BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE
PROOF. The algebra 1 can be written 1
JRA EBact JRD EBact
split extension, hence a general element reads
can use Campbell-Baker-Hausdorff formula to split it into a factor in Ro and one
are some auxiliary functions):
ERe surjective on R.'
The conclusion is that
is the group
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
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
(ii) The open 'k-orbit
endowed with a black hole structure, identifies
with the group manifold
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
THEOREM 4.6. Let u and v be smooth compactly supported functions on R
Define the following three-point functions:
cosh(a1- a2)xo, cosh(a2- ao)x1, cosh(ao- a1)x2)
sinh (2(ao- a1))z2;
A:= [cosh (2(a1- a2)) cosh(2(a2- ao)) cosh(2(ao- al))
cosh a1 - a2 cosh a2 - ao cosh ao - a1 .
[ ( ) ( ) ( )J
Then the formula
as a left-invariant associative Hilbert algebra structure. In par-
ticular, one has the strong closedness11 property:
*~ 2 )
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
:= ;:- 1 o
11The notion of strongly closed star product was introduced in
in the formal context.