Relations for annihilating fields of standard modules for affine Lie algebras

Abstract

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In a joint work with Arne Meurman this approach is developed further in the framework of vertex operator algebras. The main ingredients of that construction are defining relations for standard modules and relations among them. The arguments involve both representation theory and combinatorics, the final results hold only for affine Lie algebras $A_1^{(1)}$ and $A_2^{(1)}$. In the present paper some of those arguments are formulated and extended for general affine Lie algebras. The main result is a kind of rank theorem, guaranteeing the existence of combinatorial relations among relations, provided that certain purely combinatorial quantities are equal to dimensions of certain representation spaces. Although the result holds in quite general setting, applications are expected mainly for standard modules of affine Lie algebras.