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 $[j]\ne [k]$, where the relation $\sim$ in ${\sum }^1=\{g\in G\setminus \{1\}:L_g\ne 0\}$, defined by $g\sim h$ if and only if $g$ is connected to $h$.

14pages. arXiv admin note: text overlap with arXiv:1411.6693

References in corpus (1)

• On the structure of graded Lie superalgebras