VERTEX OPERATORS AND INTEGRAL BASES 5 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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