The $\mathfrak{sl}_2$-algebra structure of multiple Eisenstein series

We prove that the algebra of multiple Eisenstein series is an $\mathfrak{sl}_2$-algebra. In particular, we show that it is graded by weight, which also holds after taking complex linear spans. Further, we identify it as a graded $\mathbb{Q}$-algebra with the associated graded algebra of $q$-analogues of multiple zeta values with respect to the weight filtration. The main ingredient is an estimate that compares the usual defining sums of multiple Eisenstein series with sums over a lattice order depending on an integer $N$.