Vainerman, L.; Kerner, R. On special classes of \(n\)-algebras. (English) Zbl 0864.17002 J. Math. Phys. 37, No. 5, 2553-2565 (1996). The notation of \(n\)-ary algebras as a linear space is given by introducing a composition law which involves \(n\) elements: \(m:{\mathcal V}^{\otimes n}\to{\mathcal V}\). A structure theory of this algebra is developed to a large extent, establishing properties as: simple, semisimple, Abelian, nilpotent, solvable. Some detailed examples are given for the case \(\dim {\mathcal V}= 2\), \(n=3\). The relevance of \(n\)-ary algebras to physics is discussed, as their relation to Nambu-mechanics, Nambu-Lie algebras and Lie triple systems. Reviewer: B.Fauser (Tübingen) Cited in 45 Documents MSC: 17A42 Other \(n\)-ary compositions \((n \ge 3)\) 17B30 Solvable, nilpotent (super)algebras 81R05 Finite-dimensional groups and algebras motivated by physics and their representations Keywords:\(\mathbb{Z}_ 3\)-graded algebras; \(n\)-ary algebras; Nambu-Lie algebras; Lie triple systems PDFBibTeX XMLCite \textit{L. Vainerman} and \textit{R. Kerner}, J. Math. Phys. 37, No. 5, 2553--2565 (1996; Zbl 0864.17002) Full Text: DOI References: [1] DOI: 10.1103/PhysRevD.7.2405 · Zbl 1027.70503 · doi:10.1103/PhysRevD.7.2405 [2] DOI: 10.1007/BF02103278 · Zbl 0808.70015 · doi:10.1007/BF02103278 [3] Takhtajan L., Alg. Anal. 6 pp 262– (1994) [4] Kerner R., Comptes Rend. Acad. Sci. Paris Ser. II 312 pp 191– (1991) [5] DOI: 10.1063/1.529922 · doi:10.1063/1.529922 [6] DOI: 10.1090/S0002-9947-1951-0041118-9 · doi:10.1090/S0002-9947-1951-0041118-9 [7] DOI: 10.1016/0021-8693(80)90189-1 · Zbl 0425.17007 · doi:10.1016/0021-8693(80)90189-1 [8] DOI: 10.1090/S0002-9947-1952-0045702-9 · doi:10.1090/S0002-9947-1952-0045702-9 [9] DOI: 10.1112/plms/s3-6.3.366 · Zbl 0073.01704 · doi:10.1112/plms/s3-6.3.366 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.