Special and exceptional Jordan dialgebras

TL;DR

This paper investigates Jordan dialgebras, establishing analogues of classical theorems, and explores special identities, providing a bridge between properties of ordinary algebras and dialgebras.

Contribution

It introduces methods to reduce dialgebra problems to ordinary algebra cases and proves dialgebraic versions of classical algebraic theorems.

Findings

Classical theorems do not directly generalize to dialgebras.

Dialgebraic analogues of Cohn's, Shirshov's, and Macdonald's theorems are established.

A correspondence between special identities in ordinary algebras and dialgebras is identified.

Abstract

In this paper, we study the class of Jordan dialgebras. We develop an approach for reducing problems on dialgebras to the case of ordinary algebras. It is shown that straightforward generalizations of the classical Cohn's, Shirshov's, and Macdonald's Theorems do not hold for dialgebras. However, we prove dialgebraic analogues of these statements. Also, we study polylinear special identities which hold in all special Jordan algebras and do not hold in all Jordan algebras. We find a natural correspondence between special identities for ordinary algebras and dialgebras.