A maximality result for orthogonal quantum groups
Teodor Banica, Julien Bichon, Benoit Collins, Stephen Curran
aa r X i v : . [ m a t h . QA ] J un A MAXIMALITY RESULT FOR ORTHOGONAL QUANTUM GROUPS
TEODOR BANICA, JULIEN BICHON, BENOˆIT COLLINS, AND STEPHEN CURRAN
Abstract.
We prove that the quantum group inclusion O n ⊂ O ∗ n is “maximal”, where O n is the usual orthogonal group and O ∗ n is the half-liberated orthogonal quantum group,in the sense that there is no intermediate compact quantum group O n ⊂ G ⊂ O ∗ n . Inorder to prove this result, we use: (1) the isomorphism of projective versions P O ∗ n ≃ P U n ,(2) some maximality results for classical groups, obtained by using Lie algebras and somematrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras. Introduction
Quantum groups were introduced by Drinfeld [13] and Jimbo [15] in order to study“non-classical” symmetries of complex systems. This was followed by the fundamentalwork of Woronowicz [19], [20] on compact quantum groups. The key examples whichwere constructed by Drinfeld and Jimbo, and further analyzed by Woronowicz, were q -deformations G q of classical Lie groups G . The idea is as follows: consider the com-mutative algebra A = C ( G ). For a suitable choice of generating “coordinates” of thisalgebra, replace commutativity by the q -commutation relations ab = qba , where q > A q = C ( G q ), where G q is a quantumgroup. When q = 1 one then recovers the classical group G .For G = O n , U n , S n it was later discovered by Wang [17], [18] that one can also obtaincompact quantum groups by “removing” the commutation relations entirely. In this wayone obtains “free” versions O + n , U + n , S + n of these classical groups. This construction hasbeen axiomatized in [11] in terms of the “easiness” condition for compact quantum groups,and has led to several applications in probability. See [9], [10].It is clear from the construction that one has G ⊂ G + for G = O n , U n , S n . Since G + canbe viewed a “liberation” of G , it is natural to wonder whether there are any intermediatequantum groups G ⊂ G ′ ⊂ G + , which could be seen as “partial liberations” of G . For O n , S n this problem has been solved in the case of “easy” intermediate quantum groups[12], [8]. For S n there are no intermediate easy quantum groups S n ⊂ G ′ ⊂ S + n . Howeverfor O n there is exactly one intermediate easy quantum group O n ⊂ O ∗ n ⊂ O + n , called the“half-liberated” orthogonal group, which was constructed in [11]. At the level of relations Mathematics Subject Classification.
Key words and phrases.
Orthogonal quantum group. among coordinates, this is constructed by replacing the commutation relations ab = ba with the half-commutation relations abc = cba .In the larger category of compact quantum groups it is an open problem whether thereare intermediate quantum groups S n ⊂ G ⊂ S + n , or O n ⊂ G ⊂ O + n with G = O ∗ n . Thisis an important question for better understanding the “liberation” procedure of [11]. At n = 4 (the smallest value at which S n = S + n ), it follows from the results in [5] that theinclusion S n ⊂ S + n is indeed maximal, and it was conjectured in [6] that this is the case,for any n ∈ N . Likewise the inclusion O n ⊂ O ∗ n ⊂ O + n is known to be maximal at n = 2,thanks to the results of Podle´s in [16]. In general it is likely that these two problems arerelated to each other via combinatorial invariants [12] or cocycle twists [7].In this paper we make some progress towards solving this problem in the orthogonalcase, by showing that the inclusion O n ⊂ O ∗ n is maximal. A key tool in our analysis willbe the fact the “projective version” of O ∗ n is the same as that of the classical unitarygroup U n . By using a version of the five lemma for cosemisimple Hopf algebras (followingideas from [1], [3]), we are thus able to reduce the problem to showing that the inclusionof groups P O n ⊂ P U n is maximal. We then solve this problem by using some Lie algebratechniques inspired from [4], [14].The paper is organized as follows: Section 1 contains background and preliminaries.In Section 2 we prove that P O n ⊂ P U n is maximal. In Section 3 we prove a short fivelemma for cosemisimple Hopf algebras, which may be of independent interest. We thenuse this in Section 4 to prove our main result, namely that O n ⊂ O ∗ n is maximal. Acknowledgements.
Part of this work was completed during the Spring 2011 program“Bialgebras in free probability” at the Erwin Schr¨odinger Institute in Vienna, and T.B.,B.C., S.C. are grateful to the organizers for the invitation. The work of T.B., J.B., B.C.was supported by the ANR grants “Galoisint” and “Granma”, the work of B.C. wassupported by an NSERC Discovery grant and an ERA grant, and the work of S.C. wassupported by an NSF postdoctoral fellowship and by the NSF grant DMS-0900776.1.
Orthogonal quantum groups
In this section we briefly recall the free and half-liberated orthogonal quantum groupsfrom [17], [11], and the notion of “projective version” for a unitary compact quantumgroup. We will work at the level of Hopf ∗ -algebras of representative functions.First we have the following fundamental definition, arising from Woronowicz’ work [19]. Definition 1.1.
A unitary Hopf algebra is a ∗ -algebra A which is generated by elements { u ij | ≤ i, j ≤ n } such that u = ( u ij ) and u = ( u ∗ ij ) are unitaries, and such that: (1) There is a ∗ -algebra map ∆ : A → A ⊗ A such that ∆( u ij ) = P nk =1 u ik ⊗ u kj . (2) There is a ∗ -algebra map ε : A → C such that ε ( u ij ) = δ ij . (3) There is a ∗ -algebra map S : A → A op such that S ( u ij ) = u ∗ ji .If u ij = u ∗ ij for ≤ i, j ≤ n , we say that A is an orthogonal Hopf algebra. MAXIMALITY RESULT FOR ORTHOGONAL QUANTUM GROUPS 3
It follows that ∆ , ε, S satisfy the usual Hopf algebra axioms. The motivating examplesof unitary (resp. orthogonal) Hopf algebra is A = R ( G ), the algebra of representativefunction of a compact subgroup G ⊂ U n (resp. G ⊂ O n ). Here the standard generators u ij are the coordinate functions which take a matrix to its ( i, j )-entry.In fact every commutative unitary Hopf algebra is of the form R ( G ) for some compactgroup G ⊂ U n . In general we use the suggestive notation “ A = R ( G )” for any unitary(resp. orthogonal) Hopf algebra, where G is a unitary (resp. orthogonal) compact quantumgroup . Of course any group-theoretic statements about G must be interpreted in termsof the Hopf algebra A .It can be shown that shown that a unitary Hopf algebra has an enveloping C ∗ -algebra,satisfying Woronowicz’ axioms in [19]. In general there are several ways to complete aunitary Hopf algebra into a C ∗ -algebra, but in this paper we will ignore this problem andwork at the level of unitary Hopf algebras.The following examples of Wang [17] are fundamental to our considerations. Definition 1.2.
The universal unitary Hopf algebra A u ( n ) is the universal ∗ -algebra gen-erated by elements { u ij | ≤ i, j ≤ n } such that the matrices u = ( u ij ) and u = ( u ∗ ij ) in M n ( A u ( n )) are unitaries.The universal orthogonal Hopf algebra A o ( n ) is the universal ∗ -algebra generated by self-adjoint elements { u ij | ≤ i, j ≤ n } such that the matrix u = ( u ij ) ≤ i,j ≤ n in M n ( A o ( n )) isorthogonal. The existence of the Hopf algebra structural morphisms follows from the universalproperties of A u ( n ) and A o ( n ). As discussed above, we use the notations A u ( n ) = R ( U + n )and A o ( n ) = R ( O + n ), where U + n is the free unitary quantum group and O + n is the freeorthogonal quantum group .Note that we have R ( O + n ) ։ R ( O n ), in fact R ( O n ) is the quotient of R ( O + n ) by therelations that the coordinates u ij commute. At the level of quantum groups, this meansthat we have an inclusion O n ⊂ O + n .In other words, R ( O + n ) is obtained from R ( O n ) by “removing commutativity” amongthe coordinates u ij . It was discovered in [11] that one can obtain a natural orthogonalquantum group by requiring instead that the coordinates “half-commute”. Definition 1.3.
