identities use techniques different from those employed in [M]: the verifications of the
formulas are direct computations using Theorem 3.1.5 and the commutation relations
found in Chapter 2. Theorem 3.1.5 presents a new commutation identity involving
exponentials of formal power series (with zero constant term) whose coefficients are
elements of an affine Lie algebra. Although Proposition 3.2.2 is proved using a method
similar to the procedure Mitzman relied on, its statement is new; it is a generalization
of Lemma 4.3.4 (iii) in [M] for the simply-laced affines. We note that it is only in
this instance (Proposition 3.2.2) that we must resort to such tactics (i.e., a proof
which doesn't compute the formula directly)! Our treatment of the formulas for the
unequal root length affines differs completely from that of [M] and [G]. We prove the
identities in a case by case manner; the cases are divided naturally by the order of
the graph automorphism and by the inner products of the different roots. We first
derive the identities for the simply-laced affines and then use these formulas (along
with Theorem 3.1.5 and the Baker-Campbell-Hausdorff Theorem) to give constructive
proofs of the needed identities for the unequal root length affines.
In Sections 4.1 and 4.2 we present a brief account of the background material
needed to place the problem of finding an explicit description of a Z-form for the
universal enveloping algebras of the affine Lie algebras within the realm of vertex
operator theory. Section 4.2 introduces the notion of vertex operator algebra as
found in [F-L-M]. Here we also define the closely related concept of vertex algebra.
Such structures were first discussed in [B], although Borcherds' definition does not
include either the Jacobi identity or the properties of the Virasoro algebra. In the
same section we also describe the construction of a vertex operator algebra VL for
an even positive definite lattice L. The reader may wish to consult Chapters 5 - 8
of [F-L-M] for a more thorough account. Then Section 4.2 recalls the construction
of [F-K], [Seg] of the simply-laced affine Lie algebras using the central extension of
the lattice L utilized in Section 4.1 to construct VL- We also give a definition of
a Chevalley basis and a description of such a basis for the simply-laced affine Lie
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