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

TL;DR

This paper investigates identities of the pre-Jordan product in free dendriform algebras, revealing that all identities up to degree 7 are consequences of degree 4 identities, but new independent identities emerge at degree 8.

Contribution

It identifies the degrees at which new identities for the pre-Jordan product appear, using computational methods and representation theory.

Findings

All identities of degree ≤ 7 are derived from degree 4 identities.

New independent identities are found at degree 8.

There is an isomorphism between these identities and identities for the Jordan diproduct in dialgebras.

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.