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ON KAC’S JORDAN SUPERALGEBRA ALEJANDRAS.CO´RDOVA-MARTÍNEZ⋆,ABBASDAREHGAZANI⋄, ANDALBERTOELDUQUE⋆ Abstract. The group-scheme of automorphisms of the ten-dimensional ex- 7 ceptionalKac’sJordansuperalgebraisshowntobeisomorphictothesemidi- 1 rectproductofthedirectproductoftwocopiesofSL2 bytheconstantgroup 0 schemeC2. 2 This is usedto revisit, extend, andsimplify,known results onthe classifi- cationofthetwistedformsofthissuperalgebraandofitsgradings. n a J 4 Kac’s ten-dimensional superalgebra K10 is an exceptional Jordan superalgebra 2 whichappearedforthe firsttimeinKac’sclassification[Kac77]ofthe finite dimen- sionalsimpleJordansuperalgebrasoveranalgebraicallyclosedfieldofcharacteristic ] A 0. It was constructed by Lie theoretical terms from a 3-grading of the exceptional simple Lie superalgebra F(4). R A more conceptual definition was given in [BE02] over an arbitrary field F of . h characteristicnottwo(theseassumptionson thegroundfieldwill bekept throughout at the paper), using thethree-dimensionalKaplanskysuperalgebraK3. Theevenpart m (K3)¯0 is a copy of the ground field F: (K3)¯0 = Fa, with a2 = a; while the odd partis a two-dimensionalvectorspaceW endowedwith anonzeroskew-symmetric [ bilinear form (|). The multiplication is determined as follows: 1 v 1 a2 =a, av = v =va, vw =(v|w)a, 8 2 9 7 for any v,w ∈W. We extend (|) to a supersymmetric bilinear form on K3: (K3)¯0 .06 andTh(Ken3)K¯11a0re=oFrt1h⊕ogoKn3a⊗l,FwKit3h,(wa|iath)=(K2110.)¯0 =F1⊕ K3⊗FK3 ¯0 =F1⊕F(a⊗a)⊕ 01 (gWive⊗nbWy)i,mapnodsi(nKg1t0h)(cid:0)¯1at=1iKst3h⊗e(cid:1)FuKni3ty¯1, a=nd(Wfor⊗hao)m⊕og((cid:0)ean⊗eoWus)e.leT(cid:1)mheenmtsuxlt,iyp,lizc,att∈ionKis, 7 (cid:0) (cid:1) 3 1 3 : (x⊗y)(z⊗t)=(−1)yz xz⊗yt− (x|z)(y |t)1 . (0.1) v 4 i (cid:16) (cid:17) X Ifthecharacteristicis3,thenK9 :=K3⊗FK3 isasimpleidealinK10. Otherwise r K10 is simple. a It must be remarked that an ‘octonionic’ construction of K has been given in 10 [RZ15]. Date:January23,2017. 2010 Mathematics Subject Classification. Primary17C40; Secondary17C70,17C30. Keywords and phrases. Kac’sJordansuperalgebra;group-schemeofautomorphisms,twisted forms,gradings. ⋆SupportedbytheSpanishMinisteriodeEconom´ıayCompetitividad—FondoEuropeodeDe- sarrolloRegional(FEDER)MTM2013-45588-C3-2-P,andbytheDiputaciónGeneraldeAragoń— FondoSocialEuropeo(GrupodeInvestigaciońdeA´lgebra). A.S.Co´rdova-Mart´ınezalsoacknowl- edgessupportfromConsejoNacionaldeCienciayTecnolog´ıa(CONACyT,México)throughgrant 420842/262964. ⋄ This research was in part supported by a grant from IPM (No. 94160221) and partially carriedoutinIPM-IsfahanBranch. 1 2 A.S.CO´RDOVA-MARTÍNEZ,A.DAREHGAZANI,ANDA.ELDUQUE Usingthe constructionaboveofK intermsofK ,the groupofautomorphisms 10 3 Aut K wascomputedin[ELS07]. Thiswasusedin[CDM10]toclassifygradings 10 on K over algebraically closed fields of characteristic zero. 10 (cid:0) (cid:1) Our goal here is to compute the group-scheme of automorphisms of K , and 10 to use it to revisit, extend, and simplify drastically, the known results on twisted forms and on gradings on K . 