Next we examine the vertex algebra approach Borcherds uses in his announcement
[B] to give a (nonexplicit) description of an integral form of the universal enveloping
algebra of a simply-laced affine Lie algebra. Then we extend the method found in [B]
to give an analogous description of a Z-form of the remaining unequal root length
A vertex algebra consists of a Z-graded vector space V = LLGZ Vn together with
a linear map v i-» Y(v, z) from V into the vector space (End V)[[z, z'1]] and two dis-
tinguished vectors 1 and w in V so that a number of properties, such as the Jacobi
identity, hold. If in addition to the above properties the vector space V satisfies the
conditions that dim Vn oo for all n and dim Vn = 0 for sufficiently small n, then V
is called a vertex operator algebra. The formal Laurent series Y(v, z) = ]£n€Z vnZ~n~1
are called vertex operators. Since vertex operator algebras are actually vertex alge-
bras, one could use the term "restricted" vertex algebra for vertex operator algebra.
However, we choose to use the two reference terms "vertex algebra" [B] and "vertex
operator algebra" [F-L-M] to distinguish the two concepts in our work; both struc-
tures are useful. We also note that our definition of vertex algebra and the definition
of vertex operator algebra (which is the same as in [F-L-M]) are modifications of
Borcherds' definition of a vertex algebra used in [B].
An integral (or Z-) form Az of an algebra A over a field F is an algebra over Z
such that Az ®z F = A. Integral bases and integral forms for finite-dimensional Lie
algebras were first formulated in general by Chevalley [Chev]. These bases, called
Chevalley bases, have integral structure constants which are related to root strings
and are such that the Z-span of the Chevalley bases is invariant under the action
of ' q ^ , where xa is a root vector contained in the Chevalley basis and m i s a
nonnegative integer. The latter property led to the construction of Chevalley groups
(of adjoint type) ([Chev], cf. [H], [Stein]). Later Kostant [Kost], and also Cartier (cf.
[T]), discovered and gave an explicit description of a Z-form generated by the elements
*~r~ for the universal enveloping algebras associated to these finite-dimensional Lie
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