QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES 11
3.3. Examples.
3.3.1. One dimensional split extensions of Heisenberg algebras. The strategy is
as follows. First we observe that (the connected simply connected group associated
to) every such extension acts simply transitively on the symmetric space
R
0
,
and
is therefore diffeomorphic to it. Then, we remark that this diffeomorphism can be
chosen to be symplectic. The latter is almost a homomorphism, up to an action of a
subgroup of the automorphism group of the kernel on
R
0
(this is done by embedding
the extension in a subgroup of the automorphism group of the kernel containing
R
0 ).
Finally we check that the pullback of that kernel by the diffeomorphism gives
an invariant kernel on the extension.
So let us start with a Heisenberg Lie algebra
bn =
V
EB
JR.Z,
where (V, Ov) is a
2n-dimensional symplectic vector space, Z is central, and [v, v']
=
Ov( v, v')Z. One
easily sees that a split extension (with
i
o p
=
Id)
tn
of
bn
by a= JR.A:
p
0
----+
bn
----+
tn
+==!a
----+
0,
i
must be such that, for all v
E
V
and z
E
JR., [A, v
+
zZ]
=
X.v
+
p,(v)Z
+
2dzZ with
X
E
End(V), p,
E
V* and d
E
JR. such that X- did
E
.sp(V, Ov ). As symplectic
form on the associated group, we choose the left-invariant 2-form whose value at
the identity is the Chevalley 2-coboundary
n
=
-oZ*
=
Z*[·, ·],which is a natural
generalization of that on
R
0
in
[BiMs].
This 2-coboundary is nondegenerate if and
only if d
=/=-
0, which we will assume from now on9
,
and consider extensions with
parameters dX, dp, and 2d, which we will denote by (dX, dp,, 2d).
We have the following symplectic Lie algebra isomorphisms:
(i) (dX,dp,,2d)
~
(X,p,,2) for all dE JR.0
,
through the map L(a,v,z)
(da,
v, z).
(ii) (X, p,, d)
~
(X, 0, 2) for allp,
E
V*, through the map L(a, v, z)
=
(a, v
+
au,
z) with iuf2v
=
p,.
(iii) (X, 0, 2)
~(X',
0, 2) through the map L(a, v, z)
=
(a, M.v, z), if and only
if ME Sp(V, Ov) is such that MXM-
1
=X', i.e. if X- Id and X'- Id
belong to the same adjoint orbit of Sp(V, Ov).
Thus we concentrate on algebras of type
(X,
0, 2) from which we can recover the
quantization of the others.
Now let
to
=
(I,
0, 2),
t'
=
(X,
0, 2), and
R
0
,
R'
the associated groups. Ele-
ments of these algebras will be denoted respectively by aA+v+zZ and aA' +v+zZ.
The difference between the actions of
A'
and
A
on
V
is
X
:=
X-
Id which lies
in .sp(V, Ov ). Extending the action X on
to
by [X, aA
+
v
+
zZ]
:=
X.v, we can
therefore view
to
and
t'
as subalgebras of the semidirect product g
=
to
x
.s
of
to
by
.s
=
span(X)
C
.sp(V, Ov ). At the level of the groups (S corresponding to
.s),
on the one hand we can identify
R
0
with
g
IS
as manifolds, and on the other
hand as a subgroup
of(),
R'
acts on the quotient. That action is simply transi-
tive. Indeed, on R
0
and R', we have global coordinate maps I(aA
+
v
+
zZ)
=
exp(aA) exp(v
+
zZ) and I'(aA'
+
v
+
zZ)
=
exp(aA') exp(v
+
zZ) such that
I'(aA'
+
v
+
zZ)
=
exp(aX).I(aA
+
v
+
zZ), giving a decomposition g
=
sr
E
R'
with
s
E
S
and
r
E
R
0 .
Thus, acting on
eS
E
g
IS,
such an element
g
gives
9Quantizing
extensions with d = 0 requires a non exact 2-form. Since our method needs an
exact one, we shall have to apply it on central extensions of our algebras. An example of that
procedure is given in the next section.
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