On solvable Lie and Leibniz superalgebras with maximal codimension of nilradical

Journal of Algebra 591 (2022) 500–522 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra On solvable Lie and Leibniz superalgebras with maximal codimension of nilradical ✩ L.M. Camacho a,∗, R.M. Navarro b, B.A. Omirov c aDpto. Matemática Aplicada I, Universidad de Sevilla, Sevilla, Spain bDpto. de Matemáticas, Universidad de Extremadura, Cáceres, Spain cNational University of Uzbekistan, Institute of Mathematics, Tashkent, Uzbekistan a r t i c l e i n f o a b s t r a c t Article history: Received 23 June 2021 Available online 5 November 2021 Communicated by Alberto Elduque MSC: 17A32 17A70 17B30 17B40 Keywords: Solvable Lie superalgebras Solvable Leibniz superalgebras Derivations Nilpotent Lie superalgebras Nilpotent Leibniz superalgebras Along this paper we show that under certain conditions the method for describing of solvable Lie and Leibniz algebras with maximal codimension of nilradical is also extensible to Lie and Leibniz superalgebras, respectively. In particular, we totally determine the solvable Lie and Leibniz superalgebras with maximal codimension of model filiform and model nilpotent nilradicals. Finally, it is established that the superderivations of the obtained superalgebras are inner. © 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). ✩This work has been supported by Agencia Estatal de Investigación (Spain), grant PID2020-115155GB- I00 (European FEDER support included, EU) and by the European Union under the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia. Project reference: FEDER-UCA18-107643. This work has been also supported by Junta de Extremadura and Fondo Europeo de Desarrollo Regional (GR18001 and IB18032). *Corresponding author. E-mail addresses: lcamac[email protected] (L.M. Camacho), rnav[email protected] (R.M. Navarro), omiro[email protected] (B.A. Omirov). https://doi.org/10.1016/j.jalgebra.2021.10.029 0021-8693/© 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 501 1. Introduction In 1950 A.I. Malcev proved that a solvable Lie algebra is uniquely determined by its nilradical [19]. A decade later Mubarakzjanov proposed the method of description of solvable Lie algebra in terms of nilradical and its nil-independent derivations [20]. He noted that the codimension of nilradical does not exceed the number of nil-independent derivations and the number of generators of the nilradical. Since then the classification of solvable Lie algebras with the Abelian, Heisenberg, filiform, quasi-filiform nilradicals is obtained, see for instance [5,6,21,25,26,24]. Recently, the topic of research has drawn a lot of attention and in particular in [11]the authors extended Mubarakzjanov’s method to Leibniz algebras. Therefore, along the last few years have been obtained classifications with different types of Leibniz nilradicals [10–12,18]. Among all the solvable Lie and Leibniz algebras special mention deserves the maximal (in dimensional sense) solvable Lie and Leibniz algebras with given nilradical, due to its properties, they are in some cases cohomologically rigid [4]. The main goal of this paper is the study of maximal solvable Lie and Leibniz superalgebras with a given nilradical. It should be noted that the structures of solvable Lie and Leibniz superalgebras are more complex than structures of solvable Lie and Leibniz algebras [23]. In particular, Lie’s