On the structure of graded Leibniz triple systems

TL;DR

This paper investigates the structure of graded Leibniz triple systems over arbitrary fields and groups, revealing a decomposition into subspaces and ideals with specific orthogonality properties.

Contribution

It provides a detailed structural decomposition of graded Leibniz triple systems, generalizing previous results to arbitrary dimensions and base fields.

Findings

Decomposition of Leibniz triple systems into subspaces and ideals.

Identification of orthogonality conditions among ideals.

Generalization to arbitrary abelian group gradings.

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 $h$.