MAXWELL - DIRAC EQUATIONS 15

T{E). The equivalence of nonlinear representations is defined by an intertwining operator

A:

{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

h®r-p+1®iv-q-u

(i-44)

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"):

Df-h

= J2 E

fp(

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,

define

n2

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)

r

+ ^ ( - l ) V ( p i , • • . ,0i_i,0t&+i, . . . ,£r+l). (1-46)