Solvable Lie and Leibniz superalgebras with a given nilradical

Research Article Luisa María Camacho*, José Manuel Fernández Barroso and Rosa María Navarro Solvable Lie and Leibniz superalgebras with a given nilradical https://doi.org/10.1515/forum-2020-0031 Received February 7, 2020 Abstract: Throughout this paper we show that under certain conditions the method for describing solvable Leibniz (resp. Lie) algebras with given nilradical by means of non-nilpotent outer derivations of the nilradical is also applicable to the case Leibniz (resp. Lie) superalgebras. Moreover, after having established the general method for Lie and Leibniz superalgebras, we classify all the solvable superalgebras on a very important class of each of them, that is, those with nilradical of maximal nilindex. Note that for (n+m)-dimensional superalgebras this maximal nilindex is n+m−1in the Lie case and n+min Leibniz. Keywords: Solvable Lie superalgebras, solvable Leibniz superalgebras, derivations, nilpotent Lie superalgebras, nilpotent Leibniz superalgebras MSC 2010: 17A32, 17A70, 17B30, 17B40 || Communicated by: Jan Frahm 1 Introduction In Lie theory, there are many works devoted to the study of solvable Lie algebras, see for instance [4, 5, 19, 23, 24, 26], where solvable Lie algebras with diverse types of nilradical have been studied. Among these types of nilradical one find naturally graded filiform and quasi-filiform Lie algebras, abelian, triangular, etc. Nevertheless, recently the solvable extension method by means of outer and non-nilpotent derivations have drawn a great deal of attention for Leibniz algebras. As examples it can consulted the following papers [1, 8–10, 17] where null-filiform nilradical has been considered among others. Thus, the nilradicals most extensively studied are filiform and null-filiform for Lie and Leibniz algebras, respectively. Additionally these algebras are those with maximal nilindex. Remark that for n-dimensional algebras the maximal nilindex is n−1in the Lie case and nin the Leibniz one. In the present study we tackle the generalization of these classifications for Leibniz and Lie superalgebras. Studying solvable Leibniz (resp. Lie) superalgebras, on the other hand, presents more difficulties than studying solvable Leibniz (resp. Lie) algebras [21]. In particular, Lie’s theorem is not verified in general and neither its corollaries. Therefore, for a solvable Leibniz (resp. Lie) superalgebra L,L2:=[L,L]cannot be nilpotent, see [20]. Nevertheless, in our study we impose the constraint that L2– the first ideal of the descending central sequence – is nilpotent. This fact leads to [L,L]⊂n, where nis the maximal nilpotent ideal of L, so-called nilradical. We prove that under this condition any solvable Leibniz (resp. Lie) superalgebra over the real or complex field can be obtained by means of outer non-nilpotent (super)derivations of the nilradical *Corresponding author: Luisa María Camacho Departamento de Matemática Aplicada I, Universidad de Sevilla, Sevilla, Spain, e-mail: lc[email protected] José Manuel Fernández Barroso Departamento de Matemáticas, Universidad de Extremadura, Badajoz, Spain, e-mail: f[email protected]s Rosa María Navarro Departamento de Matemáticas, Universidad de Extremadura, Cáceres, Spain, e-mail: rnavarr[email protected] 1272 |L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 in the same way as occurs for Leibniz (resp. Lie) algebras. Furthermore, we apply the aforementioned method to classify all the solvable Leibniz (resp. Lie) superalgebras with nilradical the Leibniz (resp. Lie) superalgebra of maximal nilindex. Remark that this classification for Leibniz superalgebras constitutes a generalization of the corresponding classification obtained for Leibniz algebras in [9]. Throughout the present paper we will consider vector spaces and algebras over the field of real or complex numbers. We also assume that the reader is familiarized with the basics of Lie theory. 