Korean J. Math. Vol. 29 No. 2 (2021) pp.271-291
DOI: https://doi.org/10.11568/kjm.2021.29.2.271

Cohomology and deformations of Hom-Lie-Yamaguti color algebras

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Published: Jun 30, 2021
Keywords:
Hom-algebra, Color algebra, Representation, Deformation
Mathematics Subject Classification:
17D99, 17B61, 17D15

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A. Nourou Issa
Département de Mathématiques, Université d'Abomey-Calavi, Cotonou, BENIN

Abstract

Hom-Lie-Yamaguti color algebras are defined and their representation and cohomology theory is considered. The $(2,3)$-cocycles of a given Hom-Lie-Yamaguti color algebra $T$ are shown to be very useful in a study of its deformations. In particular, it is shown that any $(2,3)$-cocycle of $T$ gives rise to a Hom-Lie-Yamaguti color structure on $T \oplus V$, where $V$ is a $T$-module, and that a one-parameter infinitesimal deformation of $T$ is equivalent to that a $(2,3)$-cocycle of $T$ (with coefficients in the adjoint representation) defines a Hom-Lie-Yamaguti color algebra of deformation type.


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References

[1] K. Abdaoui, F. Ammar, and A. Makhlouf, Constructions and cohomology of Hom-Lie color algebras, Comm. Algebra 43 (11) (2015), 4581–4612. Google Scholar

[2] F. Ammar, I. Ayadi, S. Mabrouk, and A. Makhlouf, Quadratic color Hom-Lie algebras, in: Associative and Nonassociative Algebras and Applications, Springer Proceedings in Mathematics and Statistics book series, vol. 311 (2020), pp. 287–312. Google Scholar

[3] F. Ammar, Z. Ejbehi, and A. Makhlouf, Cohomology and deformations of Hom-algebras, J. Lie Theory 21 (4) (2011), 813–836. Google Scholar

[4] F. Ammar, N. Saadaoui, and A. Makhlouf, Cohomology of Hom-Lie superalgebras and q- deformed Witt superalgebra, Czechoslovak Math. J. 63 (3) (2013), 721–761. Google Scholar

[5] A. S. Dzhumadil’daev, Cohomology of colour Leibniz algebras: pre-simplicial approach, In: Lie Theory and its Applications in Physics III. Proceedings of the Third International Workshop, World Scientific, Singapore, 2000, pp. 124–136. Google Scholar

[6] D. Gaparayi, S. Attan, and A. N. Issa, Hom-Lie-Yamaguti superalgebras, Korean J. Math. 27 (1) (2019), 175–192. Google Scholar

[7] D. Gaparayi and A. N. Issa, A twisted generalization of Lie-Yamaguti algebras, Int. J. Algebra 6 (5-8) (2012), 339–352. Google Scholar

[8] D. Gaparayi and A. N. Issa, Hom-Akivis superalgebras, J. Algebra Comput. Appl. 6 (1) (2017), 36–51. Google Scholar

[9] S. Guo, X. Zhang, and S. Wang, Representations and deformations of Hom-Lie-Yamaguti superalgebras, Adv. Math. Phys. 2020, art. ID 9876738, 12 pages. Google Scholar

[10] J. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2) (2006), 314–361. Google Scholar

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

[12] A. N. Issa and P. L. Zoungrana, On Lie-Yamaguti color algebras, Mat. Vesnik 71 (3) (2019), 196–206. Google Scholar

[13] Z. Jia and Q. Zhang, Nilpotent ideals of Lie color triple systems, J. Jilin Univ. (Science Edition) 49 (4) (2011), 674–678. Google Scholar

[14] M. Kikkawa, Geometry of homogeneous Lie loops, Hiroshima Math. J. 5 (1975), 141–179. Google Scholar

[15] M. K. Kinyon and A. Weinstein, Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math. 123 (2001), 525–550. Google Scholar

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

[17] Y. Ma, L. Y. Chen, and J. Lin, One-parameter formal deformations of Hom-Lie-Yamaguti algebras, J. Math. Phys. 56 (2015), art. 011701. Google Scholar

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

[19] M. F. Ou edraogo, Sur les superalg`ebres triples de Lie [Th`ese de Doctorat de 3e Cycle], Universit e de Ouagadougou, Burkina-Faso, 1999. Google Scholar

[20] R. Ree, Generalized Lie elements, Canadian J. Math. 12 (1960), 493–502. Google Scholar

[21] V. Rittenberg and D. Wyler, Generalized superalgebras, Nuclear Phys. B 139 (1978), no. 3, 189–202. Google Scholar

[22] M. Scheunert, Generalized Lie algebras, J. Math. Phys. 20 (4) (1979), 712–720. Google Scholar

[23] Y. Sheng, Representations of Hom-Lie algebras, Algebra Repr. Theory 15 (2012), no. 6, 1081– 1098. Google Scholar

[24] S. D. Silvestrov, On the classification of 3-dimensional coloured Lie algebras, In: Quantum Groups and Quantum Spaces, Banach Center Publications, Warszawa 40 (1997), 159–170. Google Scholar

[25] K. Yamaguti, On the Lie triple system and its generalization, J. Sci. Hiroshima Univ. ser. A 21 (1957-1958), 155–160. Google Scholar

[26] K. Yamaguti, On cohomology groups of general Lie triple systems, Kumamoto J. Sci. ser. A 8 (3) (1969), 135–146. Google Scholar

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

[28] L. Yuan, Hom-Lie colour algebra structures, Comm. Alg. 40 (2) (2012), 571–592. Google Scholar

[29] T. Zhang, Cohomology and deformations of 3-Lie colour algebras, Linear Mult. Alg. 63 (2015), 651–671. Google Scholar

[30] T. Zhang and J. Li, Deformations and extensions of Lie-Yamaguti algebras, Linear Mult. Alg. 63 (11) (2015), 2212–2231. Google Scholar

[31] T. Zhang and J. Li, Representations and cohomologies of Hom-Lie-Yamaguti algebras with applications, Colloq. Math. 148 (1) (2017), 131–155. Google Scholar