2012/12/21 by Bremner, Murray R., Madariaga, Sara
#17A30 #17A50 #17C50 #18D50 #20C30 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 17C05. Secondary 17-04 #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1212.5631
Pre-Jordan algebras were introduced recently in analogy with pre-Lie algebras. A pre-Jordan algebra is a vector space A with a bilinear multiplication x ⋅ y such that the product x ∘ y = x ⋅ y + y ⋅ x endows A with the structure of a Jordan algebra, and the left multiplications L_⋅(x)\colon y ↦ x ⋅ y define a representation of this Jordan algebra on A. Equivalently, x ⋅ y satisfies these multilinear identities: [see PDF]. The pre-Jordan product x ⋅ y = x \succ y + y \prec x in any dendriform algebra also satisfies these identities. We use computational linear algebra based on the representation theory of the symmetric group to show that every identity of degree ≤ 7 for this product is implied by the identities of degree 4, but that there exist new identities of degree 8 which do not follow from those of lower degree. There is an isomorphism of S8-modules between these new identities and the special identities for the Jordan diproduct in an associative dialgebra.