MAXWELL - DIRAC EQUATIONS
5
It was proved in [28] that this is a linear map from U(p) to the vector space of differentiable
maps from E^ to E^ (see formula (1.10) of [28] and the sequel). For completeness we
recall the proof in [28]. The vector field Xx, X G p , defines a linear differential operator dx
of degree at most one operating on the space C°°(£*£,) by dxF = DF.TX, F G
C00{E00).
The fact that X H- TX is a nonlinear representation of p on E^ implies that X i— dx is a
linear representation of p on linear differential operators of degree at most one. This linear
continuous representation has a canonical extension Y i— dY to U(p) on linear differential
operators of arbitrary order operating on
COC(EPQ).
If e
y
, Ye U(p), is the part of dY of
degree not higher than one, then Y •-» eY is a linear map of U(p) into the space of linear
partial differential operators of degree at most one. Let Y e U(p). We write Y = Z + a,
where a G C I and Z has no component on C I (relative to the natural graduation of
U(p)). Then the previous definition of TY gives eYF = DF.Tz + aF, which proves that
Y t— Ty is a linear map on U(p). Moreover, it was proved in [28] that
dt
where
TY{t)(u(t))=TPoY{t)(u(t)), YeU(p), (1.10)
y(t)=exp(tadp
0
)r, adPoZ=[P0,Z], (1.11)
if
u(t)=Tfl,(u(t)). (1.12)
We note that definition (1.11) makes sense since (adp0)n, n 0, is a linear map from U(p)
to U(p) leaving invariant the subspace of elements of degree at most /, I 0, in U(p) and
since then exp(ta,dp0)Y = 2no(
n
0
- 1
(^
a
dpo)
n
^ converges absolutely. For completeness
we also recall the proof of (1.10). Since ^Y{t) = adpoy(£), it follows from (1.9) and (1.12),
that
^TY{t){u{i)) = T
Ay(t)
(u(t)) 4- (DTY(t).TPo)(u(t))
= T[Po,Y(t)]{u(t)) + TY(t)Po(u{t)) = TPoY(ty
TY, Y G U(p), denotes the n-homogeneous part of Ty and we shall identify TY with
a n-linear symmetric map and also with a continuous linear map from ® E^ into E^
where 0 is the n-fold complete tensor product endowed with the projective tensor product
topology. T¥(u), r^
n
(w) and U^{u) (resp. T$(u),
Tfn{u)
and U°(u)) is the projection
of TY(u), T$(u) and Ug(u) on M? (resp. D). We also define fY,Y G U{p) by
TY=TY+fY. (1.13)
Since it does not bring any contradiction, we shall also denote by
TM1
or
T1M
(resp.
TD1
or T1D) the linear representation of p which is the restriction of the linear representation
T1
to MP (resp. D). Similarly,
U™1
or
U]M
(resp.
U^1
or
U*D)
denotes the restriction
of U\ to MP (resp. D).
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