Leibniz triple systems admitting a multiplicative basis

Abstract

Let $(T,⟨\cdot ,\cdot ,\cdot ⟩)$ 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 $⟨e_i,e_j,e_k⟩\in \mathbb{F}{e}_r$ for some $r\in I$. We show that if $T$ admits a multiplicative basis then it decomposes as the orthogonal direct sum $T={⨁}_k{\mathfrak{I}}_k$ 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.