Naturally graded Zinbiel algebras with nilindex n - 3

NATURALLY GRADED ZINBIEL ALGEBRAS WITH NILINDEX n−3 J.Q. ADASHEV, L.M. CAMACHO, S. G ´ OMEZ-VIDAL, I.A. KARIMJANOV Abstract. We present the classification of a subclass of n-dimensional naturally graded Zinbiel algebras. This subclass has the nilindex n−3 and the characteristic sequence (n−3,2,1).In fact, this result completes the classification of naturally graded Zinbiel algebras of nilindex n−3. Mathematics Subject Classification 2010: 17A32. Key Words and Phrases: Zinbiel algebra, Leibniz algebra, nilpotency, characteristic sequence 1. Introduction. Intensive investigation on Lie algebras leads to the appearance of a new algebraic object – Leibniz algebras. The Leibniz algebras introduced by Loday in [7] are a ”non commutative” algebras analogue to Lie algebras. It should be mentioned that Leibniz algebras inherit an important Lie algebra property: the operator of right multiplication on an element of an algebra is a derivation. Leibniz algebras form a Koszul operad in the sense of V. Ginzburg and M. Kapranov [6]. Under the Koszul duality the operad of Lie algebras is dual the operad of associative and commutative algebras. The notion of dual Leibniz algebra defined by J.-L. Loday [8] is precisely the dual operad of Leibniz algebras in this sense. In this paper, we study algebras which are the dual to Leibniz algebras in Koszul sense. J.-L. Loday studied in [8] categorical properties of Leibniz algebras and considered in this connection a new object – Zinbiel algebras (Leibniz is written in reverse order). Since the category of Zinbiel algebras is Koszul dual to the category of Leibniz algebras, sometimes they are also called dual Leibniz algebras. In [2, 5, 9] some crucial properties of Zinbiel algebras were obtained. Particularly, in [5], the authors prove that every finite-dimensional Zinbiel algebra over complex numbers is nilpotent. However, the study of nilpotent algebras is too complex and should be carried out with additional conditions, such as conditions on nilindex, various types of gradations, characteristic sequence and others. The aim of this work is to continue the study of complex finite-dimensional naturally graded Zinbiel algebras. The n-dimensional Zinbiel algebras of nilindex kwith n−2≤k≤nare classified in [1, 2]. The classification of complex n-dimensional naturally graded Zinbiel algebras of nilindex n−3 is a difficult problem and it should be divided into three cases. Namely, it is necessary to consider the possibilities of the characteristic sequence of such algebras: (n−3,3), (n−3,1,1,1) and (n−3,2,1). The classification of complex naturally graded Zinbiel algebras of nilindex n−3 with characteristic sequence equal to (n−3,3) and (n−3,1,1,1) has been done in [1]. The knowledge of naturally graded algebras of a certain family offers significant information about their structural properties. In this paper we obtain the classification of naturally graded Zinbiel algebras of nilindex n−3 with characteristic sequence (n−3,2,1).Thus, we complete the study for the n−3 case. All the spaces and the algebras are considered over the field of complex numbers. We omit the products which are equal to zero for convenience. Throughout all the work we use the software Mathematica (see [3]) to compute the Zinbiel identity in low dimensions and to formulate the generalizations of the calculations, which are proved for arbitrary dimension. Moreover, the program allows us to construct new bases using some general transformation of the generators of the algebra. Since the direct sum of nilpotent Zinbiel algebras is nilpotent, we shall consider only non split algebras. 2. Preliminaries In this section we introduce some definitions, notations and results, which are necessary for the understanding of graded Zinbiel algebras. 1 2 J.Q. ADASHEV, L.M. CAMACHO, S. G´ OMEZ-VIDAL, I.A. KARIMJANOV Definition 2.1. A vector space Zover a field Kwith a bilinear operation “◦” is called Zinbiel algebra if for any x, y, z ∈ Z the following identity (2.1) (x◦y)◦z=x◦(y◦z) + x◦(z◦y) holds. Examples of Zinbiel algebras can be found in [2, 5, 8]. Z(a, b, c) denotes the following polynomial: Z(a, b, c) = (a◦b)◦c−a◦(b◦c)−a◦(c◦b). Zinbiel algebras are defined by the identity Z(a, b, c) = 0. For a given Zinbiel algebra Zthe sequence of two-sided ideals defined recursively as follow: Z1=Z,Zk+1 =Z ◦ Zk, k ≥1. is said to be the lower central series. Definition 2.2. A Zinbiel algebra Zis called nilpotent if there exists s∈Nsuch that Zs6= 0 and Zs+1 = 0.The minimal number ssatisfying this property is called the index of nilpotency or nilindex of the algebra Z. For a given Zinbiel algebra Zwe introduce denotations: R(Z) = {x∈ Z | y◦x= 0 for any y∈ Z} − − the right annihilator of Z, L(Z) = {x∈ Z | x◦y= 0 for any y∈ Z} − − the left annihilator of Z, Cent(Z) = {x, y ∈ Z | x◦y=y◦x= 0 for any y∈ Z} − − the center of