The half-liberated othogonal Hopf algebra A ∗ o ( n ) is the universal ∗ -algebragenerated by self-adjoint elements { u ij | ≤ i, j ≤ n } which half-commute in the sense that abc = cba for any a, b, c ∈ { u ij } , and such that the matrix u = ( u ij ) ≤ i,j ≤ n in M n ( A ∗ o ( n )) is orthogonal. The existence of the Hopf algebra structural morphisms again follows from the universalproperties of A ∗ o ( n ). We use the notation A ∗ o ( n ) = R ( O ∗ n ), where O ∗ n is the half-liberatedorthogonal quantum group . Note that we have R ( O + n ) ։ R ( O ∗ n ) ։ R ( O n ), i.e. O n ⊂ O ∗ n ⊂ O + n . As discussed in the introduction, our aim in this paper is to show that the TEODOR BANICA, JULIEN BICHON, BENOˆIT COLLINS, AND STEPHEN CURRAN inclusion O n ⊂ O ∗ n is maximal. A key tool in our analysis will be the projective versionof a unitary quantum group, which we now recall. Definition 1.4.
The projective version of a unitary compact quantum group G ⊂ U + n isthe quantum group P G ⊂ U + n , having as basic coordinates the elements v ij,kl = u ik u ∗ jl . In other words, P R ( G ) = R ( P G ) ⊂ R ( G ) is the subalgebra generated by the elements v ij,kl = u ik u ∗ jl . It is clearly a Hopf ∗ -subalgebra of R ( G ). In the case where G ⊂ U n isclassical we recover of course the well-known formula P G = G/ ( G ∩ T ), where T ⊂ U n isthe group of norm one multiples of the identity.The following key result was proved in [12]. Theorem 1.5.
We have an isomorphism
P O ∗ n ≃ P U n .Proof. First, thanks to the half-commutation relations between the standard coordinateson O ∗ n , for any a, b, c, d ∈ { u ij } we have abcd = cbad = cdab . Thus the standard coordi-nates on the quantum group P O ∗ n commute ( ab · cd = cd · ab ), so this quantum group isactually a classical group. A representation theoretic study, based on the diagrammaticresults in [11], allows then to show this classical group is actually P U n . See [12]. (cid:3) Note that in fact the techniques developed in the present paper enable us to give a verysimple proof of this theorem, avoiding the diagramatic techniques from [11], [12]. See thelast remark in Section 4. 2.
Classical group results
In this section we prove that the inclusion
P O n ⊂ P U n is maximal in the category ofcompact groups (we assume throughout the paper that n ≥
2, otherwise there is nothingto prove). We will see later on, in Sections 3 and 4 below, that this result can be “twisted”,in order to reach to the maximality of the inclusion O n ⊂ O ∗ n .Let ˜ O n be the group generated by O n and T · I n (the group of multiples of identityof norm one). That is, ˜ O n is the preimage of P O n under the quotient map U n ։ P U n .Let f SO n ⊂ ˜ O n be the group generated by SO n and T · I n . Note that ˜ O n = f SO n if n isodd, and if n is even then ˜ O n has two connected components and f SO n is the componentcontaining the identity.It is a classical fact that a compact matrix group is a Lie group, so f SO n is a Lie group.Let so n (resp. u n ) be the real Lie algebras of SO n (resp. U n ). It is known that u n consistsof the matrices M ∈ M n ( C ) satisfying M ∗ = − M , and so n = u n ∩ M n ( R ). It is easy tosee that the Lie algebra of f SO n is so n ⊕ i R .First we need the following lemma: Lemma 2.1. If n ≥ , the adjoint representation of SO n on the space of real symmetricmatrices of trace zero is irreducible. MAXIMALITY RESULT FOR ORTHOGONAL QUANTUM GROUPS 5
Proof.
Let X ∈ M n ( R ) be symmetric with trace zero, and let V be the span of { U XU t : U ∈ SO n } . We must show that V is the space of all real symmetric matrices of tracezero.First we claim that V contains all diagonal matrices of trace zero. Indeed, since wemay diagonalize X by conjugating with an element of SO n , V contains some non-zerodiagonal matrix of trace zero. Now if D = diag ( d , d , . . . , d n ) is a diagonal matrix in V ,then by conjugating D by − I n − ∈ SO n we have that V also contains diag ( d , d , d , . . . , d n ). By a similar argument we see thatfor any 1 ≤ i, j ≤ n the diagonal matrix obtained from D by interchanging d i and d j lies in V . Since S n is generated by transpositions, it follows that V contains any diagonal matrixobtained by permuting the entries of D . But it is well-known that this representation of S n on diagonal matrices of trace zero is irreducible, and hence V contains all such diagonalmatrices as claimed.Now if Y is any real symmetric matrix of trace zero, we can find a U in SO n such that U Y U t is a diagonal matrix of trace zero. But we then have U Y U t ∈ V , and hence also Y ∈ V as desired. (cid:3) Proposition 2.2.
