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Classification of five dimensional nilpotent Jordan 4 1 algebras 0 2 n Ahmed Hegazi, Hani Abdelwahab a J Mathematics department, Faculty of Science, Mansoura University, Egypt 6 1 [email protected] hanii [email protected] ] A R January 17, 2014 . h t a m Abstract [ We use the method which has been described in [1] to give a classification of 1 v 5-dimensional nilpotent Jordan algebras over an algebraically closed fields of char- 2 acteristic = 2. Weshowthatthereare44classesofnon-associative nilpotentJordan 9 6 algebras over an algebraically closed fields of characteristic = 2, up to isomorphism 9 6 3 each class has only one algebra except 6 classes contain infinite number of non- . isomorphic algebras. 1 0 4 Keywords and phrases: Nilpotent Jordan algebras, second cohomology, centeral exten- 1 sion, automorphism group, orbits, isomorphism. : v 2010 Mathematics Subject Classification: 17C10, 17C55, 17-08 i X r a 1 Introduction The classification of small-dimensional Jordan algebras is a classical problem. The au- thors described a computional method to construct nilpotent Jordan algebras over any field and used this method to classify nilpotent Jordan algebras of dimensions 6 3 over any field and four dimensional nilpotent algebras over an algebraically closed fields of characteristic = 2 and real field R (see [1]). The original aim of the research presented 6 in this paper was to extend the results to 5-dimensional nilpotent Jordan algebras over an algebraically closed fields of characteristic = 2. Nilpotent associative Jordan algebras 6 of dimension five were classified by Mazzola ( [2] - nilpotent commutative associative algebras of dimension 6 5, over algebraically closed fields of characteristic not 2,3), and recently, Poonen ([3] - nilpotent commutative associative algebras of dimension 6 5, over 1 algebraically closed fields). To complete the classification, we construct all 5-dimensional nilpotent non-associative Jordan algebras over an algebraically closed fields of characteristic = 2. Our methodology, which is the same as in [1] with some restrictions to avoid 6 reconstructing commutative associative algebras is explained in Section 2. Here is an outline of the paper. In Section 2 we describe the cohomological method that we use to classify nilpotent Jordan algebras, which also appeared in [1]. In Section 3 we present the main results of [1], that is the classification of the nilpotent Jordan algebras of dimension 6 4 which are needed in Section 4. Then in Section 4 the main work is performed to classify the 5-dimensional nilpotent non-associative