Lie and Jordan products in interchange algebras

TL;DR

This paper investigates polynomial identities involving Lie brackets and Jordan products derived from associative operations satisfying the interchange identity, using computational algebra to identify new identities up to degree 7.

Contribution

It systematically determines all polynomial identities of degree ≤ 7 relating the Lie and Jordan structures in interchange algebras, revealing new identities in various degrees.

Findings

Two new identities in degree 6 for Lie-Lie case

No new identities in degrees ≤ 6 for Lie-Jordan case

Multiple new identities in degrees 4 to 7 for Jordan-Jordan case

Abstract

We study Lie brackets and Jordan products derived from associative operations $\circ, \bullet$ satisfying the interchange identity $(w \bullet x ) \circ ( y \bullet z ) \equiv (w \circ y ) \bullet ( x \circ z )$. We use computational linear algebra, based on the representation theory of the symmetric group, to determine all polynomial identities of degree $\le 7$ relating (i) the two Lie brackets, (ii) one Lie bracket and one Jordan product, and (iii) the two Jordan products. For the Lie-Lie case, there are two new identities in degree 6 and another two in degree 7. For the Lie-Jordan case, there are no new identities in degree $\le 6$ and a complex set of new identities in degree 7. For the Jordan-Jordan case, there is one new identity in degree 4, two in degree 5, and complex sets of new identities in degrees 6 and 7.