Dendriform analogues of Lie and Jordan triple systems

TL;DR

This paper uses computer algebra to classify polynomial identities of degree up to 7 for certain triple operations in dendriform dialgebras, revealing new identities and their degrees.

Contribution

It determines all multilinear polynomial identities up to degree 7 for specific triple operations in dendriform dialgebras, a novel classification using computational algebra.

Findings

Identified one degree 3 identity for pre-Lie triple products.

Found three degree 5 identities for pre-Lie triple products.

Discovered ten degree 7 identities for pre-Jordan triple products.

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.