The radical of a vertex operator algebra

Each v ∈ V has a vertex operator Y (v, z) = ∑ n∈Z vnz −n−1 attached to it, where vn ∈ EndV. For the conformal vector ω we write Y (ω, z) = ∑ n∈Z L(n)z . If v is homogeneous of weight k, that is v ∈ Vk, then one knows that vn : Vm → Vm+k−n−1 and in particular the zero mode o(v) = vwtv−1 induces a linear operator on each Vm. We extend the “o” notation linearly to V, so that in general o(v) is the sum of the zero modes of the homogeneous components of v. Then we define the radical of V to be J(V ) = {v ∈ V |o(v) = 0} (1.1). The problem arose in some work of the first three authors in [DLiM] and in work of the fourth author in [M] of describing J(V ) precisely. We will essentially solve this problem in the present paper in an important special case, namely that V is a vertex operator algebra of CFT type. This means that the Z-grading on V has the shape