The universal central extension of the holomorphic current algebra

Abstract

We identify the universal differential module Ω1(A) for the Frechet algebra A of holomorphic functions on a complex Stein manifold X, and more generally on a Riemannian domain R over X and for the algebra of germs of holomorphic functions on a compact subset K⊂ℂn. It turns out to be isomorphic to the Frechet space of holomorphic 1-forms on X, resp. R, resp. to the space Ω1(K) of germs of holomorphic 1-forms in K. This determines the center of the universal central extension of the Lie algebra 𝒪(R,𝔨 of holomorphic maps from R to a finite-dimensional simple complex Lie algebra 𝔨.