*(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 $:Q(u,z):$. There is a certain derivation $D$ on $V$. The quotient space $V/DV$ is a Lie algebra, where the Lie algebra product is $[u,v]=$ the coefficient of ${z}^{-1}$ in $:Q(u,z):\left(v\right)$. 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.

##### MSC:

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

17B69 | Vertex operators; vertex operator algebras and related structures |

20D08 | Simple groups: sporadic finite groups |