10 Here we follow the functorial approach (see, for instance, [Wat79]). An affine group-schemeoverFisarepresentablegroup-valuedfunctordefinedonthecategory of unital, commutative, associative algebras over F: AlgF. Thus the group-scheme Aut K10 is the functor AlgF →Grp that takes any object R in AlgF to the group AutR(cid:0)K10(cid:1)⊗FR)(the groupofautomorphismsofthe R-superalgebraK10⊗FR, i.e., the groupofR-linearisomorphismspreservingthe multiplicationandthe grading), (cid:0) with the natural definition on morphisms. ItturnsoutthatAut K isisomorphictoasemidirectproduct SL ×SL ⋊ 10 2 2 C (Theorem 1.1), where C is the constant group scheme attached to the cyclic 2 2 group of two elements (a(cid:0)lso d(cid:1)enoted by C ). This extends the result(cid:0)in [ELS07](cid:1). 2 Moreover,the canonical projection π :Aut K →C induces an isomorphism 10 2 of pointed sets H1 F,Aut K → H1(F,C ) (Theorem 2.1). This associates 10 (cid:0)2 (cid:1) to the isomorphism class of any twisted form of K , the isomorphism class of a 10 quadraticétalealgeb(cid:0)raover(cid:0)F,w(cid:1)h(cid:1)ichallowstogeteasilytheclassificationoftwisted forms of K (Corollary 2.2). Twisted forms of K were classified, in a completely 10 10 differentway,in[RZ03]and[EO00]. We believethatourapproachismorenatural. A simple observation shows that SL ×SL ⋊C is also the automorphism 2 2 2 group-schemeofK ×K ,orofJ(W)×J(W),whereJ(W)istheJordansuperalgebra 3 3 of the ‘superform’ (|) on W = W¯1 a(cid:0)bove. That i(cid:1)s, J(W)¯0 = F1, J(W)¯1 = W and the multiplication is given by 1x=x1=x, vw =(v |w)1, for any x ∈ J(W), v,w ∈ W. This observation will be used in the proof of the bijection H1 F,Aut K ≃H1(F,C ). 10 2 Finally,givenanabeliangroupG, aG-gradingona superalgebraA corresponds (cid:0) (cid:0) (cid:1)(cid:1) to a homomorphism of group schemes GD −→Aut(A), whereGD istheCartierdualtotheconstantgroupschemegivenbyG. (See[EK13] for the basic definitions and facts on gradings.) TheclassificationofG-gradingsuptoisomorphismonK ×K (orJ(W)×J(W)) 3 3 is an easy exercise (Proposition3.2), and it follows at once fromthis the classification of G-gradings, up to isomorphism, on K (Theorem 3.5), thus extending and 10 simplifying widely the results in [CDM10]. 1. The group-scheme of automorphisms Consider the two-dimensional vector space W = (K3)¯1, which is endowed with the nonzero skew-symmetric bilinear form (|) : W ×W → F. The special linear group-scheme SL(W) coincides with the symplectic group scheme Sp(W), which is, up to isomorphism, the group-scheme of automorphisms of K (or of J(W)). 3 The constant group scheme C acts on SL(W)×SL(W) by swapping the argu- 2 ments,andhencewegetanaturalsemidirectproduct SL(W)×SL(W) ⋊C . The 2 groupisomorphismΦ in [ELS07, p. 3809]extends naturallyto a homomorphismof (cid:0) (cid:1) ON KAC’S JORDAN SUPERALGEBRA 3 affine group-schemes: Φ: SL(W)×SL(W) ⋊C −→Aut K 2 10 (cid:0) (cid:1) (cid:0) (cid:1) 1 7→ 1, a⊗a7→a⊗a, (f,g) 7→ Φ(f,g) :va⊗⊗av 7→7→fa(⊗v)g⊗(va),, (1.1) v⊗w7→f(v)⊗g(w), generator of C 7→ τ : 17→ 1, 2 (x⊗y 7→(−1)xyy⊗x, for any R in AlgF, f,g ∈ SL(W)(R) (i.e., f,g ∈ EndR(W ⊗F R) ≃ M2(R) of determinant 1), v,w ∈ WR := W ⊗FR, x,y ∈ (K3)R. (Note that a representation of a constant group scheme ρ : G → GL(V) is determined by its behavior over F: ρF :G→GL(V).) Theorem 1.1. Φ is an isomorphism of affine group-schemes. Proof. If F denotes an algebraic closure of F, then Φ is a group isomorphism F [ELS07, Theorem 3.3]. Since SL(W) × SL(W) ⋊ C ≃ SL × SL ⋊ C is 2 2 2 2 smooth, it is enough to prove that the differential dΦ is bijective (see, for instance [EK13, Theorem A.50]). The L(cid:0)ie algebra of SL((cid:1)W)×SL((cid:0)W) ⋊C is(cid:1)sl(W)× 2 sl(W),whiletheLiealgebraofAut K isthe evenpartofitsLiesuperalgebraof 10 (cid:0) (cid:1) derivations, which is again, up to isomorphism, sl(W)×sl(W) identified naturally with a subalgebra of EndF K10 (se(cid:0)e [B(cid:1)E02, Theorem 2.8]). Moreover, with the natural identifications, dΦ is the identity map. (cid:3) (cid:0) (cid:1) Thelastresultinthissectionisthesimpleobservationthat SL(W)×SL(W) ⋊ C is also the group-scheme of automorphisms of the Jordansuperalgebra K ×K 2 3 3 and J(W) × J(W). This will be instrumental in the next s(cid:0)ection. Its proof(cid:1)is straightforward. Proposition 1.2. The natural transformations defined by: Ψ1 : SL(W)×SL(W) ⋊C −→Aut K ×K 2 3 3 (cid:0) (cid:1) (cid:0) (a,(cid:1)0)7→(a,0), (f,g) 7→ Ψ1 : (0,a)7→(0,a), (f,g)  (v,w)7→ f(v),g(w) , generator of C2 7→ τ :(x,y)7→(y,x), (cid:0) (cid:1) for v,w∈W and x,y ∈K , and 3 Ψ2 : SL(W)×SL(W) ⋊C −→Aut J(W)×J(W) 2 (cid:0) (cid:1) (cid:0) (1,0)7→(cid:1)(1,0), (f,g) 7→ Ψ2 : (0,1)7→(0,1), (f,g)  (v,w)7→ f(v),g(w) , generator of C2 7→ τ :(x,y)7→(y,x), (cid:0) (cid:1) for v,w∈W and x,y ∈J(W), are isomorphisms of group-schemes. 4 A.S.CO´RDOVA-MARTÍNEZ,A.DAREHGAZANI,ANDA.ELDUQUE 2. Twisted forms The set of the isomorphism classes of twisted forms of K is identified with 10 the pointed set H1 F,Aut K . (This is H1 F/F,Aut(K in the notation of 10 10 [Wat79].) The key to the c(cid:0)lassifica(cid:0)tion(cid:1)o(cid:1)f twisted forms(cid:0)of K is the f(cid:1)o(cid:1)llowing result: 10 Theorem 2.1. The canonical projection π : SL(W)×SL(W) ⋊C →C induces 2 2 a bijection of pointed spaces π∗ :H1 F,Aut((cid:0)K10 →H1(F,C2(cid:1)). Proof. First,thenaturalsectionι:C(cid:0) → SL(W(cid:1))(cid:1)×SL(W) ⋊C satisfiesπ◦ι=id, 2 2 so π∗◦ι∗ =id, and π∗ is surjective. (cid:0) (cid:1) Second, the short exact sequence 1−→SL(W)×SL(W)−→ SL(W)×SL(W) ⋊C −→C −→1 2 2 induces an exact sequence (see [Wat(cid:0)79, §18.1]) in cohom(cid:1) ology: 1→SL(W)×SL(W)→ SL(W)×SL(W) ⋊C →C 2 2 →H1(F,SL(W)×SL(W(cid:0)))→H1 F, SL((cid:1)W)×SL(W) ⋊C2 →H1(F,C2). Since H1(F,SL ) is trivial, this gives(cid:0)an(cid:0)exact sequence (cid:1) (cid:1) 2 1−→H1 F, SL(W)×SL(W) ⋊C →H1(F,C ), 2 2 but, in principle, this do(cid:0)es(cid:0)not prove that π∗(cid:1)is a b(cid:1)ijection. It just says that the only element in H1 F, SL(W)×SL(W) ⋊C2 that is sent by π∗ to the trivial element in H1(F,C ) is the trivial element. 