2014/08/13 by Bremner, Murray, Madariaga, Sara
#17B60 #17C50 #18D50 #FOS: Mathematics #Primary 17A30 #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Secondary 17A50
paper · doi:10.48550/arxiv.1408.3069
We study Lie brackets and Jordan products derived from associative operations ∘, \bullet satisfying the interchange identity (w \bullet x ) ∘ ( y \bullet z ) ≡ (w ∘ y ) \bullet ( x ∘ z ). 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.