On post-Lie structures for free Lie algebras
Annika Burmester, Ulf Kühn
TL;DR
This work develops a comprehensive framework for post-Lie algebras and their subadjacent Hopf-algebra structures, focusing on the Grossman-Larson product and its graded dual coproducts. It specializes to post-Lie structures on free Lie algebras, providing explicit dual-coproduct formulas and magmatic viewpoints that enable concrete computations. The Ihara, ari, and uri brackets are analyzed as post-Lie structures on free Lie algebras, with explicit GLP formulas and dual coproducts, and their connections to Goncharov’s coproduct and Ecalle’s bimoulds are established. Assuming threshold-shuffle identities for certain Bernoulli-related multiplicities, the uri framework yields a conjectural post-Lie structure whose depth-graded form recovers the ari structure and links to the algebra of multiple zeta values and their $q$-analogues, suggesting deep structure for indecomposables and potential Hopf-analytic realizations.
Abstract
We study post-Lie structures on free Lie algebras, the Grossman-Larson product on their enveloping algebras, and provide an abstract formula for its dual coproduct. This might be of interest for the general theory of post-Hopf algebras. Using a magmatic approach, we explore post-Lie algebras connected to multiple zeta values and their $q$-analogues. For multiple zeta values, this framework yields an algebraic interpretation of the Goncharov coproduct. Assuming that the Bernoulli numbers satisfy the so called threshold shuffle identities, we present a post-Lie structure, whose induced Lie bracket we expect to restrict to the dual of indecomposables of multiple $q$-zeta values. Our post-Lie algebras align with Ecalle's theory of bimoulds: we explicitly identify the ari bracket with a post-Lie structure on a free Lie algebra, and conjecture a correspondence for the uri bracket.