ABSTRACT. In this work we recover the construction of Kac-Kazhdan-Lepowsky-Wilson (1981) of the basic modules for the affine Lie (K) (K) (K) algebras of types A , D and E using a new method. We start with an even lattice and an automorphism which has certain properties similar to those of the Coxeter element of the Weyl group and build the whole theory from these properties. We define the vertex operators on a certain vector space and compute their brackets directly. This computation gives rise to cocycles which satisfy the appropriate conditions to construct a finite-dimensional semisimple Lie algebra. In particular, we prove that the vertex operators (together with certain other operators) provide an irreducible representation of the twisted affinization of that finite-dimensional Lie algebra.
When the lattice is a root lattice of type A, D or E and the automorphism is a Coxeter or twisted Coxeter automorphism we obtain the "principal realization" of the affine Lie algebra of (K) (K) (K) type A , D or E and the representation we have constructed is the basic representation in the theory of Kac-Moody Lie algebras.

Download File Size:3.62 MB

---