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 $L_{\cdot }(x):y\mapsto x\cdot y$ 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.

23 pages

References in corpus (6)

• Splitting of operations, Manin products and Rota-Baxter operators
• Some results on L-dendriform algebras
• On Products and Duality of Binary, Quadratic, Regular Operads
• On special identities for dialgebras
• Pre-alternative algebras and pre-alternative bialgebras
• Gröbner-Shirshov Bases and Hilbert Series of Free Dendriform Algebras

Cited by in corpus (5)

• Jordan Trialgebras and Post-Jordan Algebras
• On tortkara triple systems
• Jordan quadruple systems
• Dendriform analogues of Lie and Jordan triple systems
• Gröbner-Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras (unabridged version)