Elkadri ABDAOUI, Sonia MASSOUD, Sami MABROUK, Abdenacer MAKHLOUF

Yau-type ternary Hom-Lie bialgebras

The purpose of this paper is to introduce and study 3-Hom-Lie bialgebras, which are a ternary version of HomLie bialgebras introduced by Yau (2015). We provide their properties, some key constructions and their 3-dimensional classification. Moreover we discuss their representation theory and their generalized derivations and coderivations. Furthermore, a more generalized notion called generalized 3-Hom-Lie bialgebra is also considered.

PDF

___

[1] Abramov V, Kerner R, Le Roy B. Hypersymmetry: a Z3 -graded generalization of supersymmetry. Journal of Mathematical Physics 1997; 38: 1650-1669.
[2] Abramov V. 3-Lie superalgebras induced by Lie superalgebras. Axioms 2019; 8 (1): 21.
[3] Abramov V, Lätt P. Induced 3-Lie algebras, superalgebras and induced representations. Proceedings of the Estonian Academy of Sciences 2020; 69 (2): 116-133.
[4] Ammar F, Mabrouk S, Makhlouf A. Constuction of quadratic n-ary Hom-Nambu algebras. In: Makhlouf A, Paal E, Silvestrov S, Stolin A (editors). Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics & Statistics, Vol. 85. Berlin, Germany: Springer, 2014, pp. 201-232.
[5] Ammar F, Mabrouk S, Makhlouf A. Representations and cohomology of n-ary multiplicative Hom-Nambu-Lie algebras. Journal of Geometry and Physics 2011; 61 (10): 1898-1913.
[6] Arnlind J, Makhlouf A, Silvestrov S. Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras. Journal of Mathematical Physic 2011; 52: 123502.
[7] Ataguema H, Makhlouf A, Silvestrov S. Generalization of n-ary Nambu algebras and beyond. Journal of Mathematical Physic 2009; 50: 083501. doi: 10.1063/1.3167801
[8] Bagger J, Lambert N. Gauge symmetry and supersymmetry of multiple M2-branes. Physical Review 2008; D77: 065008.
[9] Basu A, Harvey JA. The M2-M5 brane system and a generalized Nahm’s equation. Nuclear Physics B 2005; 713: 136-150.
[10] Bai R, Cheng Y, Li J, Meng W. 3-Lie bialgebras. Acta Mathematica Scientia Series B 2014; 34: 513-522.
[11] Bai R, Bai C, Wang J. Realizations of 3-Lie algebras. Journal of Mathematical Physic 2010; 51: 063505.
[12] Bai R, Song G, Zhang Y. On classification of n-Lie algebras. Frontiers of Mathematics in China 2011; 6 (4): 581-606.
[13] Bai C, Guo L, Sheng Y. Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie algebras. Advances in Theoretical and Mathematical Physics 2019; 23 (1): 27-74.
[14] Bai R, Guo W, Lin L, Zhang Y. n-Lie bialgebras. Linear and Multilinear Algebra 2017. doi: 10.1080/03081087.2017.1299090
[15] Ben Abdeljelil A, Elhamdadi M, Kaygorodov I, Makhlouf A. Generalized Derivations of n-BiHom-Lie Algebras. In: Silvestrov S, Malyarenko A, Rančić M (editors). Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, Vol. 317. Cham, Switzerland: Springer, 2020, pp. 81-97. doi: 10.1007/978- 3-030-41850-2-4
[16] Benayadi S, Makhlouf A. Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. Journal of Geometry and Physics 2014; 76: 38-60.
[17] Beites PD, Kaygorodov I, Popov Y. Generalized derivations of multiplicative n-ary Hom-Ω color algebras. Bulletin of the Malaysian Mathematical Sciences Society 2019; 42 (1): 315-335.
[18] Cai LQ, Sheng Y. Purely Hom-Lie bialgebras. Science China Mathematics 2018; 61. doi: 10.1007/s11425-016-9102-y [19] Daletskii Y, Takhtajan L. Leibniz and Lie algebra structures for Nambu algebra. Letters in Mathematical Physics 1997; 39: 127-141.
[20] Desmedt V. Existence of a Lie bialgebra structure on every Lie algebra. Letters in Mathematical Physics 1994; 31: 225-231.
