On the description of Leibniz algebras with nilindex n−3

ON THE DESCRIPTION OF THE LEIBNIZ ALGEBRAS WITH NILINDEX n−3 J.M. CABEZAS, L.M. CAMACHO, J.R. G´ OMEZ, B.A. OMIROV Abstract. In this paper we present the classification of a subclass of naturally graded Leibniz algebras. These n-dimensional Leibniz algebras have the characteristic sequence equal to (n−3,3).For this purpose we use the software Mathematica. AMS Subject Classifications (2000): 17A32, 17A36, 17A60, 17B70. Key words: Lie algebra, Leibniz algebra, nilpotence, natural gradation, characteristic sequence, p-filiformlicity. 1. Introduction Leibniz algebras are one of the new algebras introduced by Loday [11], [12] in connection with the study of periodicity phenomena in algebraic K-theory. Leibniz algebras have been introduced as a ”non-antisymmetric” analogue of Lie algebras. A Leibniz algebra Lis a vector space equipped with a bracket [-,-] satisfying the identity [x, [y, z]] = [[x, y], z]−[[x, z], y]. If the antisymmetric relation is assumed, this identity is equivalent to the Jacobi identity. Hence, a Lie algebra is a Leibniz algebra. It is well known that the natural gradation of nilpotent Lie and Leibniz algebras is very helpful in investigating their structural properties. A remarkable fact of the naturally graded algebras is the relative simplicity of the study of the cohomological properties, (see for example [6]- [10] and [13]). Recently, some papers are focused to the study of some interesting families of Leibniz algebras, such as p-filiform and quasi-filiform Leibniz algebras. These algebras have their characteristic sequences equal to (n−p, 1,1, ..., 1) and (n−2,2) with dim(L) = n, [4]–[5]. Naturally graded p-filiform Leibniz algebras are already classified in [2] and [4]. The classification of naturally graded nul-filiform and filiform Leibniz algebras reader can find in [1]. The quasi-filiform n-dimensional Leibniz algebras have characteristic sequence (n−2,1,1) (the case of 2-filiform) or (n−2,2) [3] and [5]. For a given Leibniz algebra Lwe define the descending central series as follows: L1=L, Lk+1 = [Lk, L], k ≥1. If there exists a natural number ssuch that Ls= 0,then the Leibniz algebra Lis said to be nilpotent and minimal such number is called the nilindex of the algebra L. Bellow we present a gradation closely related to the descending central series. Let Lbe a nilpotent Leibniz algebra with nilindex s. We put Li=Li/Li+1 for 1≤i≤s−1,and grL =L1⊕L2⊕ · · · ⊕ Ls−1.It is easy to check embedding 1 2 J.M. CABEZAS, L.M. CAMACHO, J.R. G´ OMEZ, B.A. OMIROV [Li, Lj]⊆Li+jand therefore, the algebra grL is graded algebra, which is called the naturally graded Leibniz algebra. Let xbe a nilpotent element of the set L\L2. For the nilpotent