Jordan algebras, some results relied on computer calculations (specifically, computing a Gro¨bner bases for ideals in a polynomial rings) and were inspired by Gröbner basis computations in the computational algebra system Magma [4]. Finally in section 5 we summarize our main results of classification of five dimensional nilpotent Jordan algebras. Remark 1.1. Throughout the paper K be an algebraically closed fields of characteristic = 2. Also we use some notational conventions, if J be Jordan algebra with basis ele- 6 ments a ,...,a . Then by δ we denote the symmetric bilinear map J J K with 1 n ai,aj × → δ (a ,a ) = 1 if i = k and j = l, or i = l andj = k, and it takes the value 0 otherwise. ai,aj k l 2 A summary of the method The main idea that we use here is to obtain the nilpotent Jordan algebras of dimension n as central extensions of Jordan algebras of smaller dimension. The central extensions are defined using the second cohomology space, and the isomorphism classes of the central extensions correspond to the orbits of the automorphism groupon theset of thesubspaces of this cohomology space. Let J be a Jordan algebra, for a vector space V, let Z2 (J,V) denote the set of Jor. symmetric bilinear maps θ : J J V with the property that × → θ(a,d (b c))+θ(b,d (a c))+θ(c,d (a b)) = θ(a b,c d)+θ(b c,a d)+θ(a c,b d) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ for all a,b,c,d J. The set Z2 (J,V) is viewed as a vector space over k and the elements ∈ Jor. of Z2 (J,V) are said to be Jordan cocycles. A linear map f : J V is called a 1- Jor. −→ cochain from J to V. The set of all 1-cochains from J to V is denoted by C1(J,V). Then the bilinear map δf : J J Vdefined by (δf)(x,y) := f(x y) is a Jordan cocycle, × −→ ◦ called a coboundary from J to V. The set of all coboundaries is denoted by δC1(J,V), and the second cohomology space H2 (J,V) is defined as the quotient Z2 (J,V)/δC1(J,V). Jor. Jor. Let J be a Jordan algebra and V a vector space over a field k, we denote the center of J by Z(J). For θ Z2 (J,V), define a Jordan algebra J as follows. The underlying ∈ Jor. θ space of J is J V. The product of two elements a + v, b + w J , is defined as θ θ ⊕ ∈ (a+v) (b+w) = a b + θ(a,b) where a b denotes the product in J. Then J is J J θ ◦ ◦ ◦ 2 a Jordan algebra and V is an ideal of J such that V 6 Z(J ). In addition, J = J /V, θ θ ∼ θ and hence J is a central extension of J. Further, if θ , θ Z2 (J,V) such that θ 1 2 ∈ Jor. θ θ δC1(J,V) then J = J , and so the isomorphism type of J only depends on 1 − 2 ∈ θ1 ∼ θ2 θ the element θ+δC1(J,V) of H2 (J,V). Conversely let M be a Jordan algebra such that Jor. Z(M) = 0, and set