4
CRISTIANO HUSU
The main result of the second section is the Jacobi identity for relative
Z2-twisted vertex operators, namely,
zt0
a',6' [ zl ~ z2
i
x
-*',&'
i
-t z~
*
o1s(^T^jY^z1)Y4v,z2)-
-c(a,b)z-a'}'
R - 4 - W—-)y.(V2)n(v») =
. lz-1+«',S-
(fl±i2)la'y
£
W
((i + 2)» + (-D'
*1 ' »=0.1 \ A *2 -
,(*1 ~ Zo)*
p=0,l
.y.(y
z
.(^u,
ato
)t;,^ I
(-i)p-v"
y
2
^2
(Theorem 2.1), which holds for vectors u and v in Vjr,, that are, roughly
speaking, elements of the symmetric algebra tensored with elements a and
6 of a central extension of the lattice. Several comments are necessary to
read this identity correctly. Binomial expressions are to be expanded as
power series in the second variable. A choice of a branch of the logarithm
is necessary to allow the possibly complex power a', V . We denote
by the symbol a the element of the lattice corresponding to a and by a' the
projection of the lattice element a on the fixed nondegenerate subspace of the
complexification of L\ 0 is an automorphism of the central extension of the
lattice such that 0 = 1 and
02
= 1; A is a homomorphism from the lattice to
±1 defined by the condition that, as operator on T, 0a = X(a)a. Y*i*(u, z)
is the relative vertex operator constructed in [2]; c(a,b) is the commutator
of a and 6.
In the third and last section we combine the constructions of the previ-
ous sections in the particular case of the twisted version of A^ , for which a
twisted vertex operator construction was found in [9]. We recall the defini-
tion of this affine Lie algebra in the first subsection of Section 3. Then in the
second subsection, following a particular construction of untwisted relative
vertex operators as in [2], we construct the relative (Z2-twisted) vertex op-
erator Y*(ya, z) which, acting on level-fc standard A[ -modules, is equivalent
to the Z-algebra operator which was used to determine the structure of the
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