The radical of a vertex operator algebra
  The radical $J(V)$ of a vertex operator algebra $V$ is defined to be the subspace of $V$ consisting of vectors $v$ such that the zero mode $o(v)=0$ on $V$ where $o(v)=v_{wt v-1}$ if $v$ is homogeneous. We establish various facts about $o(v),$ including the determination of $J(V)$ which is shown to be essentially equal to $(L(0)+L(-1))V.$ 
C. Dong, H. Li, G. Mason, P. Montague