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

Abstract

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 {ring operator} y = x · y + y · x endows A with the structure of a Jordan algebra, and the left multiplications L· (x) : y {mapping} x · y define a representation of this Jordan algebra on A. Equivalently, x · y satisfies these multilinear identities:((x {ring operator} y) · (z · u) + (y {ring operator} z) · (x · u) + (z {ring operator} x) · (y · u); ≡ z · [(x {ring operator} y) · u] + x · [(y {ring operator} z) · u] + y · [(z {ring operator} x) · u],; x · [y · (z · u)] + z · [y · (x · u)] + [(x {ring operator} z) {ring operator} y] · u; ≡ z · [(x {ring operator} y) · u] + x · [(y {ring operator} z) · u] + y · [(z {ring operator} 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. © 2013 Elsevier Inc. All rights reserved.