On Split $\delta$-Jordan Lie Triple Systems

Abstract:

      The aim of this article is to study the structures of arbitrary split $\delta$-Jordan Lie triple systems, which are a generalization of split Lie triple systems. By developing techniques of connections of roots for this kind of triple systems, we show that any of such $\delta$-Jordan Lie triple systems $T$ with a symmetric root system is of the form $T=U+\sum_{[\alpha]\in \Lambda^{1}/\sim} I_{[\alpha]}$ with $U$ a subspace of $T_{0}$ and any $I_{[\alpha]}$ a well described ideal of $T$, satisfying $\{I_{[\alpha]},T,I_{[\beta]}\}=\{I_{[\alpha]},I_{[\beta]},T\}=\{T,I_{[\alpha]},I_{[\beta]}\}=0$ if $[\alpha]\neq [\beta]$.