On the Homology of Free Lie Algebras

Comment.Math.Univ.Carolinae 39,4 1998661{669 661

On the homology of free Lie algebras

Calin Popescu

Abstract. Given a principal ideal domain R of characteristic zero, containing 1=2, and a

connected di erential non-negatively graded free nite typ e R-mo dule V ,we prove that

the natural arrow LFH V ! FH LV is an isomorphism of graded Lie algebras over R,

and deduce thereby that the natural arrow UFH LV ! FHU LV is an isomorphism

of graded co commutative Hopf algebras over R; as usual, F stands for free part, H for

homology, L for free Lie algebra, and U for universal enveloping algebra. Related facts

and examples are also considered.

Keywords: di erential graded Lie algebra, free Lie algebra on a di erential graded mo d-

ule, universal enveloping algebra

Classi cation : 17B55, 17B01, 17B70, 17B35

Letting H , L and U resp ectively denote the homology functor, the free Lie

algebra functor and the universal enveloping algebra functor, Quillen [10] has

shown inter alia that, given a eld K of characteristic zero, if V is a di erential

graded K -vector space, then the natural arrow L H V ! H L V is an isomor-

phism of graded K -Lie algebras, and if L is a di erential graded K -Lie algebra,

then the natural arrow UH L ! HU L is an isomorphism of graded co com-

mutative K -Hopf algebras. This is no longer the case in non-zero characteristic

[2], [8] or if the ground eld is replaced by a commutative ring of characteris-

tic zero, containing 1=2 [1], [9], [11]. However, in this latter situation, under

suitable reasonable hyp otheses, the natural arrow UH L ! HU L is still an

isomorphism up to a certain dimension, which dep ends on the rst non-invertible

prime in the ground ring and the connectivity of L [1], [9], [11]. Within an

appropriate framework, factoring torsion out in homology seems to b e a rst step

towards recovering Quillen-like results, i.e., corresp onding induced isomorphisms

in al l dimensions: if, for instance, R is a principal ideal domain of characteristic

zero, containing 1=2, and L; d is a connected di erential non-negatively graded

Lie algebra over R , with a free nite typ e underlying mo dule, then the natural

arrow UF H L ! FHU L is an isomorphism of graded co commutative Hopf

algebras, provided that ad xdx = 0, for homogeneous x in L [9]; here

ev en

F stands for free part, aduv is the Lie bracket [u; v ] of the elements u and v

of L so ad uv =[u; [u;:::[u; v ] :::]], n nested Lie brackets, and = R

denotes the least non-invertible prime in R of course, if Q R ,we set = 1

1 1

= ad = 0. If the \nilp otency" condition is removed and agree that ad

away in the preceding, then the natural arrow UF H L ! FHU L might no

662 C. Pop escu

longer b e an isomorphism, though it still is a monomorphism in al l dimensions

[9]. This failure can, however, b e remedied and Quillen-like results recovered by

considering suitably generated free Lie algebras:

1. Theorem. If R is a principal ideal domain of characteristic zero, containing

1=2, and V is a connected di erential non-negatively graded R -free mo dule of nite

typ e, then the natural morphisms L FH V ! FH L V , of graded R -Lie algebras,

and UF H L V ! FHU L V , of graded co commutative R -Hopf algebras, are

b oth isomorphisms.

Consequently, FH V emb eds into FH L V , as a submo dule, via L FH V ,

and FH L V emb eds into FHU L V , as a sub Lie algebra, via UF H L V .

Before pro ceeding, let us make some remarks up on the ingredients.

2. Remarks. 1 As already noticed in the intro duction, within the context

under consideration, the natural morphisms UF H ! FHU are always

monic [9], so, for such an arrow to b e an isomorphism it is sucient that it b e

epic.

2 The imp ortance of factoring torsion out in homology, in order to obtain

Quillen-like results, is easily shown by considering L x; dx over R = Z[1=2],

with x of degree 2: the natural morphisms L H x; dx ! H L x; dx and

UH L x; dx ! HU L x; dx are b oth trivial in p ositive dimensions, though

H L x; dx and UH L x; dx are not, their rst relevant comp onents arising in

dimension 4, where b oth equal R=3[dx; [x; dx]].