2 Preliminaries 2.1 Preliminaries for Lie superalgebras A vector space Vis said to be ℤ2-graded if it admits a decomposition in direct sum, V=V 0⊕V 1, where  0, 1∈ℤ2. An element X∈Vis called homogeneous of degree iif it is an element of V i,i∈ℤ2. In particular, the elements of V 0(resp. V 1) are also called even (resp. odd). A Lie superalgebra (see [22]) is a ℤ2-graded vector space g=g 0⊕g 1, with an even bilinear commutation operation (or “supercommutation”) [⋅,⋅], which satisfies the conditions (1) [X,Y]=−(−1) i j[Y,X], (2) (−1) k i[X,[Y,Z]]+(−1) i j[Y,[Z,X]]+(−1) j k[Z,[X,Y]]=0(Jacobi superidentity) for all X∈g i,Y∈g j,Z∈g kwith i,j, k∈ℤ2. Thus, g 0is an ordinary Lie algebra, and g 1is a module over g 0; the Lie superalgebra structure also contains the symmetric pairing S2g 1→g 0. In general, the descending central sequence of a Lie superalgebra g=g 0⊕g 1is defined in the same way as for Lie algebras: C0(g):=g,Ck+1(g):=[Ck(g),g]for all k≥0. Consequently, if Ck(g)={0}for some k, then the Lie superalgebra is called nilpotent. Then the smallest integer ksuch that Ck(g)={0}is called the nilindex of the Lie superalgebra g. Likewise, the derived sequence of gis defined by D0(g):=g,Dk+1(g):=[Dk(g),Dk(g)] for all k≥0. If this sequence is stabilized in zero, then the Lie superalgebra is said to be solvable. All nilpotent Lie superalgebras are solvable ones. Engel’s theorem and its direct consequences remaind valid for Lie superalgebras. In particular, a Lie superalgebra Lis nilpotent if and only if adLxis nilpotent for every homogeneous element xof L. Moreover, for solvable Lie superalgebras we have that a Lie superalgebras Lis solvable if and only if its Lie algebra L0is solvable. Nevertheless, we do not have the analog of Lie’s Theorem and neither its corollaries for solvable Lie superalgebras. For a nilpotent Lie algebra of dimension nthe nilindex is evidently ≤n−1, and the Lie algebras with nilindex n−1form the class of n-dimensional filiform Lie algebras [25]. Analogously, for a nilpotent Lie superalgebra of type (n,m), i.e., with dimensions of the even and odd part nand m, respectively, the nilindex is always ≤n+m−1. The function f(n,m), defined as the maximal nilindex for the Lie superalgebras of type (n,m), is, in most cases, ≤n+m−2. The only case in which is obtained the maximal nilindex n+m−1 is exactly when n=2and modd (for more details see [11, Theorem 4.17]). Furthermore, for any odd mthere is only one Lie superalgebra with this maximal nilindex [11]. This Lie superalgebra was denoted by K2,m (in honor to Professor Yu. Khakimdjanov) and can be expressed in an adapted basis {x0,x1,y1,...,ym} as follows: K2,m:{ { { { { [x0,yi]=−[yi,x0]=yi+1,1≤i≤m−1, [yi,ym+1−i]=[ym+1−i,yi]=(−1)i+1x1,1≤i≤1 2(m+1), where the omitted products are equal to zero and mis an odd positive integer greater than 1. Recall now the definition of derivations of superalgebras [15]. Call derivation of degree sof a superalgebra A,s∈ℤ2, an endomorphism D∈EndsAwith the property D(ab)=D(a)b+(−1)s⋅deg aaD(b), denote DersA⊂EndsAthe space of all derivations of degree s, assuming that Der A=Der0A⊕Der1A. It can be seen that Der Ais also a Lie superalgebra, so-called Lie superalgebra of derivations of A. As an example L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 |1273 of derivation we have ad a:b→[a,b]. Derivations of this kind are called inner; they form an ideal inner A in Der A, because [D,ad a]=ad D(a),where D∈Der A. Note that if Dis an even derivation of superalgebra A=A0⊕A1, then exp tD,t∈ℂ, is 1-parameter group of automorphisms. In particular, if Ais Lie superalgebra, then exp(ad a)is an automorphism of Afor a∈A0. The group generated by all such automorphisms is called the group of inner automorphisms. In general, for Lie superalgebras all the automorphisms and isomorphisms are considered to be even endomorphisms. 2.2 Preliminaries for Leibniz superalgebras Remark that many results and definitions of the above subsection can be extended for Leibniz superalgebras. Definition 2.1 ([2]). A ℤ2-graded vector space L=L 0⊕L 1is called a Leibniz superalgebra if it is equipped with a product [⋅,⋅]which satisfies the condition [x,[y,z]]=[[x,y],z]−(−1) i j[[x,z],y](Leibniz superidentity) for all x∈L,y∈L i,z∈L j,i,j∈ℤ2. Note in particular that if a Leibniz superalgebra Lsatisfies the identity [x,y]=−(−1) i j[y,x]for x∈L i,y∈L j, then the Leibniz superidentity becomes the Jacobi superidentity. Consequently, Leibniz superalgebras are a generalization of Lie superalgebras. Also and in the same way as