Generators of relations for annihilating fields

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For an untwisted affine Kac-Moody Lie algebra g, and a given positive integer level k, vertex operators x(z)=∑ x(n)z⁻ⁿ⁻¹, x, generate a vertex operator algebra V. For the maximal root θ and a root vector xθ of the corresponding finite-dimensional g, the field xθ(z)ᵏ⁺¹ generates all annihilating fields of level k standard g-modules. In this paper we study the kernel of the normal order product map r(z)⊗ Y(v,z):r(z) Y(v,z): for v∈ V and r(z) in the space of annihilating fields generated by the action of ddz and g on xθ(z)ᵏ⁺¹. We call the elements of this kernel the relations for annihilating fields, and the main result is that this kernel is generated, in certain sense, by the relation xθ(z)ddz(xθ(z)ᵏ⁺¹)= (k+1)xθ(z)ᵏ⁺¹ddzxθ(z). This study is motivated by Lepowsky-Wilson's approach to combinatorial Rogers-Ramanujan type identities, and many ideas used here stem from a joint work with Arne Meurman.