New identities in dendriform algebras

TL;DR

This paper introduces new combinatorial identities in dendriform dialgebras, linking algebraic structures with classical combinatorics and extending known identities like Bohnenblust-Spitzer and Chen integrals.

Contribution

It presents novel identities in dendriform algebras that generalize and connect classical combinatorial and algebraic phenomena.

Findings

New identities related to Lyndon words and Lie algebra rewriting

Generalization of Bohnenblust-Spitzer identity

Connection to Chen integrals and Malvenuto-Reutenauer algebra

Abstract

Dendriform structures arise naturally in algebraic combinatorics (where they allow, for example, the splitting of the shuffle product into two pieces) and through Rota-Baxter algebra structures (the latter appear, among others, in differential systems and in the renormalization process of pQFT). We prove new combinatorial identities in dendriform dialgebras that appear to be strongly related to classical phenomena, such as the combinatorics of Lyndon words, rewriting rules in Lie algebras, or the fine structure of the Malvenuto-Reutenauer algebra. One of these identities is an abstract noncommutative, dendriform, generalization of the Bohnenblust-Spitzer identity and of an identity involving iterated Chen integrals due to C.S. Lam.