V = Z(M) and J = M/Z(M). Let π: M J be the projection map. 6 → Choose an injective linear map f: J M such that π(f(a)) = a for all a J. Define → ∈ θ: J J V by θ(a,b) = f(a b) f(a) f(b). Then θ is a cocycle such that M = J . ∼ θ × → ◦ − ◦ Though θ depends on the choice of f, the coset θ+δC1(J,V) is independent of f. Hence the central extension M of J determines a well-defined element of H2 (J,V). Jor. Let V be m-dimensional vector space then Z2 (J,V) = Z2 (J,km) = Z2 (J,k)m Jor. Jor. Jor. and H2 (J,V) = H2 (J,km) = H2 (J,k)m. Let us now fix a basis e ,...,e of V. Jor. Jor. Jor. { 1 m} A Jordan cocycle θ Z2 (J,V) can be written as ∈ Jor. m θ(a,b) = θ (a,b)e , i i Xi=1 where θ Z2 (J,k). Furthermore, θ is a coboundary if and only if all θ are. For i ∈ Jor. i θ Z2(J,V), let θ denote the radical of θ; that is, the set of elements a J such that ⊥ ∈ ∈ θ(a,b) = 0 for all b J. Then ∈ θ = θ θ .... θ . ⊥ ⊥1 ∩ ⊥2 ∩ ∩ ⊥m For each θ Z2 (J,V) and φ Aut(J), the automorphism group of J. Define ∈ Jor. ∈ φθ(a,b) = θ(φ(a),φ(b))foranya,b J.SoAut(J)actsonZ2 (J,V), andφθ δC1(J,V) ∈ Jor. ∈ if andonly if θ δC1(J,V).Then Z2 (J,V) andδC1(J,V) areinvariant under the action ∈ Jor. of Aut(J). So Aut(J) acts on H2 (J,V) hence the quotient H2 (J,V) can be viewed as Jor. Jor. an Aut(J)-module. Lemma 2.1 (see [1]). Let θ = (θ ,θ ,.....,θ ) and η = (η ,η ,.....,η ) H2 (J,V) and 1 2 m 1 2 m ∈ Jor. θ Z(J) = η Z(J) = 0 . Then J and J are isomorphic if and only if there exist ⊥ ⊥ θ η ∩ ∩ φ Aut(J) such that φθ span the same subspace of H2 (J,k) as the η . ∈ i Jor. i A Jordan algebra J is said to be that J has a central component if J = J k where 1 ⊕ J is an ideal of J and k is viewed as a 1-dimensional Jordan algebra, in this case J 1 can be viewed as 1-dimensional centeral extension of J corresponding to θ = 0 ” Trivial 1 1-dimensional centeral extension”. Lemma 2.2 (see [1]). Let θ(a,b) = m θ (a,b)e H2 (J,V) and θ Z(J) = 0 . i=1 i i ∈ Jor. ⊥ ∩ Then Jθ has a centeral component if aPnd only if θ1,θ2,.....,θm are linearly dependent . Note that J = J k is associative if and only if J is associative, So if we need to 1 1 ⊕ construct all nilpotent non-associative Jordan algebras of dimension n with centeral components we will only consider the trivial 1-dimensional centeral extension of all nilpotent non-associative Jordan algebras of dimension n 1. − 3 Let G (H2(J,k)) be the Grassmanian of subspaces of dimension r in H2(J,k) . There r is anatural action of Aut(J) on G (H2(J,k)) defined by : r φ < θ ,θ ,.....,θ >=< φθ ,φθ ,.....,φθ > 1 2 r 1 2 r for V =< θ ,θ ,.....,θ > G (H2(J,k)) and φ Aut(J). Define 1 2 r r ∈ ∈ U (J) = V =< θ ,θ ,.....,θ > G (H2(J,k)) : θ Z(J) = 0;i = 1,2,...,r . r { 1 2 r ∈ r ⊥i ∩ } Theorem 2.3 (see [1]). Let U (J)/Aut(J) be the set of Aut(J)-orbits of U (J). Then r r there exists a canonical one-to-one correspondense from U (J)/Aut(J) onto the set of r isomorphism classes of Jordan algebras without centeral components which are central extensions of J by V and have r-dimensional center where r = dimV. Now to avoid construction of associative nilpotent Jordan algebras, define a subspace H2 (J,V) of H2 (J,V) as follows : Ass. Jor. H2 (J,V) = θ H2 (J,V) : θ(a b,c) = θ(a,b c) for all a,b,c J . Ass. { ∈ Jor. ◦ ◦ ∈ } Then J is associative if and only if J is associative and θ H2 (J,V), conversly J is θ ∈ Ass. θ non-associative if and only if J is non-associative or J is associative and θ / H2 (J,V). ∈ Ass. We will call θ a non-associative Jordan cocycle if θ H2 (J,V) and θ / H2 (J,V). ∈ Jor. ∈ Ass. All nilpotent Jordan algebras of dimensions 6 3 are associative. Hence the classifica- tionof5-dimensionalnilpotentnon-associativeJordanalgebrasrequiresthatwedetermine the trivial and non-trivial centeral extension of nilpotent non-associative algebras of dimension 4, non-trivial centeral extension of nilpotent associative algebras of dimension at most 4 by non-associative Jordan cocycles. The determination of these algebras is achieved in Section 4. 3 Nilpotent Jordan algebras of dimension 6 4 Here and in next sections the basis elements of the algebras will be denoted by the letters a,b,.... The multiplication of an algebra is specified by giving only the nonzero products among the basis elements.We denote the j-th algebra of dimension i by J . i,j There are the following nilpotent Jordan algebras of dimensions 6 4 (cf. e.g., [1]): 4 Algebra Multiplication taple Z(J) Comments J a associative 1,1 −−−−− J a,b associative 2,1 −−−−− J a2 = b b associative 2,2 J a,b,c associative 3,1 −−−−− J a2 = b b,c associative 3,2 J a b = c c associative 3,3 ◦ J a2 = b,a b = c c associative 3,4 ◦ J a,b,c,d associative 4,1 −−−−− J a2 = b b,c,d associative 4,2 J a b = c c,d associative 4,3 ◦ J a2 = b,a b = c c,d associative 4,4 ◦ J a b = d,c2 = d d associative 4,5 ◦ J a2 = b , b c = d d non-associative 4,6 ◦ J a2 = b , a b = d , c2 = d d associative 4,7 ◦ J a b = c,a c = d d non-associative 4,8 ◦ ◦ J a b = c,a c = d,b2 = d d non-associative 4,9 ◦ ◦ J a b = c,a c = d,b c = d d non-associative 4,10 ◦ ◦ ◦ J a2 = b , a b = c , a c = d , b2 = d d associative 4,11 ◦ ◦ J a2 = c , a b = d c,d associative 4,12 ◦ J a2 = c , b2 = d c,d associative 4,13 Remark3.1. Wereplacedunderisomorphismsomecertainalgebrasin[1]byJ ,J ,J ,J ,J 4,3 4,5 4,8 4,9 4,10 and J introducing as they