24
CRISTIANO HUSU
T h e o r e m 1.11 Let Vi,v2,- -vn be a family of vectors in a vertex operator
algebra and let 1 m j n; then
Y(vn,vn_1,'--v1;zn,zn_1,---z1;{zij 3 i j};m;\jj]) =
= ¥(¥(••• Y(Y(vn,znm)Y(vn^
(1.72)
V 2 , 2 2 l H , Z l ) -
where both sides of this identity are existing expressions. In particular,
%
l
» » - i r «i;*n«n-i,• *i; tei 3 * j}) =
= Y(Y(- Y(Y(vn, «„,»_i)wB_i, a^-Ln-a) t2, *2iM, *i)- (1.73)
n ^(^r^V
ltin \ *M-1 /
We call the identity (1.73) the multi-operator Jacobi identity.
Proof The existence of the right-hand side of(1.72) is a direct application
of Lemma 1.9. The existence of the left-hand side of (1.72) is a consequence
of Lemma 1.9 and the following identities, which also show the equality of
the two expressions.
Using (1.23) and the linearity of vertex operators, we obtain
Y(vn, v
n
_i,..., vi; znj z
n
_i,..., zX] {z{j\i j}; 2; [j, j]) =
= Y([Y(vn,znl) xZn2 [Y(vn^lyzn.1A) x2n_12 [Y(v3,z31) xZ32 Y(v2,z2i)]
] ] ^ i ) n ^ (
£ L
^ ) =
j=2 \ Z\ J
= r(y(r(n,,jnJ)y(vi,vy) *3, z32K *2iK *i)- (i-74)
(n^(^))(n^(^e)).
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