Z. It is easy to see that the center and the right annihilator of Zare two-sided ideals. Let us denote by Lxthe operator of left multiplication on element x, i.e. Lx:Z −→ Z such that Lx(y) = x◦yfor any y∈ Z. Let Zbe a complex n-dimensional Zinbiel algebra and xbe an element of the set Z \ Z2. For the operator Lxwe define a descending sequence C(x) = (n1, n2,...,nk) with n1+···+nk=n, which consists of the dimensions of the Jordan blocks of the operator Lx. In the set of such sequences we consider the lexicographic order, that is, C(x) = (n1, n2,...,nk)< C(y) = (m1, m2,...,ms) if there exists isuch that ni< miand nj=mjfor j < i. Taking into account the equality n1+···+nk= m1+···+mssuch comparison is always applicable. Definition 2.3. The sequence C(Z) = max{C(x) : x∈ Z \ Z2}is called the characteristic sequence of the algebra Z. In [5], the authors prove that Zinbiel algebras of finite dimension are nilpotent. Since we focused our attention on finite dimension complex nilpotent Zinbiel algebras. Let Zbe a finite-dimensional nilpotent Zinbiel algebra with nilindex equal to s. For i(1 ≤i≤s) we put Zi=Zi/Zi+1 and we obtain the graded Zinbiel algebra gr(Z) = Z1⊕ Z2⊕...⊕ Zs,where Zi◦ Zj⊆ Zi+j. An algebra Zif called naturally graded if Z∼ =gr(Z).It is not difficult to see that Zi+1 =Z1◦ Zi in the naturally graded algebra Z. Let Zbe a naturally graded Zinbiel algebra with characteristic sequence (n−3,2,1).By definition of characteristic sequence there exists a basis {e1, e2,...,en}in the algebra Zsuch that the operator Le1has one block Jn−3of size (n−3),one block J2of size 2 and one block J1of size one. Note that there will be six possibilities for the operators Le1.By a change of basis it is easy to prove that the six cases can be reduced to the following three cases: I.   Jn−30 0 0J20 0 0 J1  , II.   J20 0 0Jn−30 0 0 J1  , III.  J10 0 0Jn−30 0 0 J2  . Definition 2.4. A Zinbiel algebra Zis called either of first type (type I), second type (type II) or third type (type III) if the operator Le1has the form: I.   Jn−30 0 0J20 0 0 J1  , II.   J20 0 0Jn−30 0 0 J1  , III.  J10 0 0Jn−30 0 0 J2   NATURALLY GRADED ZINBIEL ALGEBRAS WITH NILINDEX n−3 3 respectively. From now on we denote by Cj ithe combinatorial numbers Cj i=i j. The following result holds: Lemma 2.5. [4] Let Zbe a Zinbiel algebra such that e1◦ei=ei+1 for 1≤i≤k−1,with respect to the adapted basis {e1,...,ek, ek+1,...,en}.Then ei◦ej=Cj i+j−1ei+j,for 2≤i+j≤k 3. Main Result 3.1. Type I. Algebras of type I with n≥8.So, we have the following brackets:            e1◦ei=ei+1,1≤i≤n−4, e1◦en−3= 0, e1◦en−2=en−1, e1◦en−1= 0, e1◦en= 0. It is easy to see that Zi⊇ heiiwhere 1 ≤i≤n−3. It is evident that dim(Z1)>1.In fact, if dim(Z1) = 1,then the algebra Zis one-degenerated and therefore it is a zero-filiform algebra, but it is not an algebra of nilindex n−3.Let us assume that en−2∈ Zr1and en∈ Zr2, then en−1∈ Zr1+1. We can distinguish the following cases: Case I. If r1=r2= 1. Then we have that Z1=< e1, en−2, en>, Z2=< e2, en−1>, Z3=< e3>, . . . , Zn−3=< en−3> and the following products: e1◦e1=e2, e1◦en−2=en−1, en−2◦e1=α1e2+α2en−1, en−2◦en−2=α3e2+α4en−1, en−2◦en=α5e2+α6en−1, en◦e1=β1e2+β2en−1, en◦en−2=β3e2+β4en−1, en◦en=β5e2+β6en−1, e1◦e2=e3, en−2◦e2=γ1e3, en−2◦en−1=γ2e3, en◦e2=γ3e3, en◦en−1=γ4e3. From the equality Z(e1, en, e1) = Z(e1, en, en) = 0 we have β1=β5= 0. Let us consider the equalities Z(e1, en−2, e1) = Z(e1, en−1, e1) = 0 then it follows α1= 0. From the equalities Z(e1, e1, en−2) = Z(en−2, e1, e1) = Z(en−2, en−1, e1) = Z(e1, e1, en) = 0 Z(e1, en, e2) = Z(en, en−1, e1) = Z(e1, en−2, en−2) = Z(e1, en−1, en−2) = 0 Z(e1, en−2, en) = Z(e1, en, en−1) = Z(e1, e1, en−1) = Z(e1, en−2, e2) = 0 Z(en−2, en, e1) = Z(en−2, en−2, e1) = 0 we obtain γ1=γ2=γ3=γ4=α3=α5=β3= 0, and e2◦en−2=en−2◦e2=e2◦en−1=en−1◦e2=e2◦en=en◦e2= 0. Now, by mathematical induction method, we prove that en−1◦ek= 0 and ek◦en−1= 0 with 2≤k≤n−3. •If k= 2,then we have en−1◦e2=e2◦en−1= 0. •Let us suppose that for some kthe equalities en−1◦ek= 0 and ek◦en−1= 0 are true. We prove it for k+ 1. en−1◦ek+1 =en−1◦(e1◦ek) = (en−1◦e1)◦ek−en−1◦(ek◦e1) = =−C1 ken−1◦ek+1 =−ken−1◦ek+1, en−1◦ek+1 = 0. ek+1 ◦en−1= (e1◦ek)◦en−1=e1◦(ek◦en−1) + e1◦(en−1◦ek) = = 0 4 J.Q. ADASHEV, L.M. CAMACHO, S. G´ OMEZ-VIDAL, I.A. KARIMJANOV As in previous cases, it easy to see that ek◦en−2=en−2◦ek= 0 and ek◦en=en◦ek= 0 for 2≤k≤n−3. Thus, we have obtained the following family of algebras: Z(a1, a2, a3, a4, a5, a6) :                        ei◦ej=Cj i+j−1ei+j,2≤i+j≤n−3, e1◦en−2=en−1, en−2◦e1=a1en−1, en−2◦en−2=a2en−1, en−2◦en=a3en−1, en◦e1=a4en−1, en◦en−2=a5en−1, en◦en=a6en−1, where we omit the products that are equal to zero. Theorem 3.1. An arbitrary Zinbiel algebra of the family Z(a1, a2, a3, a4, a5, a6)is isomorphic to one of the following pairwise non-isomorphic algebras: Z1(1,0,0,0,1,0), Z2(0,0,0,0,1,0), Z3(0,1,0,1,0,0), Z4(0,0,0,1,0,0), Z5(0,1,0,0,0,0), Z6(1,1,0,0,0,0), Z7(λ, 