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arXiv · 1408.3069

Lie and Jordan products in interchange algebras

Murray Bremner·Sara Madariaga
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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.

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Murray Bremner, Sara Madariaga. 2014-08-13. Lie and Jordan products in interchange algebras. https://doi.org/10.1080/00927872.2015.1085545

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