Special identities for the pre-Jordan product in the free dendriform algebra

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·yx·y such that the product x∘y=x·y+y·xx∘y=x·y+y·x endows A   with the structure of a Jordan algebra, and the left multiplications L·(x):y↦x·yL·(x):y↦x·y define a representation of this Jordan algebra on A  . Equivalently, x·yx·y satisfies these multilinear identities:(x∘y)·(z·u)+(y∘z)·(x·u)+(z∘x)·(y·u)≡z·[(x∘y)·u]+x·[(y∘z)·u]+y·[(z∘x)·u],x·[y·(z·u)]+z·[y·(x·u)]+[(x∘z)∘y]·u≡z·[(x∘y)·u]+x·[(y∘z)·u]+y·[(z∘x)·u].The pre-Jordan product x·y=x≻y+y≺xx·y=x≻y+y≺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⩽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 S8S8-modules between these new identities and the special identities for the Jordan diproduct in an associative dialgebra.