Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -79,14 +79,35 @@ :::  ::: proof-Apply $\operatorname{Res}_x\operatorname{Res}_u u^{m+1}$ to @prop:jacobi-identity for the fields $Y(\omega,u)$ and $Y(a,z)$. Expanding $(z+x)^{m+1}$ in nonnegative powers of $x$ gives--$$-[\omega_{(m+1)},Y(a,z)]-=\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(\omega_{(j)}a,z).-$$--Use $\omega_{(j)}=L_{j-1}$ and $Y(L_{-1}a,z)=\partial_zY(a,z)$.+Use @prop:jacobi-identity with first state $\omega$ and field coordinates $u,z$. Taking $\operatorname{Res}_x$ turns each of the first two delta kernels into $1$. Multiplying by $u^{m+1}$ and taking $\operatorname{Res}_u$ therefore gives++$$+\operatorname{Res}_u u^{m+1}[Y(\omega,u),Y(a,z)]+=[L_m,Y(a,z)].+$$++On the right-hand side, the $u$-residue of the delta kernel is++$$+\operatorname{Res}_u u^{m+1}u^{-1}+\delta\!\left(\frac{z+x}{u}\right)+=\iota_{z,x}(z+x)^{m+1}+=\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}x^j.+$$++Thus the remaining $x$-residue is++$$+\begin{aligned}+[L_m,Y(a,z)]+&=\operatorname{Res}_x\left(+\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}x^j\right)+\left(\sum_{n\in\mathbb Z}Y(\omega_{(n)}a,z)x^{-n-1}\right)\\+&=\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(\omega_{(j)}a,z).+\end{aligned}+$$++The residue selects $n=j$. Since $\omega_{(j)}=L_{j-1}$, this is the claimed formula; lower truncation makes the sum finite even for $m<-1$. The three special cases use $Y(L_{-1}a,z)=\partial_zY(a,z)$. :::  ## Weights and dilations