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56
N.M. Katz:
Proof
According to (3.5.1.2), we have
(3.5.5.1)
Lie (P)(a (v)) =
dr(v)
whence for any integer my > 1, we have
(3.5.5.2)
Lie (p)mv (a (v)) = Lie
(p)mv1 (df(v)) = d (pm~I (f(v))).
Let us expand the second member of (3.5.5.0), remembering that the a(v),
and hence the Lie(P)m~(a(v)), are
closed
forms.
d(P m'l(f(1))
~ ( 1) i+1
m~ em'~(f(i)) 1[ Lie(p)mv(a(v)))
i*l
v*l,i
(3.5.5.3)
=Lie(p)m'(a(1))~(1)i+lmipm'l(f(i)) ~I Lie(p)mv(a(v))
i#l
+ e=''(f(1)) E ( 1)
'+1
rn, Lie(P)"'(a(i)) 1I
Lie(p)m"(a(v))"
i.1
The first term in (3.5.5.4) is the part of the first member of (3.5.5.0) corre
sponding to i. 1. Thus it remains to see that
ml P"'x(f(1)) II
Lie(p)m'(a(v))
(3.5.5.4)
v*l
= P"' '(f(1)) Z ( 1) i+~ m, Lie (P)"'(a(i)) 1I Lie
(P)m~(a
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 Fall '11
 NormanKatz

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