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 ${\mathrm{SL}}_2(\mathbb{Z})$, which is the smallest extension of the algebra $Q{M}_{\ast }$ 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.

v2, minor changes, to appear in Algebra and Number Theory

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