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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.
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

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