Generating function identities for untwisted standard modules of affine Lie algebras

Abstract

The present work generalizes to affine Lie algebras of type A,D or E the generating function identities used to construct bases of untwisted representations of the affine Lie algebra . For this, we apply multi-operator extensions of the Jacobi identity for generalized vertex algebras to the relative vertex operators associated to a dijrect sum of copies of the weight lattice of a Lie algebra of type A D or E. Then, using the properties of the δ-function, we construct mtalti-operator extensions of the Jacobi identity for relative untwisted vertex operators, and we obtain generating function identities for standard representations of untwisted affine Lie algebras as combinations of coefficients of these multi-operator Jacobi identities.