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.