QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES 17
w' and m' are complicated functions of
(/3, ')', J)
and
l
is given by
(4.5)
l(/3
J)
=
-J((3b
+
')'C
+
Jd)
'')', 1 _
e-f3b-"fc-lid '
which is not surjective except when b
=
c
=
d
=
0. Taking the inverse a general
element of
RuH
reads [
e-ww
-mM -lM
eJ
1 h
u
J,
where the range of
l
is not the whole
JR.
Since the action of
R'
is
simply
transitive,
R
is not surjective on
RuH.
When b
=
c
=
d
=
0, the conditions for (4.3) to be an algebra are easy to solve,
leaving only two
a priori
possible two-parameter families of algebras:
Algebra 1.
1
A=J1+2.h+sV
B=W
C=M
D=L+rV
[A,B] =
B+sC
[A,C]
=
~C
1
[A, D]
=
2sB
+
2. D
[B,D]
=
-rC.
with
r =1-
0. The general symplectic form on that algebra is given by
(4.7)
-(3
0
0
0
-')')
2~r
0 '
0
Since det w
= (
2 ~r)
2
we must have
(3
=1-
0,
r
=1-
0. That algebra will be denoted by
t
1
.
The analytic subgroup of R whose Lie algebra is
t
1
is denoted by R
1
.
Algebra 2.
A=
J1
+rJ2 +sV
B=W
C=M
D=L.
[A,B]
=
B+sC
[A,C]=(r+1)C
[A,D]
=
2sB
+
(1- r)D
There is no way to get a non-degenerate symplectic form on that algebra.
REMARK
4.4. One can eliminate the two parameters in algebra
t 1
by the iso-
morphism
(4.8)
(
1 0
0 1
¢
=
0
2sr
0 0
0 0 )
0 4s
1/r
4s2 /r
0 1
which fixes
s
=
0 and
r
=
1 and transforms
t 1
into the algebra defined by [A', B']
=
B'
[A' C']
=
'lC'
[A' D']
= l
D'
[B' D']
=
-C'
' ' 2 ' ' 2 ' ' .
It is now easy to prove that
PROPOSITION
4.5.
The group
R
1
of algebra
t 1
acts transitively on U, i.e.
RuH
=
R1uH.
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