The fake monster formal group. 27 May, 31 Dec 1998 
Duke Math J. Vol 100 No. 1 (1999), 139-165. 
Richard E. Borcherds, 
D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 1SB, England. 
home page: www.dpmms.cam.ac.uk/~reb 

Contents. 1. Introduction. 
Notation and terminology. 
2. Some theorems about smooth Hopf algebras. 
3. Liftings of Lie algebra elements. 
4. The smooth fake monster Hopf algebra. 
5. A smooth Hopf algebra for the Virasoro algebra. 
6. The no-ghost theorem over Z. 
7. An application to modular moonshine. 
8. Open problems. 

1. Introduction. The main result of this paper is the construction of 
``good'' integral forms for the universal enveloping algebras of the fake 
monster Lie algebra and the Virasoro algebra. As an application we construct 
formal group laws over the integers for these Lie algebras. We also prove a 
form of the no-ghost theorem over the integers, and use this to verify an 
assumption used in the proof of the modular moonshine conjectures.