Cohomology and Abelian Extensions of 3-Bihom-Lie Algebras

Abstract

In this paper, we give the cohomology of 3-Bihom-Lie algebras and we show that an α²β⁻¹-derivation is a closed 1-Bihom-cochain with the adjoint representation. As an application, we introduce abelian extensions of 3-Bihom-Lie algebras and obtain that there is a one-to-one correspondence between equivalent classes of abelian extensions and the second cohomology group by closed 2-Bihom-cochains. Moreover, we introduce the notion of a generalized derivation of 3-Bihom-Lie algebras and we construct a new 3-Bihom-Lie algebra with a generalized derivation.

Keywords

3-Bihom-Lie algebras / cohomology / abelian extensions / generalized derivations / 17A40 / 17B61 / 17B40 / 17B56

References

[1] Ammar F, Mabrouk S, Makhlouf A. Representations and cohomology of n-ary multiplicative Hom-Nambu–Lie algebras. J. Geom. Phys.. 2011, 61(10): 1898-1913.

[2] Ben Hassine A, Mabrouk S, Ncib O. 3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras. Linear Multilinear Algebra. 2022, 70(1): 101-121.

[3] Cheng Y, Qi H. Representations of Bihom-Lie algebras. Algebra Colloq.. 2022, 29(1): 125-142.

[4] Das A., Cohomology of BiHom-Associative algebras. J. Algebra Appl., 2022, 21(1): Paper No. 2250008, 22 pp.

[5] Graziani G., Makhlouf A., Menini C., Panaite F., BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras. SIGMA Symmetry Integrability Geom. Methods Appl., 2015, 11: Paper No. 086, 34 pp.

[6] Kitouni A, Makhlouf A, Silvestrov S. On n-ary generalization of BiHom-Lie algebras and BiHom-associative algebras. Algebraic Structures and Applications. 2020, Cham, Springer99126. 317

[7] Li J, Chen L. The construction of 3-Bihom-Lie algebras. Comm. Algebra. 2020, 48(12): 5374-5390.

[8] Li J, Chen L, Cheng Y. Representations of Bihom-Lie superalgebras. Linear Multilinear Algebra. 2019, 67(2): 299-326.

[9] Li J, Chen L, Sun B. Bihom-Nijienhuis operators and T*-extensions of Bihom-Lie superalgebras. Hacet. J. Math. Stat.. 2019, 48(3): 785-799

[10] Liu L, Makhlouf A, Menini C, Panaite F. BiHom-Novikov algebras and infinitesimal BiHom-bialgebras. J. Algebra. 2020, 560: 1146-1172.

[11] Liu J, Makhlouf A, Sheng Y. A new approach to representations of 3-Lie algebras and Abelian extensions. Algebr. Represent. Theory. 2017, 20(6): 1415-1431.

[12] Mabrouk S, Makhlouf A, Massoud S. Generalized representations of 3-Hom-Lie algebras. Extracta Math.. 2020, 35(1): 99-126.

[13] Song L, Tang R. Cohomologies, deformations and extensions of n-Hom-Lie algebras. J. Geom. Phys.. 2019, 141: 65-78.

[14] Wang S., Guo S., BiHom-Lie superalgebra structures and BiHom-Yang–Baxter equations. Adv. Appl. Clifford Algebr., 2020, 30(3): Paper No. 35, 18 pp.

[15] Xu S., Cohomology, derivations and abelian extensions of 3-Lie algebras. J. Algebra Appl., 2019, 18(7): Paper No. 1950130, 26 pp.

[16] Zhang T. Cohomology and deformations of 3-Lie colour algebras. Linear Multilinear Algebra. 2015, 63(4): 651-671.

RIGHTS & PERMISSIONS

Peking University