Triple Derivations and Triple Homomorphisms of Perfect Lie Superalgebras

TL;DR

This paper investigates the structure of triple derivations and triple homomorphisms in perfect Lie superalgebras, establishing conditions under which these maps are derivations or inner derivations, and characterizing triple homomorphisms.

Contribution

It introduces the concept of triple homomorphisms for Lie superalgebras and proves their properties, extending the understanding of derivations and homomorphisms in this algebraic context.

Findings

Triple derivations of perfect Lie superalgebras with zero center are derivations.

Triple derivations of the derivation algebra are inner derivations.

Homomorphisms, anti-homomorphisms, and their sums are all triple homomorphisms under certain conditions.

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 $L,~L^{'}$ be Lie superalgebras over a commutative ring $R$, the notion of triple homomorphism from $L$ to $L^{'}$ is introduced. We proved that, under certain assumptions, homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms.