The basic representation of g(A(1)) is then defined on V by the following formulas [FrKa]:
π(u(n))=u(n),u∈h π(E(n)α)=Xn(α)cα,π(k)=1; This is called the homogeneous vertex operator construction of the basic representation.
The vertex operators were introduced in string theory around 1969, but the vertex operator construction entered string theory only at its revival in the mid 1980s. Thus, the representation theory of affine algebras became an important ingredient of string theory (see [GrScWi]).
The vertex operators turned out to be useful even in the theory of finite simple groups. Namely, a twist of the homogeneous vertex operator construction based on the Leech lattice produced the 196883-dimensional Griess algebra and its automorphism group, the famous finite simple Monster group (see Sporadic simple group) [FrLeMe].
The vertex operator constructions were, quite unexpectedly, applied to the theory of soliton equations. This was based on the observation (see [DaJiKaMi]) that the orbit of the vector vΛ0 of the basic representation under the loop group satisfies an infinite hierarchy of partial differential equations, the simplest of them being classical soliton equations, like the Korteweg-de Vries equation.