3 Let L be a connected di erential non-negatively graded R -Lie algebra,

with a free nite typ e underlying mo dule. If Q denotes the quotient eld of R ,

then, by standard identi cations and Kunneth isomorphisms, the \rationalized"

natural arrow Q UF H L ! Q FHU L is essentially Quillen's isomorphism

R R

UH Q L ! HU Q L [9], so FHU L=R if and only if UF H L=R ,

R R

and this is obviously the case, if and only if FH L=0 in UF H L=R , take

primitives b oth sides.

Back to our theorem, let L = L V in the preceding and deduce that FH L V =

0if and only if FH V =0; this is easily seen by identifying, as usual, the co-

commutative Hopf algebras U L V and T V , where T denotes the tensor algebra

functor, and noting that TF H V ! FHT V , as co commutative Hopf algebras,

by the Kunneth theorem. Our theorem is thus quite straightforward for V with

FH V =0 e.g., for acyclic V .

4 Under the hyp otheses in the theorem, we can therefore exhibit FH L V as

a free graded Lie algebra on FH V , a remarkable fact that do es not necessarily

hold for free graded Lie algebras equipp ed with a decomposable di erential i.e.,

a di erential sending the generating mo dule W into L W . This is easily seen

by considering the minimal mo del of Quillen for the complex pro jective plane

[12]: L x; y ;d over the rationals, with x of degree 1, y of degree 3, dx =0

and dy =[x; x].

On the homology of free Lie algebras 663

5 Finally,ifR is a eld of characteristic zero, weobviously recover Quillen's

cited result on the natural morphism L H V ! H L V .

The remainder of the pap er is almost entirely devoted to the pro of of the

theorem, of which the following lemma, adapted from [1], is an imp ortant step.

3. Lemma. Let R b e a principal ideal domain of characteristic zero, containing

1=2; let further L; @ b e a connected di erential non-negatively graded Lie algebra

over R , with a free nite typ e underlying mo dule; and let, nally, W; be

a connected di erential non-negatively graded free nite typ e R -mo dule, with

FH W;=0.

If either @ = 0 or W; is acyclic, then the canonical injection L; @ !

L; @ q L W; induces an isomorphism FH L; @ ! FH L; @ q L W; of

graded Lie algebras.

If, in addition, the natural arrow UF H L; @ ! FHU L; @ is an isomor-

phism of graded co commutative Hopf algebras, then so is the natural morphism

UF H L; @ q L W; ! FHU L; @ q L W;.

4. Remark. Letting, as usual, = R denote the least prime or 1 not

invertible in R see the intro duction, the condition on UF H L; @ ! FHU L; @

in the second half of the lemma is satis ed if, for instance, ad x@x = 0, for

homogeneous x in L [9], this latter b eing automatically ful lled for @ =0

ev en

on L or on all of L, or for -nilp otent L with , or for Q R .

ev en

Proof of the Lemma: The canonical injection : L; @ ! L; @ q L W;

is clearly a rightinverse for the surjection :L; @ q L W; ! L; @ , sending

L; @ identically onto itself and L W; to zero. Then H is a rightinverse for

H and there results a trivial connecting morphism in the long exact homology

sequence asso ciated to the right split short exact sequence of di erential graded

Lie algebras

{

0 0

0 ! L ;@ ! L; @ q L W; ! L; @ ! 0;

0 0

in which, of course, L ;@ is the kernel of . Consequently,

H { H

0 0

0 ! H L ;@ ! H L; @ q L W; ! H L; @ ! 0

is a short exact sequence of graded Lie algebras with right splitting H , yielding

another short exact sequence of graded Lie algebras

FH { FH

0 0

1 0 ! FH L ;@ ! FH L; @ q L W; ! FH L; @ ! 0;

with a right R -splitting induced by H .

On the other hand, under the given hyp otheses, W; is free on a nite typ e

p ositively graded R -basis fu ;v g, with u = 0 and v = a u , a 2 R nf0g;

of course, for acyclic W;, wemay and will assume all a =1. It then follows

664 C. Pop escu

0 0 0

that L ;@ =L ad w u ; adw v ;@ , with w running through an R -basis

for U L, and @ satisfying

@ adw u = adU @ w u ;

0 jw j

@ adw v = adU @ w v +1 a adw u ;

where jw j denotes the mo dulo 2 reduction of the degree of w . Letting fx g and

fy g resp ectively equal the sets fadw u g and fadw v g,we see easily that:

0 0 0 0 0 0

a for @ =0, L ;@ =L W ;@ , where W ;@ is free on the nite typ e

0 0

p ositively graded R -basis fx ;y g, with @ x = 0 and @ y = b x ,

b 2 R nf0g; and

0 0 0

b for acyclic W;, L ;@ =L y ;@ y .

