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.
Categories: math.NT