QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES 9
actually lies in the semisimple part .s[(b). For any
x
E
.s[(b), we denote by
x
xs
+
xN, xs, xN
E
.s[(b), its abstract Jordan-Chevalley decomposition. Observe
that, denoting by .s[(b) = .s[+ EB.sL the decomposition in
(±1)-~-
eigenspaces, one
has that
a
C
.s[_. Also, the algebra
aN:= {aN}aEa
is an Abelian subalgebra in .s[_
commuting with
a.
Setting
as:= {as}aEa,
we have
DEFINITION
3.3. A gAs is called weakly nilpotent if Jb(as) C 3b(aN) and7
aN
c
Der(b).
Let be =: EBaEP ba be the weight space decomposition with respect to the
action of
as.
Note that for all
a,
one has
aN.ba
C
b0
.
Moreover, for all
Xa
E
ba
and
as
E
as,
one has
a(as.Xa)
=
a(as)a(Xa)
=
aasa- 1aXa
= ~(as).a(Xa) =
-as.a(Xa)·
Therefore,
-a
E
If and aba = b-a· Note in particular that abo= bo.
DEFINITION
3.4. An involutive Lie algebra (g, a) is called weakly nilpotent
if there exists a sequence of subalgebras { ai}oir of g such that
(i)
a0
is a weakly nilpotent gAs of g with associated supplementary ideal
b(O).
(ii) ai+l is a weakly nilpotent gAs of 3bil ( ai) (0 :::; i :::;
r -
1) where, for
i ~ 1, b(i) denotes the a-stable ideal of
Jb(i-lJ
(ai-d associated with ai.
(iii) 3brl ( ar) is Abelian.
PROPOSITION
3.5.
Assume the iLa (g, a) to be weakly nilpotent. Then there
exists a (complex) subalgebra
.s
of
g
such that the restriction prp
Is :
.s -----. p
is a linear
isomorphism.
PROOF.
Let
a
be a weakly nilpotent gAs of g and set
Va
:= ba EB La for
every
a
E
If. Note that
V0
= b0 and that each subspace
Va
is then a-stable and
one sets
Va
= ta EB Pa for the corresponding eigenspace decomposition. Choose a
partition8 of lf\{0} as lf\{0} =: q+ U
q,-
with the properties that -IJ+ =
q,-
and
that if
a,
(3
E
q+ with
a+
(3
E
If then
a+
(3
E
q+. One has b = EBaEP+
Va
EB bo.
We set b+ := EBaEP+ ba and p+ := prp (b+). It turns out that the restriction map
prplb+ : b+-----. p+ is a linear isomorphism. Indeed for
X
E
be:= b n t and
a E
as
one has
a(a.X)
=
aaaaX
=
~(a).X
=
-a.X;
hence as.be
C
p. Therefore, for
all
X
E
ba n t
a
-=/=-
0, one can find
a
E
as
such that
a.X
=
X
E
p n
£;
thus
ban t = 0 as soon as
a
-=/=-
0, yielding ker(prplb+) = 0. The last condition of
Definition 3.3 implies that
as
acts by derivations on b, hence the usual argument
yields that b+ is a subalgebra normalized by
a
EB b0
.
Moreover the first condition
7The
last condition is automatic when b is Abelian. Observe also that it is satisfied when
p( a)
is contained in a Levi factor of the derivation algebra Der( b).
8Such
a partition can be defined as follows. Let ~be a Cartan subalgebra of sl(b) containing
as and let A E
~*
denote the set of weights of the representation of sl(b) on b. Note that
the restriction map
.
-+
.las
from A to
as
is surjective onto 1. Let
~
=
~IR
Ell
i~JR
be a real
decomposition such that the restriction of the Killing form to
~IR
is positive definite. Every weight
in A is then real valued when restricted to
~IR
[KnaOl].
Now, any choice of a basis of
~IR
defines a
partial ordering on A with the desired properties. To pass to the set of weights cl, consider the
~JR-
components,
a~,
of as viewed as a vector subspace
of~-
The restriction map p: cl--+ .PlaiR is then a
s
bijection. Indeed, for. E ker(p), one has, by ((linearity,
.(a+ia') =.(a) +i.(a') =OVa, a'
E a~.
Hence.= 0 as an element of 1. Therefore, an order on A induces on cl an order having the same
properties.
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