As an example, take the Virasoro algebra, i.e. $V$ is spanned by elements of the form $L\_{-2}^{k\_1} \cdots L\_{-n}^{k\_{n-1}} \Omega$ where $\Omega$ is the vacuum and $n \geq 2$. As I understand, we define


$$C\_2(V) = \{\psi^i\_{-h\_i-1}v | v \in V\}$$


where $i$ labels all the operators strongly generating the VOA, and $h\_i$ are the conformal dimensions. So, for the Virasoro algebra one has,


$$C\_2(V) = \{L\_{-3}v | v \in V\}.$$


Now it is claimed that $V/C\_2(V) \cong \mathbb C[x]$, where the isomorphism maps $L\_{-2}\Omega \mapsto x$. However, I do not see how this is the case. We are modding out all elements of the form $L\_{-3}v$, i.e. all those that contain a factor of $L\_{-3}$. Then say, $L\_{-10}\Omega$ cannot be part of an equivalence class with a representative given by a power of $L\_{-2}\Omega$, so I do not see how $V/C\_2(V)$ is just generated by $L\_{-2}\Omega$.


For reference, see page 14 of [Arakawa's paper.](https://arxiv.org/abs/1004.1492)



Zhu's $V/C_2(V)$ algebra
You have underdefined the $C\_2(V)$ subspace.


It is as you wrote but with $\psi^i$ being *any* element of $V$. You can intuitively think of this as being the subspace of iterated normally ordered products of vertex operators for which at least one is a derivative.


Thus in your Virasoro example, $L\_{-10}\Omega\in C\_2(V)$ by setting $\psi^i=\partial^7T$ where $T$ is the stress tensor.


