Moonshine Cohomology 
Bong H. Lian, Gregg J. Zuckerman 

Abstract: We construct a new cohomology functor from the a certain category 
of {\it quantum operator algebras} to the category of {\it Batalin-Vilkovisky 
algebras}. This {\it Moonshine cohomology} has, as a group of natural 
automorphisms, the Fischer-Griess Monster finite group. We prove a general 
vanishing theorem for this cohomology. For a certain commutative QOA attached 
to a rank two hyperbolic lattice, we show that the degree one cohomology is 
isomorphic to the so-called Lie algebra of physical states. In the case of a 
rank two unimodular lattice, the degree one cohomology gives a new 
construction of Borcherd's Monster Lie algebra. As applications, we compute 
the graded dimensions and signatures of this cohomology as a hermitean Lie 
algebra graded by a hyperbolic lattice. In the first half of this paper, we 
give as preparations an exposition of the theory of quantum operator algebras. 
Some of the results here were announced in lectures given by the first author 
at the Research Institute for Mathematical Sciences in Kyoto in September 94.