Leibniz triple systems admitting a multiplicative basis

Abstract

Let T be an arbitrary Leibniz triple system over an arbitrary field of scalars $\mathbb{F}$. A basis ${\left\{e_i\right\}}_{i\in I}$ of T is multiplicative if for $i,j,k\in I$ we have $〈e_i,e_j,e_k〉\in \mathbb{F}{e}_r$ for some $r\in I$. We show that if T admits a multiplicative basis then it decomposes as the orthogonal direct sum 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 under new conditions we prove that the above direct sum is by means of its minimal ideals.

Keywords:

• Infinite dimensional triple system
• Leibniz triple system
• multiplicative basis
• non-associative triple system
• structure theory

2010 MSC:

• 17A32
• 17A40
• 17A60

Acknowledgment

We would like to thank the referee for the detailed reading of this work and for the suggestions which have improved the final version of the same.