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.

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.