Factorization of formal exponentials and uniformization
Abstract
Let $\mathfrak{g}$ be a Lie algebra in characteristic zero equipped with a vector space decomposition $\mathfrak{g}=\mathfrak{g}^\oplus \mathfrak{g}^+$, and let $s$ and $t$ be commuting formal variables. We prove that the CampbellBakerHausdorff map $C:s\mathfrak{g}^ [[s,t]]\times t\mathfrak{g}^+[[s,t]]\to s\mathfrak{g}^[[s,t]]\oplus t\mathfrak{g}^+[[s,t]]$ given by $e^{sg^}e^{tg^+}=e^{C(sg^,tg^+)}$ for $g^\pm\in\mathfrak{g}^\pm[[s,t]]$ is a bijection, as is well known when $\mathfrak{g}$ is finitedimensional over $\mathbb{R}$ or $\mathbb{C}$, by geometry. It follows that there exist unique $\Psi^\pm\in\mathfrak{g}^\pm[[s,t]]$ such that $e^{tg^+}e^{sg^}= e^{s\Psi^}e^{t\Psi^+}$ (also well known in the finitedimensional geometric setting). We apply this to $\mathfrak{g}$ consisting of certain formal infinite series with coefficients in a Lie algebra $\mathfrak{p}$. For $\mathfrak{p}$ the Virasoro algebra (resp., a Grassmann envelope of the NeveuSchwarz superalgebra), the result was first proved by Huang (resp., Barron) as a step in the construction of a (super)geometric formulation of the notion of vertex operator (super)algebra. For the Virasoro (resp., N=1 NeveuSchwarz) algebra with zero central charge the result gives the precise expansion of the uniformizing function for a sphere (resp., supersphere) with tubes resulting from the sewing of two spheres (resp., superspheres) with tubes in twodimensional genuszero holomorphic conformal (resp., N = 1 superconformal) field theory. The general result places such uniformization problems into a broad formal algebraic context.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 1999
 arXiv:
 arXiv:math/9908151
 Bibcode:
 1999math......8151B
 Keywords:

 Quantum Algebra;
 Rings and Algebras;
 Representation Theory;
 17B01;
 17B65;
 17B68;
 30F10
 EPrint:
 LaTex file, 31 pages