Triple Derivations and Triple Homomorphisms of Perfect Lie Superalgebras

Abstract

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 $Der(L)$ is an inner derivation. Let $L,L^{{}^{\prime }}$ be Lie superalgebras over a commutative ring $R$, the notion of triple homomorphism from $L$ to $L^{{}^{\prime }}$ is introduced. We proved that, under certain assumptions, homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms.

12pages in Indagationes Mathematicae, 2016