Vertex algebras, Kac-Moody algebras, and the monster. (English) Zbl 0613.17012

Borcherds, Richard E.

Proc. Natl. Acad. Sci. USA 84, 3068-3071 (1986).

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:

17B67	Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B69	Vertex operators; vertex operator algebras and related structures
20D08	Simple groups: sporadic groups

Keywords:

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