On Split $\delta$-Jordan Lie Triple Systems
Received:January 23, 2019  Revised:September 04, 2019
Key Word: split $\delta$-Jordan Lie triple system   Lie triple system   root system
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11801121), the Natural Science Foundation of Heilongjiang Province (Grant No.QC2018006) and the Fundamental Research Fundation for Universities of Heilongjiang Province (Grant No.LGYC2018JC002).
 Author Name Affiliation Yan CAO Department of Mathematics, Harbin University of Science and Technology, Heilongjiang 150080, P. R. China Liangyun CHEN School of Mathematics and Statistics, Northeast Normal University, Jilin 130024, P. R. China
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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 $[\alpha]\neq [\beta]$.