Liu, L. Chen and Y. Ma, Hom-Nijienhuis operators and $T^*$-extensions of hom-Lie superalgebras, Linear Algebra Appl. 439 (7), 2131-2144, 2013.
Y. Sheng, Representations of hom-Lie algebras, Algebr. Represent. Theory 15 (6), 1081-1098, 2012.
S. Wang and S. Guo, BiHom-Lie superalgebra structures, arXiv:1610.02290.
J. Zhao, L. Chen and L. Ma, Representations and $T^*$-extensions of hom-Jordan-Lie algebras, Comm. Algebra 44 (7), 2786-2812, 2016.