On the algebraic structure of iterated integrals of quasimodular forms

Abstract

We study the algebra ${\mathcal{ℐ}}^{QM}$ of iterated integrals of quasimodular forms for ${SL}_2\left(\mathbb{Z}\right)$, 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.