4 S. PREVOST

nouncement [B] to find a (nonexplicit) description of an integral form (but not a

Z-basis) for the universal enveloping algebra of a simply-laced affine Lie algebra. To

show that the Z-subalgebra defined by Borcherds is indeed an integral form, we use

the discussion in [L-P] and [Mac] concerning Schur polynomials and a basis of the

symmetric algebra 5'(h.z). Schur polynomials also arise in calculations involving the

Virasoro operator L(—1). We first demonstrate that the Z-subalgebra defined by

Borcherds contains (with a slight alteration) the Z-form generated by the Z-basis

given by Garland [G] and Mitzman [M]. (The explicit description is found in Theo-

rem 4.2.6 below.) Then using a standard diagonal map argument, we prove that the

(slightly altered) Z-form obtained from the Z-basis in Theorem 4.2.6 must contain

the Z-subalgebra defined by Borcherds.

We define a Z-subalgebra of the enveloping algebra of an unequal root length affine

which is analogous to the Z-subalgebra described in Borcherds' announcement [B] for

the enveloping algebra of a simply-laced affine. We then show this is an integral form

of the enveloping algebra, and is (essentially) the same integral form generated by

the Z-basis of Theorem 4.2.6.

The material of this paper is organized in the following manner.

In Chapter 2 we briefly rewrite Chapter 2 and Chapter 3 of [M], i.e., the construc-

tion of the simply-laced affine algebras and the unequal root length affine algebras.

To recover the unequal root length affines, we consider these algebras as subalge-

bras of the simply-laced affines fixed by a suitable extension v of a Dynkin diagram

automorphism of order two or three (cf. [K-K-L-W], [M]).

In Chapter 3 we give our treatment of the main theorem of [G] and [M]. This

theorem (Theorem 3.1.6) gives an explicit description of an integral basis for the

universal enveloping algebra for any of the affine Lie algebras discussed in Chapter 2.

While the overall idea of the proof of this theorem is the same as that of Theorem 4.2.6

in [M], the proofs of the commutator identities needed for the straightening arguments

are all new. Except for Proposition 3.2.2 below, the arguments which prove the