Paper detail

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 $\mathcal{M}^k_m(\mathfrak g, G)$ and $\mathcal{M}^n_{m'}(\mathfrak g, G)$ of meromorphic functions depending on a number of formal complex parameters $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_n)$ 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 $\mathcal{M}^{k+n}_{m+m'}(\mathfrak g, G)$.