0 0

In either case, FH L ;@ =0 cf. Remark 23, and the rst half of the lemma

follows by 1.

To prove the second half, consider the commutative diagram

UF H L; @ ! UF H L; @ q L W;

? ?

y y

FHU L; @ ! FHU L; @ q L W;

in which the vertical arrows are the resp ective natural morphisms UF H !

FHU , while the horizontal arrows are b oth induced by . The top arrowisan

isomorphism, by the rst half of the lemma, the left one is an isomorphism, by

hyp othesis, so the pro of will b e complete if the b ottom arrow is shown an isomor-

0 0

phism. To this end, observe, by the preceding and [2], [7], that U L ;@

0 0

! U L; @ q L W; , as left U L ;@ -mo dules and right U L; @ - U L; @

como dules, under U { U followed bymultiplication. A simple glance at the

0 0

! FHU L; @ q L W;, Kunneth theorem yields FHU L ;@ FHU L; @

0 0

and since FHU L ;@ =R cf. Remark 23, the lemma follows.

The pro of of the theorem is now straightforward.

Proof of the Theorem: Let d denote the di erential on V and note that, under

0 0 0

the assumed hyp otheses, V; d splits as V; d=V ;d FH V; d; 0, where V

0 0 0 0

is a d-stable submo dule of V , d is the restriction of d to V , and FH V ;d =0.

0 0

Observing further that L V; d=L V ;d q L FH V; d; 0, nothing remains but

0 0

apply the lemma with L = L FH V; d, @ = 0 and W;=V ;d .

5. Remark. There is another argument proving that the natural arrow

UF H L V ! FHU L V is an isomorphism: by Remark 21, it suces to show

it epic, which amounts to prove the corresp onding induced morphism b etween the

resp ective indecomp osables epic [7]; and this is easily seen by noting that this

latter morphism is essentially the identityonFH V | of course, the rst part

of the theorem along with the Kunneth theorem playa key r^ole in the argument.

The theorem now provides the following completion of the lemma.

On the homology of free Lie algebras 665

6. Corollary. If R is a principal domain of characteristic zero, containing 1=2,

if L; @ is a connected di erential non-negatively graded Lie algebra over R ,

with a free nite typ e underlying mo dule, and if W; is a connected di erential

non-negatively graded free nite typ e R -mo dule, then the canonical pro jection

L; @ q L W; ! L; @ , sending L W; to zero, yields an R -spilt short exact

sequence

0 ! L fad uw g ! FH L; @ q L W; ! FH L; @ ! 0

of graded Lie algebras, with u and w running through R -bases of FHU L; @ and

FH W;, resp ectively.

And if, in addition, the natural arrow UF H L; @ ! FHU L; @ is an isomor-

phism of graded co commutative Hopf algebras, then so is the natural morphism

UF H L; @ q L W; ! FHU L; @ q L W;.

The pro of go es along the lines in the pro of of the lemma and is hence omitted.

Wenow consider some examples. In all cases, the ground ring R , of course, is

a principal ideal domain of characteristic zero, containing 1/2; as usual, = R

denotes the least prime or 1 not invertible in R , and, for integer k 1, we set

0 0

N k; =k 2, where k =2bk=2+1c is the smallest even integer exceeding k .

Our rst example is inspired by the surjective mo del of Quillen for the Hopf

3 7 4 n

bration S ! S ! S [12]; as usual, S denotes the n-sphere.

7. Example. Consider a nite typ e graded set of generators fx ;y ;z g, with

x of degree 2n , y of degree 2n + 1, and z of degree 4n + 2, all n b eing

p ositive integers. Let further L = L x ;y ;z be endowed with the di eren-

tial d given by dx = 0, dy = a x , a 2 R nf0g, and dz = 2a [x ;y ].

A mere change of generators, z = z [y ;y ], renders d \indecomp osable", so

0 0

FH L; d=L z and FHU L; d=T z .

Next, we deal with a relatively general pattern that can also be related to

Quillen minimal mo dels, as examples in the sequel will show.

