On split Leibniz triple systems

Abstract

In order to study the structure of arbitrary split Leibniz triple systems, we introduce the class of split Leibniz triple systems as the natural extension of the class of split Lie triple systems and split Leibniz algebras. By developing techniques of connections of roots for this kind of triple systems, we show that any of such Leibniz triple systems $T$ with a symmetric root system is of the form $T=U+\sum_{[j]\in \Lambda^{1}/\sim} I_{[j]}$ with $U$ a subspace of $T_{0}$ and any $I_{[j]}$ a well described ideal of $T$, satisfying $\{I_{[j]},T,I_{[k]}\} =\{I_{[j]},I_{[k]},T\}=\{T,I_{[j]},I_{[k]}\}=0$ if $[j]\neq [k]$.