Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

COHOMOLOGY AND DEFORMATIONS OF HOM-LIE-YAMAGUTI COLOR ALGEBRAS

  • Issa, A. Nourou (Departement de Mathematiques, Universite d'Abomey-Calavi)
  • Received : 2020.12.05
  • Accepted : 2021.04.30
  • Published : 2021.06.30
Download PDF
⟨ Previous Next ⟩

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⊕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

Acknowledgement

The author would like to thank the anonymous referee for the careful reading of the paper and for valuable suggestions that helped to improve the presentation of this paper.

References

  1. K. Abdaoui, F. Ammar, and A. Makhlouf, Constructions and cohomology of Hom-Lie color algebras, Comm. Algebra 43 (11) (2015), 4581-4612. https://doi.org/10.1080/00927872.2014.910797
  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. https://doi.org/10.1007/978-3-030-35256-1_16
  3. F. Ammar, Z. Ejbehi, and A. Makhlouf, Cohomology and deformations of Hom-algebras, J. Lie Theory 21 (4) (2011), 813-836.
  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. https://doi.org/10.1007/s10587-013-0049-6
  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.
  6. D. Gaparayi, S. Attan, and A. N. Issa, Hom-Lie-Yamaguti superalgebras, Korean J. Math. 27 (1) (2019), 175-192. https://doi.org/10.11568/KJM.2019.27.1.175
  7. D. Gaparayi and A. N. Issa, A twisted generalization of Lie-Yamaguti algebras, Int. J. Algebra 6 (5-8) (2012), 339-352.
  8. D. Gaparayi and A. N. Issa, Hom-Akivis superalgebras, J. Algebra Comput. Appl. 6 (1) (2017), 36-51.
  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.
  10. J. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2) (2006), 314-361. https://doi.org/10.1016/j.jalgebra.2005.07.036
  11. A. N. Issa, Hom-Akivis algebras, Comment. Math. Univ. Carolinae 52 (4) (2011), 485-500.
  12. A. N. Issa and P. L. Zoungrana, On Lie-Yamaguti color algebras, Mat. Vesnik 71 (3) (2019), 196-206.
  13. Z. Jia and Q. Zhang, Nilpotent ideals of Lie color triple systems, J. Jilin Univ. (Science Edition) 49 (4) (2011), 674-678.
  14. M. Kikkawa, Geometry of homogeneous Lie loops, Hiroshima Math. J. 5 (1975), 141-179. https://doi.org/10.32917/hmj/1206136626
  15. M. K. Kinyon and A. Weinstein, Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math. 123 (2001), 525-550. https://doi.org/10.1353/ajm.2001.0017
  16. D. Larsson and S. D. Silvestrov, Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra 288 (2005), 321-344. https://doi.org/10.1016/j.jalgebra.2005.02.032
  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.
  18. A. Makhlouf and S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), 51-64. https://doi.org/10.4303/jglta/S070206
  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.
  20. R. Ree, Generalized Lie elements, Canadian J. Math. 12 (1960), 493-502. https://doi.org/10.4153/CJM-1960-044-x
  21. V. Rittenberg and D. Wyler, Generalized superalgebras, Nuclear Phys. B 139 (1978), no. 3, 189-202. https://doi.org/10.1016/0550-3213(78)90186-4
  22. M. Scheunert, Generalized Lie algebras, J. Math. Phys. 20 (4) (1979), 712-720. https://doi.org/10.1063/1.524113
  23. Y. Sheng, Representations of Hom-Lie algebras, Algebra Repr. Theory 15 (2012), no. 6, 1081-1098. https://doi.org/10.1007/s10468-011-9280-8
  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.
  25. K. Yamaguti, On the Lie triple system and its generalization, J. Sci. Hiroshima Univ. ser. A 21 (1957-1958), 155-160.
  26. K. Yamaguti, On cohomology groups of general Lie triple systems, Kumamoto J. Sci. ser. A 8 (3) (1969), 135-146.
  27. D. Yau, Hom-algebras and homology, J. Lie Theory 19 (2009), 409-421.
  28. L. Yuan, Hom-Lie colour algebra structures, Comm. Alg. 40 (2) (2012), 571-592. https://doi.org/10.1080/00927872.2010.533726
  29. T. Zhang, Cohomology and deformations of 3-Lie colour algebras, Linear Mult. Alg. 63 (2015), 651-671. https://doi.org/10.1080/03081087.2014.891589
  30. T. Zhang and J. Li, Deformations and extensions of Lie-Yamaguti algebras, Linear Mult. Alg. 63 (11) (2015), 2212-2231. https://doi.org/10.1080/03081087.2014.1000815
  31. T. Zhang and J. Li, Representations and cohomologies of Hom-Lie-Yamaguti algebras with applications, Colloq. Math. 148 (1) (2017), 131-155. https://doi.org/10.4064/cm6903-6-2016