operator of right multiplication Rxwe define a decreasing sequence C(x) = (n1, n2,...,nk), which consists of the dimensions of Jordan blocks of the operator Rx. On the set of such sequences we consider the lexicographic order, that is, C(x) = (n1, n2,...,nk)≤ C(y) = (m1, m2, . . . , ms)⇐⇒ there exists i∈Nsuch that nj=mjfor any j < i and ni< mi. The sequence C(L) = max C(x)x∈L\L2is called characteristic sequence of the algebra L. If C(L) = (1,1,...,1) then evidently, the algebra Lis abelian. The set R(L) = {x∈L|[y, x] = 0 for any y∈L}is said to be a right annihilator of the algebra L. In this work we classify a subclass of naturally graded Leibniz algebras with nilindex n−3.In case of Leibniz algebras with nilindex equal to n−3,for the characteristic sequence we have the following tree possibilities: (n−3,1,1,1),(n−3,2,1) and (n−3,3). The first one is 3-filiform case. We will focus our attention on the study of those with characteristic sequence (n−3,3). Throughout all the work, we use the software Mathematica. Since in the case of non-Lie Leibniz algebras the skewsymmetric identity is not valid, this classification is very complex and we should overcome the difficulties, which need a lot of computations. Using computer programs is very helpful for computing the Leibniz identity in low dimension and formulate the generalizations of the calculations, which are proved for arbitrary finite dimension. The used program can be find in [5]. Some examples of the programs for various types of Leibniz algebras classes are in the following Web site: http://personal.us.es/jrgomez. 2. Naturally graded Leibniz algebras with characteristic sequence (n−3,3). Let Lbe a naturally graded n-dimensional Leibniz algebra which characteristic sequence equal to (n−3,3). From the definition of the characteristic sequence, it follows the existence of a basis {e1, e2,...,en}such that element e1∈L\L2and the operator of right multiplication Re1has one of the following forms: Jn−30 0J3,J30 0Jn−3 Definition 2.1. A naturally graded Leibniz algebra Lwhich characteristic sequence is equal to (n−3,3), is called algebra of the second type if there exists a basic element e1∈L\L2such that the operator Re1has the form: Jn−30 0J3; if Re1has the other form, then it is called algebra of the second type. Since the classification of Leibniz algebras of the second type is more complicated and it needs to use more original technics, first we present the description of the second type. ON THE DESCRIPTION OF THE LEIBNIZ ALGEBRAS WITH NILINDEX n−3 3 Theorem 2.1. Let Lbe an n-dimensional naturally graded Leibniz algebra of the second type (n≥9). Then it is isomorphic to one of the following pairwise nonisomorphic algebras: λ µ dim(L) L0,1 (0,0,0,0,0) odd or even L0,2 (0,0,0,λ,−1) λ∈ {0,1}odd or even L0,3 (1,0,0,λ,−1) λ∈Codd or even L0,4 (1,0,1/4,λ,−1) λ∈Codd or even L0,5 (0,0,1,λ,−1) λ∈Codd or even L0,6 (0,1,0,λ,−1) λ∈ {0,1}odd or even L0,6 (µ,1,0,λ,−1) λ∈Cµ∈ {1,2}odd or even L0,7 (0,1,µ,λ,−1) λ∈Cµ∈C\ {0}odd or even L0,8 (−2λ,1,−λ,2,−1) λ∈ {−2,−4/3}odd or even L0,9 (2λ,1,λ,0,−1) λ∈C\ {0,1}odd or even L0,10 (1,1,1/4,1/4,−1) odd or even L0,10 (1,1,1/4,1/2,−1) odd or even L0,10 (2,1,1,1,−1) odd or even L0,10 (2,1,1,0,−1) odd or even L0,11 (1,λ,1/4,0,−1) λ∈C\ {0,1/2}odd or even L1,2 (0,0,0,λ,−1) λ∈ {0,1}even L1,3 (1,0,0,λ,−1) λ∈Ceven L1,4 (1,0,1/4,λ,−1) λ∈Ceven L1,6 (µ,1,0,λ,−1) λ∈Cµ∈Ceven L1,7 (0,γ,µ,λ,−1) λ∈Cγ, µ ∈C\ {0}even L1,9 (−2λ,1,λ,µ,−1) λ∈C\ {0,1}µ∈Ceven L1,11 (λ,1,λ2/4,µ,−1) λ∈C\ {−2,0}µ∈Ceven L1,12 (−1,0,0,λ,−1) λ∈ {0,1}even L1,13 (−2,0,1,λ,−1) λ∈Ceven L1,14 (−4,0,2,λ,−1) λ∈Ceven L1,15 (0,0,−1,λ,−1) λ∈Ceven L1,16 (−2,0,−1,λ,−1) λ∈Ceven L1,17 (0,−1,0,λ,−1) λ∈ {0,1}even L1,18 (−1,−1,0,λ,−1) λ∈Ceven L1,19 (−2,−1,0,1,−1) even L1,20 (1,−1,0,λ,−1) λ∈C\ {−1/2}even L1,21 (1,1/3,0,λ,−1) λ∈Ceven L1,22 (−2,−1,1,λ,−1) λ∈ {0,1}even L1,23 (1,1/2,1/4,λ,−1) λ∈Ceven 4 J.M. CABEZAS, L.M. CAMACHO, J.R. G´ OMEZ, B.A. OMIROV λ γ, µ dim(L) L1,24 (−4,−1,2,λ,−1) λ∈Ceven L1,25 (−3,−4/3,2,λ,−1) λ∈Ceven L1,26 (2/5,2,2/5,λ,−1) λ∈Ceven L1,27 (2/λ,λ,1,µ,−1) λ∈C\ {−1,0,1}µ∈Ceven L1,28 (8/5,1/2,−4/5,λ,−1) λ∈Ceven L1,29 (λ,−1,λ2/4,0,−1) λ∈C\ {−2,0}even L1,30 (1,−1,1/4,λ,−1) λ∈ {−1/2,1/4}even L1,31 (−8,2,16,λ,−1) λ∈Ceven L1,32 (−2,λ,1,0,−1) λ∈C\ {−1,0}even L1,33 (−2,1,1,λ,−1) λ∈ {−1,1}even where the algebra Lǫ,j (α1,α2,α3,α4,β):ǫ∈ {0,1},1≤j≤33, β ∈ {−1,0} has the following multiplication:                            [ei, e1] = ei+1,1≤i≤n−1, i 6= 3 [e1, e4] = α1e2+βe5, [e2, e4] = α2e3, [e4, e4] = α3e2, [e5, e4] = α4e3, [e1, e5] = (α1−α2)e3−e6, [e4, e5] = (α3−α4)e3, [e1, ei] = βei+1,6≤i≤n−1, [ei, en+3−i] = ǫ(−1)ien,4≤i≤n−1. Proof. From the condition of the theorem we have the following multiplication of the basic element e1on the right side: [ei, e1] = ei+1,1≤i≤n−1, i 6= 3,[e3, e1] = [en, e1] = 0. From these products we conclude that L1=< e1, e4>, L2=< e2, e5>, L3=< e3, e6>, Li=< ei+3 >, 4≤i≤n−3 and e2, e3∈R(L). Let us introduce denotations [e1, e4] = α1e2+β1e5,[e2, e4] = α2e3+β2e6,[e3, e4] = β3e7, [e4, e4] = α3e2+β4e5,[e5, e4] = α4e3+β5e6, [ei, e4] = βiei+1,6≤i≤n−1,[en, e4] = 0. The equalities [ei, e5] = [[ei, e4], e1]−[[ei, e1], e4],1≤i≤nderive [e1, e5] = (α1−α2)e3+ (β1−β2)e6,[e2, e5] = (β2−β3)e7,[e3, e5] = β3e8, [e4, e5] = (α3−α4)e3+ (β4−β5)e6,[e5, e5] = (β5−β6)e7, [ei, e5] = (βi−βi+1)ei+2,6≤i≤n−2 [en−1, e5] = [en, e5] = 0. Using induction on jfor any value iit can be proved that [ei, ej] = j−4 X k=0 (−1)kj−4 kβi+k!ei+j−3,5≤i≤n−3,6≤j≤n+ 3 −i. In the case of e4∈R(L) we obtain the