To b egin with, supp ose we are given the following short exact sequence of

di erential graded Lie algebras

{

0 0 00 00

2 0 ! L V ;d ! L V ;d ! L V ;d ! 0;

00 00

with right splitting : L V ;d ! L V ;d, and free nite typ e connected non-

0 00

negatively graded R -mo dules V , V and V ; we do not require that dV V .

As in the pro of of the lemma, we derive the short exact sequences of graded Lie

algebras b elow:

H { H

0 0 00 00

3 0 ! H L V ;d ! H L V ;d ! H L V ;d ! 0;

with right splitting H ; and

FH { FH

0 0 00 00

4 0 ! FH L V ;d ! FH L V ;d ! FH L V ;d ! 0;

666 C. Pop escu

with a right splitting induced by H . Under obvious identi cations, the theorem

allows us to rewrite 4 as

FH { FH

0 0 00 00

4 0 ! L FH V ;d ! FH L V ;d ! L FH V ;d ! 0;

0 0

with a corresp onding right splitting, so FH L V ;d = L FH V ;d

00 00

L FH V ;d , as R -mo dules.

On the other hand, since b oth 2 and 4 involve R -free ob jects of nite

typ e, the corresp onding universal enveloping algebras form, resp ectively, short

exact sequences of homology Hopf algebras [2]. Thus, under obvious identi -

0 0 00 00 0 0

cations, 2 yields T V ;d=T V ;d T V ;d , as left T V ;d -mo dules

00 00 0 0 00 00

and T V ;d -como dules, so FH T V ;d = FHT V ;d FHT V ;d =

0 0 00 00 0

TF H V ;d TF H V ;d , by the Kunneth theorem; similarly, 4 yields

0 0 00 00 0 0

UF H L V ;d = TF H V ;d TF H V ;d , as left TF H V ;d -mo dules

00 00

and TF H V ;d -como dules, and it should now be clear that the natural ar-

row UF H L V ;d ! FHU L V ;d is an isomorphism of graded co commutative

Hopf algebras.

0 0 00 00

If either H V ;d or H V ;d is R - at e.g., if either of them is R -free,

0 0 00 00

then [2] H T V ;d = HT V ;d HT V ;d ; and if it happ ens that

0 0 00 00 0 0

H V ;d and H V ;d are both R - at, then [2] H T V ;d=THV ;d

00 00 0 0 00 00

THV ;d . Thus, if V ;d and V ;d b oth have R -free homologies, then

the homology of U L V ;d = T V ;d is R -free, as well: the natural arrow

UF H L V ;d ! HU L V ;d is an isomorphism of graded co commutative Hopf

algebras and it follows [4] that any minimal mo del W;D for the Cartan-

Chevalley-Eilenb erg complex C L V ;disdecomposable, and FH L V ;dmay

be regarded as the homotopy Lie algebra of W;D; as usual, denotes the

free commutative algebra functor. Furthermore, for k -connected V , with integer

k 1, in dimensions b elow N k; , H L V ;disR -free, and the natural arrow

UH L V ;d ! HU L V ;d is an isomorphism compatible with the Hopf alge-

bra structures [1], [9]. However, H L V ;d mighthave R -torsion in dimension

N k; orbeyond cf. Remark 2 2.

We now pro ceed to sp ecialize the preceding to several particular situations.

A rst such instance is related to the minimal mo del of Quillen for the pro duct

of two spheres [12].

8. Example. Given integers p 1, q 1 and r 0, consider L x; y ; z over

R , with x of degree p, y of degree q , and z of degree pr + q + 1, equipp ed with

the di erential d given by dx = 0, dy = 0 and dz = a ad xy , a 2 R nf0g.

