Jordan Triple Disystems

TL;DR

This paper introduces Jordan triple disystems by applying algorithmic methods to extend identities from Jordan triple systems to dialgebra contexts, verified through computer algebra and theoretical proofs.

Contribution

It develops a new class of nonassociative triple systems called Jordan triple disystems using algorithmic extensions of identities, combining computational and theoretical approaches.

Findings

Identification of Jordan triple disystems via KP and BSO algorithms

Verification of identities with computer algebra

Extension of Jordan triple products to dialgebra structures

Abstract

We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorithm to the defining identities for Jordan triple systems to obtain a new variety of nonassociative triple systems, called Jordan triple disystems. We give a generalized statement of the BSO algorithm introduced by Bremner and Sanchez-Ortega for extending multilinear operations in an associative algebra to corresponding operations in an associative dialgebra. We apply the BSO algorithm to the Jordan triple product and use computer algebra to verify that the polynomial identities satisfied by the resulting operations coincide with the results of the KP algorithm; this provides a large class of examples of Jordan triple disystems. We formulate a general conjecture expressed by a commutative diagram relating the output of the KP and BSO algorithms. We conclude by generalizing the Jordan triple product in a Jordan algebra to operations in a Jordan dialgebra; we use computer algebra to verify that resulting structures provide further examples of Jordan triple disystems. For this last result, we also provide an independent theoretical proof using Jordan structure theory.