Leibniz triple systems admitting a multiplicative basis

TL;DR

This paper studies Leibniz triple systems with a special basis called multiplicative basis, showing they decompose into orthogonal sums of ideals with similar bases and characterizing minimal systems.

Contribution

It introduces the concept of multiplicative bases in Leibniz triple systems and describes their structural decomposition into minimal ideals with such bases.

Findings

Decomposition of Leibniz triple systems with multiplicative bases into orthogonal sums of ideals.

Characterization of minimal Leibniz triple systems via multiplicative bases.

Under mild conditions, the decomposition aligns with minimal ideals.

Abstract

Let $(T,\langle \cdot, \cdot, \cdot \rangle)$ be a Leibniz triple system of arbitrary dimension, over an arbitrary base field ${\mathbb F}$. A basis ${\mathcal B} = \{e_{i}\}_{i \in I}$ of $T$ is called multiplicative if for any $i,j,k \in I$ we have that $\langle e_i,e_j,e_k\rangle\in {\mathbb F}e_r$ for some $r \in I$. We show that if $T$ admits a multiplicative basis then it decomposes as the orthogonal direct sum $T= \bigoplus_k{\mathfrak I}_k$ of well-described ideals ${\mathfrak I}_k$ admitting each one a multiplicative basis. Also the minimality of $T$ is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by means of the family of its minimal ideals.