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Triple derivations and triple homomorphisms of perfect Lie superalgebras. (English) Zbl 1420.17019

Summary: In this paper, we study triple derivations and triple homomorphisms of perfect Lie superalgebras over a commutative ring \(R\). It is proved that, if the base ring contains \(\frac{1}{2}\), \(L\) is a perfect Lie superalgebra with zero center, then every triple derivation of \(L\) is a derivation, and every triple derivation of the derivation algebra \(\operatorname{Der}(L)\) is an inner derivation. Let \(L\), \(L'\) be Lie superalgebras over a commutative ring \(R\), the notion of triple homomorphism from \(L\) to \(L'\) is introduced. We prove that, under certain assumptions, homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms.

MSC:

17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
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