0,0,0,0,0), λ ∈C, Z8(0, λ, 1,0,0,1), λ ∈C\ {0}, Z9(α, −α (α−1)2,1,0,0,1), α ∈C\ {0,1}, Z10(0,0,1,0,1,1), Z11(1,0,1,0,1,1), Z12(0,0,1,1,0,0), Z13(0,0,1,0,0,0), Z14(λ, 1,1,0,1,1), λ ∈C, Z15(0,1,1,−1,1,1), Z16(1,1,1,0,1,1). Proof. Let Zbe satisfying to the hypothesis of the theorem. Due to the property of natural gradation of the algebra it is enough to consider the following change of generators: e′ 1=P1e1+Pn−2en−2+Pnen, e′ n−2=Q1e1+Qn−2en−2+Qnen, e′ n=R1e1+Rn−2en−2+Rnen. Making the general change of basis in the family Z(a1, a2, a3, a4, a5, a6),we derive the expressions of the new parameters in the new basis (1): a′ 1=a1P1Qn−2+a2Pn−2Qn−2+a3PnQn−2+a4P1Qn+a5Pn−2Qn+a6PnQn P1Qn−2+a2Pn−2Qn−2+a3Pn−2Qn+a5PnQn−2+a6PnQn , a′ 2=a2Q2 n−2+a3Qn−2Qn+a5Qn−2Qn+a6Q2 n P1Qn−2+a2Pn−2Qn−2+a3Pn−2Qn+a5PnQn−2+a6PnQn , a′ 3=a2Qn−2Rn−2+a3Qn−2Rn+a5QnRn−2+a6QnRn P1Qn−2+a2Pn−2Qn−2+a3Pn−2Qn+a5PnQn−2+a6PnQn , a′ 4=a1P1Rn−2+a2Pn−2Rn−2+a3PnRn−2+a4P1Rn+a5Pn−2Rn+a6PnRn P1Qn−2+a2Pn−2Qn−2+a3Pn−2Qn+a5PnQn−2+a6PnQn , a′ 5=a2Qn−2Rn−2+a3QnRn−2+a5Qn−2Rn+a6QnRn P1Qn−2+a2Pn−2Qn−2+a3Pn−2Qn+a5PnQn−2+a6PnQn , a′ 6=a2R2 n−2+a3Rn−2Rn+a5Rn−2Rn+a6R2 n P1Qn−2+a2Pn−2Qn−2+a3Pn−2Qn+a5PnQn−2+a6PnQn , and the following restrictions: (2)          Q1=R1= 0, P1Rn−2+a2Pn−2Rn−2+a3Pn−2Rn+a5PnRn−2+a6PnRn= 0, P1Qn−2+a2Pn−2Qn−2+a3Pn−2Qn+a5PnQn−2+a6PnQn6= 0, P1(Qn−2Rn−QnRn−2)6= 0. We can distinguish two cases: Case 1. Let en∈R(Z) be, then a3=a6= 0. NATURALLY GRADED ZINBIEL ALGEBRAS WITH NILINDEX n−3 5 From the restrictions, (P1+a2Pn−2+a5Pn)Rn−2= 0, (P1+a2Pn−2+a5Pn)Qn−26= 0, P1(Qn−2Rn−QnRn−2)6= 0.    ⇒Rn−2= 0. it follows that P1Qn−2Rn6= 0.Thus, the new parameters are: a′ 1=a1P1Qn−2+a2Pn−2Qn−2+a4P1Qn+a5Pn−2Qn Qn−2(P1+a2Pn−2+a5Pn), a′ 2=a2Qn−2+a5Qn P1+a2Pn−2+a5Pn , a′ 4=Rn(a4P1+a5Pn−2) Qn−2(P1+a2Pn−2+a5Pn), a′ 5=a5Rn P1+a2Pn−2+a5Pn , We observe that the nullity of a5is invariant. Moreover, it is easy to check that the nullity of the following expression a′ 2a′ 4−a′ 1a′ 5=(a2a4−a1a5)P1Rn (P1+a2Pn−2+a5Pn)2 is invariant. Thus, we can distinguish the following non-isomorphic cases: Case 1.1. Let a56= 0 be. Then choosing Rn=P1+a2Pn−2+a5Pn a5 , Pn−2=−a4P1 a5 , Qn=−a2Qn−2 a5 we have a′ 5= 1, a′ 4= 0, a′ 2= 0, a′ 1=(a2a4−a1a5)P1 (a2a4−a5)P1−a2 5Pn and the determinant is formed by the potencies of the following non-zero factors: P1Qn−2a5((a2a4− a5)P1−a2 5Pn). a) If a2a4−a1a56= 0, choosing Pn=P1(a1−1) a5we receive a′ 1= 1.It follows the algebra Z1(1,0,0,0,1,0). b) If a2a4−a1a5= 0, then we obtain a′ 1= 0 and