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Vertex Algebras, Kac-Moody Algebras, and the Monster

Richard E. Borcherds
Proceedings of the National Academy of Sciences of the United States of America
Vol. 83, No. 10 (May 15, 1986), pp. 3068-3071
Stable URL: http://www.jstor.org/stable/27441
Page Count: 4
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Vertex Algebras, Kac-Moody Algebras, and the Monster
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Abstract

It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.

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