The extension of cochain complexes of meromorphic functions to multiplications

Let $\mathfrak{g}$ be an infinite-dimensional Lie algebra and $G$ be the algebraic completion of its module. Using a geometric interpretation in terms of sewing two Riemann spheres with a number of marked points, we introduce a multiplication between elements of two spaces ${\mathcal{M}}_m^k(\mathfrak{g},G)$ and ${\mathcal{M}}_{m^{\mathrm{\prime }}}^n(\mathfrak{g},G)$ of meromorphic functions depending on a number of formal complex parameters $(x_1,\dots ,x_k)$ and $(y_1,\dots ,y_n)$ with specific analytic and symmetry properties, and associated to $\mathfrak{g}$-valued series. These spaces form a chain-cochain complex with respect to a boundary-coboundary operator. The main result of the paper shows that the multiplication is defined by an absolutely convergent series and takes values in the space ${\mathcal{M}}_{m+m^{\mathrm{\prime }}}^{k+n}(\mathfrak{g},G)$.