for Lie superalgebras, isomorphisms are assumed to be consistent with the ℤ2-graduation. Let us now denote by Rxthe right multiplication operator, i.e., Rx:L→Lgiven as Rx(y):=[y,x]for y∈L, then the can be expressed as R[x,y]=RyRx−(−1) i jRxRy, where x∈L i,y∈L j. If we denote by R(L)the set of all right multiplication operators, then it can be seen that R(L)with the multiplication ⟨Ra,Rb⟩:=RaRb−(−1) i jRbRa(2.1) for Ra∈R(L) i,Rb∈R(L) j, is a Lie superalgebra. Note that Rais a derivation. In fact, the condition for being a derivation of a Leibniz superalgebra (for more details see [16]) is d([x,y])=(−1)|d||y|[d(x),y]+[x,d(y)]. Since the degree of Rzas homomorphism between ℤ2-graded vector spaces is the same as the degree of the homogeneous element z, that is, |Rz|=|z|, the condition for Rzto be a derivation is exactly Rz([x,y])=(−1)|z||y|[Rz(x),y]+[x,Rz(y)]. This last condition can be rewritten [[x,y],z]=(−1)|z||y|[[x,z],y]+[x,[y,z]] which is nothing but the graded Leibniz identity. Let us note that the concepts of descending central sequence, nilindex, the variety of Leibniz superalgebras and Engel’s theorem are natural extensions from Lie theory. Let V=V 0⊕V 1be the underlying vector space of L,L=L 0⊕L 1∈Leibn,m, being Leibn,mthe variety of Leibniz superalgebras, and let G(V)be the group of the invertible linear mappings of the form f=f 0+f 1 such that f 0∈GL(n,ℂ)and f 1∈GL(m,ℂ)(then G(V)=GL(n,ℂ)⊕GL(m,ℂ)). The action of G(V)on Leibn,m induces an action on the Leibniz superalgebras variety: two laws λ1,λ2are isomorphic if there exists a linear mapping f=f 0+f 1∈G(V)such that λ2(x,y)=f−1  i+ j(λ1(f i(x),f j(y))) for any x∈V i,y∈V j. Furthermore, the description of the variety of any class of algebras or superalgebras is a difficult problem. Different papers (for example, [3, 6, 12, 13]) are regarding the applications of algebraic groups theory to the description of the variety of Lie and Leibniz algebras. 1274 |L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 Definition 2.2. For a Leibniz superalgebra L=L 0⊕L 1we define the right annihilator of Las the set Ann(L):={x∈L:[L,x]=0}. It is easy to see that Ann(L)is a two-sided ideal of Land [x,x]∈Ann(L)for any x∈L 0. This notion is nice and compatible with the right annihilator in Leibniz algebras. If we consider the ideal generated as I:=ideal⟨[x,y]+(−1) i j[y,x]:x∈L i,y∈L j⟩, then I⊂Ann(L). Let L=L 0⊕L 1be a nilpotent Leibniz superalgebra with dimL  0=nand dimL  1=m.From equation (2.1) we have that R(L)is a Lie superalgebra, and in particular R(L 0)is a Lie algebra. As L 1has L 0-module structure we can consider R(L 0)as a subset of GL(V 1), where V 1is the underlying vector space of L 1. So, we have a Lie algebra formed by nilpotent endomorphisms of V 1. Applying the Engel’s theorem [14] we have the existence of a sequence of subspaces of V 1,V0⊂V1⊂V2⊂⋅⋅⋅⊂Vm=V 1,with R(L 0)(Vi+1)⊂V i.Then it can be defined the descending sequences Ck(L 0)and Ck(L 1)and the super-nilindex in the same way as for Lie superalgebras. That is, C0(L i):=L i,Ck+1(L i):=[Ck(L i),L 0],k≥0,i∈ℤ2.If L=L 0⊕L 1is a nilpotent Leibniz superalgebra, then Lhas super-nilindex or s-nilindex (p,q)if satisfies (Cp−1(L 0)) =0,(Cq−1(L 1)) =0,Cp(L 0)=Cq(L 1)=0. Note that similar to the case of null-filiform Leibniz algebras, a Leibniz superalgebra is null-filiform if and only if it is single-generated. Moreover, a null-filiform superalgebra has both maximal super-nilindex and maximal nilindex, being the only one verifying the latter condition. Note also that this maximal nilindex is n+m, exactly a unit greater than the maximal nilindex for Lie superalgebras. 3 Solvable extension method for Lie and Leibniz superalgebras We will start by Lie superalgebras which are a particular case of Leibniz superalgebras. Note that dealing with solvable Lie superalgebras is by far much complicated than dealing with solvable Lie algebras. Thus, we do not have the analog of Lie’s theorem and neither its corollaries. Therefore, for a solvable Lie superalgebra L, L2:=[L,L]– the first ideal of the descending central sequence – cannot be nilpotent. However, if we impose the constraint that L2is nilpotent, then we have the following result. Theorem 3.1. Let Lbe a non-nilpotent solvable Lie superalgebra over the real or complex field, with L2nilpo- tent and fixed nilradical n. Then Lcan always be obtained by means of outer non-nilpotent derivations of the nilradical n. Proof. If Lis a solvable Lie superalgebra with L2nilpotent, then we have [L,L]⊂n, where nis the maximal nilpotent ideal of L, so-called nilradical. Under this condition any solvable Lie superalgebra over the real or complex field admits a decomposition L=t ⊕n, where  ⊕denotes the semidirect sum and satisfying the relations [t,n]⊂n,[n,n]⊂n,[t,t]⊂n. By means of the super Jacobi identity it can be seen that ad zfor any z∈t, acts as a derivation of the nilpotent superalgebra n. Since the elements z∉n, these derivations are not nilpotent, and given a basis {z1,...,zn} of tand arbitrary non-null scalars α1,...,αnit is satisfied that (α1ad z1+⋅⋅⋅+αnad zn)k=0,k≥1, i.e., the matrix α1ad z1+⋅⋅⋅+αnad znis not nilpotent. In the same way as for Lie algebras we say that the elements z1,...,znare nil-independent [5, 18]. Remark 3.1. Consequently, we have a first restriction on the dimension of a solvable Lie superalgebra having a given nilradical, more concretely, dim Lis bounded by the maximal number of nil-independent derivations of the nilradical. L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 |1275 Moreover, we have a more general result: Proposition 3.1. If L=L0⊕L1is a non-nilpotent solvable Lie superalgebra over the real or complex field with nilradical n=n0⊕n1, then dim(L0)>dim(n0). Proof. The result derives from the fact that Lis solvable (resp. nilpotent) iff L0is a solvable (resp. nilpotent) Lie algebra. Therefore, we have that n0is a nilpotent Lie algebra and L0is a non-nilpotent solvable Lie algebra, which leads to the result of the statement of the proposition. In fact, L0is non-nilpotent because otherwise L is a nilpotent superalgebra which is a contradiction. On the other hand and with respect to Leibniz superalgebras, we extend in a natural way the above results in the same way that it has been already done for Leibniz algebras [8–10, 17]. Theorem 3.2. Let Lbe a non-nilpotent solvable Leibniz superalgebra over the real or complex field, with L2 nilpotent and fixed nilradical n. Then Lcan always be obtained by means of outer non-nilpotent derivations of the nilradical n. Proposition 3.2. If L=L0⊕L1is a non-nilpotent solvable Leibniz superalgebra over the real or complex field with nilradical n=n0⊕n1, then dim(L0)>dim(n0). The latter result means that in our both kind of extensions, Lie and Leibniz superalgebras, we add always at least one even basis vector. 4 Classification of solvable Lie superalgebras with nilradical K2,m Throughout the present section we apply the aforedescribed method to classify all the solvable Lie superalgebras L, verifying L2is nilpotent and with the Lie superalgebra K2,mas nilradical. For being able to apply this method we need to obtain first the superalgebra of derivations of K2,m. For the effective computation of both derivations, even and odd ones, we generalize a method used for Lie algebras (for more details see for instance [7]), which consists mainly in find a suitable gradation for each superalgebra K2,mand then obtain all the homogeneous derivations associated with the gradation. These derivations span the superalgebra of the derivations. Note that this method presents a truly advantage in terms of being able to distinguish inner from outer derivations. We consider now the following gradation associated with K2,m: ⟨x0⟩ ⏟⏟⏟⏟⏟⏟⏟ g−1⊕{0} ⏟⏟⏟⏟⏟⏟⏟ g0⊕⟨ym⟩ ⏟⏟⏟⏟⏟⏟⏟⏟⏟ g1⊕⟨ym−1⟩ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ g2⊕⋅⋅⋅⊕⟨y1⟩ ⏟⏟⏟⏟⏟⏟⏟ gm⊕⟨x1⟩ ⏟⏟⏟⏟⏟⏟⏟ gm+1 , K2,m=⨁m+1 i=−1giwith [gi,gj]⊂gi+j. Note that actually each giis ℤ2-graded, i.e., gi=gi 0⊕gi 1, thus for instance g−1=g−1 0⊕g−1 1=⟨x0⟩⊕{0}. Suppose then d∈Der0(K2,m)(resp. Der1(K2,m), so it can expressed as d=∑j∈ℤdj, where each djis an even (resp. odd) homogeneous derivation of order j, i.e., dj(gi)⊂gi+jbeing gk={0}if k∉{−1,0,1,...,m+1}. It can easily be seen that we have in fact the following expression for both even and odd derivations d=∑m+2 j=−m−2dj. Remark that for obtaining djwe have taken into account that the ideals of the descending central sequence Ci(K2,m)=Ci 0(K2,m)⊕Ci 1(K2,m)are characteristics, that is, stable under any derivation. In particular, if djis an even derivation, we have dj(Ci 0(K2,m))⊂Ci 0(K2,m)and dj(Ci 1(K2,m))⊂Ci 1(K2,m),0≤i≤m+1, whereas if djis an odd derivation we have dj(Ci 0(K2,m))⊂Ci 1(K2,m)and dj(Ci 1(K2,m))⊂Ci 0(K2,m),0≤i≤m+1. Recall that the endomorphisms ad xi, ad yiare always derivations, so-called inner derivations with ad xieven derivations and ad yiodd ones. 1276 |L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 Remark 4.1. It can easily be seen that the endomorphisms of K2,mdenoted by t1 0,t2 0,t1,t3,t5,...,tm−2and defined as follows: t1 0(x0)=x0,t1 0(x1)=(m−1)x1,t1 0(yi)=(i−1)yi,2≤i≤m, t2 0(x1)=2x1,t2 0(yi)=yi,1≤i≤m, t1(x0)=x1, tk(yi)=yi+k,1≤i≤m−k,with kodd and 3≤k≤m−2, are even derivations of K2,m. Remark 4.2. Note that ad x0is an even derivation, ad x1is the null derivation and ad yiwith 1≤i≤mare non-vanishing odd derivations of K2,m. Thus, a straightforward computation leads to the following result. Theorem 4.1. The derivations {ad x0,t1 0,t2 0,t1,t3,t5,...,tm−2,ad y1,ad y2,...,ad ym}constitute a basis of Der(K2,m)=Der0(K2,m)⊕Der1(K2,m), being {ad x0,t1 0,t2 0,t1,t3,t5,...,tm−2}a basis of Der0(K2,m)and {ad y1,ad y2,...,ad ym}a basis of Der1(K2,m). Remark 4.3. Note that all the odd derivations of the basis are inner. Corollary 4.1. Any non-nilpotent outer derivation fof K2,mis of the form f=α1t1 0+α2t2 0+∑ k kodd 3≤k≤m−2 βktk+β1t1, where either α1=0or α2=0. Next, from Corollary 4.1 we have the following: Proposition 4.1. Any solvable Lie superalgebra Lover the real or complex field, with L2nilpotent and nilradical of maximal nilindex, i.e., isomorphic to K2,m, has dimension m+3or m+4. Also from Corollary 4.1 we have the following result after replacing fby ad z. Proposition 4.2. Let Lbe an (m+3)-dimensional solvable Lie superalgebra over the real or complex field, with L2nilpotent and nilradical isomorphic to K2,m. Then there exists a basis, namely {x0,x1,z,y1,...,ym}with {x0,x1,z}as a basis of L0and {y1,...,ym}as a basis of L1, in which Lcan be expressed by the only non-null bracket products that follow [x0,yi]=−[yi,x0]=yi+1,1≤i≤m−1, [yi,ym+1−i]=[ym+1−i,yi]=(−1)i+1x1,1≤i≤1 2(m+1), [z,x0]=−[x0,z]=α1x0+β1x1, [z,x1]=−[x1,z]=((m−1)α1+2α2)x1, [z,y1]=−[y1,z]=α2y1+m−2 ∑ k=3 βky1+k,kodd, [z,yi]=−[yi,z]=((i−1)α1+α2)yi+m−2 ∑ k=3 βkyi+k,2≤i≤m,kodd, where either α1=0or α2=0and yi+kvanishes whenever i+k∉{1,...,m} Remark 4.4. Note that since fis composed by a linear combination of even derivations then zis a basis vector of L0. Remark 4.5. Notice that all the isomorphisms π:L→L󸀠restricted to the nilradical are considered to be automorphisms, i.e., the nilradical remains invariant under any isomorphism. Consequently, always from π(z)=z󸀠,π(x0)=x󸀠 0and π(y1)=y󸀠 1– so-called the generators of the superalgebra – we obtain recursively L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 |1277 y󸀠 i+1:=[x󸀠 0,y󸀠 i]for 1≤i≤m−1and after x󸀠 1:=(−1)i+1[y󸀠 i,y󸀠 m+1−i]=(−1)i+1[y󸀠 m+1−i,y󸀠 i]for any iverifying 1≤i≤1 2(m+1). Therefore any isomorphism will be determined giving only the values of the generators {z󸀠,x󸀠 0,y󸀠 1}. Next, we are going to study separately the cases of low dimensions, m=3,5,7and after we generalize the result for any arbitrary m,m≥9. 4.1 Total dimension m+3and either m=3or m=5 Proposition 4.3. Let Lbe under the conditions of Proposition 4.2 with m=3. Then there exists a basis, namely {x0,x1,z,y1,y2,y3}with {x0,x1,z}a basis of L0and {y1,y2,y3}a basis of L1, in which Lis isomorphic either to L6 0,1,0or L6 1,−1 2,1or L6 1,α2,0with α2a parameter belonging to ℝor ℂaccording to the case considered. The aforementioned superalgebras are pairwise non-isomorphic, even the 1-parametric family for two distinct values of the parameter. Being L6 α1,α2,β1, [x0,yi]=−[yi,x0]=yi+1,[yi,y4−i]=[y4−i,yi]=(−1)i+1x1,1≤i≤2, [z,x0]=−[x0,z]=x0α1+β1x1,[z,x1]=−[x1,z]=(2α1+2α2)x1, [z,y1]=−[y1,z]=α2y1,[z,yi]=−[yi,z]=((i−1)α1+α2)yi,2≤i≤3. Proof. The superalgebras according to Proposition 4.2 remain as L6 α1,α2,β1with either α1=0or α2=0. Thus, we consider two cases. Case (1). If α1=0, then there is not loss of generality in supposing α1=1. In fact, we obtain the new α1, α󸀠 1equal 1after applying the change of scale (isomorphism) defined by {z󸀠=1 α1z,x󸀠 0=x0,y󸀠 1=y1}. Next, it can be seen that the isomorphisms which can simplify the above multiplication table are defined by {z󸀠=z+bx1,x󸀠 0=Ax0+Bx1,y󸀠 1=Cy1}with AC =0, otherwise instead of simplifying we are adding new structure constants. After applying any of these isomorphisms the new structure constants are β󸀠 1=Aβ1+B(1+2α2) A2C2,α󸀠 2=α2. Therefore, the parameter α2remains invariant and β󸀠 1can be made zero whenever (1+2α2)=0by choosing B as B=−Aβ1 (1+2α2). Thus, if α2=−1 2, we obtain then L6 1,α2,0for α2=−1 2. If α2=−1 2, on the other hand, we have two possibilities: β1=0or β1=0. For the first possibility β1=0, we get L6 1,α2,0for α2=−1 2and for the second one after the change of scale defined by {z󸀠=z,x󸀠 0=β1x0,y󸀠 1=y1}we get L6 1,−1 2,1. Case (2). We suppose α1=0and then α2=0. The change of scale {z󸀠=1 α2z,x󸀠 0=x0,y󸀠 1=y1}leads to α2=1. Moreover, by means of the isomorphism {z󸀠=z,x󸀠 0=x0−β1 2x1,y󸀠 1=y1}we obtain β󸀠 1=0and then L6 0,1,0. The following table shows that the superalgebra L6 0,1,0is not isomorphic to any of the others. Lie superalgebra gdim(C1(g)0) L6 1,α2,0,α2∈ℝor ℂ2 L6 1,−1 2,12 L6 0,1,01 Furthermore, L6 1,α2,0and L6 1,−1 2,1derive from the case α1=0where after any generic change of basis (isomorphism) α2remains invariant. Only rest to see L6 1,−1 2,0≇L6 1,−1 2,1, in fact if we apply any generic change of basis to L6 1,−1 2,1which verifies β1=1, it is never obtained L6 1,−1 2,0which verifies β1=0and that concludes the proof of the proposition. 1278 |L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 Proposition 4.4. Let Lbe under the conditions of Proposition 4.2 with m=5. Then there exists a basis, namely {x0,x1,z,y1,...,y5}with {x0,x1,z}a basis of L0and {y1,...,y5}a basis of L1, in which Lis isomorphic either to L8 1,−3 2,1,0or L8 0,1,0,δwith δ∈{0,1}, or L8 1,α2,0,0with α2a parameter belonging to ℝor ℂaccording to the case considered. The aforementioned superalgebras are pairwise non-isomorphic, even the 1-parametric family for two distinct values of the parameter. Being L8 α1,α2,β1,β3, [x0,yi]=−[yi,x0]=yi+1,1≤i≤4, [yi,y6−i]=[y6−i,yi]=(−1)i+1x1,1≤i≤3, [z,x0]=−[x0,z]=α1x0+β1x1, [z,x1]=−[x1,z]=(4α1+2α2)x1, [z,y1]=−[y1,z]=α2y1+β3y4, [z,y2]=−[y2,z]=(α1+α2)y2+β3y5, [z,yi]=−[yi,z]=((i−1)α1+α2)yi,3≤i≤5. Proof. It is similar to the previous case. 4.2 Total dimension m+3and m=7 In a similar way it can be obtained the result for m=7. For simplicity we call by L10 α1,α2,β1,β3,β5the generic family obtained from Proposition 4.2. Thus, being L10 α1,α2,β1,β3,β5, [x0,yi]=−[yi,x0]=yi+1,1≤i≤6, [yi,y8−i]=[y8−i,yi]=(−1)i+1x1,1≤i≤4, [z,x0]=−[x0,z]=α1x0+β1x1, [z,x1]=−[x1,z]=(6α1+2α2)x1, [z,y1]=−[y1,z]=α2y1+β3y4+β5y6, [z,y2]=−[y2,z]=(α1+α2)y2+β3y5+β5y7, [z,y3]=−[y3,z]=(2α1+α2)y3+β3y6, [z,y4]=−[y4,z]=(3α1+α2)y4+β3y7, [z,yi]=−[yi,z]=((i−1)α1+α2)yi,5≤i≤7, with either α1=0or α2=0. Proposition 4.5. Let Lbe under the conditions of Proposition 4.2 with m=7. Then there exists a basis, namely {x0,x1,z,y1,...,y7}with {x0,x1,z}a basis of L0and {y1,...,y7}a basis of L1, in which Lis isomorphic to one superalgebra of the listed below. Notice that unless otherwise is stated, all the superalgebras are pairwise non-isomorphic, even the 1-parametric families for different values of the parameter. Note, also that all the superalgebras L10 ... are over ℝor ℂaccording to the case considered, nevertheless L10,ℝ ... only appears over ℝ. List of Lie superalgebras (where always δ∈{0,1}and the parameters belong to ℝor ℂdepending on the case considered): ∙L10 1,α2,0,δ,0(α2is a parameter), ∙L10 1,−5 2,1,δ,0, ∙L10,ℝ 1,−5 2,−1,1,0(appears only over ℝ), ∙L10 0,1,0,δ,0, ∙L10 0,1,0,0,1, ∙L10 0,1,0,1,β5with β5a non-null parameter. They are pairwise non-isomorphic over ℝbut over ℂwe have L10 0,1,0,1,β5≅L10 0,1,0,1,β󸀠 5 if and only if there exists A∈ℂsuch that β󸀠 5=β5 A2and A3=1. Note that this last relationship between the two set of parameters corresponding to two isomorphic superalgebras is very frequent from now on. Therefore, let us give the following definition. L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 |1279 Definition 4.1. If m≡1(mod 4), we define the following relation on ℂ1 2(m−3−2i). We say (βi+1,βi+3,...,βm−3 2)∼(β󸀠 i+1,β󸀠 i+3,...,β󸀠m−3 2) if (βi+1,βi+3,...,βm−3 2)=(Ai+1β󸀠 i+1,Ai+3β󸀠 i+3,...,Am−3 2β󸀠m−3 2)for some i-th root of unity A. One may easily check that ∼is an equivalence relation on ℂ1 2(m−3−2i). Analogously, if m≡3(mod 4), we define the following relation on ℂ1 2(m−1−2i). We say (βi+1,βi+3,...,βm−1 2)∼(β󸀠 i+1,β󸀠 i+3,...,β󸀠m−1 2) if (βi+1,βi+3,...,βm−1 2)=(Ai+1β󸀠 i+1,Ai+3β󸀠 i+3,...,Am−1 2β󸀠m−1 2)for some i-th root of unity A. 4.3 Total dimension m+3and m≥9 Next, we study the general case m≥9and we name Lm+3 α1,α2,β1,β3,... the generic family from Proposition 4.2. Thus, Lm+3 α1,α2,β1,β3,... being [x0,yi]=−[yi,x0]=yi+1,1≤i≤m−1, [yi,ym+1−i]=[ym+1−i,yi]=(−1)i+1x1,1≤i≤1 2(m+1), [z,x0]=−[x0,z]=α1x0+β1x1, [z,x1]=−[x1,z]=((m−1)α1+2α2)x1, [z,y1]=−[y1,z]=α2y1+m−2 ∑ k=3 βky1+k,kodd, [z,yi]=−[yi,z]=((i−1)α1+α2)yi+m−2 ∑ k=3 βkyi+k,2≤i≤m,kodd, where either α1=0or α2=0and yi+kvanishes whenever i+k∉{1,...,m} Theorem 4.2. Let Lbe an (m+3)-dimensional solvable Lie superalgebra, m≥9, over the real or complex field, with L2nilpotent and nilradical isomorphic to K2,m. Then there exists a basis, namely {x0,x1,z,y1,...,ym} with {x0,x1,z}as a basis of L0and {y1,...,ym}as a basis of L1, in which Lis isomorphic to one superalgebra of the listed below. Notice that unless otherwise is stated, all the superalgebras are pairwise non-isomorphic, even the parametric families for different values of the parameters. Note also that all the superalgebras Lm+3 ... are over ℝor ℂaccording to the case considered and so the parameters, nevertheless Lm+3,ℝ ... only appears over ℝ. ∙If m≡1(mod 4): (1) Lm+3 1,α2,0,β3,...,β(m−3)/2with α2,β3,...,β(m−3)/2parameters. Note that the first βi=0, if exists, can be normalized to 1. Over ℝthe superalgebras are pairwise non-isomorphic for distinct values of the parameters, but over ℂwe get Lm+3 1,α2,0,...,1,βi+1,... ≅Lm+3 1,α2,0,...,1,β󸀠 i+1,... if and only if (βi+1,...,βm−3 2)∼(β󸀠 i+1,...,β󸀠m−3 2) with ∼the equivalence relation defined in Definition 4.1. (2) Lm+3 1,(2−m)/2,1,0,...,0. This superalgebra verifies β3=⋅⋅⋅=βm−3 2=0. (3) Lm+3 1,(2−m)/2,1,0,...,1,βi+1,...,β(m−3)/2with some βi=0,i≥3. The first βi=0has been normalized to 1. Over ℝ the superalgebras are pairwise non-isomorphic for distinct values of the parameters, but over ℂwe get Lm+3 1,2−m 2,1,0,...,1,βi+1,... ≅Lm+3 1,2−m 2,1,0,...,1,β󸀠 i+1,... 1286 |L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 From the Leibniz superidentity for the triples {x1,z,x1}we obtain α1=−2a1and δ=0.Thus, the transformation z󸀠=z−n ∑ k=2 αk+1xk permits to get [z,x1]=−2a1x1+α2x2. Next, it is easy to prove that [z,xi]=0with 2≤i≤nconsidering the Leibniz superidentity for the triples {z,xi,x1}and the induction method over i.Let [z,y1]=m ∑ k=1 βkyk be (to simplify we take β1k=βk). Application of the Leibniz superidentity constraint in the following cases we get the results given in the table. Leibniz superidentity Constraint {x1,z,y1}β1= −a1[z,y1]= −a1y1+∑m k=2βkyk {z,y1,x1} [z,y2]=(β2−1 2α2)y3+∑m k=4βk−1yk {z,y2,y1}β2=1 2α2,[z,y1]= −a1y1+1 2α2+βm−1ym−1+βmym, βk=0,3≤k≤m−2[z,y2]=βm−1ym {z,y2,x1} [z,y3]=0 By the induction method, it is easy to prove that [z,yi]=0for 4≤i≤m.From the Leibniz superidentity for the triples {z,x1,y1}we get βm−1=0.Let us take the following transformation of basis: x󸀠 1=x1+n ∑ i=3 Aixi,y󸀠 1=y1+n ∑ i=3 Aiyi, x󸀠 j=xj+n ∑ i=j+2 Ai−j+1xi,2≤j≤n,y󸀠 j=yj+n ∑ i=j+2 Ai−j+1yi,2≤j≤m, (5.1) with A3=− a3 4a4 ,A4=− a4 6a1 ,Ai=− 1 2(i−1)a1(ai+i−2 ∑ k=3 Akai+1−k),5≤i≤n. Checking all the products, we can assume that ai=0with 3≤i≤n. In the last step, we consider [z,z]=n ∑ k=2 γkxk. From the Leibniz superidentity for {z,z,x1}we obtain α2=γ2=γ3=⋅⋅⋅=γn−1=0 and from {z,z,y1}we get γn=4nβma1. Finally, making the change of bases z󸀠=1 a1z−2nβm a1xn, we can assume that a1=1and βm=0.For all that, we get to the family of statement. Case m=n.Analogously to the previous case and follow the same steps it was concluded that [z,x1]=−2a1x1+α2x2,[z,xi]=0,2≤i≤n, [z,y1]=−a1y1+1 2α2y2+βmym,[z,yi]=0,2≤i≤m. L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 |1287 Next, from the Leibniz superidentity for the triples {z,y1,y1}we get that βm=0.The change of basis (5.1) permits to suppose ai=0with 3≤i≤n.Now, considering [z,z]=n ∑ k=2 γkxk and from the Leibniz superidentity {z,z,x1}, we get [z,z]=γnxn. Finally, the change of basis given by z󸀠=1 a1z−γn 2na2 1 xnallows to determine the superalgebra of statement. Funding: The authors are partially supported by Junta de Extremadura and Fondo Europeo de Desarrollo Regional (GR18001 and IB18032) and also by Agencia Estatal de Investigación (Spain), grant MTM2016- 79661-P (European FEDER support included, EU). References [1] J. K. Adashev, L. M. Camacho and B. A. Omirov, Central extensions of null-filiform and naturally graded filiform non-Lie Leibniz algebras, J. Algebra 479 (2017), 461–486. [2] S. Albeverio, S. A. Ayupov and B. A. Omirov, On nilpotent and simple Leibniz algebras, Comm. Algebra 33 (2005), no. 1, 159–172. [3] S. Albeverio, B. A. Omirov and I. S. Rakhimov, Varieties of nilpotent complex Leibniz algebras of dimension less than five, Comm. Algebra 33 (2005), no. 5, 1575–1585. [4] J. M. Ancochea Bermúdez, R. Campoamor-Stursberg and L. García Vergnolle, Indecomposable Lie algebras with nontrivial Levi decomposition cannot have filiform radical, Int. Math. Forum. 1(2006), 309–316. [5] J. M. Ancochea Bermúdez, R. Campoamor-Stursberg and L. García Vergnolle, Classification of Lie algebras with naturally graded quasi-filiform nilradicals., J. Geom. Phys. 61 (2011), no. 11, 2168–2186. [6] D. Burde and C. Steinhoff, Classification of orbit closures of 4-dimensional complex Lie algebras, J. Algebra 214 (1999), 729–739. [7] L. M. Camacho, J. R. Gómez and R. M. Navarro, Cohomolgy of some nilpotent Lie algebras, Extracta Math. 15 (2000), no. 1, 155–174. [8] L. M. Camacho, B. A. Omirov and K. K. Masutova, Solvable Leibniz algebras with filiform nilradical., Bull. Malays. Math. Sci. Soc. 39 (2016), no. 1, 283–303. [9] J. M. Casas, M. Ladra, B. A. Omirov and I. A. Karimjanov, Classification of solvable Leibniz algebras with null-filiform nilradical, Linear Multilinear Algebra 61 (2013), no. 6, 758–774. [10] J. M. Casas, M. Ladra, B. A. Omirov and I. A. Karimjanov, Classification of solvable Leibniz algebras with naturally graded filiform nilradical, Linear Algebra Appl. 438 (2013), no. 7, 2973–3000. [11] J. R. Gómez, Y. Khakimdjanov and R. M. Navarro, Some problems concerning to nilpotent Lie superalgebras, J. Geom. Phys. 51 (2004), no. 4, 473–486. [12] M. Goze and Y. Khakimdjanov, Nilpotent Lie Algebras, Math. Appl. 361, Kluwer Academic, Dordrecht, 1996. [13] F. Grunewald and J. O’Halloran, Varieties of nilpotent Lie algebras of dimension less than six, J. Algebra 112 (1988), no. 2, 315–326. [14] N. Jacobson, Lie Algebras, Tracts Pure Appl Math. 10, Interscience, New York, 1962. [15] V. G. Kac, A sketch of Lie superalgebra theory, Comm. Math. Phys. 53 (1977), 31–64. [16] A. K. Khudoyberdiyev and B. A. Omirov, Infinitesimal deformations of null-filiform Leibniz superalgebras, J. Geom. Phys. 74 (2013), 370–380. [17] M. Ladra, K. K. Masutova and B. A. Omirov, Corrigendum to “Classification of solvable Leibniz algebras with naturally graded filiform nilradical” [Linear Algebra Appl. 438 (7) (2013) 2973–3000], Linear Algebra Appl. 507 (2016), 513–517. [18] G. M. Mubarakzjanov, On solvable Lie algebras (in Russian), Izv. Vyssh. Uchebn. Zaved.Mat. 1963 (1963), 114–123. [19] J. C. Ndogmo and P. Winternitz, Solvable Lie algebras with abelian nilradicals, J. Phys. A. 27 (1994), 405–423. [20] D. D. Repovš and M. V. Zaicev, Codimension growth of solvable lie superalgebras, J. Lie Theory 28 (2018), no. 4, 1189–1199. [21] M. C. Rodríguez-Vallarte and G. Salgado, On indecomposable solvable Lie superalgebras having a Heisenberg nilradical, J. Algebra Appl. 15 (2016), no. 10, Article ID 1650190. [22] M. Scheunert, The Theory of Lie Superalgebras. An Introduction, Lecture Notes in Math. 716, Springer, Berlin, 1979. [23] L. Šnobl and D. Karásek, Classification of solvable Lie algebras with a given nilradical by means of solvable extensions of its subalgebras, Linear Algebra Appl. 432 (2010), no. 7, 1836–1850. 1288 |L.M. Camacho et al., Solvable Lie and Leibniz superalgebras 2 [24] S. Tremblay and P. Winternitz, Solvable Lie algebras with triangular nilradicals, J. Phys. A. 31 (1998), 789–806. [25] M. Vergne, Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes, (in French), Bull. Soc. Math. France 98 (1970), 81–116. [26] Y. Wang, J. Lin and S. Deng, Solvable Lie algebras with quasifiliform nilradicals, Comm. Algebra. 36 (2008), 4052–4067.