Factorization of formal exponentials and uniformization
| dc.creator | Barron, Katrina | |
| dc.creator | Huang, Yi-Zhi | |
| dc.creator | Lepowsky, James | |
| dc.date | 1999-08-27 | |
| dc.date.accessioned | 2026-07-07T10:58:55Z | |
| dc.date.available | 2026-07-07T10:58:55Z | |
| dc.description | 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 Campbell-Baker-Hausdorff 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 finite-dimensional over $\mathbb{R}$ or $\mathbb{C}$, by geometry. It follows that there exist unique $Ψ^\pm\in\mathfrak{g}^\pm[[s,t]]$ such that $e^{tg^+}e^{sg^-}= e^{sΨ^-}e^{tΨ^+}$ (also well known in the finite-dimensional 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 Neveu-Schwarz 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 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. | |
| dc.description | LaTex file, 31 pages | |
| dc.identifier | https://arxiv.org/abs/math/9908151 | |
| dc.identifier | http://arxiv.org/abs/math/9908151 | |
| dc.identifier | J.Algebra.228:551-579,2000 | |
| dc.identifier | doi:10.1006/jabr.2000.8285 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/187274 | |
| dc.subject | Quantum Algebra | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | Rings and Algebras | |
| dc.subject | Representation Theory | |
| dc.subject | 17B01; 17B65, 17B68, 30F10 | |
| dc.title | Factorization of formal exponentials and uniformization | |
| dc.type | text |