4 P. BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE
when the category it applies to is that of formal power series in a formal parameter
with coefficients in associative algebras.
For instance, Drinfel'd twisting elements in elementary quantum group theory
constitute examples of formal UDF's (see e.g.
[CP95]).
Other formal examples
in the Hopf algebraic context have been given by Giaquinto and Zhang
[GZ98].
In
[Zag94],
Zagier produced a formal example from the theory of modular forms.
The latter has been used and generalized by Connes and Moscovici in their work
on codimension-one foliations
[CM04].
In
[Rie93],
Rieffel proves that von Neumann's oscillatory integral formula
[vN31]
for the composition of symbols in Weyl's operator calculus actually consti-
tutes an example of a non-formal UDF for the actions of
!Rd
on associative Frechet
algebras. The latter has been extensively used for constructing large classes of ex-
amples of noncommutative manifolds (in the framework of Connes' spectral triples
[Co94])
via Dirac isospectral deformations1 of compact spin Riemannian mani-
folds
[CLOl]
(see also
[CDV]).
Some Lorentzian examples have been investigated
in
[BDSR]
and
[PS06].
Other very interesting related approaches can be found in
[HNW, Ga05]
and references quoted therein.
Oscillatory integral UDF's for proper actions of non-Abelian Lie groups have
been given in
[Bie02, BiMs, BiMa, BBM].
Several of them were obtained
through geometrical considerations on solvable symplectic symmetric spaces. Nev-
ertheless, the geometry underlying the one in
[BiMs]
remained unclear.
In the present work we build on these works in the AdS context, with when
needed reminders of their main features so as to remain largely self-contained.
First we show that the latter geometry is that of a solvable symplectic symmetric
space which can be viewed as a curvature deformation of the rank one non-compact
Hermitian symmetric space.
[It
can also be viewed as a curvature deformation of
the AdS space-time, as we shall see in the last section of the article.]
Next, we develop some generalities on UDF's for groups which act strictly
transitively on a symplectic symmetric space. We give some precise criteria. We
end the section by providing new examples with exact symplectic forms such as
UDF's for solvable one-dimensional extensions of Heisenberg groups, as well as
examples with non-exact symplectic forms.
In the last section we apply these developments to noncommutative Lorentzian
geometry. In anti de Sitter space AdSn23, every open orbit
Mo
of the Iwasawa
component
AN
of 80(2, n) is canonically endowed with a causal black hole struc-
ture
[CD07]
(generalizing the BTZ-construction in dimension n
=
3).
We define
the analog of a Dirac-isospectral noncommutative deformation for a triple built on
M
0

The deformation is maximal in the sense that its underlying Poisson structure
is symplectic on the open AN-orbit
M
0

In particular, it does not come from an
application of Rieffel's deformation machinery for isometric actions of Abelian Lie
groups. Moreover, via the group action, the black hole structure is encoded in the
deformed spectral triple, with no other additional geometrical data, in contradis-
tinction with the commutative level2

1
A deformation triple (
Ao, Ho, Do)
is said isospectral when
Ho
and
Do
are the same for
all values of
0.
2
An interesting challenge would be to analyze which operator algebraic notions attached to
the triple are responsible for the singular causality. That is not investigated in the present article.
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