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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 $$V=V(R)$$ 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 $$:Q(u,z):(v)$$. 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 $$U(A)$$, 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.