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Keywords:
nilpotent $n$-Lie superalgebra; Engel's theorem; $S^{\ast }$ algebra; Frattini subalgebra
nilpotent $n$-Lie superalgebra; Engel's theorem; $S^{\ast }$ algebra; Frattini subalgebra
Summary:
The paper studies nilpotent $n$-Lie superalgebras over a field of characteristic zero. More specifically speaking, we prove Engel's theorem for $n$-Lie superalgebras which is a generalization of those for $n$-Lie algebras and Lie superalgebras. In addition, as an application of Engel's theorem, we give some properties of nilpotent $n$-Lie superalgebras and obtain several sufficient conditions for an $n$-Lie superalgebra to be nilpotent by using the notions of the maximal subalgebra, the weak ideal and the Jacobson radical.
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