Some structure theories of Leibniz triple systems

TL;DR

This paper explores the structural properties of Leibniz triple systems and their envelopes, establishing key relationships and extending classical theorems like Levi's theorem to this algebraic context.

Contribution

It introduces the involutive automorphism for Leibniz triple systems, characterizes their $ ext{Z}_2$-grading, and extends fundamental structural theorems such as Levi's theorem.

Findings

Relationship between solvable radicals of T and U(T) established

Levi's theorem extended to Leibniz triple systems

Introduced the notion of representations for Leibniz triple systems

Abstract

In this paper, we investigate the Leibniz triple system $T$ and its universal Leibniz envelope $U(T)$. The involutive automorphism of $U(T)$ determining $T$ is introduced, which gives a characterization of the $\Z_2$-grading of $U(T)$. We give the relationship between the solvable radical $R(T)$ of $T$ and $Rad(U(T))$, the solvable radical of $U(T)$. Further, Levi's theorem for Leibniz triple systems is obtained. Moreover, the relationship between the nilpotent radical of $T$ and that of $U(T)$ is studied. Finally, we introduce the notion of representations of a Leibniz triple system.