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 $x \cdot y$ such that the product $x \circ y = x \cdot y + y \cdot x$ endows $A$ with the structure of a Jordan algebra, and the left multiplications $L_\cdot(x)\colon y \mapsto x \cdot y$ define a representation of this Jordan algebra on $A$. Equivalently, $x \cdot y$ satisfies these multilinear identities: [see PDF]. The pre-Jordan product $x \cdot y = x \succ y + y \prec x$ 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 $\le 7$ 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 $S_8$-modules between these new identities and the special identities for the Jordan diproduct in an associative dialgebra.