The author constructs a realization of an algebra that is usually slightly larger than a Kac-Moody algebra and equal to if is of finite or affine type. Let be a Fock space associted with an even lattice R. This space has a structure of a vertex algebra. Products on are defined through the generalized vertex operator . There is a certain derivation on . The quotient space is a Lie algebra, where the Lie algebra product is the coefficient of in . If is the root lattice of a Kac-Moody algebra , then contains as a subalgebra. To reduce to a smaller subalgebra, the Virasoro algebra is used.
The author constructs an integral form for the universal enveloping algebra , some new irreducible integrable representation of , and a sort of affinization of . Finally a relation between vertex algebras and the Frenkel-Lepowsky-Meurman representation of the monster is discussed.