18
P.
BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE
PROOF. The algebra 1 can be written 1
=
JRA EBact JRD EBact
Span{B,
C},
a
split extension, hence a general element reads
r
1
(a,
{3, "(,
8)
=
eAeoD e!3W+"YM.
One
can use Campbell-Baker-Hausdorff formula to split it into a factor in Ro and one
in
R'
(where
f
and
g
are some auxiliary functions):
(4.9)
r1(a, {3,"(,
8)
=
~ehe(f(o)+!3)w(g(o)+"Y)M
ERe surjective on R.'
0
The conclusion is that
R
1
is the group
R
that we were searching for. To
summarize, the structure is as follows.
(i) The AdS space is decomposed into a family of cells: the orbits of a sym-
plectic solvable Lie group
R
as in Proposition 4.2 above. Note that these
cells may be viewed as the symplectic leaves of the Poisson generalized
foliation associated with the left-invariant symplectic structure on
n.
(ii) The open 'k-orbit
M
0
,
endowed with a black hole structure, identifies
with the group manifold
'k.
4.2. Deformation triples for Mo.
4.2.1. Left-invariant Hilbert function algebras on Ro. In this section, we present
a modified version of the oscillatory integral product (2.2) leading to a left-invariant
associative algebra structure on the space of square integrable functions on
R
0 .
THEOREM 4.6. Let u and v be smooth compactly supported functions on R
0
.
Define the following three-point functions:
(4.10)
and
S :=Sv(
cosh(a1- a2)xo, cosh(a2- ao)x1, cosh(ao- a1)x2)
- &
sinh (2(ao- a1))z2;
0,1,2
A:= [cosh (2(a1- a2)) cosh(2(a2- ao)) cosh(2(ao- al))
1
cosh a1 - a2 cosh a2 - ao cosh ao - a1 .
[ ( ) ( ) ( )J
dimR.o-2] 2
Then the formula
(4.11)
(2)
1
1
liS
u
*
0
v :
=
Odim Ro
A e
e
u
0
v
R.o xR.o
extends to
L
2
(R0
)
as a left-invariant associative Hilbert algebra structure. In par-
ticular, one has the strong closedness11 property:
J
u
*~ 2 )
v
=
J
uv .
PROOF. The oscillatory integral product (2.2) may be obtained by intertwining
the Weyl product on the Schwartz spaceS (in the Darboux global coordinates (2.1))
by the following integral operator
[BiMs]:
T
:= ;:- 1 o
(¢01)*
oF,
11The notion of strongly closed star product was introduced in
[CFS]
in the formal context.
Previous Page Next Page