The inclusion g SO n ⊂ U n is maximal in the category of connectedcompact groups.Proof. Let G be a connected compact group satisfying f SO n ⊂ G ⊂ U n . Then G is a Liegroup, let g denote its Lie algebra, which satisfies so n ⊕ i R ⊂ g ⊂ u n .Let ad G be the action of G on g obtained by differentiating the adjoint action of G onitself. This action turns g into a G -module. Since SO n ⊂ G , g is also an SO n -module.Now if G = f SO n , then since G is connected we must have so n ⊕ i R = g . It follows fromthe real vector space structure of the Lie algebras u n and so n that there exists a non-zerosymmetric real matrix of trace zero X such that iX ∈ g .But by Lemma 2.1 the space of symmetric real matrices of trace zero is an irreduciblerepresentation of SO n under the adjoint action. So g must contain all such X , and hence g = u n . But since U n is connected, it follows that G = U n . (cid:3) Our aim is to extend this result to the category of compact groups. To do this we needto compute the normalizer of f SO n in U n , i.e. the subgroup of U n consisting of unitary U for which U − XU ∈ f SO n for all X ∈ f SO n . For this we need two lemmas. Lemma 2.3.
The commutant of SO n in M n ( C ) , denoted SO ′ n , is as follows: (1) SO ′ = { (cid:18) α β − β α (cid:19) , α, β ∈ C } . (2) If n ≥ , SO ′ n = { αI n , α ∈ C } . TEODOR BANICA, JULIEN BICHON, BENOˆIT COLLINS, AND STEPHEN CURRAN
Proof. At n = 2 this is a direct computation. For n ≥
3, an element in X ∈ SO ′ n commutes with any diagonal matrix having exactly n − −
1. Hence X is a diagonal matrix. Now since X commutes with any evenpermutation matrix and n ≥
3, it commutes in particular with the permutation matrixassociated with the cycle ( i, j, k ) for any 1 < i < j < k , and hence all the entries of X are the same: we conclude that X is a scalar matrix. (cid:3) Lemma 2.4.
The set of matrices with non-zero trace is dense in SO n .Proof. At n = 2 this is clear since the set of elements in SO having a given trace is finite.Assume that n > T ∈ SO n ≃ SO ( R n ) with T r ( T ) = 0. Let E ⊂ R n be a2-dimensional subspace preserved by T and such that T | E ∈ SO ( E ). Let ǫ > S ǫ ∈ SO ( E ) with || T | E − S ǫ || < ǫ and T r ( T | E ) = T r ( S ǫ ) ( n = 2 case). Now define T ǫ ∈ SO ( R n ) = SO n by T ε | E = S ǫ and T ǫ | E ⊥ = T | E ⊥ . It is clear that || T − T ǫ || ≤ || T | E − S ǫ || < ǫ and that T r ( T ǫ ) = T r ( S ǫ ) + T r ( T | E ⊥ ) = 0. (cid:3) Proposition 2.5. ˜ O n is the normalizer of f SO n in U n .Proof. It is clear that ˜ O n normalizes f SO n , so we must show that if U ∈ U n normalizes f SO n then U ∈ ˜ O n . First note that U normalizes SO n . Indeed if X ∈ SO n then U − XU ∈ f SO n ,so U − XU = λY for λ ∈ T and Y ∈ SO n . If T r ( X ) = 0, we have λ ∈ R and hence λY = U − XU ∈ SO n . The set of matrices having non-zero trace is dense in SO n byLemma 2.4, so since SO n is closed and the matrix operations are continous, we concludethat U − XU ∈ SO n for all X ∈ SO n .Thus for any X ∈ SO n , we have ( U XU − ) t ( U XU − ) = I n and hence X t U t U X = U t U .This means that U t U ∈ SO ′ n . Hence if n ≥
3, we have U t U = αI n by Lemma 2.3, with α ∈ T since U is unitary. Hence we have U = α / ( α − / U ) with α − / U ∈ O n , and U ∈ f O n . If n = 2, Lemma 2.3 combined with the fact that ( U t U ) t = U t U gives againthat U t U = αI , and we conclude as in the previous case. (cid:3) We can now extend Proposition 2.2 as follows.
