Dendriform analogues of Lie and Jordan triple systems

Abstract

We use computer algebra to determine all the multilinear polynomial identities of degree $\le 7$ satisfied by the trilinear operations $(a\cdot b)\cdot c$ and $a\cdot (b\cdot c)$ in the free dendriform dialgebra, where $a\cdot b$ is the pre-Lie or the pre-Jordan product. For the pre-Lie triple products, we obtain one identity in degree 3, and three independent identities in degree 5, and we show that every identity in degree 7 follows from the identities of lower degree. For the pre-Jordan triple products, there are no identities in degree 3, five independent identities in degree 5, and ten independent irreducible identities in degree 7. Our methods involve linear algebra on large matrices over finite fields, and the representation theory of the symmetric group.

Some references corrected; final version (to appear in Communications in Algebra)

References in corpus (5)

• On embedding of dendriform algebras into Rota---Baxter algebras
• Special identities for the pre-Jordan product in the free dendriform algebra
• Gröbner-Shirshov Bases and Hilbert Series of Free Dendriform Algebras
• The partially alternating ternary sum in an associative dialgebra
• Higher identities for the ternary commutator

Cited by in corpus (2)

• Operads of decorated trees and their duals
• Veronese powers of operads and pure homotopy algebras