10 P.

BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE

of the same definition implies [a,

b0

]

=

0. The proposition follows by induction.

One starts applying the above considerations to

a

=

a0

and b

=

b(o). This yields

a subalgebra .s0 supplementary to Jb(o) (a0

)

and normalized by it. One then sets

g1

:=

JboJ(a0), considers a weakly nilpotent

a1

in g1 and gets a subalgebra .s1

that is now supplementary to and normalized by

JWl

(a1). Applying this procedure

inductively, one gets a sequence a subalgebras .So, ... ,.Sr-l..Sr

:=

Jbr(nr) n p such

that .Si+l normalizes .Si and .Si n

.Sj

=

0, and defines .s

:=

EBi.Si·

D

REMARK 3.6. In the exact symplectic case, one has Po

..l

a

since .;[a, bo]

=

0.

3.2. Darbou:x charts and kernels: an extension lemma. Let (.si, ni) with

i

=

1, 2 be symplectic Lie algebras, and denote by (Si, wi) the corresponding simply

connected symplectic Lie groups (wi being left-invariant). Given a homomorphism

p: .s1 --+ Der(.s2) n.sp(fl2), form the corresponding semi-direct product .s

:=

.s1 Xp.S2

and consider the associated simply connected Lie group S endowed with the left-

invariant symplectic structure w defined by

We

:=

n1 EB n2 (

e

denotes the unit

element in

S).

LEMMA 3.7. If Pi: (.si,ni)--+ (S;,wi) (i

=

1,2) are Darboux charts, the map

¢ : (.s, n

:=

fl1 EB fl2)

---+

(S, w) : (X

1.

X2)

~

¢2(X2) .¢1 (X1) is Darboux.

PROOF. For

X

E .s1 and

Y

E .s2 one has:

L-~..-1¢*(X) = £-~..-1

(Rq,1*

(¢2.x))

= £-~..-1 £-~..-1

Rq,

1

¢2*X

'~-'*

q;,

"P*

4 c/2

'+'1

*

'+'2

* *

=

Ad¢11

(Lr1 /J2*x)

2 *¢2

while for Y E .s1 one has

L-~..-1

¢*(Y)

=

£-~..-1 £-~..-1

Lq,2*¢1*Y

=

£-~..-1

¢1*Y.

'I'

*P

'1'1 * '1'2 * '1'1 *

Hence

wq,(X, Y)

= n(Ad¢1 1 (£-~..-1

¢2*X),

£-~..-1

¢1*Y)

=

0,

lf'2 *¢2 lf'l

*

because the first (resp. the second) argument belongs to .s2 (resp. to .s1), and for

X, X'

E

.s2,

wq,(X,X')

=

n(Ad¢11(Lq,-1

¢2*X),

Ad¢11(Lq,-1

¢2*X'))

2 *¢2 2 *¢2

=

Ad¢11*n2(Lr1

¢2*X, Lq,-1 ¢2*x')

2 *c/2 2 *P2

=

n2 (Lr1 ¢2*X, Lr1 ¢2*x')

2 *¢2 2 *¢2

= w2*"'2(x, x') =

n2(X,

x').

A similar (and simpler) computation applies for two elements of .s1

.

D

A direct computation shows

LEMMA 3.8. Let Ki E Fun((Si)

3

)

be a left-invariant three point kernel on Si

(i

=

1, 2}. Assume K

2 ®

1 E

Pun((S)3)

is invariant under conjugation by elements

of S1. Then K

:=

K

1

®

K2 E

Pun((S)3)

is left-invariant (under S).

In particular, given associative kernels satisfying the above hypotheses, their

tensor product defines an associative invariant kernel on the semidirect product S.

We will call it extension product of

K

2

by

K

1

•