we have the algebra Z2(0,0,0,0,1,0). Case 1.2. Let a5= 0 be. Then, a′ 5= 0 and we have a′ 1=a1P1Qn−2+a2Pn−2Qn−2+a4P1Qn Qn−2(P1+a2Pn−2), a′ 2=a2Qn−2 P1+a2Pn−2 , a′ 4=a4P1Rn Qn−2(P1+a2Pn−2). with P1Qn−2Rn(P1+a2Pn−2)6= 0. We observe that the nullities of a2and a4are invariant, so we can distinguish the following cases: a) Let a46= 0 be. Then, choosing Rn=Qn−2(P1+a2Pn−2) a4P1 , Qn=−Qn−2(a1P1+a2Pn−2) a4P1 , we get a′ 4= 1 and a′ 1= 0. a.1) If a26= 0, then choosing Qn−2=P1+a2Pn−2 a2 ,we obtain a′ 2= 1 and the algebra Z3(0,1,0,1,0,0). a.2) If a2= 0, then we have a′ 2= 0 and the algebra Z4(0,0,0,1,0,0). 6 J.Q. ADASHEV, L.M. CAMACHO, S. G´ OMEZ-VIDAL, I.A. KARIMJANOV b) Let a4= 0 be. Then a′ 4= 0 and we have a′ 1=a1P1+a2Pn−2 P1+a2Pn−2 , a′ 2=a2Qn−2 P1+a2Pn−2 . We have that the nullity of the following expression: a′ 1−1 = P1(a1−1) P1+a2Pn−2 . is invariant. b.1) Let a26= 0 be. Then, choosing Qn−2=P1+a2Pn−2 a2 ,we obtain a′ 2= 1. •If a1−16= 0,then putting Pn−2=−a1P1 a2,we have a′ 1= 0 and the algebra Z5(0,1,0,0,0,0). The determinant of change of basis consists of the potencies of the following non-zero factors a2(a1−1)P1Rn. •If a1−1 = 0,then a′ 1= 1 and we obtain Z6(1,1,0,0,0,0). b.2) Let a2= 0 be. Then, we have a′ 2= 0, a′ 1=a1=λ∈Cand the family Z7(λ, 0,0,0,0,0),with λ∈C. Case 2. Let en/∈R(Z) be, then (a3, a6)6= (0,0).We can suppose that a36= 0,in another case, a3= 0 and a66= 0 we make the following change of basis f′ 1=f1+f3.Thus, a36= 0.Taking into account the expressions given in (1), the restrictions (2) and the following expression: ∆ = a3 3a4+a2 3a4a5−a1a2 3a4a5+a2a3a2 4a5−a1a3a4a2 5− −a1a2 3a6−3a2a3a4a6+a1a2a3a4a6−a2 2a2 4a6+ +a3a5a6+a2 1a3a5a6+a2a4a5a6+a1a2a4a5a6− −a1a2 5a6−a2a2 6+ 2a1a2a2 6−a2 1a2a2 6, the nullity of the following expressions are invariant a′2 3−a′ 3a′ 5+a′2 5−a′ 2a′ 6=(a2 3−a3a5+a2 5−a2a6)(Qn−2Rn−QnRn−2)2 P1Qn−2+a2Pn−2Qn−2+a5PnQn−2+a3Pn−2Qn+a6PnQn , a′ 3a′ 5−a′ 2a′ 6=(a3a5−a2a6)(Qn−2Rn−QnRn−2)2 P1Qn−2+a2Pn−2Qn−2+a5PnQn−2+a3Pn−2Qn+a6PnQn , ∆′=∆P2 1(Qn−2Rn−QnRn−2)4 (a2Qn−2Rn−2+a5QnRn−2+a3Qn−2Rn+a6QnRn)2, a′ 3−a′ 5=(a3−a5)(Qn−2Rn−QnRn−2) P1Qn−2+a2Pn−2Qn−2+a5PnQn−2+a3Pn−2Qn+a6PnQn . We can distinguish the following non isomorphic cases: Case 2.1. Let a3a5−a2a66= 0 be. Then, choosing Pn−2=−(a6P1QnRn−2+a2a5Qn−2R2 n−2+a2 5QnR2 n−2−a6P1Qn−2Rn+ +a3a5Qn−2Rn−2Rn+a2a6Qn−2Rn−2Rn+ 2a5a6QnRn−2Rn+a3a6Qn−2R2 n+ +a6QnR2 n)1 (a3a5−a2a6)(Qn−2Rn−QnRn−2) Pn=−(−a3P1QnRn−2−a2 2Qn−2Rn−2−a2a5QnR2 n−2+a3P1Qn−2Rn− −2a2a3Qn−2Rn−2Rn−a3a5QnRn−2Rn−a2a6QnRn−2Rn−a2 