Table of Contents
1 Introduction 1
2 Construction of the affine Lie algebras 7
2.1 A\1] (I 1), D\x) (I 1) and E\X) {I = 6,7,8) 7
2.2 B™ (n 2), # (n 2), F4(1) 11
2.3 4 1 i (n 2), D& (n 2), E^ 15
2.4 G{2] 17
2.5 Zi3) 21
2.6 4 n (^ 1) 23
3 The Main Theorem 29
3.1 Integral bases of the universal enveloping algebras of the affine Lie
algebras 29
3.2 Exponential identities for the simply-laced affine Lie algebras 38
3.3 Exponential identities for B£\ C^ and F4(1) 47
3.4 Exponential identities for G\,' 49
3.5 Exponential identities for A^n-n ^i-j-i a n d EQ 51
3.6 Exponential identities for Z4 53
3.7 Exponential identities for A^ 55
4 Vertex algebras and integral forms of the universal enveloping alge-
bras of the affine Lie algebras 58
4.1 Vertex operator algebras and vertex algebras 58
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