QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES
13
operator
Teas
in [BiMs] works here, giving thus rise to the same kernel (2.2). This
reflects again the fact that our groups are all subgroups of the automorphism group
of a symplectic symmetric space on which they act simply transitively.
3.3.2.
Non exact example.
As mentioned before, all the examples of symplectic
symmetric Lie groups shown up to now were endowed with an exact symplectic
form, as our method requires exactness in order for the left translations to be
strongly Hamiltonian. We present here an example with a non exact symplectic
form showing that, as expected, considering a central extension allows to apply the
same method.
Lett= span (A, V1, V2, W1, W2, Z) be the Lie algebra defined
[A,
Wi] = -Wi,
[A,
Z] = 2Z, [V1, V2] = Z, and choose the non
exact1°Chevalleylti,2-=lti][A,by
cocycle 0: O(A, Z) = 1, O(lti, Wi) = 1, O(VI. V2) = 1/2. We define g as the central
extension oft by the element
E
with commutators [X,
Y]
9
= [X,
Y]~
+
O(X,
Y).E
for all
X, Y
E
g, where we extended [·,
·]~
and 0 by zero on
E.
Then 0 =
-JE*
is
a 2-coboundary.
Now the connected simply connected Lie group
n
whose Lie algebraist can
be realized as the coadjoint orbit 0 of
E*
in g*, and a global Darboux chart
J
from
JR6
to 0 is given by:
J(qi,PI.
q2,p2,
q3,p3)
=
exp(q3H)
exp((p3
+
q1P1
+
q2P2- PIP2/2)Z)
exp((q2
+
pi/4)WI) exp(p2V1)
exp((q1- P2/4)W2) exp(p1 V2) ·
E*.
In this chart, the (dual) moment maps are linear in the
Pi,
so that the Moyal
product is covariant. From now on, the method outlined above can be carried on
the same way as in [BiMs], with the same integral operator
Te
as before.
3.3.3.
The Iwasawafactor
ofsp(2,
JR.)~
so(2, 3). The Lie algebra go :=
sp(n,
JR.)
of the group
90
:=
Sp(n,R)
is defined as the set of 2n
x
2n real matrices
X
such
that
T
X F
+
F X
=
0
where
F
:= ( _
~n
1
0
) .
One has
(3.1) go= {(
~ ___;~
) where
A
E
Mat(n
X
n,R)
and
si
=
Tsi
;i
= 1,2}.
In particular,
F E
90
and a Cartan involution of go is given by (} :=
Ad(F).
The
corresponding Cartan decomposition go = to
EB
Po is then given by
(3.2) (
S S' )
to~
u(n) and Po= { S' -S },
where the matrices S and S' are symmetric. For n = 2, a maximal Abelian subal-
gebra
a
in Po is generated by
H1 =En- E33
and H2 = E22-
E44
where as usual
Eij
denotes the matrix whose component are zero except the element
ij
which is
one. The restricted roots
\I
w.r.t
a
are then given by
(3.3)
with
ao
:= 2H2, a1 := Hi -
H2,
hence a2 := Hi
+
H2
and a3 := 2Hi, where
Ht(Hj)
:= Jij· The corresponding root spaces ga; (i = 0, ... , 3) are one-dimensional,
generated respectively by
No= E24,
N1 = E12-
E43,
N2 = E14
+
E23,
N3
= E13·
100ne
can actually show that every symplectic 2-cocycle on
1:
is non exact and that what
follows can be applied to any one of them.
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