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:
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