Symmetrization of Jordan dialgebras

TL;DR

This paper investigates polynomial identities of symmetrized products in Jordan dialgebras, revealing that identities up to degree 5 are consequences of commutativity, with new identities emerging at degrees 6 and 7.

Contribution

It provides a complete characterization of identities in low degrees and identifies new identities at higher degrees using computational and representation theory methods.

Findings

Identities of degree ≤5 are consequences of commutativity.

Complete set of degree 6 identities not derived from commutativity.

Existence of new identities at degree 7.

Abstract

A basic problem for any class of nonassociative algebras is to determine the polynomial identities satisfied by the symmetrization and the skew-symmetrization of the original product. We consider the symmetrization of the product in the class of special Jordan dialgebras. We use computational linear algebra to show that every polynomial identity of degree $n \le 5$ satisfied by the symmetrized Jordan diproduct in every diassociative algebra is a consequence of commutativity. We determine a complete set of generators for the polynomial identities in degree 6 which are not consequences of commutativity. We use a constructive version of the representation theory of the symmetric group to show that there exist further new identities in degree 7.