3-filiform Leibniz algebras of maximum length

3-FILIFORM LEIBNIZ ALGEBRAS OF MAXIMUM LENGTH. L.M. CAMACHO, E.M. CA˜ NETE, J.R. G´ OMEZ, B.A. OMIROV Abstract. This work completes the study of the solvable Leibniz algebras, more precisely, it completes the classification of the 3-filiform Leibniz algebras of maximum length [4]. Moreover, due to the good structure of the algebras of maximum length, we also tackle some of their cohomological properties. Our main tools are the previous result of Cabezas and Pastor [3], the construction of appropriate homogeneous basis in the considered connected gradation and the computational support provided by the two programs implemented in the software Mathematica. AMS Subject Classifications (2010): 17A32, 17A36, 17A60, 17B70. Key words: Lie algebra, Leibniz algebra, nilpotence, natural gradation, characteristic sequence, p-filiform, maximum length, cohomology. 1. Introduction Leibniz algebras appear from the cohomology study done by Loday in 1993 [12] and they are further investigated by several authors as Ayupov, Casas and others ([1], [7]). In the cohomology study there is an important family of Leibniz algebras: those whose length of the gradation is maximum. The remarkable fact that an algebra can be decomposed into direct sum of subspaces of dimension 1 makes easier the calculations of the derivations since they induce the corresponding gradation of the group of cohomologies. The main goal of this paper is to continue the study of the p-filiform Leibniz algebras of maximum length. These algebras play a main role in mathematics over the last years, either in the classification theory or in geometrical, analytical and physical applications. In early works we have already closed the classification of the p-filiform Leibniz algebras of maximum length for 0 ≤p≤2 (see [1], [2]). Here we study the 3-filiform Leibniz algebras of maximum length, their spaces of derivations and their first cohomology group. Moreover, we will use three programs very helpful to obtain the classification of maximum length algebras and their space of derivations. Recall [12] that an algebra Lover a field Fis called a Leibniz algebra if it satisfies the following Leibniz identity: [x, [y, z]] = [[x, y], z]−[[x, z], y],∀x, y, z ∈ L where [., .] denotes the multiplication in L. Consider an arbitrary algebra Lin the set of n-dimensional Leibniz algebras over a field F. Let B={e1, e2,· · · en}be a basis of L. Then Lis determined, un to isomorphisms, by the multiplication rule for the basis elements; namely, [ei, ej] = n X k=1 γk ijek where γk ij are the structure constants. Therefore, fixing a basis, we can regard each algebra of dimension n over a field F as a point in the n3-dimensional space of structure constants endowed with the Zariski topology. From now on the Leibniz algebras will be considered over the field of complex numbers C, and with finite dimension. Let Lbe a Leibniz algebra, then Lis naturally filtered by the descending central sequence L1=L,Lk+1 = [Lk,L] with k≥1.Thus, a nilpotent algebra Lhas nilindex equal to sif s is the minimum integer such that Ls6={0}and Ls+1 ={0}. A Leibniz algebra Lis Z-graded if L=⊕i∈ZVi, where [Vi, Vj]⊆Vi+jfor any i, j ∈Zwith a finite number of non null spaces Vi. L.M. Camacho and E.M. Ca˜nete would like to thank to the Institute of Mathematics and Information Technologies of Uzbekistan for their hospitality. The second author was partially supported by a Grant of the Junta de Andaluc´ıa (Spain). 