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.

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.