2(cid:0) (cid:0) (cid:1) (cid:1) Through the isomorphisms in the previous section, it is enough to prove that π induces a bijection of pointed sets π∗ :H1 F,Aut J(W)×J(W) →H1(F,C2). Write J = J(W) × J(W) and identify SL(W) × SL(W) ⋊ C with Aut(J) (cid:0) (cid:0) (cid:1)(cid:1) 2 by means of Ψ2 in Proposition 1.2. The(cid:0)n J¯0 = F × F, an(cid:1)d π corresponds to the restriction map Aut(J) → Aut(J¯0), and the induced map in cohomology π∗ : H1(F,Aut(J)) → H1(F,Aut(J¯0)) corresponds to the map that takes the isomorphism class of a twisted form H of J to the isomorphism class of H¯0. If H¯0 is trivial (H¯0 ≃F×F) then, by the above, H is isomorphic to J. Otherwise, H¯0 is a quadratic separable field extension of F, so that H¯1 is a two-dimensional vector space over H¯0, and the multiplication H¯1×H¯1 →H¯0 is H¯0-bilinear and anticom- mutative, and hence uniquely determined. It turns out that H is isomorphic to J(W)⊗FH¯0. This proves the injectivity of π∗. (cid:3) Corollary 2.2. Any twisted form of K splits after a quadratic separable field 10 extension, and for any such field extension K of F, there is a unique (up to isomorphism) nontrivial twisted form of K which splits over K. 10 If K is a quadratic separable field extension of F, the unique nontrivial twisted formofK thatsplitsoverKcorrespondstothenontrivialelementinH1(K/F,C ), 10 2 and hence if the Galois group Gal(K/F) is generated by σ (σ2 = id), this twisted form is given by K10,K ={X ∈K10⊗FK:(τ ⊗id)(X)=(id⊗σ)(X)} (2.1) ={X ∈K10⊗FK:(τ ⊗σ)(X)=X}, where τ is given in (1.1). ON KAC’S JORDAN SUPERALGEBRA 5 Hence, with K = F(α), α2 ∈ F, σ(α) = −α, so if {u,v} is a symplectic basis of W =(K3)¯1, K10,K ¯0 =F-span 1⊗1,(a⊗a)⊗1,(u⊗u)⊗α,(v⊗v)⊗α, (cid:0) (cid:1) D (u⊗v−v⊗u)⊗1,(u⊗v+v⊗u)⊗α , (2.2) E K10,K ¯1 =F-span (a⊗x+x⊗a)⊗1,(a⊗x+x⊗a)⊗α:x∈{u,v} . (cid:0) (cid:1) D E Remark 2.3. • BothK andJ(W)arerigid: anytwistedformofanyofthesesuperalgebras 3 is isomorphic to it. This is because Aut(K ) ≃ Aut(J(W)) ≃ SL(W) ≃ 3 SL . 2 • The proofofTheorem2.1showsthatthe twistedformsofK ×K (respec- 3 3 tively J(W)×J(W)) are, up to isomorphism, the superalgebras K3 ⊗F K (resp. J(W)⊗F K), where K is quadratic étale algebra over F, and any two such forms are isomorphic if and only if so are the correspondingétale algebras. 3. Gradings Given an abelian group G, a G-grading on a superalgebra A = A¯0 ⊕A¯1 is a decomposition into a direct sum of subspaces: A = A , such that A A ⊆ g∈G g g h A foranyg,h∈G,andeachhomogeneouscomponentisasubspaceinthe‘super’ gh L senseofA: Ag =(Ag∩A¯0)⊕(Ag∩A¯1). Wewillwritedeg(x)=gincase06=x∈Ag. Two G-gradings Γ : A = A and Γ′ = A′, are isomorphic if there is g∈G g g∈G g an automorphism ϕ∈Aut(A) such that ϕ(A )=A′ for any g ∈G. L gL g AgradingbyGisequivalenttoahomomorphismofaffinegroupschemesGD → Aut(A)(see[EK13]),andthisshowsthattwosuperalgebraswithisomorphicgroup- schemes of automorphisms have equivalent classifications of G-gradings up to isomorphism. Therefore, in order to classify gradings on K it is enough to classify 10 gradings on K ×K (or J(W)×J(W)), and this is straightforward. 