16 INTRODUCTION

The coboundary operator satisfies 62 = 0. The set of r-cocycles, Zr(G,K) is the subset

of elements c G

Cr(G1K)

such that 6c = 0. The set of r-coboundaries,

Br(G,K)

is the

subset of elements c G Cr(G,K) such that c = 6p for some p G Cr~l{G,K). We have

Br{G,K) C Z r (G, if), since S2 = 0. The r-cohomology with coefficients in if is the

quotient

tfr(G, if) = Z r (G, K)/Br{G, if). (1.47)

1.6.2.b An extension (^,F) of (it, if) by (u'^K') is an exact sequence of G-modules 0 —•

if — » F — if' —• 0 and is given by a 1-cocycle of G for the representation t in C(K', if)

defined by C{K', if) B A H £yi4 = ugAu'_

1

G £(if',if) . The representation is equivalent

to the direct sum if this cocycle is a coboundary.

1.6.2.c Nonlinear representations as successive extensions. Let (V,K) be a nonlinear rep-

resentation of G, V = YlnLi Vn- The group law gives:

* & = W + W ' ® ^), (1-48)

i.e. Rg = V ? ^ 1 . ! ® KLi) is a 1-cocycle of G in C(S2K,K) for the representation given

by:

£(®2SK, K)3A^tgA= VfAiV,1-! ® V^)- (1.49)

Therefore (by A) on if, F is obtained by successive extensions of V1 by its (symmetric)

tensorial powers (g^V1, n 2: first F 1 by V 1 ® F 1 , then the result by ^V1 etc. Thus

one has [11]:

1.6.2.d T H E O R E M: If (V,if) is a nonlinear representation of a Lie group G in a Frechet

space of differentiable vectors, and if the cohomology spaces H1 (G, £(gJif', if)) — {0}

for all n2, then V is linearizable.

Scheme of proof: Rg = V^fV^ g V ^ ) is a 1-cocycle in Zl{G,C(®2SK, K)), therefore it

is the coboundary of some B2 G C(g2K, if) :

Vg2 = V]B2 - B2{®2Vgx) (1.50)

and V is equivalent to a representation

g*-+{I- B2)'1 oVgo(I- B2) = Vg1 + [terms of order 3] (1.51)

and so on, so that we get the linearizing operator J ^ ^

2

( / — Bk).

We make the following remarks concerning the application of this theorem:

i) The fact that the infrared cocycle R^2, defined by (1.26), is nontrivial shows that in

the case of the Poincare group the cohomology space H1 is nontrivial for n = 2, when the

linear part "describes" both massive and zero mass particles. This is in contrast with the

case where only massive particles are present, for which H1 = {0} for n 2, as was proved

in [31].

ii) It was proved in [12] that the time translation subgroup is formally linearizable in the

case of the Yang-Mills equations.