Of course, for r =0, d is \indecomp osable" dz = ay , and we fall rightin the

context of the theorem. On the other hand, for r = 1, R = Q and a = 1, we

p+1 q +1 p+1

recover Quillen's minimal mo del for the pro duct S S of the spheres S

q +1

and S [12]. A direct computation of the Euler-Poincar e series [2] reveals

that the kernel of the natural pro jection L x; y ; z ;d ! L x; 0, with obvious

rightinverse L x ; 0 ,! L x; y ; z ;d, is the free di erential graded Lie algebra

On the homology of free Lie algebras 667

n n n

0 0

on fad xy ; ad xz g, n 2 N , whose di erential d satis es d ad xy =0

n n+r

0 np 0

and d ad xz =1 a ad xy , whatever n in N . Thus, d is \indecom-

0 0

p osable" i.e., the mo dule of generators is d -stable, so d preserves wordlength

and the preceding pattern applies: the graded co commutative Hopf algebras

UF H L x; y ; z ;d and FHU L x; y ; z ;d are isomorphic under the natural ar-

row, b oth b eing corresp ondingly identi ed to T fad xy g T x.

k =0;::: ;r 1

As for FH L x; y ; z ;d, it ts in the R -split short exact sequence of graded

Lie algebras

0 ! L fad xy g ! FH L x; y ; z ;d ! L x ! 0;

k =0;::: ;r 1

corresp onding to 4 . Consequently,

FH L x; y ; z ;d=L fad xy g L x ;

k =0;::: ;r 1

as R -mo dules, the Lie algebra structure for FH L x; y ; z ;d b eing sub ject to

r r r 2

ad xy =0 e.g.,ad xy = 0, [[x; x]; ad xy ] = 0 etc.. Thus,

L x; for r =0;

FH L x; y ; z ;d=

L x L y ; for r =1;

as Lie algebras, as well.

It seems worth remarking that

n n 0

H T x; y ; z ;d= HT fad xy ; ad xz g ;d T x;

n2N R

so, if a is a unit in R , then H T x; y ; z ;d = T fad xy g

k =0;::: ;r 1 R

T x=UF H L x; y ; z ;d, whichis R -free, and it follows that any minimal mo del

W;D for C L x; y ; z ;d is decomp osable, and FH L x; y ; z ;d maybe re-

garded as the homotopy Lie algebra of W;D; also, R -torsion cannot o ccur in

H L x; y ; z ;d in dimensions less than N min fp; q g;.

The situation describ ed b elow is another instance related to Quillen mo dels.

9. Example. Given integers p 1, q 1, r 1 and s 0, consider

L u; v ; x; y ; z over R , with u of degree p, v of degree p +1, x of degree q , y of de-

gree r and z of degree qs + r + 1, equipp ed with the di erential d given by du =0,

dv = au, dx =0, dy = 0 and dz = b ad xy , where a; b 2 R nf0g. For p =2,

q =1, r =4, s =1, R = Q and a = b =1,we recover the main ingredientofthe

2 5 7 4

surjective mo del of Quillen for the comp osition S S ! S ! S , where the rst

arrow is the pro jection smashing the 6-skeleton down, and the second is the Hopf

bration [12]. As in Example 7, a direct computation of the Euler-Poincar e series

reveals that the kernel of the natural pro jection L u; v ; x; y ; z ;d ! L x; 0,

with obvious rightinverse L x ; 0 ,! L u; v ; x; y ; z ;d, is the free di erential

n n n n

graded Lie algebra on fad xu; ad xv ; ad xy ; ad xz g, n 2 N , whose

668 C. Pop escu

n n n

0 0 0 nq

di erential d satis es d ad xu = 0, d ad xv = 1 a ad xu,

n n n+s

0 0 nq

d ad xy = 0 and d ad xz = 1 b ad xy , whatever n in N .

Mutatis mutandis, the considerations in the preceding example now rep eat verba-

tim.

We end with two more examples inspired by Quillen minimal mo dels. Recall

that the ab elian R -Lie algebra on a graded set fx g, denoted by hx i, is the free

graded R -mo dule generated by fx g.

10. Example. Fix integers n 2 and p 1 and consider L x ;::: ;x over

R , with x of degree 2ip 1, i = 1;::: ;n, equipp ed with the di erential d

given by dx = a=2 [x ;x ], i = 1;::: ;n, where a 2 R nf0g. For

i j

j +k =i

p =1, R = Q and a =1,we recover the minimal mo del of Quillen for the com-

plex pro jective n-space, CP [12]. A direct computation of the Euler-Poincar e

series [2] reveals that the kernel of the natural pro jection L x ;::: ;x ;d !

