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Vertex algebras, Kac-Moody algebras, and the monster. (English) Zbl 0613.17012

The author constructs a realization of an algebra that is usually slightly larger than a Kac-Moody algebra $A$ and equal to $A$ if $A$ is of finite or affine type. Let $V=V\left(R\right)$ be a Fock space associted with an even lattice R. This space has a structure of a vertex algebra. Products on $V$ are defined through the generalized vertex operator $:Q\left(u,z\right):$. There is a certain derivation $D$ on $V$. The quotient space $V/DV$ is a Lie algebra, where the Lie algebra product is $\left[u,v\right]=$ the coefficient of ${z}^{-1}$ in $:Q\left(u,z\right):\left(v\right)$. If $R$ is the root lattice of a Kac-Moody algebra $A$, then $V/DV$ contains $A$ as a subalgebra. To reduce $V/DV$ to a smaller subalgebra, the Virasoro algebra is used.

The author constructs an integral form for the universal enveloping algebra $U\left(A\right)$, some new irreducible integrable representation of $A$, and a sort of affinization of $A$. Finally a relation between vertex algebras and the Frenkel-Lepowsky-Meurman representation of the monster is discussed.