Associative triple trisystems and standard embeddings

TL;DR

This paper introduces associative triple trisystems, extending existing algebraic frameworks, and explores their relationships with Jordan and Lie triple systems, providing new examples and a standard embedding construction.

Contribution

It develops the theory of associative triple trisystems, connecting them with classical systems and defining di-endomorphisms for standard embeddings.

Findings

Established relationships between associative, Jordan, and Lie triple systems

Provided a matrix example with a non-trivial associative dialgebra structure

Defined di-endomorphisms enabling standard embeddings

Abstract

Building on the established theories of Jordan triple disystems and Leibniz triple systems, we introduce and develop the theory of associative triple trisystems, filling a significant gap in the existing framework. We establish the classical relationships between associative, Jordan, and Lie triple systems within the context of trisystems. We present a significant example by equipping the space of matrices with a non-trivial associative dialgebra structure. We conclude defining the concept of di-endomorphisms of any module, which enables the construction of the standard embedding for any associative triple trisystem.