3Qn−2R2 n− −a3a6QnR2 n)1 (a3a5−a2a6)(Qn−2Rn−QnRn−2) and using the restriction (2), we obtain a′ 3= 1. NATURALLY GRADED ZINBIEL ALGEBRAS WITH NILINDEX n−3 7 a) Let a3−a56= 0 be. Then, choosing Qn=−a2Qn−2Rn−2+a5Qn−2Rn a3Rn−2+a6Rn , Rn−2=a3Qn−2−a5Qn−2−a6Rn a3 we get a′ 5= 0, a′ 6= 1 and a′ 2=λ∈C\ {0}.The determinant of the change of basis is formed by the potencies of the following non-zero factors: (a3Rn−2+a6Rn)P1Qn−2(a2R2 n−2+a3Rn−2Rn+a5Rn−2Rn+a6R2 n). a.1) Let ∆ 6= 0 be. Then, we choose Rn=−(a2a3a4−a3a5−a2a4a5+a1a2 5+a2a6−a1a2a6)P1 (a3−a5)(a3a5−a2a6), Qn−2= (a2 3a4−a1a2 3a6−2a2a3a4a6+a3a5a6+a1a3a5a6+ +a2a4a5a6−a1a2 5a6−a2a2 6+a1a2a2 6)P1 (a3−a5)2(a3a5−a2a6), and we get a′ 1=a′ 4= 0 and the family Z8(0, λ, 1,0,0,1), λ ∈C\{0}.The determinant of change of basis is formed by the non-zero potencies of the following factors (a3−a5)(a3a5−a2a6)∆P1. a.2) Let ∆ = 0 be. Then, we have ∆′=a′ 5= 0, a′ 3=a′ 6= 1, a′ 4−a′ 1−3a′ 2a′ 4+a′ 2a′ 1a′ 4−a′2 2a′2 4−a′ 2(a′ 1−1)2= 0. Thus, we obtain the following family ei◦ej=Cj i+j−1ei+j,2≤i+j≤n−3, e1◦en−2=en−1, en−2◦e1=αen−1, en−2◦en=en−1, en−2◦en−2=βen−1,with β6= 0 en◦en=en−1, en◦e1=γen−1,with γ−α−3βγ +αβγ −β2γ2−β(1 −α)2= 0 Now, we make the generic change of basis e′ 1=P1e1+Pn−2en−2+Pnen, e′ n−2=Q1e1+Qn−2en−2+Qnen, e′ n=R1e1+Rn−2en−2+Rnen. and we have the expressions of the new parameters and the new restrictions: α′=αP1Qn−2+βPn−2Qn−2+γP1Qn+PnQn−2+Pn−2Qn+PnQn P1Qn−2+βPn−2Qn−2+Pn−2Qn+PnQn−2+PnQn , β′=βQ2 n−2+ 2Qn−2Qn+Q2 n P1Qn−2+βPn−2Qn−2+Pn−2Qn+PnQn−2+PnQn , γ′=αP1Rn−2+βPn−2Rn−2+γP1Rn+PnRn−2+Pn−2Rn+PnRn P1Qn−2+βPn−2Qn−2+Pn−2Qn+PnQn−2+PnQn , 1 = βQn−2Rn−2+Qn−2Rn+QnRn−2+QnRn P1Qn−2+βPn−2Qn−2+Pn−2Qn+PnQn−2+PnQn , 1 = βR2 n−2+ 2Rn−2Rn+R2 n P1Qn−2+βPn−2Qn−2+Pn−2Qn+PnQn−2+PnQn , 0 = P1Rn−2+βPn−2Rn−2+Pn−2Rn+PnRn−2+PnRn= 0.(∗∗) 8 J.Q. ADASHEV, L.M. CAMACHO, S. G´ OMEZ-VIDAL, I.A. KARIMJANOV Putting Pn−2=−P1QnRn−2−P1Qn−2Rn+βQn−2Rn−2Rn+Qn−2R2 n+QnR2 n β(QnRn−2−Qn−2Rn), Pn= (P1QnRn−2+β2Qn−2R2 n−2−P1Qn−2Rn+ 2βQn−2Rn−2Rn+ +βQnRn−2Rn+Qn−2R2 n+QnR2 n)1 β(QnRn−2−Qn−2Rn), Qn−2=Rn−2+Rn, Qn=−βQn−2Rn−2 Rn−2+Rn =−βRn−2, we get α′=−(−P1Rn−2+ 2βP1Rn−2−αβP1Rn−2+β2γP1Rn−2+ +βR2 n−2−β2R2 n−2−P1Rn+βP1Rn−αβP1Rn+ +Rn−2Rn−βRn−2Rn+R2 n−βR2 n)1 β(βR2 n−2+Rn−2Rn+R2 n), γ′= (P1Rn−2−βP1Rn−2+αβP1Rn−2−βR2 n−2+P1Rn+ +βγP1Rn−Rn−2Rn−R2 n)1 β(βR2 n−2+Rn−2Rn+R2 n), β′=β6= 0 with P1(βR2 n−2+Rn−2Rn+R2 n)6= 0.It is easy to prove that: γ′−α′−3β′γ′+α′β′γ′−β′2γ′2−β′(α′−1)2=γ−α−3βγ +αβγ −β2γ2−β(α−1)2 βR2 n−2+Rn−2Rn+R2 