Algebras, dialgebras, and polynomial identities

TL;DR

This survey reviews recent advances in the theory of associative and nonassociative dialgebras, focusing on polynomial identities, algorithms for identity conversion, and various algebraic structures like Lie and Jordan systems.

Contribution

It compiles and discusses recent developments, including the KP and BSO algorithms, and introduces new conjectures and research directions in dialgebra theory.

Findings

Analysis of polynomial identities in dialgebras

Introduction of KP and BSO algorithms for identity conversion

Proposal of a conjecture relating the KP and BSO algorithms

Abstract

This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc-bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raul Felipe, Luiz A. Peresi, and Juana Sanchez-Ortega.