algebra L0,1 (0,0,0,0,0). ON THE DESCRIPTION OF THE LEIBNIZ ALGEBRAS WITH NILINDEX n−3 5 Let now e4/∈R(L).Then we consider the following cases: e5∈R(L) Then ei∈R(L) for 2 ≤i≤n, i 6= 4. From the equalities [[ei, e1], e4] = [[ei, e4], e1],1≤i≤n, we have α2=α1, α4=α3, β3=β2=β1, βi=β4,5≤i≤n−1. For n≥8 we have also β1= 0. The change of basis taken as e′ i=ei,1≤i≤n, i 6= 4,5,6, e′ j=ej−β4ej−3,4≤j≤6 deduces β4= 0. If we take the change of basis in the following way: e′ 1=Ae1+Be4, e′ n−2=e1, e′ j= [e′ j−1, e′ 1],2≤j≤n, j 6=n−2 with condition AB(A+α1B)6= 0, then we obtain the algebra of the first type. Therefore, this case is impossible for the algebra of the second type. e5/∈R(L) The embedding [e4, e4]∈R(L) implies β4= 0 and from [ei,[e4, e1]] = −[ei,[e1, e4]], with 1 ≤i≤nwe obtainβ1=−1. If e6∈R(L), then for n≥9 it follows β1= 0,which is a contradiction with the condition β1=−1. Therefore, e6/∈R(L). It is easy to check that [ei, ej] + [ej, ei]∈R(L) for any values of i, j. Applying this for i= 1 and j= 5 we obtain β2= 0. The following equalities: [e1, ei] = −ei+1,[e2, ei] = [e3, ei] = 0,6≤i≤n−1 are proved by induction on i. From [e1,[e4, e2j+1]] = −[e5, e2j+1] + [e2j+2, e4], j ≥2, we have that 2β2j+2 =β5+β2j+1 + 2j−4 X k=1 (−1)k2j−3 k(β5+k−β4+k), j ≥2. Similar as in [5] we derive βj=β5,6≤j≤n−1,for n odd, βj=β5,6≤j≤n−2,for n even and [e4, en−1] = −β5enfor nodd, [e4, en−1] = (βn−1−2β5)enfor neven, [ei, en+3−i] = (−1)i(βn−1−β5)en,5≤i≤n−2, for neven. If βn−1=β5, then by the change of basis defined as e′ i=ei,1≤i≤n, i 6= 4,5,6, and e′ i=ei−β5ei−3,4≤i≤6 we can assume β5= 0. If βn−16=β5(the case of neven), then by using the change of basis:            e′ i= (βn−1−β5)iei,1≤i≤3, e′ 4=e4−β5e1, e′ 5= (βn−1−β5)(e5−β5e2), e′ 6= (βn−1−β5)2(e6−β5e3), e′ i= (βn−1−β5)i−4ei,7≤i≤n 6 J.M. CABEZAS, L.M. CAMACHO, J.R. G´ OMEZ, B.A. OMIROV we obtain [ei, en+3−i] = (−1)ienfor 4 ≤i≤n−1.Thus, multiplication in Lis as follows:                            [ei, e1] = ei+1,1≤i≤n−1, i 6= 3, [e1, e4] = α1e2−e5, [e2, e4] = α2e3, [e4, e4] = α3e2, [e5, e4] = α4e3, [e1, e5] = (α1−α2)e3−e6, [e4, e5] = (α3−α4)e3, [e1, ei] = −ei+1,6≤i≤n−1, [ei, en+3−i] = ǫ(−1)ien,4≤i≤n−1, ǫ ∈ {0,1}. Case 1. ǫ= 0 (nodd or even) Applying the general change of generators of the basis: e′ 1= n X i=1 Aiei, e′ n−2= n X i=1 Biei, we determine the other elements of the new basis and the products in this basis. Then the new parameters are the following: α′ 1=(α1A1+ 2α3A4)B4 A2 1+α1A1A4+α3A2 4 , α′ 2=α2B4 A1+α2A4 , α′ 3=α3B2 4 A2 1+α1A1A4+α3A2 4 , α′ 4=(α4A1+α2α3A4)B2 4 (A1+α2A4)(A2 1+α1A1A4+α3A2 4), satisfying the restriction A1(A1+α2A4)(A2 1+α1A1A4+α3A2 4)B46= 0. Note that for new parameters we have α′2 1−4α′ 3=(α2 1−4α3)A2 1B2 4 (A2 1+α1A1A4+α3A2 4)2, α′ 1α′ 2−2α′ 3=(α1α2−2α3)A1B2 