• DOI:10.1006/jabr.2000.8285
• Corpus ID: 14786624

Factorization of Formal Exponentials and Uniformization

@article{Barron2000FactorizationOF,
  title={Factorization of Formal Exponentials and Uniformization},
  author={Katrina Deane Barron and Yi-Zhi Huang and James Lepowsky},
  journal={Journal of Algebra},
  year={2000},
  volume={228},
  pages={551-579}
}

• K. Barron, Yi-Zhi Huang, J. Lepowsky
• Published 27 August 1999
• Mathematics
• Journal of Algebra

Let g be a Lie algebra over a field of characteristic zero equipped with a vector space decomposition g = g − ⊕ g + , and let s and t be commuting formal variables commuting with g. We prove that the map C: sg − [[s, t]] × tg + [[s, t]] → sg − [[s, t]] ⊕ tg + [[s, t]] defined by the Campbell–Baker–Hausdorff formula and given by esg − etg + = eC(sg − , tg + ) for g ± ∈ g ± [[s, t]] is a bijection, as is well known when g is finite-dimensional over R or C, by geometry. It follows that there exist unique Ψ ± ∈ g ± [[s, t]] such that etg + esg − = esΨ − etΨ + (also well known in the finite-dimensional geometric setting). We apply this to a Lie algebra g consisting of certain formal infinite series with coefficients in a Z-graded Lie algebra p, for instance, an affine Lie algebra, the Virasoro algebra, or a Grassmann envelope of the N = 1 Neveu–Schwarz superalgebra. For p the Virasoro algebra, the result was first proved by Huang as a step in the construction of a geometric formulation of the notion of vertex operator algebra, and for p a Grassmann envelope of the Neveu–Schwarz superalgebra, it was first proved by Barron as a corresponding step in the construction of a supergeometric formulation of the notion of vertex operator superalgebra. In the special case of the Virasoro (resp., N = 1 Neveu–Schwarz) 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 two-dimensional genus-zero holomorphic conformal (resp., N = 1 superconformal) field theory. The general result places such uniformization problems into a broad formal algebraic context.