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Zbl 0613.17012
Borcherds, Richard E.
Vertex algebras, Kac-Moody algebras, and the monster. (English)
[J] Proc. Natl. Acad. Sci. USA 84, 3068-3071 (1986). ISSN 0027-8424; ISSN 1091-6490/e

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 $z\sp{-1}$ 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. \par 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.
[H.Yamada]

MSC 2000:: *17B67 Kac-Moody algebras
17B69 Vertex operators
20D08 Simple groups: sporadic finite groups

Keywords: Kac-Moody algebra; Fock space; universal enveloping algebra; representation; vertex algebras; monster

Cited in: Zbl 1146.17308 Zbl 1055.17001 Zbl 1043.17016 Zbl 1037.17036 Zbl 0981.17022 Zbl 0987.17012 Zbl 0963.14006 Zbl 0956.17019 Zbl 0884.17021 Zbl 1032.17048 Zbl 0861.17017 Zbl 0823.17041 Zbl 0813.17018 Zbl 0906.17022 Zbl 0848.17032 Zbl 0704.17001 Zbl 0674.17001 Zbl 0657.17011

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