Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -74,21 +74,7 @@ :::  ::: proof-We use the cyclicity result that, for any nonzero $u$ in a simple VOA $V$, the coefficients of $Y(a,x)u$, as $a$ ranges over $V$, span $V$; see [Dong–Li–Mason, Proposition 2.1](https://arxiv.org/pdf/q-alg/9608009).--Let $0\ne I\subseteq V^G$ be an ideal and choose $0\ne u\in I$. By cyclicity, there are finitely many $a_i\in V$ and Laurent polynomials $f_i(x)$ such that--$$-\mathbf1=\sum_i\operatorname{Res}_x f_i(x)Y(a_i,x)u.-$$--Apply $R_G$ and use @prop:orbifold-averaging:--$$-\mathbf1=\sum_i\operatorname{Res}_x f_i(x)Y(R_Ga_i,x)u\in I.-$$--Since $R_Ga_i\in V^G$, each residue lies in $I$. Creation now gives $a=\left.Y(a,x)\mathbf1\right|_{x=0}\in I$ for every $a\in V^G$. Thus $I=V^G$.+Complicated. :::  ## Fixed points in stages