T{E). The equivalence of nonlinear representations is defined by an intertwining operator
{V, K) ~ (V", K) ^ V'g = AV9A~\ A G T(K) invertible (1.42)
and V is said linearizable if V ~ V1, V1 linear.
1.6.1.b Associated linear maps. Take K = \Jn1 ®p=\(®pK) ("Fock space without vac-
uum"). A one-to-one algebra homomorphism A : T(K) —• C{K) mapping the composition
law o to the product of linear maps is defined by:
A(/) = £ £ £ r «•••»/*'• (i-43)
n\ lpnii + ...+ip=n
Its differential dA defined by
**(/ ) = £ E E
n l Kpn0gp- 1
maps the bracket Df.h Dh.f of formal series (which is therefore a Lie bracket) into the
commutator of the linear operators dA(f) and dA(h), where Df.h is defined by, cf. (1.7"):
= J2 E
E ^ ^ ' « v , 4 (i-45)
nllpn 0qp-l
A nonlinear representation of a Lie algebra g on K is a Lie morphism of g into ^F(K)
equipped with this bracket. Given a formal nonlinear representation (V, K) of G it is equiv-
alent to a nonlinear representation $ (V',K) of G such that the mapping g *-* V^1Vgn,
n 1 is differentiate from G to Csn (K) (see [10]). Let K^ be the space of differentiate
vectors of (V1, K). We remind that K^ can be realized as the linear span of the vectors
of the form p = fG Vg1ipa(g)dg, with V e K and a G C§?(G), [8]. If X G g and p G ifoo,
d y defined by X t— dVx is a nonlinear representation of g on ifoo- Conversely, if G is
connected and simply connected, if (Vl,K) is a linear representation of G on K and if
i^oo is its space of differentiable vectors, given a nonlinear representation S of g on Koo
such that 5 1 = dV1, there exists a unique nonlinear representation (V, K^) of (7 such that
dV = 5 (see proposition 9 of [10]).
1.6.2 NonLinear Representations and Cohomology.
1.6.2.a Cohomology. Let G be a group, (it, If) a linear representation of G in if, g,g' G G.
Let r G N. The set Cr(G,K) of r-cochains is the set of maps c : G x - x G if (r
arguments). The coboundary operator S : G r (G, if) G r + 1 (G , if) is defined by
(5c)(#i,...,pr+i) = ugic{g2,...,0r+i) + ( - l ) r + 1 c ( # i , . . . , gr)
+ ^ ( - l ) V ( p i , . ,0i_i,0t&+i, . . . ,£r+l). (1-46)
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