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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.

Reviewer: H.Yamada

MSC:
17B67Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B69Vertex operators; vertex operator algebras and related structures
20D08Simple groups: sporadic finite groups