20 P.

BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE

• 'Yi and

r;

denote respectively the Dirac "(-endomorphism and the spin-

connection element associated with X;.

In that expression the elements 'Yi's and f;'s are constant. However, already at the

formal level, a left-invariant vector field

X

as infinitesimal generator of the right

regular representation does not in general act on the deformed algebra. In order to

cure this problem, we twist the spinor module in the following way.

DEFINITION 4.9. Let dr g be a right-invariant Haar measure on

R

0

and consider

the associated space of square integrable functions L;ight(Ro). Set

1i

:=

L;ight(Ro) 0

C2

;

and denote by

1i00

the space of smooth vectors in 1i of the natural right represen-

tation of

R

0

on 1i. Then intertwining

*~

2

)

by the inverse mapping

~: L;ight(Ro)--+ L

2

(Ro): ~(u)(g)

:=

u(g-

1

)

yields a right invariant noncommutative L;ight (Ro)-bi-module structure (respec-

tively a (L;ight(R0

)) 00

-bi-module structure) on 1i (resp.

1i00

).

The latter will be

denoted 1io (resp. 1i(J).

We summarize the main results of this paper in the following:

THEOREM 4.10. The Dirac operator D acts in 1i(J as a derivation of the non-

commutative hi-module structure. In particular, for all a E (L;ight (Ro) )

00

,

the

commutator

[D,

a]

extends to 1i as a bounded operator. In other words, the triple

(L;ight(Ro) )

00

,

1i(J, D) induces on

Mo

a pseudo-Riemannian deformation triple.

5. Conclusions, remarks and further perspectives

To the AdS space we associated a symplectic symmetric space ( M, w, s). That

association is natural by virtue of the uniqueness property mentioned in Propo-

sition 4.2. The data of any invariant (formal or not) deformation quantization

on

(M,

w,

s) yields then canonically a UDF for the actions of a non-Abelian solv-

able Lie group. Using it we defined the noncommutative Lorentzian spectral triple

(A00

,

1i, D) where

A

00

:=

(L;ight(Ro))

00

is a noncommutative Frechet algebra mod-

elled on the space of smooth vectors of the regular representation on square inte-

grable functions on the group

R

0 .

The underlying commutative limit is endowed

with a causal black hole structure encoded in the R 0-group action. A first ques-

tion that this construction raises is that of defining within the present Lorentzian

context the notion of causality at the operator algebraic level.

Another direction of research is to analyze the relation between the present ge-

ometrical situation and the corresponding one within the quantum group context.

Indeed, our universal deformation formulas can be used at the algebraic level to

produce nonstandard quantum groups 80(2, n- l)q via Drinfeld twists. An inter-

esting challenge would then be to study the behaviour of the representation theory

under the deformation process.

More generally the somewhat elliptic sentence with which we started the paper

may now be better understood if we remark that the physical motivation section

and the quantum group framework suggest to study a number of questions related

to (noncommutative) singleton physics, in particular: