Generalized Derivations of Lie triple systems

Abstract

In this paper, we present some basic properties concerning the derivation algebra $\mathrm{Der}(T)$, the quasiderivation algebra $\mathrm{QDer}(T)$ and the generalized derivation algebra $\mathrm{GDer}(T)$ of a Lie triple system $T$, with the relationship $\mathrm{Der}(T)\subseteq \mathrm{QDer}(T)\subseteq \mathrm{GDer}(T)\subseteq \mathrm{End}(T)$. Furthermore, we completely determine those Lie triple systems $T$ with condition $\mathrm{QDer}(T)=\mathrm{End}(T)$. We also show that the quasiderivations of $T$ can be embedded as derivations in a larger Lie triple system.