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On special classes of \(n\)-algebras. (English) Zbl 0864.17002
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.

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
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[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
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