3 3 Actually, fix a symplectic basis {u,v} of W =(K3)¯1. Definition 3.1. Given an abelian group G, consider the following gradings (e denotes the neutral element of G): • For g ,g ∈G, denote by Γ1(G;g ,g ) the G-grading given by: 1 2 1 2 deg(x)=e for any x∈ K3×K3)¯0, deg(u,0)=g1 =deg(v,(cid:0)0)−1, deg(0,u)=g2 =deg(0,v)−1. • For g,h ∈ G with h2 = e 6= h, denote by Γ2(G;g,h) the G-grading given by: deg(a,a)=e, deg(a,−a)=h, deg(u,u)=g =deg(v,v)−1, deg(u,−u)=gh=deg(v,−v)−1. Proposition 3.2. Any grading on K ×K by the abelian group G is isomorphic 3 3 to either Γ1(G;g ,g ) or to Γ2(G;g,h) (for some g ,g or g,h in G). 1 2 1 2 Moreover, no grading of the first type (Γ1(G;g ,g )) is isomorphic to a grading 1 2 of the second type (Γ2(G;g,h)), and • Γ1(G,g ,g ) is isomorphic to Γ1(G;g′,g′) if and only if the sets 1 2 1 2 {g ,g−1,g ,g−1} and {g′,(g′)−1,g′,(g′)−1} coincide. 1 1 2 2 1 1 2 2 6 A.S.CO´RDOVA-MARTÍNEZ,A.DAREHGAZANI,ANDA.ELDUQUE • Γ2(G;g,h) is isomorphic to Γ2(G;g′,h′) if and only if h′ = h and g′ ∈ {g,gh,g−1,g−1h}. Proof. Any G-gradingonJ:=K3×K3 inducesa G-gradingonJ¯0,whichis isomorphic to F×F, and hence we are left with two cases: (1) ThegradingonJ¯0istrivial,i.e.,J¯0iscontainedinthehomogeneouscompo- nentJe. Then,withW =(K3)¯1,bothW×0=(a,0)J¯1and0×W =(0,a)J¯1 aregradedsubspacesofJ¯1. Hencewecantakebases{ui,vi}ofW,i=1,2, such that {(u1,0),(v1,0),(0,u2),(0,v2)} is a basis of J¯1 consisting of homogeneous elements. We can adjust v , i = 1,2, so that (u | v ) = 1. If i i i deg(u ,0)=g , i=1,2, the grading is isomorphic to Γ1(G;g ,g ). i i 1 2 (2) The grading on J¯0 is not trivial. Then there is an element h ∈ G of order 2 such that deg(a,a) = e and deg(a,−a) = h. (Note that (a,a) is the unity element of J¯0(≃ F×F), so it is always homogeneous of degree e.) As J¯0 = (J¯1)2, there are homogeneous elements (u1,u2),(v1,v2) ∈ J¯1 such that (u ,u )(v ,v )=(a,a). If g =deg(u ,u ), this grading is isomorphic 1 2 1 2 1 2 to Γ2(G;g,h). The conditions for isomorphism are clear. (cid:3) Any grading Γ1(G;g ,g ) is a coarsening of the grading Γ1 Z2;(1,0),(0,1) , 1 2 whileanygradingΓ2(G;g,h)isacoarseningofthegradingΓ2 Z×Z ;(1,¯0),(0,¯1) , (cid:0) 2 (cid:1) where Z =Z/2Z. As an immediate consequence, we obtain the next result. 2 (cid:0) (cid:1) Corollary 3.3. Up to equivalence, there are exactly two fine gradings on K ×K : 3 3 Γ1 Z2;(1,0),(0,1) and Γ2 Z×Z ;(1,¯0),(0,¯1) . 2 (cid:0)Proposition3.2a(cid:1)ndCoro(cid:0)llary3.3havecomple(cid:1)telysimilarcounterpartsforJ(W)× J(W). In order to transfer these results to Kac’s superalgebra K , take into account 10 that K10 is generated by its odd part, as (K10)¯1 2 = (K10)¯0, so any grading is determined by its restriction to the odd part, and use the commutativity of the (cid:0) (cid:1) diagram Aut K ×K oo Ψ1 SL(W)×SL(W) ⋊C Φ // Aut K 3(cid:127)_ 3 2 (cid:127)_ 10 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:15)(cid:15) (cid:15)(cid:15) GL(W ×W) ≃ //GL (W ⊗a)⊕(a⊗W) where the vertical arrows are given by the restrictions to(cid:0)the odd parts, and t(cid:1)he bottom isomorphism is given by the natural identification W ×W → (W ⊗a)⊕ (a⊗W), (v,w)7→v⊗a+a⊗w. Thus Definition 3.1 transfers to K as follows: 10 Definition 3.4. Given an abelian group G, consider the following G-gradings on K : 10 • For g ,g ∈G, denote by Γ1 (G;g ,g ) the G-grading determined by: 1 2 K10 1 2 deg(u⊗a)=g =deg(v⊗a)−1, deg(a⊗u)=g =deg(a⊗v)−1. 