On the algebraic structure of iterated integrals of quasimodular forms

TL;DR

This paper investigates the algebraic structure of iterated integrals of quasimodular forms, revealing it as a polynomial algebra generated by Lyndon words, and extends the analysis to modular forms.

Contribution

It establishes that the algebra of iterated integrals of quasimodular forms is a polynomial algebra in infinitely many variables, characterized by Lyndon words on Eisenstein series.

Findings

The algebra of iterated integrals of quasimodular forms is a polynomial algebra.

Lyndon words on Eisenstein series generate the algebra.

An analogous structure is proven for modular forms.

Abstract

We study the algebra $\mathcal{I}^{QM}$ of iterated integrals of quasimodular forms for $\operatorname{SL}_2(\mathbb{Z})$, which is the smallest extension of the algebra $QM_{\ast}$ of quasimodular forms, which is closed under integration. We prove that $\mathcal{I}^{QM}$ is a polynomial algebra in infinitely many variables, given by Lyndon words on certain monomials in Eisenstein series. We also prove an analogous result for the $M_{\ast}$-subalgebra $\mathcal{I}^{M}$ of $\mathcal{I}^{QM}$ of iterated integrals of modular forms.