EXTENSIONS OF THE JACOBI IDENTITY 13
The coefficient of z2
1
*z3 *
3,
i,j Z, in (1.34) equals the formal expression
(zi + z2i)\zl + z3iyz£6 P 2 1 + Z 3 2 ) . (1.35)
Extracting then the coefficient of z\, k G Z, in (1.35), we see that (1.35)
and therefore (1.30) exist. The existence of (1.31) is seen the same way, by
means of the coefficient of z^1"1 z^1"3z^i"k, i,j, k £ Z (and without the need
of Proposition 8.8.5 of [6]).
For the equality of the two expressions note that
- U
fZl
+
Zn
+
Z32\
-l*
fZ21
+
Z32\
/i
QZ\
z3 8 [ ) z31 8 (-^-J (1-36)
can be obtained by replacing Z2\ by Z21 + 2:32 in
^(^)*,f©- (L37)
Applying Proposition 2.1.1 of [6], (1.37) can be written as
^ ( £ L ± * ) ^ (
S
) , (1,S)
and (1.36) can be written as
•1* { ^ ^ ) z£S ( £ » ± ? » ) . (1.39)
This shows that (1.31) is equal to (1.34) and consequently to (1.30). Q
Lemma 1.3 The expression on the right-hand side of (1.25), the result of
multiplying it by
-1 o fz3i-Z2i\
and the expression (1.28) all exist.
Z3
Proof The existence of the right-hand side of (1.25) and of the result of
the multiplication are seen by extracting the coefficient of
z\z2z
and, respec-
tively, ^2^3^32^21? hhk,l G Z (cf. [6], Chapter 8). Analogous statements
hold for the right-hand side of (1.26). Since (1.27) is the sum of two such
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