Special Identities for the Pre-jordan Product in the Free Dendriform Algebra

Pre-Jordan algebras were introduced recently in analogy with preLie 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) : y 7→ x · y define a representation of this Jordan algebra on A. Equivalently, x ·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 ≺ 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.