On the structure of graded Leibniz triple systems

Abstract

We study the structure of a Leibniz triple system $\mathcal{E}$ graded by an arbitrary abelian group $G$ which is considered of arbitrary dimension and over an arbitrary base field $\mathbb{K}$. We show that $\mathcal{E}$ is of the form $\mathcal{E}=U+\sum_{[j]\in \sum^{1}/\sim} I_{[j]}$ with $U$ a linear subspace of the 1-homogeneous component $\mathcal{E}_{1}$ and any ideal $I_{[j]}$ of $\mathcal{E}$, satisfying $\{I_{[j]},\mathcal{E},I_{[k]}\} =\{I_{[j]},I_{[k]},\mathcal{E}\}=\{\mathcal{E},I_{[j]},I_{[k]}\}=0$ if $[j]\neq [k]$, where the relation $\sim$ in $\sum^{1}=\{g \in G \setminus \{1\} : L_{g}\neq 0\}$, defined by $g \sim h$ if and only if $g$ is connected to $h$.