1 2 • For g,h ∈ G with h2 = e 6= h, denote by Γ2 (G;g,h) the G-grading K10 determined by: deg(u⊗a+a⊗u)=g =deg(v⊗a+a⊗v)−1, deg(u⊗a−a⊗u)=gh=deg(v⊗a−a⊗v)−1. And Proposition 3.2 and Corollary 3.3 are now transferred easily to K : 10 ON KAC’S JORDAN SUPERALGEBRA 7 Theorem 3.5. Any grading on K by the abelian group G is isomorphic to either 10 Γ1 (G;g ,g ) or to Γ2 (G;g,h) (for some g ,g or g,h in G). K10 1 2 K10 1 2 No grading of the first type is isomorphic to a grading of the second type, and • Γ1 (G;g ,g ) is isomorphic to Γ1 (G;g′,g′) if and only if the sets K10 1 2 K10 1 2 {g ,g−1,g ,g−1} and {g′,(g′)−1,g′,(g′)−1} coincide. 1 1 2 2 1 1 2 2 • Γ2 (G;g,h) is isomorphic to Γ2 (G;g′,h′) if and only if h′ =h and g′ ∈ K10 K10 {g,gh,g−1,g−1h}. Moreover, there are exactly two fine gradings on K up to equivalence, namely 10 Γ1 Z2;(1,0),(0,1) and Γ2 Z×Z ;(1,¯0),(0,¯1) . K10 K10 2 (cid:0) (cid:1) (cid:0) (cid:1) Remark 3.6. Any grading on K by an abelian group G extends naturally to a 10 gradingbyeitherZ×GorZ2×GontheexceptionalsimpleLiesuperalgebraF(4), 2 whichisobtainedfromK usingthewell-knownTits-Kantor-Koecherconstruction. 10 However, not all gradings on F(4) are obtained in this way. References [BE02] G.BenkartandA.Elduque,AnewconstructionoftheKacJordansuperalgebra,Proc. Amer.Math.Soc.130(2002), no.11,3209–3217. [CDM10] A.J.CalderónMart´ın,C.DraperFontanals, andC.Mart´ınGonzález, Gradings on the Kac superalgebra, J.Algebra324(2010), no.12,3249–3261. [EK13] A.ElduqueandM.Kochetov, Gradings on simple Lie algebras,MathematicalSurveys andMonographs189,AmericanMathematicalSociety, Providence,RI,2013. [EO00] A. Elduque and S. Okubo, Pseudo-composition superalgebras, J. Algebra 227 (2000), no.1,1–25. [ELS07] A.Elduque,J.Laliena,andS.Sacristán,TheKacJordansuperalgebra: automorphisms and maximal subalgebras Proc.Amer.Math.Soc.135(2007), no.12,3805–3813. [Kac77] V.G. Kac, Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras, Comm.Algebra5(1977), no.13,1375–1400. [RZ03] M.L. Racine and E.I. Zelmanov, Simple Jordan superalgebras with semisimple even part, J.Algebra270(2003), no.2,374–444. [RZ15] M.L. Racine and E.I. Zelmanov, An octonionic construction of the Kac superalgebra K10,Proc.Amer.Math.Soc.143(2015), no.3,1075–1083. [Wat79] W.C.Waterhouse,Introductiontoaffinegroupschemes,GraduateTextsinMathemat- ics66.Springer-Verlag,NewYork-Berlin,1979. Departamento de Matema´ticas e Instituto Universitario de Matema´ticas y Aplica- ciones, Universidadde Zaragoza,50009Zaragoza,Spain E-mail address: [email protected] Department of Mathematics, University of Isfahan, Isfahan, Iran, P.O.Box: 81746- 73441 and School of Mathematics, Institute for Research in Fundamental Sciences (IPM),P.O. Box: 19395-5746 E-mail address: [email protected] Departamento de Matema´ticas e Instituto Universitario de Matema´ticas y Aplica- ciones, Universidadde Zaragoza,50009Zaragoza,Spain E-mail address: [email protected]