## A moonshine module for the monster.(English)Zbl 0556.17008

Vertex operators in mathematics and physics, Proc. Conf., Berkeley/CA 1983, Publ., Math. Sci. Res. Inst. 3, 231-273 (1985).
[For the entire collection see Zbl 0549.00013.]
The list of finite simple groups consists of the groups of Lie type, the alternating groups and the 26 sporadic groups. The largest sporadic group is called the Monster and denoted $$F_ 1$$. It contains 20 or 21 of the sporadic groups. This group gave rise to many mysteries even before its actual appearance, promising deep connections with different areas of mathematics. ”Moonshine” refers to connections between $$F_ 1$$ and the modular function j.
The introduction gives the history of ”Monstrous Moonshine”. Then the main ideas and steps of the construction of a natural module for $$F_ 1$$ with character j are given. In spite of the fact that the module is infinite-dimensional, the authors claim that it is the most natural and simplest representation of $$F_ 1$$. They hope that it is possible to develop a unified theory of the finite simple groups, based on lattices and vertex operators.
Reviewer: L.N.Vaserstein

### MSC:

 17B65 Infinite-dimensional Lie (super)algebras 20D08 Simple groups: sporadic groups 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 11F03 Modular and automorphic functions

Zbl 0549.00013