  We describe a unification of several apparently unrelated factorizations
arisen from quantum field theory, vertex operator algebras, combinatorics and
numerical methods in differential equations. The unification is given by a
Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff
formula in our study of the Hopf algebra approach of Connes and Kreimer to
renormalization in perturbative quantum field theory. There we showed that the
Birkhoff decomposition of Connes and Kreimer can be obtained from a certain
Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter
operator. We will explain how the same decomposition generalizes the
factorization of formal exponentials and uniformization for Lie algebras that
arose in vertex operator algebra and conformal field theory, and the even-odd
decomposition of combinatorial Hopf algebra characters as well as to the Lie
algebra polar decomposition as used in the context of the approximation of
matrix exponentials in ordinary differential equations.
