The extension of cochain complexes of meromorphic functions to multiplications

TL;DR

This paper introduces a new multiplication operation for spaces of meromorphic functions associated with infinite-dimensional Lie algebras, extending cochain complexes through geometric sewing of Riemann spheres, with proven convergence and well-defined algebraic structure.

Contribution

It defines a multiplication on meromorphic function spaces linked to Lie algebras via geometric sewing, extending cochain complexes with convergence guarantees.

Findings

Multiplication is given by an absolutely convergent series.

The operation maps to a higher-degree meromorphic function space.

The structure extends the cochain complex framework.

Abstract

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 $\mathcal{M}^{k+n}_{m+m'}(\mathfrak g, G)$.