Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -232,38 +232,43 @@  ## The weight-one Lie algebra -::: proposition [Lie algebra and derivations] {@prop:voa-weight-one}+::: proposition [The weight-one Lie algebra] {@prop:voa-weight-one} If $V$ is of CFT type, then $V_1$ is a Lie algebra with bracket  $$-[a,b]=a_{(0)}b.-$$--Its zero modes act on $V$ by derivations:--$$-[a_{(0)},Y(b,z)]=Y(a_{(0)}b,z),-\qquad-[a_{(0)},b_{(0)}]=(a_{(0)}b)_{(0)}.-$$--The zero-mode action need not be faithful.-:::--::: proof-Weight compatibility gives $a_{(0)}b\in V_1$ for $a,b\in V_1$. Taking the $z^{-1}$ coefficient of skew-symmetry gives+[a,b]:=a_{(0)}b,\qquad a,b\in V_1.+$$+:::++::: proof+The bracket is bilinear, and weight compatibility gives $a_{(0)}b\in V_1$. Taking the $z^{-1}$ coefficient of skew-symmetry gives  $$ a_{(0)}b=-b_{(0)}a+T(b_{(1)}a). $$ -Here higher terms vanish by nonnegative weights, while $b_{(1)}a\in V_0=\mathbb C\mathbf1$, so the last term is zero. Taking $\operatorname{Res}_x\operatorname{Res}_z$ in @prop:jacobi-identity gives the field commutator, and then taking its zero mode gives the second identity. Acting on $d\in V_1$ yields--$$-[a,[b,d]]-[b,[a,d]]=[[a,b],d],-$$--which is the Jacobi identity. The field commutator also proves the derivation property on all of $V$.+Higher terms vanish by nonnegative weights. Since $b_{(1)}a\in V_0=\mathbb C\mathbf1$ and $T\mathbf1=0$, the bracket is antisymmetric.++Now we prove the Jacobi identity holds for the Lie algebra. Taking $\operatorname{Res}_x\operatorname{Res}_z\operatorname{Res}_w$ in @prop:jacobi-identity gives++$$+[a_{(0)},b_{(0)}]=(a_{(0)}b)_{(0)},+$$++where the bracket on the left is the commutator of endomorphisms. Applying this identity to $d\in V_1$ yields++$$+a_{(0)}(b_{(0)}d)-b_{(0)}(a_{(0)}d)+=(a_{(0)}b)_{(0)}d,+$$++or, in terms of the bracket on states,++$$+[a,[b,d]]-[b,[a,d]]=[[a,b],d].+$$++Together with antisymmetry, this is the Jacobi identity. :::  ## References