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3-Lie algebras with an ideal \(N\). (English) Zbl 1178.17004

Summary: We define the hypo-nilpotent ideal in \(n\)-Lie algebras and obtain all solvable 3-Lie algebras with an \(m\)-dimensional simplest filiform 3-Lie algebra as a maximal hypo-nilpotent ideal. We prove that the dimension of such solvable 3-Lie algebras is at most \(m+2\), and there is no solvable 3-Lie algebra with the simplest filiform 3-Lie algebra as the nilradical.

MSC:

17A42 Other \(n\)-ary compositions \((n \ge 3)\)
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17B30 Solvable, nilpotent (super)algebras
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[1] Nambu, Y., Generalized Hamiltonian dynamics, Phys. rev. D, 7, 2405-2412, (1973) · Zbl 1027.70503
[2] A. Vinogradov, M. Vinogradov, On multiple generalizations of lie algebras and poisson manifolds, American Mathematical Society, Contemp. Math., vol. 219, 1998, pp. 273-287. · Zbl 1074.17501
[3] Marmo, G.; Vilasi, G.; Vinogradov, A., The local structure of \(n\)-Poisson and \(n\)-Jacobi manifolds, J. geom. phys., 25, 141-182, (1998) · Zbl 0978.53126
[4] Takhtajan, L., On foundation of the generalized Nambu mechanics, Commun. math. phys., 160, 295-315, (1994) · Zbl 0808.70015
[5] Gautheron, P., Some remarks concerning Nambu mechanics, Lett. math. phys., 37, 103-116, (1996) · Zbl 0849.70014
[6] Panov, A., Multiple Poisson brackets, Vestn. samar. GoS. univ. mat. mekh. fiz. khim. biol., 2, 33-42, (1996), (in Russian) · Zbl 0986.37048
[7] Filippov, V., \(n\)-Lie algebras, Sibirsk. mat. zh., 26, 6, 126-140, (1985) · Zbl 0585.17002
[8] Kasymov, S., On a theory of \(n\)-Lie algebras, Algebra log., 26, 3, 277-297, (1987) · Zbl 0647.17001
[9] Bai, R.; Chen, L.; Meng, D., The Frattini subalgebra of \(n\)-Lie algebras, Acta math. sin. (engl. ser.), 23, 5, 847-856, (2007) · Zbl 1152.17004
[10] Bai, R.; Meng, D., The strong semi-simple \(n\)-Lie algebras, Comm. algebra, 31, 11, 5331-5341, (2003) · Zbl 1038.17003
[11] Bai, R.; An, H.; Li, Z., Centroid structures of \(n\)-Lie algebras II, Linear algebra appl., 430, 229-240, (2009) · Zbl 1221.17004
[12] W. Ling, On the structure of \(n\)-Lie algebras, Dissertation, University-GHS-Siegen, Siegn, 1993. · Zbl 0841.17002
[13] Pozhidaev, A.P., Simple factor algebras and subalgebras of Jacobians, Sibirsk. math. J., 39, 3, 512-517, (1998) · Zbl 0936.17006
[14] Pozhidaev, A.P., Monomial n-Lie algebras, Algebra log., 37, 5, 307-322, (1998) · Zbl 0936.17007
[15] Pozhidaev, A.P., On simple n-Lie algebras, Algebra log., 38, 3, 181-192, (1999) · Zbl 0930.17004
[16] D. Barnes, On \((n + 2)\) dimensional \(n\)-Lie algebras, arXiv: 0704.1892.
[17] Bai, R.; Wang, X.; Xiao, W.; An, H., Structure of low dimensional \(n\)-Lie algebras over a field of characteristic 2, Linear algebra appl., 428, 1912-1920, (2008) · Zbl 1205.17007
[18] M. Williams, Nilpotent \(n\)-Lie algebras, Comm. Algebra, Dissertation, North Carolina State University, in press. · Zbl 1250.17003
[19] Kasymov, S., Solvability in representations of \(n\)-Lie algebras, Sibirsk. mat. zh., 39, 2, 289-291, (1998)
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