4 (A1+α2A4)(A2 1+α1A1A4+α3A2 4), α′ 1α′ 2−2α′ 4=(α1α2−2α4)A1B2 4 (A1+α2A4)(A2 1+α1A1A4+α3A2 4). Consequently, the nullity of α2 1−4α3is invariant in the following sense: if α2 1−4α3= 0,then α′2 1−4α′3= 0 and if α2 1−4α36= 0,then α′2 1−4α′36= 0. Analogously, the expressions α1α2−2α3and α1α2−2α4are nullity invariants. Consider the following subcases: α2= 0,α3= 0 Then, α′ 1=α1B4 A1+α1A4 , α′ 2= 0, α′ 3= 0 and α′ 4=α4B2 4 A1(A1+α1A4). •α1= 0. If α4= 0,then the algebra L0,2 (0,0,0,λ,−1) with λ= 0 is obtained. If α46= 0,then we obtain the algebra L0,2 (0,0,0,λ,−1) with λ= 1. •α16= 0. If α4= 0,then we easily obtain α′ 1= 1. Thus, we have the algebra L0,3 (1,0,0,λ,−1) with λ= 0. ON THE DESCRIPTION OF THE LEIBNIZ ALGEBRAS WITH NILINDEX n−3 7 If α46= 0,then choosing appropriate values of A4and B4we derive α′ 1=α′ 4= 1. Hence, the algebra L0,3 (1,0,0,λ,−1) with λ= 1 is obtained. α2= 0,α36= 0 Then, α′ 1=(α1A1+ 2α3A4)B4 A2 1+α1A1A4+α3A2 4 , α′ 2= 0, α′ 3=α3B2 4 A2 1+α1A1A4+α3A2 4 , α′ 4=α4B2 4 A2 1+α1A1A4+α3A2 4 . •If α2 1−4α3= 0,then taking adequate value of B4we obtain α′ 1= 1, α′ 3= 1/4 and α′ 4=α4 α2 1 =λ. So, we obtain the family of algebras L0,4 (1,0,1/4,λ,−1) with λ∈C. •If α2 1−4α36= 0,then taking suitable values of A4and B4we deduce α′ 1= 0, α′ 3= 1 and α′ 4=α4 α3 =λ. The family L0,5 (0,0,1,λ,−1),λ∈Cis obtained. α26= 0,α3= 0 Then, α′ 1=α1B4 A1+α1A4 , α′ 2=α2B4 A1+α2A4 , α′ 3= 0, α′ 4=α4B2 4 (A1+α1A4)(A1+α2A4). •α1= 0. If α4= 0,then the choosing appropriate B4leads α′ 2= 1. Thus, we obtain L0,6 (0,1,0,λ,−1), λ = 0. If α46= 0,then taking adequate A4and B4we derive α′ 2=α′ 4= 1. The algebra L0,6 (0,1,0,λ,−1), λ = 1 is obtained. •α16= 0. Xα4= 0. If α1−α2= 0,then for suitable B4we have α′ 1=α′ 2= 1,i.e. we obtain the algebra L0,6 (µ,1,0,λ,−1) with µ= 1, λ = 0. If α1−α26= 0,then for adequate A4and B4it follows that α′ 1= 2, α′ 2= 1. The algebra L0,6 (µ,1,0,λ,−1),with µ= 2, λ = 0 is obtained. Xα46= 0. If α1−α2= 0,then for appropriate value of B4we have α′ 1=α′ 2= 1 and α′ 4=α4 α2 1 =λ. Therefore, we obtain the family of algebras L0,6 (µ,1,0,λ,−1), where µ= 1, λ∈C\ {0}. If α1−α26= 0,then taking suitable values of A4and B4we obtain α′ 1= 2, α′ 2= 1, α′ 4=2α4 α1α2 =λ, i.e., the family L0,6 (µ,1,0,λ,−1), µ= 2, λ ∈C\ {0}is obtained. α26= 0,α36= 0 •α2 1−4α36= 0, α1α2−2α36= 0. Taking appropriate A4and B4we derive α′ 1= 0, α′ 2= 1, α′ 3=−(α1α2−2α3)2 α2 2(α2 1−4α3)=µ, 8 J.M. CABEZAS, L.M. CAMACHO, J.R. G´ OMEZ, B.A. OMIROV α′ 4=−(α1α2−2α3)(α1α2−2α4) α2 2(α2 1−4α3)=λ. Hence, we obtain the family of algebras L0,7 (0,1,µ,λ,−1), where µ∈C\{0}, λ ∈ C. •α2 1−4α36= 0, α1α2−2α3= 0. It yields α′ 3−α′ 4=(α3−α4)α2A1B2 4 (A1+α2A4)(α2A2 1+ 2α3A1A4+α2α3A2 4), 2α′ 3α′ 4−α′2 2α′ 3−α′2 4=(2α3α4−α2 2α3−α2 4)α2 2A2 1B4 4 (A1+α2A4)2(α2A2 1+ 