were in the previous table. To be somewhat more concrete, 4,13 let J ,J ,J ,J ,J and J are four dimensional nilpotent Jordan algebras defined as follow 1 2 3 4 5 6 J : a2 = c,b′2 = c. 1 ′ ′ ′ • J : a2 = c,b′2 = c,a c = d. 2 ′ ′ ′ ′ ′ ′ • ◦ J : a2 = c,b′2 = c,a c = d,b c = d. 3 ′ ′ ′ ′ ′ ′ ′ ′ ′ • − ◦ ◦ J : a2 = c,b′2 = c,a c = d,b c = d,a b = d. 4 ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ • − ◦ ◦ ◦ J : a2 = d,b′2 = d,c2 = d. 5 ′ ′ ′ ′ ′ • J : a2 = c,b′2 = c,a b = d. 6 ′ ′ ′ ′ ′ ′ • ◦ Define ϕ ,ϕ ,ϕ ,ϕ and ϕ as follow : 1 2 3 4 5 5 ϕ (a) = a + b,ϕ (b) = σ a b ,ϕ (c) = c,ϕ (d) = d where σ2 = 1, then • 1 ′ 2 1 ′ − 2 1 ′ 1 ′ − J ≅ J and J ≅ J . (cid:0) (cid:1) 1 4,3 5 4,5 ϕ (a) = a+b,ϕ (b) = a b,ϕ (c) = 1c,ϕ (d) = 1d, then J ≅ J . • 2 ′ 2 2 ′ −2 2 ′ 2 2 ′ 2 2 4,10 ϕ (a) = a+ b,ϕ (b) = a b,ϕ (c) = c,ϕ (d) = d, then J ≅ J . • 3 ′ 2 3 ′ − 2 3 ′ 3 ′ 3 4,8 ϕ (a) = a b c,ϕ (b) = a+b c,ϕ (c) = 2c,ϕ (d) = 2d, then J ≅ J . • 4 ′ − − 2 4 ′ − 2 4 ′ − 4 ′ − 4 4,9 ϕ (a) = a+b,ϕ (b) = a b,ϕ (c) = c+d,ϕ (d) = c d, then J ≅ J . • 5 ′ 5 ′ − 5 ′ 5 ′ − 6 4,13 NowwedescribeH2 (J,k)onlyfornilpotentassociativeJordanalgebrasandH2 (J,k) Ass. Jor. for both associative and non-associative nilpotent Jordan algebras of dimension 6 4 to de- terminealgebrasandtheircocycleswhosecenteralextensionsarenilpotentnon-associative Jordan algebras. We get the following : Algebra H2 (J,k) H2 (J,k) Ass. Jor. J < δ > H2 (J ,k) 1,1 a,a Ass. 1,1 J < δ ,δ ,δ > H2 (J ,k) 2,1 a,a a,b b,b Ass. 2,1 J < δ > H2 (J ,k) 2,2 a,b Ass. 2,2 J < δ ,δ ,δ ,δ ,δ ,δ > H2 (J ,k) 3,1 a,a b,b c,c a,b a,c b,c Ass. 3,1 J < δ ,δ ,δ > H2 (J ,k) < δ > 3,2 a,b a,c c,c Ass. 3,2 ⊕ b,c J < δ ,δ > H2 (J ,k) < δ ,δ > 3,3 a,a b,b Ass. 3,2 ⊕ a,c b,c J < δ +δ > H2 (J ,k) 3,4 a,c b,b Ass. 3,4 J < δ ,δ ,δ ,δ ,δ ,δ ,δ ,δ ,δ ,δ > H2 (J ,k) 4,1 a,a b,b c,c d,d a,b a,c a,d b,c b,d c,d Ass. 4,1 J < δ ,δ ,δ ,δ ,δ ,δ > H2 (J ,k) < δ ,δ > 4,2 c,c d,d a,b a,c a,d c,d Ass. 4,2 ⊕ b,c b,d J < δ ,δ ,δ ,δ ,δ > H2 (J ,k) < δ ,δ ,δ > 4,3 a,a b,b d,d a,d b,d Ass. 4,3 ⊕ a,c b,c c,d J < δ +δ ,δ ,δ > H2 (J ,k) < δ > 4,4 a,c b,b a,d d,d Ass. 4,4 ⊕ b,d J < δ ,δ ,δ ,δ ,δ ,δ > / < δ +δ > H2 (J ,k) < δ ,δ ,δ > 4,5 a,a b,b c,c a,b a,c b,c c,c a,b Ass. 4,4 ⊕ a,d b,d c,d J < δ ,δ ,δ ,δ > 4,6 a,b a,c c,c b,d −−−−− J < δ ,δ ,δ > / < δ +δ > H2 (J ,k) < δ +δ > 4,7 a,c a,b c,c a,b c,c Ass. 4,7 ⊕ b,b a,d J < δ ,δ ,δ > 4,8 a,a b,b b,c −−−−− J < δ ,δ ,δ ,δ > / < δ +δ > 4,9 a,a b,b a,c b,c b,b a,c −−−−− J < δ ,δ ,δ ,δ > / < δ +δ > 4,10 a,a b,b a,c b,c a,c b,c −−−−− J < δ +δ > H2 (J ,k) 4,11 a,d b,b Ass. 4,11 J < δ ,δ ,δ +δ > < δ ,δ ,δ ,δ ,δ > 4,12 a,c b,b a,d b,c a,c b,b a,d b,c b,d J < δ ,δ ,δ > H2 (J ,k) < δ ,δ > 4,13 a,b a,c b,d Ass. 4,13 ⊕ b,c a,d From the previous table we have the following lemma: 6 Lemma 3.2. Every nilpotent non-associative Jordan algebra of dimension five is one of the following : 1. 