Proposition 2.6.
The inclusion ˜ O n ⊂ U n is maximal in the category of compact groups.Proof. Suppose that ˜ O n ⊂ G ⊂ U n is a compact group such that G = U n . It is a wellknown fact that the connected component of the identity in G is a normal subgroup,denoted G . Since we have f SO n ⊂ G ⊂ U n , by Proposition 2.2 we must have G = f SO n .But since G is normal in G , G normalizes f SO n and hence G ⊂ ˜ O n by Proposition 2.5. (cid:3) We are now ready to state and prove the main result in this section.
Theorem 2.7.
The inclusion
P O n ⊂ P U n is maximal in the category of compact groups.Proof. It follows directly from the observation that the maximality of ˜ O n in U n implies themaximality of P O n in P U n . Indeed, if P O n ⊂ G ⊂ P U n were an intermediate subgroup, MAXIMALITY RESULT FOR ORTHOGONAL QUANTUM GROUPS 7 then its preimage under the quotient map U n ։ P U n would be an intermediate subgroupof ˜ O n ⊂ U n , contradicting Proposition 2.6. (cid:3) A short five lemma
In this section we prove a short five lemma for cosemisimple Hopf algebras (Theorem3.4 below), which is a result having its own interest, to be used in Section 4 below.
Definition 3.1.
A sequence of Hopf algebra maps C → B i → A p → L → C is called pre-exact if i is injective, p is surjective and i ( A ) = H cop , where: A cop = { a ∈ A | ( id ⊗ p )∆( a ) = a ⊗ } The example that we are interested in is as follows.
Proposition 3.2.
Let A be an orthogonal Hopf algebra with generators u ij . Assume thatwe have surjective Hopf algebra map p : A → CZ , u ij → δ ij g , where < g > = Z . Let P A be the projective version of A , i.e. the subalgebra generated by the elements u ij u kl withthe inclusion i : P A ⊂ A . Then the sequence C → P A i ′ → A p → CZ → C is pre-exact.Proof. We have:( id ⊗ p )∆( u i j . . . u i m j m ) = ( u i j . . . u i m j m ⊗ m is even u i j . . . u i m j m ⊗ g if m is oddThus H cop is the span of monomials of even length, which is clearly P H . (cid:3) A pre-exact sequence as in Definition 3.1 is said to be exact [2] if in addition we have i ( A ) + H = ker( π ) = Hi ( A ) + , where i ( A ) + = i ( A ) ∩ ker( ε ). The pre-exact sequence inProposition 3.2 is actually exact, but we only need its pre-exactness in what follows.In order to prove the short five lemma, we use the following well-known result. We givea proof for the sake of completness. Lemma 3.3.
Let θ : A → A ′ be a Hopf algebra morphism with A, A ′ cosemisimple andlet h A , h A ′ be the respective Haar integrals of A, A ′ . Then θ is injective iff h A ′ θ = h A .Proof. For a ∈ A , we have: θ ( h A ′ ( θ ( a )) a ) = h A ′ ( θ ( a ) ) θ ( a ) = θ ( h A ′ θ ( a )1)Thus if θ is injective then h A ′ θ is a Haar integral on A , and the result follows from theuniqueness of the Haar integral. TEODOR BANICA, JULIEN BICHON, BENOˆIT COLLINS, AND STEPHEN CURRAN
Conversely, assume that h A = h A ′ θ . Then for all a, b ∈ A , we have h A ( xy ) = h A ′ ( θ ( a ) θ ( b )), so if θ ( a ) = 0, we have h A ( ab ) = 0 for all b ∈ H . It follows from theorthogonality relations that a = 0, and hence θ is injective. (cid:3) Theorem 3.4.