2α3A1A4+α2α3A2 4)2. Xα3−α4= 0. Therefore, 2α3α4−α2 2α3−α2 46= 0 and taking the suitable values of A4 and B4we obtain α′ 1= 4, α′ 2= 1, α′ 3= 2, α′ 4= 2. Thus, the algebra L0,8 (−2λ,1,−λ,2,−1) with λ=−2 is obtained. Xα3−α46= 0. If 2α3α4−α2 2α3−α2 4= 0,then α46= 0, α3=α2 4 2α4−α2 2 ,α46=α2 2 2. Choosing adequate values of A4and B4we obtain α′ 1= 8/3, α′ 2= 1, α′ 3= 4/3, α′ 4= 2, i.e., we derive the algebra L0,8 (−2λ,1,−λ,2,−1) with λ=−4/3. If 2α3α4−α2 2α3−α2 46= 0,then as before we deduce α′ 1= 2α′ 3, α′ 2= 1, α′ 3=−(α3−α4)2 2α3α4−α2 2α3−α2 4 =λ,α′ 4= 0 and the family L0,9 (2λ,1,λ,0,−1) with λ∈C\ {0,1}is obtained. •α2 1−4α3= 0, α1α2−2α36= 0. Then, α16= 2α2,α′2 1−4α′ 4=(α2 1−4α4)4A1B2 4 (2A1+α1A4)2(A1+α2A4). Xα2 1−4α4= 0. Then, α1α2−2α46= 0 and from the above we deduce α′ 1= 1, α′ 2= 1, α′ 3= 1/4, α′ 4= 1/4.So, we obtain the algebra L0,10 (λ,1,λ2/4,µ,−1) with λ= 1, µ = 1/4. Xα2 1−4α46= 0, α1α2−2α4= 0 ⇒α′ 1= 1, α′ 2= 1, α′ 3= 1/4, α′ 4= 1/2, i.e., we obtain L0,10 (λ,1,λ2/4,µ,−1) with λ= 1, µ = 1/2. Xα2 1−4α46= 0, α1α2−2α46= 0 ⇒α′ 1= 1, α′ 2=α1α2−2α4 α2 1−4α4 , α′ 3= 1/4, α′ 4= 0. The family L0,11 (1,λ,1/4,0,−1), where λ∈C\ {0,1/2}is obtained. •α2 1−4α3= 0, α1α2−2α3= 0. ON THE DESCRIPTION OF THE LEIBNIZ ALGEBRAS WITH NILINDEX n−3 9 Then, α1= 2α2, α3=α2 2, α′2 2−α′ 4=(α2 2−α4)A1B2 4 (A1+α2A4)3. Xα2 2−α4= 0. Taking an appropriate value of B4it follows that α′ 1= 2, α′ 2= 1, α′ 3= 1, α′ 4= 1. Hence, we obtain L0,10 (λ,1,λ2/4,µ,−1) with λ= 2, µ = 1. Xα2 2−α46= 0. Choosing adequate A4and B4=(α2 2−α4)A1 α3 2 yields α′ 1= 2, α′ 2= 1, α′ 3= 1 and α′ 4= 0. Thus, the algebra L0,10 (λ,1,λ2/4,µ,−1) with λ= 2, µ = 0 is obtained. Now, we consider the other case. Case 2. ǫ= 1 (neven) Similar to the case 1, we apply the general change of generators of basis. Then, we obtain all products and the following expressions for α′ i,1≤i≤4: α′ 1=(A1−A4)(α1A1+ 2α3A4) A2 1+α1A1A4+α3A2 4 , α′ 2=α2(A1−A4) A1+α2A4 , α′ 3=α3(A1−A4)2 A2 1+α1A1A4+α3A2 4 , α′ 4=(A1−A4)2(α4A1+α2α3A4) (A1+α2A4)(A2 1+α1A1A4+α3A2 4), verifying the restriction A1(A1−A4)(A1+α2A4)(A2 1+α1A1A4+α3A2 4)6= 0. Note that for these parameters we have α′2 1−4α′ 3=(α2 1−4α3)A2 1(A1−A4)2 (A2 1+α1A1A4+α3A2 4)2, α′ 1α′ 2−2α′ 3=−(α1α2−2α3)A1(A1−A4)2 (A1+α2A4)(A2 1+α1A1A4+α3A2 4), α′ 1α′ 2−2α′ 4=(α1α2−2α4)A1(A1−A4)2 (A1+α2A4)(A2 1+α1A1A4+α3A2 4), α′ 1+ 2α′ 3=(α1+ 2α3)(A1−A4)A1 A2 1+α1A1A4+α3A2 4 . Consequently, the nullity of the expressions α2 1−4α3, α1α2−2α3, α1α2−2α4, α1+ 2α3are invariants. Applying arguments as in the case 1 for the following subcases: α2= 0 α3= 0 , α2= 0,α36= 0 , α26= 0 α3= 0 , α26= 0,α36= 0 we obtain the rest algebras and families of the theorem.  The next theorem completes the classification of naturally graded Leibniz algebras with characteristic sequence (n−3,3).