1-dimensionalcenteralextensionof nilpotentnon-associativeJordanalgebrasJ ,J ,J 4,6 4,8 4,9 and J . 4,10 2. 1-dimensional centeral extension of J ,J ,J ,J J ,J and J by non- 4,2 4,3 4,4 4,5, 4,7 4,12 4,13 associative Jordan cocycles. 3. 2-dimensional centeral extension of J and J by non-associative Jordan cocycles. 3,2 3,3 4 Non-associative nilpotent Jordan algebras Inthissection wecomputecenteral extensions foralgebrasstatedinlemma 3.2. Wehavea number of subsections; in each subsection the central extensions of one particular Jordan algebra are considered excpet at subsection 4.2 we consider Jordan algebras J J and 4,8, 4,9 J . 4,10 4.1 1-dimensional centeral extension of J 4,6 TheJordanalgebraJ isnon-associative, firstlyweconsider1-dimensionaltrivialcenteral 4,6 extension of J corresponding to θ = 0. We get algebra : 4,6 J = J J . 5,1 4,6 1,1 • ⊕ Next we consider non-trivial 1-dimensional centeral extension of J . Let θ = α δ + 4,6 1 a,b α δ + α δ + α δ H2 (J ,k), the center of J , Z(J ), is spanned by d. 2 a,c 3 b,d 4 c,c ∈ Jor. 4,6 4,6 4,6 Furthermore the automorphism group,Aut(J ), consists of : 4,6 a 0 0 0 11  a a2 0 0  φ = 21 11 ,a a = 0. a 0 a 0 11 33 6 31 33    a 2a a a a2 a   41 21 31 43 11 33  The automorphism group acts as follows : α a2 (α a +α a )+2α a2 a , 1 −→ 11 1 11 3 41 3 21 31 α a (α a +α a )+α a a , 2 33 2 11 4 31 3 21 43 −→ α α a4 a , 3 −→ 3 11 33 α α a2 . 4 −→ 4 33 We need α = 0 in order to have that d do not lie in the radical of θ. Choose 3 6 a = 0,a = α1a11,a = a a and a = 1 we obtain that [α ,α ,α ,α ] 31 41 − α3 21 11 33 43 −α3 1 2 3 4 −→ [0,0,α a4 a ,α a2 ]. If α = 0, choose a = 1 and a = 1 we get a cocycle θ = δ , 3 11 33 4 33 4 11 33 α3 1 b,d 7 on the other hand if α = 0, choose a = 4 √α4 and a = 1 we get a cocycle 4 6 11 q α3 33 √α4 θ = δ + δ . It easy to see that θ and θ are not in the same Aut(J )-orbit, so we 2 b,d c,c 1 2 4,6 get two algebras : J : a2 = b,b c = d,b d = e. 5,2 • ◦ ◦ J : a2 = b,b c = d,b d = e,c2 = e. 5,3 • ◦ ◦ 4.2 1-dimensional centeral extension of J J and J . 4,8, 4,9 4,10 The Jordan algebras J J and J are non-associative and have no non-trivial 1- 4,8, 4,9 4,10 dimensional centeral extensions since the radical of θ,s always intersect with the center of these algebras. So we only consider 1-dimensional trivial centeral extensions of J J 4,8, 4,9 and J . We get algebras : 4,10 J = J J . 5,4 4,8 1,1 • ⊕ J = J J . 5,5 4,9 1,1 • ⊕ J = J J . 