A generalization of formal multiple zeta values related to multiple Eisenstein series and multiple q-zeta values

We present the $\tau$-invariant balanced quasi-shuffle algebra $\mathcal{G}^{\operatorname{f}}$, whose elements formalize (combinatorial) multiple Eisenstein series as well as multiple q-zeta values. In particular, $\mathcal{G}^{\operatorname{f}}$ has natural maps into these two algebras, and we expect these maps to be isomorphisms. Racinet studied the algebra $\mathcal{Z}^f$ of formal multiple zeta values by examining the corresponding affine scheme DM. Similarly, we present the affine scheme BM corresponding to the algebra $\mathcal{G}^{\operatorname{f}}$. We show that Racinet's affine scheme DM embeds into our affine scheme BM. This leads to a projection from the algebra $\mathcal{G}^{\operatorname{f}}$ onto $\mathcal{Z}^f$. Via the above natural maps, this projection corresponds to extracting the constant terms of multiple Eisenstein series or the limit $q\to1$ of multiple q-zeta values.