1 2 L.M. CAMACHO, E.M. CA ˜ NETE, J.R. G ´ OMEZ, B.A. OMIROV We will say that a Z-graded Leibniz algebra Ladmits a connected gradation if L=Vk1⊕Vk1+1 ⊕ · · · ⊕ Vk1+tand Vk1+i6=<0>for any i(0 ≤i≤t). Let us define the naturally graded algebras as follows: Definition 1.1. Let us take Li=Li/Li+1,1≤i≤kand grL=L1⊕ L2⊕ · · · ⊕ Lk. Then [Li,Lj]⊆ Li+jand we obtain the graded algebra grL. If grLand Lare isomorphic, in notation grL∼ =L, we say that Lis a naturally graded algebra. The above constructed gradation is called natural gradation. Definition 1.2. The number l(⊕L) = l(Vk1⊕Vk1+1 ⊕ · · · ⊕ Vk1+t) = t+ 1 is called the length of the gradation, where ⊕L is a connected gradation. The gradation ⊕L has maximum length if l(⊕L) = dim(L). We define the length of an algebra Lby: l(L) = max{l(⊕L) such that ⊕ L =Vk1⊕ · · · ⊕ Vktis a connected gradation}. An algebra Lis called of maximum length if l(L) = dim(L). The set R(L) = {x∈ L : [y, x] = 0,∀y∈ L} is called the right annihilator of L.Rxdenotes the operator Rx:L→Lsuch that Rx(y)=[y, x],∀y∈ L and it is called the right operator. The set Cent(L) = {z∈ L : [x, z]=[z, x]=0,∀x∈ L} is called the center of L. Let xbe a nilpotent element of the set L\L2. For the nilpotent operator Rxwe define a descending sequence C(x) = (n1, n2, . . . , nk), which consists of the dimensions of the Jordan blocks of the operator Rx. In the set of such sequences we consider the lexicographic order, that is, C(x)=(n1, n2, . . . , nk)< C(y)=(m1, m2, . . . , ms) if and only if there exists i∈Nsuch that nj=mjfor any j < i and ni< mi. Definition 1.3. The sequence C(L) = max C(x)x∈L\L2is called the characteristic sequence of the algebra L. Let Lbe an n-dimensional nilpotent Leibniz algebra and pa non negative integer (p<n). Definition 1.4. The Leibniz algebra Lis called p-filiform if C(L) = (n−p, 1,...,1 | {z } p ). If p= 0,Lis called null-filiform and if p= 1 it is called filiform. Therefore, an algebra with the characteristic sequence (n−2,1,1) is called 2-filiform, whereas a nilpotent algebra with nilindex n−2 is called quasi-filiform. Note that in the Lie algebras case both definitions coincide. Definition 1.5. A linear transformation dof a Leibniz algebra Lis called a derivation of Lif d([x, y]) = [d(x), y]+[x, d(y)] for any x, y ∈ L. Denote by Der(L)the set of all derivations. It is clear that the right operator Rxis a derivation for any x∈ L. Derivations of this type are called inner derivations. Similar to the Lie algebras case the set of the inner derivations forms an ideal of the algebra Der(L). Since our algebra is Z−graded, i.e L=⊕i∈ZVi, this gradation induces a gradation of the algebra Der(L) = ⊕i∈ZWiin the following way: Wi={di∈Der(L) : di(x)∈Vi+jfor any x∈Vj}. For an n-dimensional algebra of maximum length it is easy to see that Der(L) = W−n⊕ · · · ⊕ Wn (see [14]). For more details see the definition of the cohomology groups for Leibniz algebras introduced in [13]. 2. 3-filiform non-Lie Leibniz algebras of maximum length In this section we are going to continue the classification of the p-filiform Leibniz algebras of maximum length. The study of the filiform and 2-filiform cases has been already done in [2], so we are going to continue with the 3-filiform Leibniz algebras case. The used technique in this section is as follows: we will extend the naturally graded 3-filiform Leibniz algebras by using the natural gradations. In this way, we can distinguish two cases: the natural graded Lie algebras and the natural graded non-Lie algebras. The study of the first case was closed in [4], 3-FILIFORM LEIBNIZ ALGEBRAS OF MAXIMUM LENGTH 3 so we explain the results obtained in the second family. After that we will work with a homogeneous basis and we will assume that the associated gradation has maximum length. Finally, we will use some programs implemented in the software Mathematica (which will be explained below) as well as properties of the gradation and of the nilpotence to arrive at a contradiction or at the classification. I would like to stress in the fact that using computer programs is very helpful to achieve the presented classification. Two programs will be used in this section: the program of the Leibniz identity and the program of isomorphisms. The first program computes the Leibniz identity of a Leibniz algebra and was presented in [5]. The second one establishes when two algebras are isomorphic, moreover we have added some subroutines to know if two algebras are isomorphic or not, when one of them is an uniparameter family. It returns the value of the parameter for which would be isomorphics. The algorithmic method can be found on [8]. The implementation of these programs are presented in low and fixed dimension. Then we will formulate the generalizations, proving by induction the results for arbitrary fixed dimension. Finally, point out that the algorithmic method of these programs are presented with a step-by-step explanation in the following Web site: http://personal.us.es/jrgomez. 2.1. Non split case. In this section we will restrict our study to classify the 3-filiform Leibniz algebras of maximum length, which are the extension of the non split and naturally graded non-Lie Leibniz algebras. The Lie case has been closed in [4], where there is not any non split 3-filiform Leibniz algebra of maximum length. First of all, let us see the classifications of the naturally graded 3-filiform Leibniz algebras ([6]). Theorem 2.1. Let Lbe a complex n-dimensional non-split naturally graded 3-filiform non-Lie Leibniz algebra and n≥7. Then there exists a basis {e1, e2, . . . , en−3, f1, f2, f3}of the algebra, such that Lis isomorphic to L1:     [ei, e1] = ei+1,1≤i≤n−4, [e1, f1] = f3, [ei, f2] = ei+1,1≤i≤n−4. Theorem 2.2. Let Lbe a complex n-dimensional non split 3-filiform non-Lie Leibniz algebra and n≥7. Then l(L)< n. Proof: The natural gradation of L1is: L1⊕ · ·· ⊕ Ln−3where L1=< e1, f1, f2>,L2=< e2, f3>and Li=< ei>for 3 ≤i≤n−3. We are going to study the length of its extension, which is denoted by f L1.Point out that we call the extension of the algebra as the natural generalization of the structural constants of the algebra, using the information of its associated natural gradation. Note that {e2, e3, . . . , en−3, f3}belong to the ideal R(L1) and en−3∈Cent(L1). Moreover [e1, f1] + [f1, e1]∈R(L1), then we conclude f3∈R(L1). Finally, by taking the change of basis e0 1=e1, e0 i+1 = [e0 i, e0 1] for 1 ≤i≤n−4, f0 1=f1and f0 2=f2, we can write the law of f L1as:                              [ei, e1] = ei+1,1≤i≤n−4, [e1, f1] = f3+ (∗)e3+· · · + (∗)en−3, [ei, f2] = ei+1 + (∗)ei+2 · · · + (∗)en−3,1≤i≤n−4, [fi, e1]=(∗)e3+· · · + (∗)en−3,1≤i≤2, [f3, e1]=(∗)e4+· · · + (∗)en−3, [fi, fj] = (∗)e3+· · · + (∗)en−3,1≤i, j ≤2, [f3, fi]=(∗)e4+· · · + (∗)en−3,1≤i≤2 [ei, f1] = (∗)ei+2 +· · · + (∗)en−3,2≤i≤n−5, 4 L.M. CAMACHO, E.M. CA ˜ NETE, J.R. G ´ OMEZ, B.A. OMIROV where the asterisks (∗) denote the corresponding coefficients in the products. A crucial tool in the proof is the construction of a homogeneous basis, which generators are: exs=e1+ n−3 X i=2 aiei+ 3 X j=1 an−3+jfj, ext=f1+ n−3 X i=1 biei+ 3 X j=2 bn−3+jfj, fxu=f2+ n−3 X i=1 ciei+ 3 X j=1,j6=2 cn−3+jfj. Therefore the products of the generators of f L1can be defined in the new basis as follows: [exs,exs] = (1 + an−1)e2+ (∗)e3+· · · + (∗)en−3+an−2f3, [ext,ext] = b1(b1+bn−1)e2+ (∗)e3+· · · + (∗)en−3+b1f3, [fxu,fxu] = c1(1 + c1)e2+ (∗)e3+· · · + (∗)en−3+c1cn−2f3, [exs,ext] = (b1+bn−1)e2+ (∗)e3+· · · + (∗)en−3+f3, [ext,exs] = b1(1 + an−1)e2+ (∗)e3+· · · + (∗)en−3+b1an−2f3, [exs,fxu] = (1 + c1)e2+ (∗)e3+· · · + (∗)en−3+cn−2f3, [fxu,exs] = c1(1 + an−1)e2+ (∗)e3+· · · + (∗)en−3+c1an−2f3, [ext,fxu] = b1(1 + c1)e2+ (∗)e3+· · · + (∗)en−3+b1cn−2f3, [fxu,ext] = c1(b1+bn−1)e2+ (∗)e3+· · · + (∗)en−3+c1f3. Since {exs,ext,fxu}are linearly independent, then det   1an−2an−1 b11bn−1 c1cn−21 6= 0. •Case 1: If 1 + an−16= 0,we have the following subcases: Case 1.1: If [ exs,exs] and [ exs,ext] are linearly independent, we take the homogeneous basis y1=exs, yi= [yi−1, y1] for 2 ≤i≤n−3, z1=ext, z2=fxuand z3= [ exs,ext]=[y1, z1], where [[ exs,exs], ..., exs | {z } i-times ] = (1 + an−1)i−1ei+ (∗)ei+1 +· · · + (∗)en−3for 3 ≤i≤n−3, obtaining the gradation: Vks⊕V2ks⊕ · · · ⊕ V(n−3)ks⊕Vkt⊕Vku⊕Vks+kt.Let us assume that the gradation has maximum length, therefore ks, kt, kuare pairwise different. It is enough to consider the products [z2, y1] and [y1, z2] to prove that it is not possible that the gradation has maximum length. Consider [z2, y1]=[fxu,exs] = c1[exs,exs] = c1y2. Since [z2, y1]∈Vks+ku,y2∈V2ksand ks6=ku, then we conclude c1= 0. On the other hand [y1, z2] = (1+c1)e2+(∗)e3+· · ·+(∗)en−3+cn−2f3=e2+(∗)e3+· · ·+(∗)en−3+ β1f3=Ay2with A6= 0. We also have [y1, z2]∈Vks+kuand y2∈V2ks, therefore ku=ks, which is a contradiction with the assumption of maximum length. Hence there is no maximum length algebra in this subcase. Case 1.2: If [ exs,exs] and [ exs,ext] are linearly dependent, i.e., (b1+bn−1)an−2= 1 + an−1.Note that we can assert that an−26= 0 and b1+bn−16= 0 from the assumption 1 + an−16= 0. Case 1.2.1: If [ exs,ext] and [ exs,fxu] are linearly independent, we distinguish two possibilities: If cn−26= 0,let us take the basis composed of the following vectors y1=exs, yi= [yi−1, y1] for 2≤i≤n−3, z1=ext, z2=fxuand z3= [ exs,fxu]=[y1, z2], and the maximum length gradation Vks⊕V2ks⊕· · ·⊕V(n−3)ks⊕Vkt⊕Vku⊕Vks+ku. A contradiction will be obtained by computing [y1, z1]. 3-FILIFORM LEIBNIZ ALGEBRAS OF MAXIMUM LENGTH 5 Since [y1, z1]=(b1+bn−1) | {z } 6=0 e2+ (∗)e3+· · · + (∗)en−3+cn−2 |{z} 6=0 f3∈Vks+kt, y2= [exs,exs] = (1 + an−1) | {z } 6=0 e2+ (∗)e3+· · · + (∗)en−3+an−2 |{z} 6=0 f3∈V2ks, z3= [ exs,fxu] = (1 + c1) | {z } 6=0 e2+ (∗)e3+· · · + (∗)en−3+cn−2 |{z} 6=0 f3∈Vks+ku, then either [y1, z1] = Ay2with A6= 0 or [y1, z1] = Bz3with B6= 0. By properties of the gradation, we achieve ks=ktin the first case and ku=ktin the other case. Both equalities contradict the maximum length of f L1. If cn−2= 0,we can assume that 1 + c16= 0.Otherwise we would have: [exs,exs] = (1 + an−1)e2+ (∗)e3+· · · + (∗)en−3+an−2f3, [exs,ext] = (b1+bn−1)e2+ (∗)e3+· · · + (∗)en−3+f3=α[exs,exs], [exs,fxu] = (∗)e3+· · · + (∗)en−3, and the others products would be linear combinations of these. Therefore, it would not be possible to generate the element y2or z3in the new basis. Hence, by taking the new basis y1=exs, y2= [ exs,fxu], yi= [yi−1, y1] for 3 ≤i≤n−3, z1=ext, z2=fxuand z3= [ exs,ext] = [y1, z1], we get a contradiction as above, by calculating the product [y1, y1]. Since [y1, y1] = (1 + an−1) | {z } 6=0 e2+ (∗)e3+· · · + (∗)en−3+an−2 |{z} 6=0 f3, there are two possibilities, either [y1, y1] = Ay2with A6= 0 or [y1, y1] = Bz3with B6= 0. Analogously to the above case, both equalities contradict the hypothesis of maximum length of f L1. Case 1.2.2: If [ exs,ext] and [ exs,fxu] are linearly dependent it is not possible to construct a homogeneous basis, because all the products of the generators can be written as follows: [ext,ext] = b1[exs,ext] = b1α[exs,exs], [fxu,fxu] = c1[exs,fxu] = c1β[exs,exs], [exs,ext] = α[exs,exs], [ext,exs] = b1[exs,exs], [exs,fxu] = β[exs,exs], [fxu,exs] = c1[exs,exs], [ext,fxu] = b1[exs,fxu] = b1β[exs,exs], [fxu,ext] = c1[exs,ext] = c1α[exs,exs], such that, all of them are linearly dependent of [ exs,exs]. •Case 2: If 1 + an−1= 0. Case 2.1: If 1 + c16= 0. Case 2.1.1: If [ exs,ext] and [ exs,fxu] are linearly independent, we take the new basis y1=exs,z1=ext, z2=fxu, y2= [ exs,fxu]=[y1, z2], yi= [yi−1, z2] for 3 ≤i≤n−3,and z3= [ exs,ext] = [y1, z1],where yi+1 = [[ exs,fxu], ..., fxu | {z } i−times ] = (1 + c1)iei+1 +· · · + (∗)en−3 with 2 ≤i≤n−4,giving rise to the following gradation: Vks⊕Vks+ku⊕Vks+2ku⊕ · · ·⊕Vks+(n−4)ku⊕ Vkt⊕Vku⊕Vkt+ksof maximum length. 