theorem is not true (neither in its general or its reduced forms) for a solvable Leibniz (respectively, Lie) superalgebras L. Moreover, the square of a solvable superalgebra is not necessary to be nilpotent [22]. Nevertheless, throughout this paper we show that under certain conditions Mubarakzjanov’s method of description of maximal solvable Lie and Leibniz algebras with given nilradical is also applicable for Lie and Leibniz superalgebras, respectively. In particular, we determine the maximal dimensional solvable Lie and Leibniz superalgebras with model filiform and model nilpotent nilradicals. Additionally, we compute the space of superderivations on the maximal-dimensional solvable Lie and Leibniz superalgebras with filiform and model nilpotent nilradical and show that all of these superderivations are inner. These results constitute an extension of the results obtained for similar Lie and Leibniz algebras. Finally, let us note that all Lie and Leibniz superalgebras obtained in this paper are excellent candidates to cohomologycally rigid superalgebras, we leave this study for a further work though. 2. Preliminary results 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). For a homogeneous element x ∈Vwe denote |x|the degree of x(either ¯ 0or ¯ 1). A Lie superalgebra (see [15]) is a Z2-graded vector space g =g¯ 0⊕g¯ 1together with an even bilinear mapping [·, ·] :g ×g −→ g([g¯ 0, g¯ 0] ⊂g¯ 0, [g¯ 0, g¯ 1] ⊂g¯ 1and [g¯ 1, g¯ 1] ⊂g¯ 0), which for any arbitrary homogeneous elements x, y, zsatisfies the conditions: 502 L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 1. [x, y] =−(−1)|x||y|[y, x], 2. (−1)|z||x|[x, [y, z]] +(−1)|x||y|[y, [z, x]] +(−1)|y||z|[z, [x, y]] =0(super Jacobi identity). 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 remain 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 superalgebra 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. At the same time, there are also defined two other crucial sequences denoted by Ck(g¯ 0) and Ck(g¯ 1)which will play an important role in our study. They are defined as follows: C0(g¯ i):=g¯ i,Ck+1(g¯ i):=[g¯ 0,Ck(g¯ i)],k≥0,¯ i∈Z2. This last sequences allow us to introduce filiform Lie superalgebras. Thus, a Lie superalgebra g =g¯ 0⊕g¯ 1is said to be filiform if g¯ 0is a filiform Lie algebra (those algebras with nilindex dim(g¯ 0) −1) and the action of g¯ 0over g¯ 1has the structure of filiform g¯ 0-module, i.e. dim Ci−1(g¯ 1)/Ci(g¯ 1)=1for 1 ≤i ≤dim(g¯ 1). In the study nilpotent Lie algebras we have the invariant called characteristic sequence that can be naturally extended for Lie superalgebras. Definition 2.1. For an arbitrary element x ∈g¯ 0, the adjoint operator adxis a nilpotent endomorphism of the space gi, where i ∈{ ¯ 0, ¯ 1}. We denote by gzi(x)the descending sequence of dimensions of Jordan blocks of adx. Then, we define the invariant of a Lie superalgebra gas follows: gz(g)=max x∈g¯ 0\[g¯ 0,g¯ 0]gz¯ 0(x)max x∈g¯ 0\[g¯ 0,g¯ 0]gz¯ 1(x), where gziis in lexicographic order. The couple gz(g)is called characteristic sequence of the Lie superalgebra g. L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 503 Let us recall the definition of superderivations for Lie superalgebras [15]. A superderivation of degree sof a Lie superalgebra L, s ∈Z2, is an endomorphism D∈EndsL with the property D(ab)=D(a)b+(−1)s·degaaD(b). If we denote DersL ⊂EndsLthe space of all superderivations of degree s, then