5,6 4,10 1,1 • ⊕ 4.3 1-dimensional centeral extension of J 4,2 Let θ = α δ + α δ + α δ + α δ + α δ + α δ + α δ + α δ such that 1 c,c 2 d,d 3 a,b 4 a,c 5 a,d 6 c,d 7 b,c 8 b,d (α ,α ) = (0,0). The center of 7 8 6 J ,Z(J ), isspannedbyb,candd. Furthermoretheautomorphismgroup,Aut(J ), 4,2 4,2 4,2 consists of : a 0 0 0 11  a a2 a a  φ = 21 11 23 24 ,a (a a a a ) = 0. a 0 a a 11 33 44 − 34 43 6 31 33 34    a 0 a a  41 43 44   The automorphism group acts as follows : α a (α a +α a +α a )+a (α a +α a +α a )+a (α a +α a ) 1 33 1 33 7 23 6 43 43 2 43 6 33 8 23 23 7 33 8 43 −→ α a (α a +α a +α a )+a (α a +α a +α a )+a (α a +α a ) 2 34 1 34 7 24 6 44 44 2 44 6 34 8 24 24 7 34 8 44 −→ α a2 (α a +α a +α a ) 3 −→ 11 3 11 7 31 8 41 α a (α a +α a +α a +α a )+a (α a +α a +α a +α a ) 4 33 1 31 4 11 7 21 6 41 43 2 41 5 11 6 31 8 21 −→ +a (α a +α a +α a ) 23 3 11 7 31 8 41 α a (α a +α a +α a +α a )+a (α a +α a +α a +α a ) 5 34 1 31 4 11 7 21 6 41 44 2 41 5 11 6 31 8 21 −→ +a (α a +α a +α a ) 24 3 11 7 31 8 41 α a (α a +α a +α a )+a (α a +α a +α a )+a (α a +α a ) 6 34 1 33 7 23 6 43 44 2 43 6 33 8 23 24 7 33 8 43 −→ 8 α a2 (α a +α a ) 7 −→ 11 7 33 8 43 α a2 (α a +α a ) 8 −→ 11 7 34 8 44 No generality is lost by assuming that (α ,α ) = (1,0). To fix (α ,α ) (1,0) 7 8 7 8 −→ choose a = a = 1 and a = 0 then : 11 33 34 α α +2a +a (α a +2α ) 1 1 23 43 2 43 6 −→ α α a2 2 −→ 2 44 α α +a 3 3 31 −→ α (α a +α +a +α a )+a (α a +α +α a )+a (α +a ) 4 1 31 4 21 6 41 43 2 41 5 6 31 23 3 31 −→ α a (α a +α +α a )+a (α +a ) 5 44 2 41 5 6 31 24 3 31 −→ α a (α a +α )+a 6 44 2 43 6 24 −→ α 1 7 −→ α 0 8 −→ Choose a = 0,a = 1α ,a = α and a = α a we obtain that : 43 23 −2 1 31 − 3 24 − 6 44 α 0,α 0,α 0,α 1,α 0, 1 3 6 7 8 −→ −→ −→ −→ −→ α α a2 , 2 −→ 2 44 α α α +α +a +α a , 4 1 3 4 21 6 41 −→ − α a (α a +α α α ). 5 44 2 41 5 3 6 −→ − Thus no generality is lost by assuming that [α ,α ,α ,α ,α ] = [0,0,0,1,0]. 1 3 6 7 8 To fix [α ,α ,α ,α ,α ] [0,0,0,1,0] take a = a = 1 and a = a = a = 1 3 6 7 8 11 33 34 43 31 −→ a = a = 0, then choose a = α ,we get that : 24 23 21 4 − [α ,α ,α ,α ,α ,α ,α ,α ] [0,α a2 ,0,0,a (α a +α ),0,1,0]. 1 2 3 4 5 6 7 8 −→ 2 44 44 2 41 5 If α = 0 choose a = α5, so we ge a cocycle θ = δ +δ . If α = 0 then α = 0 2 6 41 −α2 1 d,d b,c 2 5 6 otherwise θ Z(J ) = 0, choose a = 1 , so we get a cocycle θ = δ +δ . A Gro¨bner ⊥∩ 4,2 6 41 α5 2 a,d b,c basis computation shows that θ and θ are not in the same Aut(J )-orbit, so we get 1 2 4,2 two algebras : J : a2 = b,d2 = e,b c = e. 5,7 • ◦ J : a2 = b,a d = e,b c = e. 5,8 • ◦ ◦ 4.4 1-dimensional centeral extension of J 4,3 Letθ = α δ +α δ +α δ +α δ +α δ +α δ +α δ +α δ and(α ,α ,α ) = 0. 