Consider a commutative diagram of cosemisimple Hopf algebras k −−−→ B i −−−→ A π −−−→ L −−−→ k (cid:13)(cid:13)(cid:13) y θ (cid:13)(cid:13)(cid:13) k −−−→ B i ′ −−−→ A ′ π ′ −−−→ L −−−→ k where the rows are pre-exact. Then θ is injective.Proof. We have to show that h A = h A ′ θ , where h A , h A ′ are the respective Haar integralsof A, A ′ . Let Λ be the set of isomorphism classes of simple L -comodules and consider thePeter-Weyl decomposition of L : L = M λ ∈ Λ L ( λ )We view A as a right L -comodule via ( id ⊗ π )∆. Then A has a decomposition intoisotypic components as follows, where A λ = { a ∈ A | ( id ⊗ π ) ◦ ∆( a ) ∈ A ⊗ L ( λ ) } : A = M λ ∈ Λ A λ It is clear that A = A coπ . Then if λ = 1, we have h A ( A λ ) = 0. Indeed for a ∈ A λ , wehave: a ⊗ π ( a ) ∈ H ⊗ L ( λ ) = ⇒ h A ( a )1 = π ( h H ( a ) a ) ∈ L ( λ ) = ⇒ h H ( a ) = 0Since π ′ θ = π , it is easy to see that θ ( A λ ) ⊂ A ′ λ and hence for λ = 1, h A ′ | A ′ λ = h A ′ θ | A λ =0 = h A | A λ . For λ = 1, we have i ( A ) = A and θ is injective on i ( A ) since θi = i ′ . Henceby Lemma 3.3 we have h A ′ θ | A = h A = h A | A . Since A = ⊕ λ ∈ Λ A λ we conclude h A = h A ′ θ and by Lemma 3.3 we get that θ is injective. (cid:3) The main result
We have now all the ingredients for stating and proving our main result in this paper.
Theorem 4.1.
The inclusion O n ⊂ O ∗ n is maximal in the category of compact quantumgroups.Proof. Consider a sequence of surjective Hopf ∗ -algebra maps as follows, whose composi-tion is the canonical surjection: A ∗ o ( n ) f → A g → R ( O n ) MAXIMALITY RESULT FOR ORTHOGONAL QUANTUM GROUPS 9
By Proposition 3.2 we get a commutative diagram of Hopf algebra maps with pre-exactrows: C −−−→ P A ∗ o ( n ) i −−−→ A ∗ o ( n ) p −−−→ CZ −−−→ C y f | y f (cid:13)(cid:13)(cid:13) C −−−→ P A i −−−→ A p −−−→ CZ −−−→ C y g | y g (cid:13)(cid:13)(cid:13) C −−−→ P R ( O n ) i −−−→ R ( O n ) p −−−→ CZ −−−→ C Consider now the following composition, with the isomorphism on the left coming fromTheorem 1.5: R ( P U n ) ≃ P A ∗ o ( n ) f | → P A g | → P R ( O n ) ≃ R ( P O n )This induces, at the group level, the embedding P O n ⊂ P U n . By Theorem 2.7 f | or g | is an isomorphism. If f | is an isomorphism we get a commutative diagram of Hopf algebramorphisms with pre-exact rows: C −−−→ P A ∗ o ( n ) i −−−→ A ∗ o ( n ) p −−−→ CZ −−−→ C (cid:13)(cid:13)(cid:13) y f (cid:13)(cid:13)(cid:13) C −−−→ P A ∗ o ( n ) i ◦ f | −−−→ A p −−−→ CZ −−−→ C Then f is an isomorphism by Theorem 3.4. Similarly if g | is an isomorphism, then g isan isomorphism. (cid:3) Observe that the technique in the proof of Theorem 4.1 also enables us to prove that
P O ∗ n ≃ P U n independently from [12]. Indeed, since P A ∗ o ( n ) is commutative, there existsa compact group G with P A ∗ o ( n ) ≃ R ( G ) and P O n ⊂ G ⊂ P U n . Then Theorem 2.7gives G = P O n or G = P U n . If G = P O n , then as in the proof of Theorem 4.1, Theorem3.4 gives that A ∗ o ( n ) ։ R ( O n ) is an isomorphism, which is false since A ∗ o ( n ) is a notcommutative if n ≥
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Invent. Math. (1988), 35–76. T.B.: Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise,France. [email protected]
J.B.: Department of Mathematics, Clermont-Ferrand University, Campus des Cezeaux,63177 Aubiere Cedex, France. [email protected]
B.C.: Department of Mathematics, Lyon 1 University, and University of Ottawa, 585King Edward, Ottawa, ON K1N 6N5, Canada. [email protected]