Leibniz triple systems admitting a multiplicative basis

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.