Lie theory for quasi-shuffle bialgebras

TL;DR

This paper develops a systematic Lie-theoretic framework for quasi-shuffle bialgebras, extending classical Lie theory concepts to a broader algebraic context related to Hopf operads and multizeta values.

Contribution

It introduces a comprehensive Lie theory for quasi-shuffle algebras, unifying previous partial results and establishing new structural insights.

Findings

Develops a Lie-theoretic framework for quasi-shuffle bialgebras

Establishes relations between quasi-shuffle algebras and free Lie algebras

Provides a systematic foundation for future research in the area

Abstract

Many features of classical Lie theory generalize to the broader context of algebras over Hopf operads. However, this idea remains largely to be developed systematically. Quasi-shuffle algebras provide for example an interesting illustration of these phenomena, but have not been investigated from this point of view.The notion of quasi-shuffle algebras can be traced back to the beginings of the theory of Rota--Baxter algebras, but was developed systematically only recently, starting essentially with Hoffman's work, that was motivated by multizeta values (MZVs) and featured their bialgebra structure. Many partial results on the fine structure of quasi-shuffle bialgebras have been obtained since then but, besides the fact that each of these articles features a particular point of view, they fail to develop systematically a complete theory.This article builds on these various results and develops the analog theory, for quasi-shuffle algebras, of the theory of descent algebras and their relations to free Lie algebras for classical enveloping algebras.