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

A. Nourou Issa

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.

Keywords


Hom-algebra, Color algebra, Representation, Deformation

Subject classification

17D99, 17B61, 17D15

Sponsor(s)



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References


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

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)

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

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)

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)

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

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

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

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)

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

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

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

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

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

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)

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)

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)

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

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)

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

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

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

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

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)

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

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

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

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

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

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

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


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