Lie and Jordan Products in Interchange Algebras

Abstract

We study Lie brackets and Jordan products derived from associative opera-We use computational linear algebra, based on the representation theory of the symmetric group, to determine all polynomial identities of degree ≤ 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 ≤ 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.

Distributive laws between the Three Graces

“…The calculations are similar to those discussed in detail in previous sections, so we provide only a brief outline. The number of distinct association types in arity n for two commutative operations is sequence OEIS A226909; see also [10]:…”