Cohomology and formal deformations of $n$-Hom–Lie color algebras
Abstract
UDC 512.5
The aim of this paper is to provide a cohomology of $n$-Hom–Lie color algebras, in particular, a cohomology governing one-parameter formal deformations. Then we also study formal deformations of the $n$-Hom–Lie color algebras and introduce the notion of Nijenhuis operator on a $n$-Hom–Lie color algebra, which may give rise to infinitesimally trivial $(n-1)$-order deformations. Furthermore, in connection with Nijenhuis operators, we introduce and discuss the notion of product structure on $n$-Hom–Lie color algebras.
References
E. Abdaoui, S. Mabrouk, A. Makhlouf, Cohomology of Hom–Leibniz and $n$-ary Hom–Nambu–Lie superalgebras}; arXiv: 1406.3776 (2014).
F. Ammar, I. Ayadi, S. Mabrouk, A. Makhlouf, Quadratic color Hom–Lie algebras, Moroccan Andalusian Meeting on Algebras and their Applications, Springer, Cham (2018), p. 287–312.
F. Ammar, S. Mabrouk, A. Makhlouf, Representations and cohomology of $n$-ary multiplicative Hom–Nambu–Lie algebras, J. Geom. and Phys., 61, № 10, 1898–1913 (2011).
F. Ammar, N. Saadaoui, Cohomology of $n$-ary-Nambu–Lie superalgebras and super $omega_infty$ 3-algebra}; arXiv:1304.5767 (2013).
J. Arnlind, A. Makhlouf, S. Silvestrov, Ternary Hom–Nambu–Lie algebras induced by Hom–Lie algebras, J. Math. Phys., 51, № 4, Article 043515 (2010).
J. Arnlind, A. Makhlouf, S. Silvestrov, Construction of $n$-Lie algebras and $n$-ary Hom–Nambu–Lie algebras}, J. Math. Phys., 52, № 12, Article 123502 (2011).
J. Arnlind, A. Kitouni, A. Makhlouf, S. Silvestrov, Structure and cohomology of $3$-Lie algebras induced by Lie algebras, Algebra, Geometry and Mathematical Physics, Springer Proc. Math. and Stat., 85, (2014).
A. Armakan, S. Silvestrov, M. Farhangdoost, Enveloping algebras of color Hom–Lie algebras, Turkish J. Math., 43, 316–339 (2019).
H. Ataguema, A. Makhlouf, S. Silvestrov, Generalization of $n$-ary Nambu algebras and beyond, J. Math. Phys., 50, № 8, Article 083501 (2009).
I. Bakayoko, S. Silvestrov, Multiplicative $n$-Hom–Lie color algebras, International Conference on Stochastic Processes and Algebraic Structures, 22, 159–187 (2017).
P. D. Beites, I. Kaygorodov, Y. Popov, Generalized derivations of multiplicative $n$-ary Hom-w color algebras, Bull. Malays. Math. Sci. Soc., 42, 315–335 (2019).
J. Bergen, D. S. Passman, Delta ideal of Lie color algebras, J. Algebra, 177, 740–754 (1995).
J. M. Casas, J.-L. Loday, T. Pirashvili, Leibniz $n$-algebras, Forum Math., 14, 189–207 (2002).
Y. L. Daletskii, L. A. Takhtajan, Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys., 39, 127–141 (1997).
V. T. Filippov, $n$-Lie algebras, Sib. Math. J., 26, 879–891 (1985) (Transl. from Russian: Sib. Mat. Zh., 26, 126–140 (1985)).
J. Feldvoss, Representations of Lie color algebras, Adv. Math., 157, 95–137 (2001).
P. Gautheron, Some remarks concerning Nambu mechanics, Lett. Math. Phys., 37, 103–116 (1996).
Sh. M. Kasymov, Theory of $n$-Lie algebras, Algebra and Logic, 26, 155–166 (1987) (Transl. from Russian: Algebra i Logika, 26, № 3, 277–297 (1987)).
I. Kaygorodov, Y. Popov, Generalized derivations of (color) $n$-ary algebras, Linear and Multilinear Algebra, 64, 1086–1106 (2016).
J. Liu, Y. Sheng, Y. Zhou, C. Bai, Nijenhuis operators on $n$-Lie algebras, Commun. Theor. Phys., 65, № 6, 659–670 (2016).
S. Montgomery, Constructing simple Lie superalgebras from associative graded algebras, J. Algebra, 195, 558–579 (1997).
R. Ree, Generalized Lie elements, Can. J. Math., 12, 493–502 (1960).
M. Rotkiewicz, Cohomology ring of $n$-Lie algebras, Extracta Math., 20, 219–232 (2005).
Y. Sheng, R. Tang, Symplectic, product and complex structures on 3-Lie algebras, J. Algebra, 508, 256–300 (2018).
M. Scheunert, Generalized Lie algebras, J. Math. Phys., 20, № 4, 712–720 (1979).
Y. Su, K. Zhao, L. Zhu, Classification of derivation-simple color algebras related to locally finite derivations, J. Math. Phys., 45, 525–536 (2004).
Y. Su, K. Zhao, L. Zhu, Simple color algebras of Weyl type, Israel J. Math., 137, 109–123 (2003).
L. A. Takhtajan, On foundation of the generalized Nambu mechanics, Commun. Math. Phys., 160, № 2, 295–315 (1994).
L. A. Takhtajan, Higher order analog of Chevalley-Eilenberg complex and deformation theory of $n$-algebras, St.,Petersburg Math. J., 6, № 2, 429–438 (1995).
T. Zhang, Cohomology and deformations of 3-Lie colour algebras, Linear and Multilinear Algebra, 63, № 4, 651–671 (2015).
M. C. Wilson, Delta methods in enveloping algebras of Lie color algebras, J. Algebra, 75, 661–696 (1995).
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