6 L.M. CAMACHO, E.M. CA ˜ NETE, J.R. G ´ OMEZ, B.A. OMIROV In order to prove that there is no maximum length algebra in this subcase it is enough to study the values of ku, ktand kssuch as the gradation considered previously has maximum length. From properties of the gradation it is easy to check that the gradation is connected if and only if ku=±1. Without loss of generality we can assume ku= 1.Let us study the values of ktand ks. Case a: If ks>0. (a) Subcase a.1 (b) Subcase a.2 (c) Subcase a.3 Subcase a.1: If ks= 2.Under these hypothesis we have kt=n−1, Vkt+ks=Vn+1 and Vn=<0> . Hence the considered gradation is not connected. Subcase a.2: If ks= 3.Then kt= 2 and Vkt+ks=V5=< z3, y3>, so the length of the gradation in not maximum. Subcase a.3: If ks>3.Then there is some Vpwith 2 ≤p≤kssuch that Vp=<0>, which contradicts the connectedness of the gradation. Case b: If ks<0. (d) Subcase b.1 and Subcase b.2 Subcase b.1: ks= 4 −n. Under these hypothesis and by connectedness we have either kt= 3−nor kt= 2. If kt= 3−nthen Vkt+ks=V7−2n.Since n≥7, hence V7−2nis vanish and the gradation is not connected. On the other hand, if kt= 2 then Vkt+ks=V6−n=< y3, z3>, which gives a contradiction with the assumption of maximum length. Subcase b.2: ks6= 4 −n. This subcase never gives a maximum length gradation because the subspace V0is always vanish (z1/∈V0). Case 2.1.2: If [ exs,ext] and [ exs,fxu] are linearly dependent we have: (1) :          [exs,exs]=(∗)e3+· · · + (∗)en−3+an−2f3 [exs,ext] = (b1+bn−1)e2+ (∗)e3+· · · + (∗)en−3+f3=α[exs,fxu], [exs,fxu] = (1 + c1) | {z } 6=0 e2+ (∗)e3+· · · + (∗)en−3+cn−2f3. and the others products are linearly dependent of these. Therefore we can assume an−26= 0,otherwise it would not be possible to get a basis because det b1+bn−11 1 + c1cn−2= 0 and det   1 0 −1 b11bn−1 c1cn−21 = 0, which implies that exs,extand fxuare linearly dependent. Let us take the new basis y1=exs, z1=ext, z2=fxu, y2= [ exs,fxu] = [y1, z2], yi= [yi−1, z2] for 3≤i≤n−3 and z3= [y1, y1],where yi= [[ exs,fxu], ..., fxu | {z } (i−1)−times ] = (1 + c1)i−1ei+ (∗)ei+1 +· · · + (∗)en−3 3-FILIFORM LEIBNIZ ALGEBRAS OF MAXIMUM LENGTH 7 with 3 ≤i≤n−3.Its associated maximum length gradation is: Vks⊕Vks+ku⊕Vks+2ku⊕ · · · ⊕ Vks+(n−4)ku⊕Vkt⊕Vku⊕V2ks. Since [ exs,ext] is linearly dependent of [ exs,fxu] and from properties of the gradation we conclude that kt=ku,which is a contradiction with the assumption of maximum length. Case 2.2: If 1 + c1= 0,it is not possible to construct a basis because all the products of the generators can be written as follows: [exs,exs]=(∗)e3+· · · + (∗)en−3+an−2f3 [ext,ext] = b1[exs,ext]=(∗)e3+· · · + (∗)en−3+b1an−2f3, [fxu,fxu] = c1[exs,fxu] = (∗)e3+· · · + (∗)en−3+c1cn−2f3, [exs,ext]=(∗)e3+· · · + (∗)en−3+f3, [ext,exs] = b1[exs,exs] = (∗)e3+· · · + (∗)en−3+b1an−2f3, [exs,fxu]=(∗)e3+· · · + (∗)en−3+cn−2f3, [fxu,exs] = c1[exs,exs]=(∗)e3+· · · + (∗)en−3+c1an−2f3, [ext,fxu] = b1[exs,fxu] = (∗)e3+· · · + (∗)en−3+b1cn−2f3, [fxu,ext] = c1[exs,ext] = (∗)e3+· · · + (∗)en−3+c1f3, such that, the element y2can not be generated in this case. Therefore the proof is closed.  