14 P. BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE
With this choice of generators, the minimal parabolic subalgebra .s :=
a
EB n with
n :=
Ef)~=O 9ai
has the following multiplication table:
(3.4a) [H1, N1] = N1, [H1, N2] = N2,
(3.4b) [H2, N1] = -N1, [H2, N2] = N2,
(3.4c)
(3.4d)
[H1, N3] = 2N3,
[No,N1] = -N2,
the other brackets being zero. Setting
[H2, No] = 2No,
[N1, N2] = 2N3;
(3.5) .s1 := Span{H2, No} and .s2 := Span{H1, N1, N2, N3},
one observes that .s is a split extension of .s2 by .s1:
(3.6)
Note that .s1 is a minimal parabolic subalgebra of .su(l, 1) while .s2 is a minimal
parabolic subalgebra of .su(l, n). In particular, the Lie algebra .sis exact symplectic
w.r.t. the
element~:=
6
EB
6
of .s* with
~i
E
.st
(i
= 1, 2) defined as
6
:=
N0
and
6
:=
N;,.
One therefore obtains a UDF for proper actions of
AN
by direct application
of the above extension lemma 3.8.
4. Isospectral deformations of anti de Sitter black holes
4.1.
Anti de Sitter black holes.
Anti de Sitter (AdS) black holes have
been introduced by Baiiados, Teitelbaum, Zannelli and Henneaux
[BTZ, BHTZ]
as connected locally AdS space-times
M
(possibly with boundary and corners)
admitting a
singular
causal structure in the following sense:
CONDITION BH. There exists a closed subset
S
in
M
called the
singularity
such that the subset Mbh constituted by all the points x such that every light like
geodesic issued from
x
ends in
S
within a finite time is a proper open subset of
Mphys :=
M\S.
Originally such solutions were constructed in space-time dimension 3, but they
exist in arbitrary dimension
n ;::::
3 (see
[BDSR, CD07]).
More precisely the
structure may be described as follows. Take
9
:= 80(2,
n
-1) (the AdS group), fix
a Cartan involution
e
and a B-commuting involutive automorphism
a
of
9
such that
the subgroup
1i
of
9
of the elements fixed by a is locally isomorphic to 80(1, n-1).
The quotient space
M
:=
9
/1i
is an n-dimensional Lorentzian symmetric space, the
anti
de
Sitter space-time.
It is a solution of the Einstein equations without source.
Let g denote the Lie algebra of
9
and denote by g =
~
EB q the ±!-eigenspace
decomposition with respect to the differential at e of a that we denote again by a.
Denote by g = t EB
j)
the Cartan decomposition induced by
e,
consider a a-stable
maximally Abelian subalgebra
a
in
j)
and choose accordingly a positive system of
roots. Denote by n the corresponding nilpotent subalgebra. Set
n
:= B(n),
t
:= nEBn
and
t
:=
a
EB
n.
Finally denote by R :=
AN
and R := AN the corresponding
analytic subgroups of
Q.
One then has
PROPOSITION 4.1.
[BDSR, CD07J The groups R and R admit open orbits
and finitely many closed orbits in the AdS space M. Prescribing as singular the
union of all closed orbits (of R and R) defines a structure of causal black hole
on an open subset Mphys in M (in the sense of the above condition (BH)). In
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