Leibniz triple systems

TL;DR

This paper introduces Leibniz triple systems as a natural extension of Lie triple systems within dialgebra theory, providing their defining identities, universal envelopes, and concrete 2D examples.

Contribution

It defines Leibniz triple systems using functorial methods, establishes their relation to Leibniz and Jordan dialgebras, and constructs their universal envelopes.

Findings

Leibniz triple systems are the dialgebra analogues of Lie triple systems.

Universal Leibniz envelopes are constructed for these systems.

Examples of 2-dimensional Leibniz triple systems are provided.

Abstract

We define Leibniz triple systems in a functorial manner using the algorithm of Kolesnikov and Pozhidaev which converts identities for algebras into identities for dialgebras. We verify that Leibniz triple systems are the natural analogues of Lie triple systems in the context of dialgebras by showing that both the iterated bracket in a Leibniz algebra and the permuted associator in a Jordan dialgebra satisfy the defining identities for Leibniz triple systems. We construct the universal Leibniz envelopes of Leibniz triple systems and prove that every identity satisfied by the iterated bracket in a Leibniz algebra is a consequence of the defining identities for Leibniz triple systems. To conclude, we present some examples of 2-dimensional Leibniz triple systems and their universal Leibniz envelopes.