2.2. Split case. This section is devoted to the study of the 3-filiform Leibniz algebras of maximum length, whose naturally graded algebras are split. Furthermore we will focus our attention in the non standard families. The definitions of standard and non standard algebras are the following: Definition 2.1. Let Lbe a split maximum length algebra and let kbe an integer where L=N1⊕N2⊕ · · · ⊕ Nk. The algebra Lis called standard if N1,N2,. . . and Nkare algebras of maximum length. Otherwise the algebra Lis called non standard. Example 2.1. The list of standard 3-filiform Leibniz algebras of maximum length consists of the following algebras: maximum length null-filiform Leibniz algebras ⊕C3, maximum length filiform Leibniz algebras ⊕C2and maximum length 2-filiform Leibniz algebras ⊕C. Note that these algebras have already been studied in [1] and [2]. Due to the previous example, we reduce our study to the non standard families, i.e., we study the extension of the naturally graded filiform non-Lie Leibniz algebras ⊕C2and the naturally graded 2-filiform non-Lie Leibniz algebras ⊕C. It should be remarked that the null-filiform case will not be studied because its extension always gives a standard algebra. The Lie case has already been done in [4], where the classification is presented in the following theorem: Theorem 2.3. Let Lbe a (n+1)-dimensional non standard 3-filiform Leibniz algebra whose associated naturally graded algebra is a Lie algebra. Then nis odd and the algebra Lis isomorphic to the maximum length Lie algebra: N:                [ei−1, e0] = ei,2≤i≤n−2, [en−3, e1] = −en−1, [en−4, e2] = en−1, [ei, en−2−i]=(−1)i−1en−1,3≤i≤ bn−3 2c, [f1, e0] = en−1. 2-Filiform case Cabezas, Camacho and Rodr´ıguez gave the classification of the naturally graded 2-filiform non-Lie Leibniz algebras in [2]. They proved that, up to isomorphisms, there are two algebras under these 8 L.M. CAMACHO, E.M. CA ˜ NETE, J.R. G ´ OMEZ, B.A. OMIROV hypothesis, which are not split. These algebras are defined by the following table of multiplications: KF4:                [ei, e1] = ei+1,1≤i≤n−3, [e1, en−1] = en+α3e3+· · · +αn−2en−2, [en−1, en−1] = β3e3+β4e4+· · · +βn−2en−2, [ei, en−1] = βi,i+2ei+2 +βi,i+3ei+3 +· · · +βi,n−2en−2,2≤i≤n−4, [en, en−1] = γ4e4+· · · +γn−2en−2. KF5:                [ei, e1] = ei+1,1≤i≤n−3, [e1, en−1] = e2+en+α3e3+· · · +αn−2en−2, [en−1, en−1] = β3e3+β4e4+· · · +βn−2en−2, [ei, en−1] = ei+1 +βi,i+2ei+2 +βi,i+3ei+3 +· · · +βi,n−2en−2,2≤i≤n−4, [en, en−1] = γ4e4+· · · +γn−2en−2. Due to the above classification we obtain the following theorem: Theorem 2.4. Let Lbe a (n+1)−dimensional 3-filiform non-Lie Leibniz algebra of maximum length, whose associated naturally graded algebra is KF4⊕C.Then Lis isomorphic to either Mor one of the algebra of the family M1,α: M:     [yi, y1] = yi+1,1≤i≤n−3, [y1, yn−1] = yn, [z1, yn−1] = yn−2, M1,α :         [yi, y1] = yi+1,1≤i≤n−3, [y1, yn−1] = yn, [yn−1, z1] = yn−2, [z1, yn−1] = αyn−2, α ∈C. Proof: The extension of the algebra KF4⊕C, via the natural gradation, is: e L:                                                [ei, e1] = ei+1 + (∗)ei+2 +· · · + (∗)en−2,1≤i≤n−3, [e1, en−1] = en+ (∗)e3+· · · + (∗)en−2, [en−1, en−1]=(∗)e3· · · + (∗)en−2, [ei, en−1]=(∗)ei+2 +· · · + (∗)en−2,2≤i≤n−4, [en, en−1]=(∗)e4+· · · + (∗)en−2, [ei, f1]=(∗)ei+2 +· · · + (∗)en−2,1≤i≤n−4, [en−1, f1]=(∗)e3+· · · + (∗)en−2, [en, f1]=(∗)e4+· · · + (∗)en−2, [f1, ei]=(∗)ei+2 +· · · + (∗)en−2,1≤i≤n−4, [f1, en−1]=(∗)e3+· · · + (∗)en−2, [f1, en]=(∗)e4+· · · + (∗)en−2, [f1, f1]=(∗)e3+· · · + (∗)en−2, where the asterisks (∗) denote the corresponding coefficients in the products. We are going to get the homogenous basis by considering the generators: exs=e1+ n X i=2 aiei+b1f1, ext=en−1+ n X i=1,i6=n−1 Aiei+B1f1, fxu=f1+ n X i=1 αiei. 3-FILIFORM LEIBNIZ ALGEBRAS OF MAXIMUM LENGTH 9 Let us consider the following products, since they will be very useful in the rest of the proof: [exs,exs] = e2+ (∗)e3+· · · + (∗)en−2+an−1en, [exs,ext] = A1e2+ (∗)e3+· · · + (∗)en−2+en, [[ exs,exs],..., exs | {z } i-times ] = ei+ (∗)ei+1 +· · · + (∗)en−2,with 3 ≤i≤n−2. Let us take the homogeneous basis y1=exs, yi= [yi−1, y1] for 2 ≤i≤n−2, yn−1=ext, yn= [y1, yn−1], z1=fxuand the associated maximum length gradation Vks⊕V2ks⊕ · · · ⊕ V(n−2)ks⊕Vkt⊕ Vkt+ks⊕Vku.This gradation is connected if and only if ks=±1.Without loss of generality we can assume ks= 1 (the case ks=−1 is analogous). We are going to continue the proof studying the possible values that the subindices ktand kucan get to obtain a maximum length gradation. •Case 1: If kt>0, there are the following possibilities: (e) Subcase 1.1 (f) Subcase 1.2 (g) Subcase 1.3 Subcase 1.1: If kt=n−1,then from the connectedness of the gradation we obtain ku= 0 or ku=n+ 1.But ku= 0 is not possible because z1∈Vkuand z1is a generator. If ku=n+ 1 then [yi, z1]∈Vn+1+i=<0>, [z1, yi]∈Vn+1+i=<0>for 1 ≤i≤n−2. Moreover [z1, yn−1] and [yn−1, z1] belong to the subspace V2n=<0>. On the other hand [z1, yn],[yn, z1]∈V2n+1 =<0> and [z1, z1]∈V2n+2 =<0> . Then z1∈Cent(e L),giving rise to a standard algebra. Subcase 1.2: If kt=n. By a similar previous reason, it is clear that [yi, z1]=[z1, yi] = 0 for 3 ≤ i≤nbecause those products belong to Vn−1+i=<0> . Moreover [z1, y1],[y1, z1]∈Vn=< yn−1>, but this is not possible since yn−1is a generator, such that yn−1∈e L \ e L2, while [z1, y1],[y1, z1]∈e L2. Since y2∈R(e L) and from the Leibniz identity we affirm that [z1, y2] = [y2, z1] = 0.Finally, since [z1, z1]∈V2n−2=<0>, we conclude that z1∈Cent(e L),such that, the obtained algebra is standard. Subcase 1.3: If kt> n, the gradation is not connected because either Vn−1=<0>or Vn=<0>. •Case 2: If kt<0, we distinguish: (h) Subcase 2.1 (i) Subcase 2.2 Subcase 2.1: If kt=−1,then from the connectedness of the gradation either ku=−2 or ku=n−1. In the first case we are going to prove that the obtained algebra is standard because z1∈Cent(e L). From properties of the gradation we have [yi, z1],[z1, yi]∈Vi−2=< yi−2>for 2≤i≤n−2,but from the law of e Lwe know [yi, z1] = α1ei+1 + (∗)ei+2 +· · · + (∗)en−2and [z1, yi]=(∗)ei+2 +· · · + (∗)en−2.Therefore [yi, z1] = 0 and [z1, yi] = 0 are concluded for 2 ≤i≤n−2. On the other hand [y1, z1],[z1, y1]∈V−1=< yn−1>, which is not possible because yn−1is a generator of e L.Finally, since [z1, yn−1],[yn−1, z1]∈V−3=<0>and yn∈Cent(e L),we have finally proved that z1∈Cent(e L), such that, the algebra is standard.