Triple derivations and triple homomorphisms of perfect Lie superalgebras

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 12, 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 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.