Special identities for quasi-Jordan algebras

TL;DR

This paper investigates the polynomial identities of semispecial quasi-Jordan algebras, showing that all identities up to degree 7 derive from basic identities, but new identities appear at degree 8, revealing complex algebraic structure.

Contribution

It demonstrates that all identities up to degree 7 are consequences of fundamental identities, and identifies new identities at degree 8 using computer algebra.

Findings

All identities in degree ≤ 7 follow from initial identities.

Six new identities are found in degree 8.

Some new identities relate to noncommutative preimages of the Glennie identity.

Abstract

Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities $a(bc) = a(cb)$, $(ba)a^2 = (ba^2)a$, and $(b,a^2,c) = 2(b,a,c)a$. These identities are satisfied by the product $ab = a \dashv b + b \vdash a$ in an associative dialgebra. We use computer algebra to show that every identity for this product in degree $\le 7$ is a consequence of the three identities in degree $\le 4$, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.