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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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