[21] Drinfeld VG. Hamiltonian Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equation. Soviet Mathematics - Doklady 1983; 27 (1): 68-71.
[22] Du C, Bai C, Guo L. 3-Lie bialgebras and 3-Lie classical Yang-Baxter equations in low dimensions. Journal of Linear and Multilinear Algebra 2018; 66 (8).
[23] Faddeev LD. Lectures on quantum inverse scattering method. In: Rasetti MG (editor). New Problems, Methods and Techniques in Quantum Field Theory and Statistical Mechanics. Series on Advances in Statistical Mechanics, Vol. 6. River Edge, NJ, USA: World Scientific Publishing Company, 1990, pp. 7-54.
[24] Filippov V. n-Lie algebras. Siberian Mathematical Journal 1985; 26: 126-140.
[25] Filippov V. δ−Derivations of Lie algebras. Siberian Mathematical Journal 1998; 39 (6): 1218-1230.
[26] Filippov V. δ -Derivations of prime Lie algebras. Siberian Mathematical Journal 199; 40: 174-184.
[27] Filippov V. δ -Derivations of prime alternative and Malcev algebras. Algebra and Logic 2000; 39: 354-358.
[28] Gustavsson A. Algebraic structures on parallel M2-branes. Nuclear Physics 2009; B811: 66-76.
[29] Guo S, Zhang X, Wang S. The constructions of 3-Hom-Lie bialgebras. arXiv 2019; arXiv:1902.03917.
[30] Ho P, Chebotar M, Ke W. On skew-symmetric maps on Lie algebras. Proceedings of the royal society Edinburgh A 2003; 113: 1273-1281.
[31] Kaygorodov I, Popov Y. Generalized derivations of (color) n-ary algebras. Linear Multilinear Algebra 2016; 64 (6): 1086-1106.
[32] Kasymov S. On a theory of n-Lie algebras. Algebra and Logika 1987; 26: 277-297. [33] Kaygorodov I, Popov Yu. Alternative algebras admitting derivations with invertible values and invertible derivations. Izvestiya: Mathematics 2014; 78: 922-935.
[34] Leger G, Luks E. Generalized derivations of Lie algebras. Journal of Algebra 2000; 228: 165-203.
[35] Liu Y, Chen L, Ma Y. Representations and module-extensions of 3-hom-Lie algebras. Journal of Geometry and Physics 2015; 98: 376-383.
[36] Ling W. On the structure of n-Lie algebras. PhD dissertation, University-GHS-Siegen, Siegen, Germany, 1993.
[37] Makhlouf A, Silvestrov S. Hom-algebra structures. Journal of Generalized Lie Theory and Applications 2008; 2 (2): 51-64.
[38] Michaelis W. Lie coalgebra. Advances in Mathematics 1980; 38: 1-54.
[39] Nambu Y. Generalized Hamiltonian dynamics. Physical Review D 1973; 7 (3): 2405-2412.
[40] Pojidaev A, Saraiva P. On derivations of the ternary Malcev algebra M8. Communications in Algebra 2006; 34: 3593-3608.
[41] Rezaei-Aghdam A, Sedghi-Ghadim L. 3-Leibniz bialgebras (3-Lie bialgebras). International Journal of Geometric Methods in Modern Physics 2017; 14 (10): 1750142.
[42] Semenov-Tian-Shansky MA. Poisson-Lie groups. The quantum duality principle and the twisted quantum double. Teoreticheskaya i Matematicheskaya Fizika 1992; 93 (2): 302-329.
[43] Sheng Y, Bai C. A new approach to hom-Lie bialgebras. Journal of Algebra 2014; 399: 232-250.
[44] Sklyanin EK. Quantum variant of the method of the inverse scattering problem. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR (LOMI) 1980; 95: 55-128 (in Russian).
[45] Takhtajan LA. Higher order analog of Chevalley-Eilenberg complex and deformation theory of n-algebras. St. Petersburg Mathematical Journal 1995; 6 (2): 429-438.
[46] Yau D. The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras. International Electronic Journal of Algebra 2015; 17: 11-45.
[47] Walter M. Lie Coalgebras. Advances in Mathematics 1980; 38: 1-54.
[48] Williams MP. Nilpotent n-Lie algebras. Communications in Algebra 2009; 37: 1843-1849.