DerL =Der0L ⊕Der1Lis the Lie superalgebra of superderivations of L, with Der0L composed by even superderivations and Der1Lby odd ones. 2.1. Preliminaries for Leibniz superalgebras Remark that many results and definitions of the above section can be extended for Leibniz superalgebras. Definition 2.2. [2]. A Z2-graded vector space L =L¯ 0⊕L¯ 1is called a Leibniz superalgebra if it is equipped with a product [·, ·]which for an arbitrary element xand homogeneous elements y, zsatisfies the condition [x, [y,z]] = [[x, y],z]−(−1)|y||z|[[x, z],y](super Leibniz identity). Note that if a Leibniz superalgebra Lsatisfies the identity [x, y] =−(−1)|x||y|[y, x] for any homogeneous elements x, y∈L, then the super Leibniz identity becomes the super Jacobi identity. 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 super Leibniz identity can be expressed as R[x,y]= RyRx−(−1)|x||y|RxRy. If we denote by R(L)the set of all right multiplication operators, then R(L)with respect to the following multiplication <R a,R b>:= RaRb−(−1)¯ i¯ jRbRa(2.1) for Ra∈R(L)¯ i, Rb∈R(L)¯ j, forms a Lie superalgebra. Note that Rais a derivation. In fact, the condition for being a derivation of a Leibniz superalgebra (for more details see [17]) 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|, then 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 super (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. 504 L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 Let V=V¯ 0⊕V¯ 1be the underlying vector space of L, L =L¯ 0⊕L¯ 1∈Leibn,m, being Leibn,m the 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, [3,8,13,14]) are regarding the applications of algebraic groups theory to the description of the variety of Lie and Leibniz algebras. Definition 2.3. 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 compatible with the right annihilator in Leibniz algebras. If we consider the ideal I:= ideal < [x, y] +(−1)|x||y|[y, x] >, then I⊂Ann(L). Let L =L¯ 0⊕L¯ 1be a nilpotent Leibniz superalgebra with dim L¯ 0=nand dim L¯ 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 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 supernilindex 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 L has 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. We have for Lie superalgebras the invariant called characteristic sequences that can be naturally extended for Leibniz superalgebras. Thus, we have the following definition. Definition 2.4. For an arbitrary element x ∈L¯ 0, the operator Rxis a nilpotent endomorphism of the space Li, where i ∈{ ¯ 0, ¯ 1}. We denote by gzi(z)the descending sequences of dimensions of Jordan blocks of Rx. Then, we define the invariant of a Leibniz superalgebra Las follows: gz(L)=max x∈L¯ 0\[L¯ 0,L¯ 0]gz¯ 0(x)max x∈L¯ 0\[L¯ 0,L¯ 0]gz¯ 1(x), L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 505 where gziis in lexicographic order. The couple gz(L)is called characteristic sequences of a Leibniz superalgebra L. 3. Maximal-dimensional solvable Lie superalgebras with filiform nilradical Throughout this section we study solvable Lie superalgebras with maximal dimension of the complementary space to nilradical, being the nilradical the model filiform Lie superalgebra Ln,m. Let us recall that in [9]the authors proved that under the condition of being L2nilpotent, any solvable Lie superalgebra over the real or complex numbers field can be obtained by means of outer non-nilpotent superderivations of the nilradical. Therefore, for any solvable Lie superalgebra rwith r2nilpotent, we have a decomposition into semidirect sum: r=t−→ ⊕n, [t,n]⊂n,[n,n]⊂n,[t,t]⊂n. Along the present section we consider as nilradical the model filiform Lie superalgebra Ln,m, that is, the simplest filiform Lie superalgebra which is defined by the only non-zero products Ln,m :[x1,x i]=−[xi,x 1]=xi+1,2≤i≤n−1 [x1,y j]=−[yj,x 1]=yj+1,1≤j≤m−1 where {x1, ..., xn}be a basis of (Ln,m)¯ 0and {y1, ..., ym}be a basis of (Ln,m)¯ 1. Note that Ln,m is the most important filiform Lie superalgebra, in complete analogy to Lie algebras, since all the other filiform Lie superalgebras can be obtained from it by deformations [7]. These infinitesimal deformations are given by the even 2-cocycles Z2 ¯ 0(Ln,m, Ln,m). On the other hand, t =span{t1, t2, t3}corresponds with the maximal torus of derivations of Ln,m, which is composed, in particular, by even superderivations. Then tis Abelian ([t, t] =0) and the operators adti(ti∈t)are diagonal. A straightforward computation leads to the following action of tover Ln,m: [t1,x i]=ixi,1≤i≤n; [t1,y j]=jyj,1≤j≤m; [t2,x i]=xi,2≤i≤n; [t3,y j]=yj,1≤j≤m. Thus, the solvable Lie superalgebra that we are going to consider, and henceforth named SLn,m, is defined in a basis {x1, ..., xn, t1, t2, t3, y1, ..., ym}by the only nonzero bracket products 506 L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 SLn,m : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [x1,x i]=−[xi,x 1]=xi+1,2≤i≤n−1; [x1,y j]=−[yj,x 1]=yj+1,1≤j≤m−1; [t1,x i]=−[xi,t 1]=ixi,1≤i≤n; [t1,y j]=−[yj,t 1]=jyj,1≤j≤m; [t2,x i]=−[xi,t 2]=xi,2≤i≤n; [t3,y j]=−[yj,t 3]=yj,1≤j≤m; with {x1, ..., xn, t1, t2, t3}a basis of (SLn,m)¯ 0and {y1, ..., ym}a basis of (SLn,m)¯ 1. The purpose now is to find out whether SLn,m is the unique solvable Lie superalgebra with maximal codimension of nilradical Ln,m. Theorem 3.1. An arbitrary complex maximal-dimensional solvable Lie superalgebra L with L2nilpotent and nilradical Ln,m is isomorphic to SLn,m. Proof. It is easy to see that superalgebra Ln,m can be considered as nilpotent Lie algebra (a Z2-graded Lie algebra), because of the fact that there is no symmetric bracket products in the law of Ln,m. Note also, that under the condition of being L2nilpotent the techniques used in Lie superalgebras are rather similar to the ones used in Lie algebras, i.e. solvable extension by means of (super)derivations of nilradical, for more details it can be consulted [9]. Considering now Ln,m as a Lie algebra, the results of [16] allow us to assert that there is a unique solvable Lie algebra with maximal codimension of nilradical, i.e. maximal dimension of the complementary space to nilradical. Moreover this maximal codimension is equal to the number of generators of the nilradical, 3in our case. It can be easily seen that this unique solvable Lie algebra described in [16]is isomorphic to SLn,m, considered the latter as a Lie algebra. Indeed, following to Theorem 3.2[16]we have a solvable Lie algebra ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [x1,x i]=−[xi,x 1]=xi+1,2≤i≤n−1; [x1,y j]=−[yj,x 1]=yj+1,1≤j≤m−1; [z1,x 1]=−[x1,z 1]=x1, [z1,x i]=−[xi,z]=(i−2)xi,3≤i≤n; [z1,y j]=−[yj,z 1]=(j−1)yj,2≤j≤m; [z2,x i]=−[xi,z 2]=xi,2≤i≤n; [z3,y j]=−[yj,z 3]=yj,1≤j≤m; and the isomorphism defined by {t1=z1+2z2+z3, t2=z2, t3=z3}shows that this Lie algebra is isomorphic to SLn,m. Now, we consider Ln,m as a Lie superalgebra. L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 507 The even superderivations Dof Ln,m are in particular derivations of the Z2-graded Lie algebra Ln,m verifying D(Ln,m 0) ⊂Ln,m 0and D(Ln,m 1) ⊂Ln,m 1. The odd superderivations Dof Ln,m, on the other hand, verify the following three conditions: D([xi,x j]) = [D(xi),x j]+[xi,D(xj)],if xi,x j∈Ln,m 0(3.1) D([xi,y j]) = [D(xi),y j]+[xi,D(yj)],if xi∈Ln,m 0,y j∈Ln,m 1(3.2) D([yi,y j]) = [D(yi),y j]−[yi,D(yj)],if yi,y j∈Ln,m 1(3.3) The equations (3.1)and (3.2) correspond with the Lie derivation condition. Thus, it remains to study the equation (3.3). Taking into account now the law of Ln,m it can be easily seen that the only possibility for having at least one non-null term in the equation (3.3) corresponds with the existence of yisuch that x1∈D(yi). Next, we explore such possibilities. For i ≥2, we get D(yi)=D([x1,y i−1]) = [D(x1),y i]+[x1,D(yi−1)] = [x1,D(yi−1)] Since [x1, D(yi−1)] ∈span{x3, ..., xn}we can exclude ifor i ≥2. Suppose i =1, and then D(y1)=α1x1+ n  k=2 αkxk,with α1=0. From equation (3.3)we get in particular D([y1,y 1]) = 0 = [D(y1),y 1]−[y1,D(y1)] = 2α1y2 and then α1=0. Therefore the equation (3.3)vanishes over Ln,m and consequently all the odd superderivations of Ln,m are in particular derivations of the Z2-graded Lie algebra Ln,m verifying D(Ln,m 0) ⊂Ln,m 1and D(Ln,m 1) ⊂Ln,m 0. Thus, all the superderivations of the Lie superalgebra Ln,m are particular cases of derivations of the Z2-graded Lie algebra Ln,m. Therefore, we can assert that SLn,m is not only the unique maximal-dimensional solvable Lie algebra with nilradical the Lie algebra Ln,m, but also is the unique maximal-dimensional solvable Lie superalgebra L with L2nilpotent and nilradical the model filiform Lie superalgebra Ln,m. 4. Maximal-dimensional solvable Lie superalgebras with model nilpotent nilradical Throughout this section firstly we extend the definition of model nilpotent to Lie superalgebras and after that we obtain the description of maximal-dimensional solvable Lie superalgebra with model nilpotent nilradical. Next we recall the definition of model nilpotent Lie algebra, for more details it can be consulted for instance [4]. Thus, the model nilpotent Lie algebra with arbitrary characteristic sequence (n1, n2, ··· , nk, 1) is composed by the Lie algebras admitting a basis 508 L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 {x1, ··· , xn1+1, ··· , xn1+n2+1, ··· , xn1+···nk+1}such that the only non-null brackets (expanded by skew-symmetry) are exactly the following [x1,x j]=xj+1,2≤j≤n1, [x1,x n1+j]=xn1+1+j,2≤j≤n2, . . . [x1,x n1+···+nk−2+j]=xn1+···+nk−2+1+j,2≤j≤nk−1, [x1,x n1+···+nk−1+j]=xn1+···+nk−1+1+j,2≤j≤nk. Therefore, we present the next definition in a natural way. Definition 4.1. The model nilpotent Lie superalgebra with arbitrary characteristic sequences (n1, ··· , nk, 1|m1, ··· , mp)is nothing but the Lie superalgebra admitting a basis {x1, ···, xn1+···nk+1, y1, ··· , ym1+···+mp}with xieven basis vectors and yjodd basis vectors, such that the only non-null brackets are exactly the following N(n1, ··· , nk, 1|m1, ··· , mp): ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [x1,x j]=−[xj,x 1]=xj+1,2≤j≤n1, [x1,x n1+···+nj+i] =−[xn1+···+nj+i,x 1]=xn1+···+nj+i+1,1≤j≤k−1,2≤i≤nj+1, [x1,y j]=−[yj,x 1]=yj+1,1≤j≤m1−1, [x1,y m1+···+mj+i]=−[ym1+···+mj+i,x 1] =ym1+···+mj+i+1,1≤j≤p−1,1≤i≤mj+1 −1. Remark 4.1. Note that the model filiform Lie superalgebra Ln,m is exactly N(n −1, 1|m). Let us consider now t =span{t1, ..., tk+1, t 1, ..., t p}the maximal torus of derivations of N(n1,··· ,nk,1|m1,··· ,mp), which is composed in particular by even superderivations. A straightforward computation leads to the following action of tover N(n1, ··· , nk, 1|m1, ··· , mp): [t1,x i]=ixi,1≤i≤n1+···+nk+1; [t1,y j]=jyj,1≤j≤m1+···+mp; [t2,x i]=xi,2≤i≤n1+1; [tj+2,x n1+···+nj+i]=xn1+···+nj+i,1≤j≤k−1,2≤i≤nj+1 +1; [t 1,y i]=yi,1≤i≤m1; [t j+1,y m1+···+mj+i]=ym1+···+mj+i,1≤j≤p−1,1≤i≤mj+1. Thus, the solvable Lie superalgebra that we are going to consider and denoted by SN(n1, ··· , nk, 1|m1, ··· , mp)is defined in a basis {x1, ..., xn1+···nk+1, t1, ..., tk+1, L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 515 [t2,x 2]=(b2−1)x2, [x2,t 2]=x2, [tj+2,x n1+···+nj+2]=(bj+2 −1)xn1+···+nj+2, [xn1+···+nj+2,t j+2]=xn1+···+nj+2, [y1,t  1]=y1, [t 1,y 1]=(b 1−1)y1, [t j+1,y m1+···+mj+1]=(b j+1 −1)ym1+···+mj+1, [ym1+···+mj+1,t  j+1]=ym1+···+mj+1 Similar to previous case and applying Leibniz identity we obtain the remaining products and b1=0, bi=1with 2 ≤i ≤k+1and b j=1with 1 ≤j≤p. Thus, we get SNP(n1, ··· , nk, 1|m1, ··· , mp). Now, we consider NP(n1, ··· , nk, 1|m1, ··· , mp)as a superalgebra. The even superderivations of this superalgebra are in particular Leibniz derivations. Then, we go to prove that the odd superderivations are also Leibniz derivations. For that purpose, it is sufficient to verify that the equations (5.2)and (5.4)vanish. Let dbe an odd superderivation. Then, we have d(x1)= m1+···+mp  k=1 akyk,d(x2)= m1+···+mp  k=1 bkyk,d(xn1+···+nj+2) = m1+···+mp  k=1 αkjyk,1≤j≤k−1, d(y1)= n1+···+nk+1  t=1 ctxt,d(ym1+···+mj+1)= n1+···+nk+1  t=1 βtjxt,1≤j≤p. From the equation (5.1)for the pair [x1, x1]we have that d(x1)=am1ym1+···+am1+···+mpym1+···+mp and for the equation (5.4)for the pair [y1, y1]we get to c1=0. Analogously, if we consider the pairs [ym1+···+mj+1, ym1+···+mj+1]with 1 ≤j≤p −1we have that β1j=0. Thus, it proves that the equation (5.4)is always zero. On the other hand, we put the equation (5.2). The products [xi, d(yj)] are always zero because of c1=β1j=0for 1 ≤j≤p −1in d(yj). The other products are trivially zero. Finally, we conclude that the odd superderivations are in particular Leibniz derivations. We can assert that SNP(n1, ··· , nk, 1|m1, ··· , mp)is the unique maximaldimensional solvable Leibniz superalgebra Lwith L2nilpotent and nilradical the model nilpotent Leibniz superalgebra NP(n1, ··· , nk, 1|m1, ··· , mp).  516 L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 7. Superderivations of the maximal-dimensional solvable Lie and Leibniz superalgebras In this section we establish that the spaces of superderivations for the superalgebras obtained in previous sections (that is, SLn,m, SN(n1, ··· , nk, 1|m1, ··· , mp), SLPn,m, SNP(n1, ··· , nk, 1|m1, ··· , mp) consist of inner superderivations. These results extend analogously results for similar Lie and Leibniz algebras. Theorem 7.1. Any superderivation on the Lie superalgebra SLn,m is inner. Proof. Our goal is to prove that the following inner superderivations {adx1, ..., adxn, adt1, adt2, adt3, ady1, ..., adym}form a basis of the space of superderivations Der(SLn,m) =Der0(SLn,m) ⊕Der1(SLn,m), with {adx1, ..., adxn, adt1, adt2, adt3} a basis of Der0(SLn,m)and {ady1, ..., adym}a basis of Der1(SLn,m). Let Dbe an even superderivation of SLn,m. Then taking into account the embeddings D(SLn,m 0) ⊂SLn,m 0and D(SLn,m 1) ⊂SLn,m 1we set D(x1)= 3  s=1 αsts+ n  k=1 akxk,D(x2)= 3  s=1 βsts+ n  k=1 bkxk,D(y1)= m  r=1 pryr, D(t1)= 3  s=1 γsts+ n  k=1 ckxk,D(t2)= 3  s=1 δsts+ n  k=1 dkxk, D(t3)= 3  s=1 νsts+ n  k=1 ekxk. According to the even superderivation condition, we can summarize the computation in the following table: Pairs Constraints {x1,t 1}αi=0,1≤i≤3, γ1=0,α 2=0,ck=kak+1,2≤k≤n−1 {x1,t 2}δ1=0,dk=ak+1,2≤k≤n−1 {x1,t 3}ν1=0,ek=0,2≤k≤n−1 {x1,y j−1} 2≤j≤m−1d(yj)=((j−1)a1+p1)yj+ m  k=j+1 pk−j+1yk,2≤j≤m {y1,x 2}b1=0,β 3=−β1 {x1,x 2}d(x3)=−β1x1+(a1+b2)x3+ n  k=4 bk−1xk {x3,y 1}β1=0⇒β3=0 {x1,x i−1} 4≤j≤n−1d(xi)=((i−2)a1+b2)xi+ n  k=i+1 bk−i+2xk,3≤i≤n {t3,t 1}e1=en=0 {t3,x 2}ν2=0 {t1,x 2}β2=γ2=0,c1=−b3,bk=0,4≤k≤n {x2,t 2}δ2=d1=0 {t1,t 2}cn=ndn {t1,y 1}γ3=0,p2=b3,pk=0,3≤k≤m {t2,y 1}δ3=0 {t3,y 1}ν3=0 L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 517 Therefore, we get D(t1)=−b3x1+ n−1  k=2 kak+1xk+ndnxn,D(t2)= n−1  k=2 ak+1xk+dnxn,D(t3)=0, D(x1)=a1x1+ n  k=3 akxk,D(x2)=b2x2+b3x3, D(xi)=((i−2)a1+b2)xi+b3xi+1,3≤i≤n, D(y1)=p1y1+b3y2, D(yj)=((j−1)a1+p1)yj+b3yj+1,2≤j≤m. Thus, we conclude dim(Der0(SLn,m)) =n +3. On the other hand, the (n +3) inner superderivations {adx1, ..., adxn, adt1, adt2, adt3}are in particular even superderivations. Hence, we obtain a basis of the space Der0(SLn,m)composed by inner even superderivations. Moreover, Dcan be expressed via inner as follows D=b3(adx1)−n−2  k=2 ak+1(adxk)−dn(adxn)+a1(adt1−2adt2−adt3)+b2(adt2)+p1(adt3). Analogously, we are going to compute the odd superderivations. Let Dnow be an odd superderivation. We put D(x1)= m  k=1 akyk,D(x2)= m  k=1 bkyk,D(y1)= 3  s=1 psts+ n  r=1 cryr, D(t1)= m  k=1 dkyk,D(t2)= m  k=1 gkyk,D(t3)= m  k=1 hkxk. According to the odd superderivation conditions we have the following computations: Pairs Constraints {x1,x i−1} 3≤j≤n−1d(xi)= m  k=i−1 bk−i+2yk,3≤i≤n {x1,y 1}d(y2)=−p1x1+ n  k=3 ck−1xk {x1,y j−1} 3≤j≤m−1d(yj)= n  k=j+1 ck−j+1xk,2≤j≤m {x2,y 2}p1=0 {x2,y 1}p2=c1=0 {t1,y 1}p3=0,ck=0,2≤k≤n {x1,t 1}dk=kak+1,1≤k≤m−1 {x1,t 2}gk=0,1≤k≤m−1 {x1,t 3}hk=ak+1,1≤k≤m−1 {x2,t 1}b1=bk=0,3≤k≤m {x2,t 2}b2=0 {t1,t 2}gm=0 {t1,t 3}dm=mhm 518 L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 Thus, we get D(t1)= m−1  k=1 kak+1yk+mhmym,D(t2)=0,D(t3)= m−1  k=1 ak+1yk+hmym, D(x1)= m  k=2 akyk,D(xi)=0,2≤i≤n, D(yj)=0,1≤j≤m. This implies that dim(Der1(SLn,m)) =m. On the other hand, we have modd inner superderivations (they are {ady1, ..., adym}). Consequently, a basis of Der1(SLn,m) form by inner odd superderivations. In particular, Dcan be expressed via inner superderivations as follows: D=−m−1  k=1 ak+1(adyk)−hm(adym). Note that all the computations have been duplicated by using the software Mathematica. Consider now the maximal-dimensional solvable Lie superalgebra with model nilpotent nilradical SN(n1, ··· , nk, 1|m1, ··· , mp). Theorem 7.2. Any superderivation of the Lie superalgebra SN(n1, ··· , nk, 1|m1, ··· , mp) is inner. Proof. We are going to prove that the following inner superderivations {adx1,...,adx n1+···nk+1,adt 1,...,adt k+1,adt  1,...,adt  p,ady 1,...,ady m1+···+mp} form a basis of the space Der(SN(n1, ··· , nk, 1|m1, ··· , mp)) with {adx1, ..., adxn1+···nk+1, adt1, ..., adtk+1, adt 1, ..., adt p}a basis for even superderivations, Der0(SN(n1, ··· , nk, 1|m1, ··· , mp)), and {ady1, ..., adym1+···+mp}a basis for the odd ones, Der1(SN(n1, ··· , nk, 1|m1, ··· , mp)). Let Dbe an even superderivation of SN(n1, ··· , nk, 1|m1, ··· , mp). Then from superderivation property we derive D(x1)=α1x1+ n1+1  s=3 αsxs+ k−1  j=1 ⎛ ⎝ n1+···+nj+1  s=n1+···+nj+3 αsxs⎞ ⎠, D(xi)=((i−2)α2+β)xi,2≤i≤n1+1, D(xn1+···+nj+i)=((i−2)α1+aj)xn1+···+nj+i,1≤j≤k−1,2≤i≤nj+1 +1, D(t1)= n1  s=2 sαs+1xs+ k−1  j=1 ⎛ ⎝ n1+···+nj+1  s=n1+···+nj+2 sαs+1xs⎞ ⎠+ L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 519 + k  j=1 (n1+···+nj+1)bjxn1+···+nj+1, D(t2)= n1  s=2 αs+1xs+b1xn1+1, D(tj+2)= n1+···+nj+1  s=n1+···+nj+2 αs+1xs+bj+1xn1+···+nj+1+1,1≤j≤k−1, D(t j)=0,1≤j≤p, D(yi)=((i−1)α1+γ)yi,1≤i≤m1, D(ym1+···+mj+i)=((i−1)α1+qj)ym1+···+mj+i,1≤j≤p−1,1≤i≤mj+1. Let Dbe an odd superderivation of SN(n1, ··· , nk, 1|m1, ··· , mp). Then the superderivation property implies D(x1)= m1  s=2 αsys+ p−1  j=1 ⎛ ⎝ m1+···+mj+1  s=m1+···+mj+2 αsys⎞ ⎠, D(xi)=0,2≤i≤n1+···+nk+1, D(t1)= m1−1  s=1 sαs+1ys+ p−1  j=1 ⎛ ⎝ m1+···+mj+1−1  s=m1+···+mj+1 sαs+1ys⎞ ⎠ + p  j=1 (m1+···+mj)δjym1+···+mj, D(ti)=0,2≤i≤k+1, D(t 1)= m1−1  s=1 αs+1ys+ p  j=1 δjym1+···+mj, D(t i)=0,2≤i≤p, D(yj)=0,1≤j≤m1+···+mp. Therefore, we have dim(Der0(SN(n1,··· ,n k,1|m1,··· ,m p)) = (n1+···nk+1)+k+1+p, dim(Der1(SN(n1,··· ,n k,1|m1,··· ,m p)) = m1+···+mp. Now, since both sets of inner superderivations of the statement of the Theorem, even and odd, are linearly independent we conclude that the set {adx1,...,adx n1+···nk+1,adt 1,...,adt k+1,adt  1,...,adt  p,ady 1,...,ady m1+···+mp} 520 L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 constitutes a basis of the superalgebra of superderivations Der(SN(n1, ··· , nk, 1|m1, ··· , mp)).  Since the proofs of the following results based on the application the same arguments and similar computations as in the proofs of Theorems 7.1 and 7.2 we present summaries of their proofs. Theorem 7.3. Any superderivation of the Leibniz superalgebra SLPn,m is inner. Proof. As a result of computing of the odd and even superderivations properties of the Leibniz superalgebra SLPn,m we obtain Der¯ 1(SLPn,m) ={0}and for an arbitrary d ∈Der¯ 0(SLPn,m)we get d(x1)=αx1,d(xi)=(β+(i−2)α)xi−γxi+1,2≤i≤n, d(t1)=γx1,d(t2)=d(t3)=0, d(y1)=δy1−γy2,d(yj)=(δ+(j−1)α)yj−γyj+1,2≤j≤m, for some parameters α, β, γ, δ. Consequently, dim(Der(SLPn,m)) =4and hence, we obtain Der(SLPn,m) = span{Rx1, Rt1, Rt2, Rt3}. In particular, dcan be expressed via inner derivations as follows: d=−γRx1+αRt1+βRt2+δRt3. Theorem 7.4. Any superderivation of the Leibniz superalgebra SNP(n1, ··· , nk, 1|m1, ··· , mp)is inner. Proof. Analogously to the previous superalgebra, we obtain that Der¯ 1(SLP(n1, ··· , nk, 1|m1, ··· , mp)) ={0}and for an arbitrary even superderivation dwe derive the following: d(x1)=αx1, d(xi)=(β+(i−2)α)xi−γxi+1,3≤i≤n1, d(xn1+1)=(β+(n1−1)α)xn1+1, d(xn1+···+nj+2)=μjxn1+···+nj+2 −γxn1+···+nj+3,1≤j≤k−1, d(xn1+···+nj+i) =(μj+(i−2)α)xn1+···+nj+i−γxn1+···+nj+i+1,1≤j≤k−1,3≤j≤nj+1, d(xn1+···+nj+1+1)=(μj+(nj+1 −1)α)xn1+···+nj+1+1,1≤j≤k−1, d(t1)=γx1, d(ti)=d(t j)=0,2≤i≤k+1,1≤j≤p, d(y1)=δy1−γy2, d(yj)=(δ+(j−1)α)yj−γyj+1,2≤j≤m1−1, L.M. Camacho et al. / Journal of Algebra 591 (2022) 500–522 521 d(ym1)=(δ+(m1−1)α)ym1, d(ym1+···+ms+1)=νsym1+···+ms+1 −γym1+···+ms+2,1≤s≤p−1, d(ym1+···+ms+i) =(νs+(i−1)α)ym1+···+ms+i−γym1+···+ms+i+1,1≤s≤p−1,2≤i≤ms+1 −1, d(ym1+···+ms+1 )=(νs+(ms+1 −1)α)ym1+···+ms+1 ,1≤s≤p−1. Then, dim(Der(SLP(n1, ··· , nk, 1|m1, ··· , mp))) =k+p +2. 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