hx i; 0, with obvious rightinverse hx i; 0 ,! L x ;::: ;x ;d, has the form

1 1 1

0 0 0

L V; d, with V = hx ;x ;::: ;x ;x ;xi, where x = 1=2 [x ;x ],

j +k =i

0 0

i =2;::: ;n, x =1=2 [x ;x ], and, of course, dx = ax , dx =0,

i j i

i+j =n+1

i i

i = 2;::: ;n, and dx = 0. Thus, FH V; d = hxi, so FH L V; d = L x

and FHT V; d = T x, by the theorem. On the other hand, it should now

be clear that we can, mutatis mutandis, go through the previous general

pattern to obtain: FH L x ;::: ;x ;d = L x hx i, as Lie algebras |

1 1

note that a[x ;x] = d1=2 [x ;x ]; UF H L x ;::: ;x ;d =

1 1

i+1 j +1

i+j =n

T x x , as left T x-mo dules and right x -como dules; and

1 1

T x ;::: ;x ;d = T V; d x ; 0, as left T V; d-mo dules and right

1 1

x ; 0-como dules. Consequently, H T x ;::: ;x ;d=HT V; d x ,

1 1 1

so FH T x ;::: ;x ;d=T x x , which shows that the natural arrow

1 1

UF H L x ;::: ;x ;d ! FHU L x ;::: ;x ;d is an isomorphism of graded

n n

1 1

co commutative Hopf algebras. And if, in addition, a is invertible in R , then

H T x ;::: ;x ;d = T x x is R -free, so [4] any minimal mo del

1 1

W;D for C L x ;::: ;x ;d is decomp osable, and FH L x ;::: ;x ;d=

n n

1 1

L x hx i may b e regarded as the homotopy Lie algebra of W;D; it might once

again b e worth remarking that H L x ;::: ;x ;d is not necessarily R -torsion

free: e.g., for n = 2, letting y =[x ; [x ;x ]] and z =[x ;x ], R=3[y; [y; z ]] is a

1 1 2 2 2

direct summand in H L x ;x ;d .

1 2 24p+16

11. Example. Fix integer p 1 and consider L fx g over R , with

n=1;2;:::

x of degree 2np 1, n = 1; 2;:::, equipp ed with the di erential d given by

dx =a=2 [x ;x ], n =1; 2;:::, where a 2 R nf0g. For p =1, R = Q

i j

i+j =n

and a = 1, we recover the minimal mo del of Quillen for CP [12]. The ker-

nel of the natural pro jection L fx g ;d ! hx i; 0, with obvious right

n=1;2;::: 1

inverse hx i; 0 ,! L fx g ;d, has again the form L V; d, this time

1 n=1;2;:::

0 0

V b eing hfx ;x g i, where x = 1=2 [x ;x ], n = 2; 3;:::,

n=2;3;:::

i j

n n

i+j =n

0 0

and dx = 0, n = 2; 3;::: . Clearly, FH V; d = and, of course, dx = ax

n n

0, so FH L V; d = 0 and FHT V; d = R cf. Remark 2. 3. As in the

On the homology of free Lie algebras 669

previous example, we now obtain successively: FH L fx g ;d = hx i;

n=1;2;::: 1

UF H L fx g ;d=x ; T fx g ;d=T V; d x ; 0,

n n

n=1;2;::: 1 n=1;2;::: 1

as left T V; d-mo dules and right x ; 0-como dules, so H T fx g ;d

1 n=1;2;:::

= HT V; d x , and FH T fx g ;d=x . The natural arrow

1 n=1;2;::: 1

UF H L fx g ;d ! FHU L fx g ;d

n n

n=1;2;::: n=1;2;:::

is thus an isomorphism of graded co commutative Hopf algebras. As b efore,

if a is a unit in R , then H T fx g ;d = x is R -free, so [4] any

n=1;2;::: 1

minimal mo del W;D for C L fx g ;d is decomp osable, and

n=1;2;:::

FH L fx g ;d = hx i may be regarded as the homotopy Lie algebra

n=1;2;::: 1

of W;D.

12. Remark. Quite similar examples can analogously be derived from other

2 3

Quillen minimal mo dels, e.g., from that for the reduced pro duct CP ^ CP ,or

from free mo dels for total spaces of brations, e.g., from that for the total space

E of the bration CP ! E ! S [12].

Acknowledgment. We are indebted to Professor Yves F elix and Canon Ray-

mond Van Schoubro eck for their supp ort.

References

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Universite Catholique de Louvain, Institut de Mathematiques, 2 chemin du Cy-

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Received Octob er 13, 1997, revised February 27, 1998