16
CRISTIANO HUSU
[Y(v4,z4) xZil [Y{v3,z3) xZ31 \Y{v2,z2) xZ21 y(vj,*i)]]] =
= Y(Y(v4,z41)Y(v3,z31)Y(v2,z2l)vl,z1) f[ z^6 (2—&\ (1.47)
and similar identities (with the same product of ^-functions) obtained by
permuting 2,3,4. The next step is to consider
Y([Y(v4,z41) xZi2 [Y(v3,z31) xZ32 YfaznW^zJ f [ z^8 ( 5 i 2
(1.48)
which, again using (1.23), equals
Y(Y(Y(v4,z42)Y(v3,z32)v2lz2l)v1,zl) l j [
z
- ^ ( ^ ^ j J
Similar expressions are obtained permuting the indices 3 and 4 in (1.48) and
(1.49) (again with the same products of ^-functions). Then, in the same
spirit, we form
Y(Y([Y(v4,z42)xZi3Y(v3,z32)]v2,z2l)vuZl)l[ f[ z^S f ^
1
"
ZA
,
(1.50)
where we set
^to
=
2i,
1 i 4.
Applying the Jacobi identity, on the one hand, we write (1.50) as
Y(Y(Y(Y(v4,z43)v3,z32)v2,z21)v1,z1) JJ 'v-i* fr'*'1 ~ **) , (1-51)
ltj4 V *t,i-l /
while, on the other hand, using (1.47) (1-49), we see that (1.50) equals
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