1 a,a 2 b,b 3 d,d 4 a,c 5 a,d 6 b,c 7 b,d 8 c,d 4 6 8 6 The center of J , Z(J ), is spanned by c and d. Furthermore the automorphism group,Aut(J ), 4,3 4,3 4,3 consists of : a a 0 0 11 12  a a 0 0  φ = 21 22 ,a a2 a2 a2 a2 = 0,a a = a a = 0. a a a a +a a a 44 11 22 − 12 21 6 11 21 12 22 31 32 11 22 12 21 34   (cid:0) (cid:1)  a a 0 a  41 42 44   9 The automorphism group acts as follows : α a (α a +α a +α a +α a )+a (α a +α a +α a ) 1 41 5 11 3 41 7 21 8 31 11 1 11 4 31 5 41 −→ +a (α a +α a +α a )+a (α a +α a +α a ) 21 2 21 6 31 7 41 31 4 11 6 21 8 41 α a (α a +α a +α a +α a )+a (α a +α a +α a ) 2 42 5 12 3 42 7 22 8 32 12 1 12 4 32 5 42 −→ +a (α a +α a +α a )+a (α a +α a +α a ) 22 2 22 6 32 7 42 32 4 12 6 22 8 42 α a (α a +2α a ) 3 44 3 44 8 34 −→ α (a a +a a )(α a +α a +α a ) 4 11 22 12 21 4 11 6 21 8 41 −→ α a (α a +α a )+a (α a +α a )+a (α a +α a )+α a a 5 11 4 34 5 44 21 6 34 7 44 41 3 44 8 34 8 31 44 −→ α (a a +a a )(α a +α a +α a ) 6 11 22 12 21 4 12 6 22 8 42 −→ α a (α a +α a )+a (α a +α a )+a (α a +α a )+α a a 7 12 4 34 5 44 22 6 34 7 44 42 3 44 8 34 8 32 44 −→ α α a (a a +a a ) 8 8 44 11 22 12 21 −→ Case (1) If α = 0 : 8 6 Take a = a = 0,a = a = a = 1,a = α3 ,a = α4 and a = α6 then 12 21 11 22 44 34 −2α8 41 −α8 42 −α8 [α ,α ,α ] [0,0,0], so we can assume that α = α = α = 0. Choose a = a = 3 4 6 3 4 6 34 41 −→ a = 0,a = α5a11+α7a21 and a = α5a12+α7a22, it follows that : 42 31 − α8 31 − α8 α α a2 +α a2 , 1 −→ 1 11 2 21 α α a2 +α a2 , 2 −→ 1 12 2 22 α 0,α 0,α 0,α 0,α 0, 3 4 5 6 7 −→ −→ −→ −→ −→ α α a (a a +a a ). 8 8 44 11 22 12 21 −→ Case (1.1) If α = α = 0 : Choose a = a = 1 and a = 1 we get a cocycle 1 2 11 22 44 α8 θ = δ . 1 c,d Case (1.2) If α = 0,α = 0 : Choosea = a = 0,a = 1 ,a = √α anda = 1 1 6 2 12 21 11 √α1 22 1 44 α8 we get a cocycle θ = δ +δ . 2 a,a c,d Case (1.3) If α = 0,α = 0 : Choosea = a = 0,a = √α ,a = 1 anda = 1 1 2 6 11 22 12 2 21 √α2 44 α8 we get again a cocycle θ = δ +δ . 2 a,a c,d Case (1.4) If α = 0,α = 0 : Choose a = a = 0,a = 1 ,a = 1 and 1 6 2 6 12 21 11 √α1 22 √α2 a = √α1√α2 we get a cocycle θ = δ +δ +δ . 44 α8 3 a,a b,b c,d Case (2) If α = 0 : 8 α a (α a +α a +α a )+a (α a +α a +α a ) 1 41 5 11 3 41 7 21 11 1 11 4 31 5 41 −→ +a (α a +α a +α a )+a (α a +α a ) 21 2 21 6 31 7 41 31 4 11 6 21 α a (α a +α a +α a )+a (α a +α a +α a ) 2 42 5 12 3 42 7 22 12 1 12 4 32 5 42 −→ +a (α a +α a +α a )+a (α a +α a ) 22 2 22 6 32 7 42 32 4 12 6 22 α α a2 3 −→ 3 44 α (a a +a a )(α a +α a ) 4 11 22 12 21 4 11 6 21 −→ α a (α a +α a )+a (α a +α a )+α a a 5 11 4 34 5 44 21 6 34 7 44 3 44 41 −→ α (a a +a a )(α a +α a ) 6 11 22 12 21 4 12 6 22 −→ α a (α a +α a )+a (α a +α a )+α a a 7 12 4 34 5 44 22 6 34 7 44 3 44 42 −→ α 0 8 −→ 10