n P2 1= 0. Now, if we choose Rn=1 2(P1+βγP1−Rn−2)− −q(P1+βγP1−Rn−2)2+ 4(P1Rn−2−βP1Rn−2+αβP1Rn−2−βR2 n−2) 2 we get γ′= 0, β′=β6= 0, α′+β′(α′−1)2= 0 with α′∈C\ {0,1}, thus β′=−α′ (α′−1)2 and we have the family Z9(α, −α′ (α′−1)2,1,0,0,1),with α∈C\ {0,1}. b) Let a3−a5= 0 be. Then, we have a′ 3=a′ 5= 1, a2 3−a2a66= 0 and a′ 1=((a1−1)Qn−2+a4Qn)P1 a2Qn−2Rn−2+a3QnRn−2+a3Qn−2Rn+a6QnRn + 1, a′ 2=a2Q2 n−2+ 2a3Qn−2Qn+a6Q2 n a2Qn−2Rn−2+a3QnRn−2+a3Qn−2Rn+a6QnRn , a′ 4=((a1−1)Rn−2+a4Rn)P1 a2Qn−2Rn−2+a3QnRn−2+a3Qn−2Rn+a6QnRn , a′ 6=a2R2 n−2+ 2a3Rn−2Rn+a6R2 n a2Qn−2Rn−2+a3QnRn−2+a3Qn−2Rn+a6QnRn , with P1(QnRn−2−Qn−2Rn)(a2Qn−2Rn−2+a3QnRn−2+a3Qn−2Rn+a6QnRn)6= 0, P1Rn−2+a2Pn−2Rn−2+a3Pn−2Rn+a5PnRn−2+a6PnRn= 0. We can suppose a′ 2= 0 , •a26= 0,if we choose Qn−2=−a3±pa2 3−a2a6 a2 Qn,we get a′ 2= 0, •a2= 0,if we choose Qn−2=−a6Qn 2a3 we have a′ 2= 0. NATURALLY GRADED ZINBIEL ALGEBRAS WITH NILINDEX n−3 9 Analogously, we can suppose that a′ 4= 0 , using Rn. Now, we have the new family: ei◦ej=Cj i+j−1ei+j,2≤i+j≤n−3, e1◦en−2=en−1, en−2◦e1=a′ 1en−1, en−2◦en=en−1, en◦en−2=en−1, en◦en=a′ 6en−1. and we make a generic change of basis. Choosing Pnand Pn−2(as in previous cases) we get a′′ 3=a′′ 5= 1. Putting Rn−2= 0, Qn−2=−a6Qn 2we have a′′ 4=a′′ 2= 0 and a′′ 1=(1 −a′ 1)P1+Rn Rn , a′′ 6=2Rn Qn It is easy to check that the nullity of a′′ 1−1 is invariant because a′′ 1−1 = (a′ 1−1)P1Qn−2 (Qn−2+a6Qn)Rn Moreover, choosing Qn= 2Rnwe obtain a′ 6= 1.The determinant of change of basis is formed by the non-zero potencies of the following factors a6P1QnRn. Now, we can distinguish two cases: b.1) If a′ 1−16= 0,choosing Rn= (a′ 1−1)P1we get a′′ 1= 0 and the algebra Z10(0,0,1,0,1,1). b.2) If a′ 1−1 = 0,then a′′ 1= 1 and we have Z11(1,0,1,0,1,1). Case 2.2. Let a3a5−a2a6= 0 be. As a36= 0 ⇒a5=a2a6 a3.We substitute in (1) and in (2) and we choose Pn−2=−a3P1Rn−2+a2a6PnRn−2+a3a6PnRn a3(a2Rn−2+a3Rn). Now, taking into account the new parameters (1), we get to: a′ 2=(a2Qn−2+a3Qn)(a3Qn−2+a6Qn)(a2Rn−2+a3Rn) a2 3P1(Qn−2Rn−QnRn−2), a′ 3=(a3Qn−2+a6Qn)(a2Rn−2+a3Rn)2 a2 3P1(Qn−2Rn−QnRn−2)6= 0. with a3P1(QnRn−2−Qn−2Rn)(a2R1+a3Rn)6= 0 and that the nullity of the following expressions: a′ 1a′ 6−a′ 3a′ 4=a3(a1a6−a3a4)(a2Rn−2+a3Rn)(Qn−2Rn−QnRn−2)P1 (a3a6PnQn+a3P1Qn−2+a2a3Pn−2Qn−2+a2a6PnQn−2+a2 3Pn−2Qn)2, a′2 3−a′ 2a′ 6=(a2 3−a2a6)(a3Qn−2+a6Qn)(a2Rn−2+a3Rn)(Qn−2Rn−QnRn−2) (a3a6PnQn+a3P1Qn−2+a2a3Pn−2Qn−2+a2a6PnQn−2+a2 3Pn−2Qn)2, are invariant. Thus, we can distinguish the non isomorphic cases: