Local Superderivations on Solvable Lie and Leibniz Superalgebras

Mediterr. J. Math. (2023) 20:76 https://doi.org/10.1007/s00009-023-02284-7 1660-5446/23/020001-22 published online January 21, 2023 c The Author(s) 2023 Local Superderivations on Solvable Lie and Leibniz Superalgebras Luisa Mar´ıa Camacho , Rosa Mar´ıa Navarro and Bakhrom Omirov Abstract. Throughout this paper, we show on one hand, that there are nilpotent and solvable Lie superalgebras with infinitely many local superderivations which are not standard superderivations. On the other hand, we show that every local superderivation is a superderivation on the maximal-dimensional solvable Lie superalgebras with model filiform or model nilpotent nilradical. Moreover, we extend the latter result for Leibniz superalgebras by showing that every local superderivation is a superderivation on the maximal-dimensional solvable Leibniz superalgebras with model filiform or model nilpotent non-Lie nilradical. Mathematics Subject Classification. 16W10, 17A70, 17B30, 17B40. Keywords. Lie superalgebra, Leibniz superalgebra, solvable Lie superalgebra, solvable Leibniz superalgebra, nilradical, superderivation, local superderivation. 1. Introduction Local derivations were considered for the first time in 1990 by Kadison [22] and also by Larson and Sourour [24]. In particular, Kadison showed that each continuous local derivation from a von Neumann algebra into the dual bimodule is a derivation. Let us note then, that the main problem studied in relation to this research topic is to determine when a local derivation is a derivation, see for instance [9,20]. Additionally, other problem that has been largely studied is to find types of algebras containing local derivations which are not derivations [1]. More recently, in [4,5,13] the authors studied the This work has been supported by Agencia Estatal de Investigaci´on (Spain), grant PID2020- 115155GB-I00 (European FEDER support included, EU) and by the European Union under 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). 76 Page 2 of 22 L. M. Camacho et al. MJOM aforementioned problems for Lie algebras, proving in particular that every local derivation of a semi-simple Lie algebra is a derivation and giving examples of solvable Lie algebras with local derivations which are not derivations. Likewise, an analogous study has been developed for Leibniz algebras, see for instance [6] and references therein. Recently, studying local superderivations on semi-simple Lie superalgebras have drawn a lot of attention [14–16,27], however none of them tackle local superderivations on solvable Lie superalgebras nor Leibniz superalgebras. Thus, this is the context of our work, studying local superderivations on solvable Lie and Leibniz superalgebras. Note that studying solvable Lie superalgebras presents more difficulties than studying solvable Lie algebras [26]. In particular, Lie’s theorem is not verified in general and neither its corollaries. Therefore, for a solvable Lie superalgebra L,L2:= [L, L] can not be nilpotent, see [25]. Nevertheless, in [11] the authors proved that under the condition of being L2nilpotent, any solvable Lie and Leibniz superalgebra over the real or complex field can be obtained by means of outer non-nilpotent superderivations of the nilradical in the same way as occurs for Lie and Leibniz algebras. In this frame, we investigate local superderivations of solvable Lie and Leibniz superalgebras. First, we prove that there are nilpotent and solvable Lie superalgebras with infinitely many local superderivations which are not ordinary superderivations (see Sects. 3and 5). Second, we prove on the maximal-dimensional solvable Lie superalgebras with model filiform or model nilpotent nilradical that every local superderivation is a superderivation (see Sect. 4). Finally, we extend this last result for the maximal-dimensional solvable Leibniz superalgebras with model filiform and model nilpotent non-Lie nilradical (see Sect. 6). 2. Preliminary Results 2.1. Preliminary for Lie Superalgebras A vector space Vis said to be Z2-graded if it admits a decomposition into a direct sum, V=V¯ 0⊕V¯ 1, where ¯ 0,¯ 1∈Z2. An element X∈Vis called homogeneous of degree |x|if it is an element of V|x|,|x|∈Z2. In particular, the elements of V¯ 0(resp. V¯ 1) are also called even (resp. odd). A Lie superalgebra (see [21]) 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)|x||y|[y,x], 2. (−1)|x||z|[x, [y,z]] + (−1)|x||y|[y,[z,x]] + (−1)|y||z|[z,[x, y]] = 0 (super Jacobi identity) for all homogeneous elements x, y, z ∈g. 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. Let us note that both the descending central sequence and the derived sequence of a Lie superalgebra g=g¯ 0⊕g¯ 1are defined in the same way as for MJOM Local Superderivations on Solvable Page 3 of 22 76 Lie algebras: C0(g):=g,Ck+1(g):=[Ck(g),g]andD0(g):=g,Dk+1(g):= [Dk(g),Dk(g)] respectively, for all k≥0. Thus, if Ck(g)={0}(resp. Dk(g)= {0}) for some k, then the Lie superalgebra is called nilpotent (resp. solvable). Note that nilpotent Lie superalgebras are in particular solvable. Remark also, that Engel’s theorem and its corollaries are still valid for Lie superalgebras. Then, a Lie superalgebra Lis nilpotent if and only if adLxis nilpotent for every homogeneous element xof L. Additionally, a Lie superalgebra Lis solvable if and only if its even part L0(a Lie algebra) is solvable. However, Lie’s Theorem does not hold for solvable Lie superalgebras. At the same time, there are also defined two other crucial sequences denoted by Ck(g¯ 0)andCk(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. Let us recall now, the definition of superderivations of superalgebras [21]. A superderivation of degree sof a superalgebra L,s∈Z2, is an endomorphism D∈EndsLwith the property D(ab)=D(a)b+(−1)s·degaaD(b) denote Ders(L)⊂EndsLthe space of all superderivations of degree s. Then Der(L)=Der0(L)⊕Der1(L) is the Lie superalgebra of superderivations of L, with Der0(L) composed by even superderivations and Der1(L)byodd ones. On the other hand, recall also that a homogeneous linear mapping Δ:L−→ Lof degree sis called a local homogeneous superderivation of degree sif for any element x∈L, there exists a superderivation Dx:L−→ L (depending on x) such that Δ(x)=Dx(x). Then, the set of all local superderivations can be expressed LocDer(L)=LocDer0(L)⊕LocDer1(L) with LocDer0(L)(resp.LocDer1(L)) composed by even (resp. odd) local superderivations. For more details it can be consulted [14]. 2.2. Preliminaries for Leibniz Superalgebras Let us note that many results and definitions of the above sub-section 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 for an arbitrary element xand homogeneous elements y, z satisfies 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. 76 Page 4 of 22 L. M. Camacho et al. MJOM Let us now denote by Rxthe right multiplication operator, i.e., Rx: L→Lgiven as Rx(y):=[y,x]fory∈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 Ra,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 [23]) 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 remark that the definition of local superderivation is a natural extension from Lie theory. Let us note also 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,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)andf¯ 1∈GL(m, C) (then G(V)=GL(n, C)⊕GL(m, C)). The action of G(V)onLeibn,m induces an action on the Leibniz superalgebras variety: two laws λ1,λ 2are 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 works (for example, [3,7,10,18, 19]) are regarding the applications of algebraic groups theory to the description of the variety of Lie and Leibniz algebras. 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 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=n and dim L¯ 1=m. From Equation (2.1)wehavethatR(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 Engel’s theorem we have the existence of a sequence of subspaces of V¯ 1,V0⊂V1⊂V2⊂ ··· ⊂ Vm=V¯ 1,with MJOM Local Superderivations on Solvable Page 5 of 22 76 R(L¯ 0)(Vi+1)⊂Vi.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. 3. Local Superderivations of the Model Filiform Lie Superalgebra We start our study with one case of nilpotent Lie superalgebra. Among all of them one that has been proved to be very relevant due to its properties is the model filiform Lie superalgebra since all the other filiform Lie superalgebras can be obtained from it by means of infinitesimal deformations [8]. These infinitesimal deformations are given by the even 2-cocycles Z2 0(Ln,m,L n,m). We consider then, the model filiform Lie superalgebra Ln,m,that is, the simplest filiform Lie superalgebra which is defined by the only non-zero bracket products that follow 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 with a basis {x1,...,x n}of (Ln,m)¯ 0and a basis {y1,...,y m}of (Ln,m)¯ 1. For an even superderivation Dof Ln,m we have D(Ln,m 0)⊂Ln,m 0and D(Ln,m 1)⊂Ln,m 1. Then we set D(x1)= n  k=1 akxk,D(x2)= n  k=1 bkxk,D(y1)= m  t=1 ctyt. Applying induction and the even superderivation condition for the products [x1,x i] we derive D(xi)=((i−2)a1+b2)xi+ n  k=i+1 bk−i+2xk,3≤i≤n. Similarly, from the products [x1,y j]weget D(yj)=((j−1)a1+c1)yj+ m  t=j+1 ct−j+1yt,2≤j≤m. Finally, from the product [x2,y 1] we obtain b1=0.Thus, we conclude 76 Page 6 of 22 L. M. Camacho et al. MJOM Der0(Ln,m) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ a1a2a3... a n00... 0 0b2b3... b n00... 0 00a1+b2... b n−100... 0 . . . . . . . . ..... . . . . . . . ..... . . 00 0 ... (n−2)a1+b200... 0 00 0 ... 0c1c2... c m 00 0 ... 00a1+c1... c m−1 . . . . . . . . ..... . . . . . . . ..... . . 00 0 ... 000... (m−1)a1+c1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . Let now Dbe an odd superderivation of Ln,m.Then we have D(x1)= m  k=1 akyk,D(x2)= m  k=1 bkyk,D(y1)= n  t=1 ctxt. According to the odd superderivation condition on the products of Ln,m and induction, similar to even superderivation case, we obtain D(xi)= m  k=i−1 bk−i+2yk,3≤i≤n, D(yj)= n  t=j+1 ct−j+1xt,2≤j≤m. Considering superderivation property for the products [x2,y 1] and [x1, xn], we get c1and bi=0,1≤i≤m−n+ 1 with m≥n. Thus, we conclude Der1(Ln,m)= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 00 0... 0a1a2... a m 00 0... 0b1b2... b m 00 0... 00b1... b m−1 . . .. . .. . ..... . .. . .. . ..... . . 00 0... 000... b 1 . . .. . .. . ..... . .. . .. . ..... . . 00 0... 000... 0 0c2c3... c n00... 0 00c2... c n−100... 0 . . .. . .. . ..... . .. . .. . ..... . . 00 0... c 200... 0 . . .. . .. . ..... . .. . .. . ..... . . 00 0... 000... 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , where bi=0,1≤i≤m−n+ 1 with m≥n. Theorem 3.1. Der(Ln,m)LocDer(Ln,m). Proof. Consider the homogeneous linear mappings Δt:Ln,m −→ Ln,m,t=2 of degree 0 defined on the basis vectors of Ln,m by Δt(x1)=x1,Δt(x2)=x2,Δt(x3)=tx3,Δt(xi)=Δ t(yj) =0,4≤i≤n, 1≤j≤m. MJOM Local Superderivations on Solvable Page 7 of 22 76 Clearly, Δtis not an even superderivation (because its matrix does not fit with the general matrix of even superderivations). Consider the following superderivations: •d1is the resultant even superderivation after replacing a1by 1 and all the rest of parameters by 0 on the general matrix of Der0(Ln,m), •d2is the resultant even superderivation after replacing b2by 1 and all the rest of parameters by 0 on the general matrix of Der0(Ln,m), •dtis the even superderivations defined by dt:= d1+(t−1)d2. Clearly, Δt(x1)=d1(x1),Δt(x2)=d2(x2),Δt(x3)=dt(x3), Δt(xi)=d0(xi)=Δ t(yj)=d0(yj),4≤i≤n, 1≤j≤m being d0the null superderivation. For an arbitrary element e=α1x1+···+αnxn+β1y1+···+βmymof Ln,m we have Δt(e)=α1x1+α2x2+α3tx3. Define superderivation deas follows de(xi)=de(yj) = 0 with 4 ≤i≤n and 1 ≤j≤mand de(x1)=β1x1+β2x2+β3x3,d e(x2)=γ2x2+γ3x3, de(x3)=(β1+γ2)x3, where β1,β 2,β 3,γ 2,γ 3are some unknowns parameters. From Δt(e)=de(e) we derive β1= 1 and the following linear system of equations α10α20 0α1α3α2⎛ ⎜ ⎜ ⎝ β2 β3 γ2 γ3 ⎞ ⎟ ⎟ ⎠ =α2 α3(t−1) , which always has a solution with respect to unknowns β2,β 3,γ 2,γ 3.Thus, we obtain the existence a superderivation desuch that de(e)=Δ t(e). The proof is complete.  Remark 3.1. In fact, in the proof of Theorem 3.1 we show the existence of infinitely many local superderivations on the model filiform Lie superalgebra Ln,m (n≥3) which are not superderivations. Note also, that analogously it can be found infinitely many odd local superderivations which are not odd superderivations. Along the next sections, we consider non-nilpotent solvable Lie and Leibniz superalgebras with different types of nilradical, starting with abelian nilradical. 76 Page 8 of 22 L. M. Camacho et al. MJOM 4. Local Superderivations of Maximal-Dimensional Solvable Lie Superalgebras with Model Filiform and Model Nilpotent Nilradical In this section first, we consider the maximal-dimensional solvable Lie superalgebra with model filiform nilradical [12]. This superalgebra is unique for each pair of dimensions (n, m) and can be expressed by the only non-null bracket products that follow: SLn,m : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [x1,x i]=−[xi,x 1]=xi+1,2≤i≤n−1,[t1,y j]=−[yj,t 1]=jyj,1≤j≤m, [x1,y j]=−[yj,x 1]=yj+1,1≤j≤m−1,[t2,x i]=−[xi,t 2]=xi,2≤i≤n, [t1,x i]=−[xi,t 1]=ixi,1≤i≤n, [t3,y j]=−[yj,t 3]=yj,1≤j≤m, with {x1,...,x n,t 1,t 2,t 3}a basis of (SLn,m)¯ 0and {y1,...,y m}a basis of (SLn,m)¯ 1. Its superalgebra of superderivations was obtained in [12]. Next, we prove the following result. Theorem 4.1. On the maximal-dimensional solvable Lie superalgebra with model filiform nilradical, every local superderivation is a superderivation. Proof. First, we are going to express in a more suitable way for our purpose the solvable Lie superalgebra SLn,m. Thus, after applying an elementary basis transformation one can express the table of multiplications SLn,m in a new basis {e1,...,e n,e n+1,...,e n+m}as follows: SLn,m : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [e1,e i]=−[ei,e 1]=ei+1,2≤i≤n−1, [e1,e n+j]=−[en+j,e 1]=en+j+1,1≤j≤m−1, [t1,e i]=−[ei,t 1]=iei,1≤i≤n+m, [t2,e i]=−[ei,t 2]=ei,2≤i≤n, [t3,e n+j]=−[en+j,t 3]=en+j,1≤j≤m. In [12] the authors proved every superderivation is exactly the adjoint operator of an element of SLn,m. Let us fix an arbitrary element z=γ1t1+ γ2t2+γ3t3+n+m p=1 βpepof the superalgebra SLn,m,then for its adjoint operator adzwe obtain adz(t1)=− n+m  p=1 pβpep,ad z(t2)=− n  p=2 βpep,ad z(t3)=− n+m  p=n+1 βpep, adz(e1)=γ1e1− n−1  p=2 βpep+1 − n+m−1  p=n+1 βpep+1, adz(ei)=(iγ1+γ2)ei+β1ei+1,2≤i≤n, adz(en+j)=((n+j)γ1+γ3)en+j+β1en+j+1,1≤j≤m. MJOM Local Superderivations on Solvable Page 9 of 22 76 Later on, when needed, from this general expression we will distinguish between even and odd superderivations. Let us consider now an arbitrary local superderivation Δ : SLn,m −→ SLn,m. Since the value of a local superderivation on any vector coincides with the value on this vector of a superderivation, in particular on the basis vectors we have the following expression: Δ(t1)=− n+m  p=1 pβ1,pep,Δ(t2)=− n  p=2 β2,pep,Δ(t3)=− n+m  p=n+1 β3,pep, Δ(e1)=γ1,1e1− n−1  p=2 δ1,pep+1 − n+m−1  p=n+1 δ1,pep+1, Δ(ei)=(iγi,1+γi,2)ei+δi,1ei+1,2≤i≤n, Δ(en+j)=((n+j)γn+j,1+γn+j,3)en+j+δn+j,1en+j+1,1≤j≤m. The goal now is to show that the expressions for adzand Δ coincide. Firstly, we will show this coincidence on the generators of the basis vectors, i.e. t1,t 2,t 3,e 1,e 2and en+1. Let us consider Δ(st2−t1) with a fixed sverifying 2≤s≤n. Thus as Δ is linear we obtain Δ(st2−t1)=sΔ(t2)−Δ(t1)= n+m  p=1 pβ1,pep−s n  p=2 β2,pep =β1,1e1+ n  p=2 (pβ1,p −sβ2,p)ep+ n+m  p=n+1 pβ1,pep, on the other hand and by definition the above coincides with the value of a superderivation, named deon the vector st2−t1,thus Δ(st2−t1)=de(st2−t1)=βe 1e1+ n  p=2 (pβe p−sβe p)ep+ n+m  p=n+1 pβe pep, comparing the coefficients of eson both expressions it follows that s(β1,s − β2,s)=0andβ1,s =β2,s. Repeating this process for all the possible swith 2≤s≤n, leads to β1,s =β2,s,2≤s≤n. Let us consider now Δ(st3−t1) with a fixed sverifying n+1 ≤s≤n+m. Since Δ is linear map, we obtain sΔ(t3)−Δ(t1)= n+m  p=1 pβ1,pep−s n+m  p=n+1 β3,pep=β1,1e1+ n  p=2 pβ1,pep + n+m  p=n+1 (pβ1,pep−sβ3,p)ep, on the other hand and by definition the above coincides with the value of a superderivation, named deon the vector st3−t1,thus Δ(st3−t1)=de(st3−t1)=βe 1e1+ n  p=2 pβe pep+ n+m  p=n+1 (pβe p−sβe p)ep, 76 Page 16 of 22 L. M. Camacho et al. MJOM Analogously to proof of Theorem 3.1 for a fixed element e=α0z+ α1x1+α2x2+β1y1+···+βmymwe have Δt(e)=α1x1+α2tx2. Now our goal is to prove the existence of a derivation desuch that Δt(e)=de(e). Let de(z)=de(yj) = 0 be with 1 ≤i≤mand de(x1)=a1x1+a2x2,d e(x2)=a3x2 where a1,a 2,a 3are unknowns. From the constraint Δt(e)=de(e)wehave that a1= 1 and the following equation α1a2+α2(a3−t)=0.This equation always has solution with respect to unknowns a2,a 3.Replacing one of theses solutions of dewegetthatΔ tis a local superderivation.  Remark 5.1. In fact, in the proof of Theorem 5.1 we show the existence of infinitely many local superderivations on the (m+ 3)-dimensional solvable Lie superalgebra Lm+3 which are not superderivations. Note also, that analogously it can be found infinitely many odd local superderivations which are not odd superderivations. 6. Local Superderivations of Maximal-Dimensional Solvable Leibniz Superalgebras with Model Filiform and Model Nilpotent Non-Lie Nilradical We consider the maximal-dimensional solvable Leibniz superalgebra with filiform nilradical [12]. This superalgebra is unique for each pair of dimensions (n, m) and can be expressed by: SLPn,m : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [xi,x 1]=xi+1,2≤i≤n−1; [yj,x 1]=yj+1,1≤j≤m−1; [t1,x 1]=−x1, [x1,t 1]=x1, [xi,t 1]=(i−2)xi,3≤i≤n; [yj,t 1]=(j−1)yj,2≤j≤m; [xi,t 2]=xi,2≤i≤n; [yj,t 3]=yj,1≤j≤m; with {x1,x 2,...,x n,t 1,t 2,t 3}a basis of (SLPn,m)0and {y1,y 2,...,y m}a basis of (SLPn,m)1.Its superalgebra of superderivations was obtained in [12]. Next, we prove the following result. Theorem 6.1. On the maximal-dimensional solvable Leibniz superalgebra with model filiform nilradical, every local superderivation is a superderivation. MJOM Local Superderivations on Solvable Page 17 of 22 76 Proof. We rewrite the table of multiplications of the superalgebra SLPn,m as follows: SLPn,m : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ei,e 1]=ei+1,2≤i≤n−1; [en+j,e 1]=en+j+1,1≤j≤m−1; [t1,e 1]=−e1, [e1,t 1]=e1, [ei,t 1]=(i−2)ei,3≤i≤n; [en+j,t 1]=(j−1)en+j,2≤j≤m; [ei,t 2]=ei,2≤i≤n; [en+j,t 3]=en+j,1≤j≤m; The superderivations of SLPn,m are inner [12]. Let us fix an arbitrary element z=γ1t1+γ2t2+γ3t3+ n+m  p=1 βpep of SLPn,m. Then for its right operator Rzwe obtain Rz(t1)=−β1e1Rz(t2)=Rz(t3)=0, Rz(e1)=γ1e1 Rz(ei)=((i−2)γ1+γ2)ei+β1ei+1,2≤i≤n−1, Rz(en)=((n−2)γ1+γ2)en, Rz(en+j)=((j−1)γ1+γ3)en+j+β1en+j+1,1≤j≤m−1, Rz(en+m)=((m−1)γ1+γ3)en+m. In [12], we prove that all superderivations are even. Let us consider an arbitrary local superderivation Δ : SLPn,m −→ SLPn,m.Similar to Lie superalgebra we have that: Δ(t1)=−β1,1e1Δ(t2)=Δ(t3)=0, Δ(e1)=γ1,1e1 Δ(ei)=((i−2)γi,1+γi,2)ei+βi,1ei+1,2≤i≤n−1, Δ(en)=((n−2)γn,1+γn,2)en, Δ(en+j)=((j−1)γn+j,1+γn+j,3)en+j+βn+j,1en+j+1,1≤j≤m−1, Δ(en+m)=((m−1)γn+m,1+γn+m,3)en+m. The goal now is to show that the expressions for Rzand Δ coincide. Firstly, we will show this coincidence on the generators of the basis vectors, i.e. t1,t 2,t 3,e 1,e 2and en+1. Let us consider Δ(t1+e2). Thus as Δ is linear we obtain Δ(t1+e2)=Δ(t1)+Δ(e2)=−β1,1e1+γ2,1e2+β2,1e3, on the other hand and by definition the above coincides with the value of a superderivation, named Reon the vector t1+e2,thus Δ(t1+e2)=Re(t1+e2)=−βe 1e1+γe 2e2+βe 1e3, comparing the coefficients of e1and e3leads to β2,1=β1,1. Let us consider Δ(t1+ej)=−β1,1e1+((j−2)γj,1+γj,2)ej+βj,1ej+1 for 3 ≤j≤n−1 76 Page 18 of 22 L. M. Camacho et al. MJOM and on the other hand and by the definition of local derivation we get that Re(t1+ej)=−βe 1e1+((j−2)γe 1+γe 2)ej+βe 1ej+1.Comparing the coefficients of e1and ej+1 we obtain βj,1=β1,1for 3 ≤j≤n−1. We consider now Δ(t1+en+1) and similar as the above we have that βn+1,1=β1,1. Analogously, if we take Δ(t1+en+j) with 2 ≤j≤m−1we have that βn+j,1=β1,1with 2 ≤j≤m−1. Consequently, there is no loss of generality in supposing Δ(t1)=−β1e1,Δ(t2)=Δ(t3)=0, Δ(e1)=γ1e1Δ(e2)=γ2e2+β1e3, Δ(ei)=((i−2)γi,1+γi,2)ei+β1ei+1,3≤i≤n−1,Δ(en)=((n−2)γn,1+γn,2)en, Δ(en+1)=γ3en+1 +β1en+2, Δ(en+j)=((j−1)γn+j,1+γn+j,3)en+j+β1en+j+1,2≤j≤m−1, Δ(en+m)=((m−1)γn+m,1+γn+m,3)en+m. Let us consider Δ(t1+e1+e2+e3)=(γ1−β1)e1+γ2e2+(β1+γ3,1+ γ3,2)e3+β1e4on the other hand and by definition the above coincides with the value of a superderivation, named Reon the vector t1+e1+e2+e3,thus Δ(t1+e1+e2+e3)=Re(t1+e1+e2+e3)=(γe 1−βe 1)e1+γe 2e2 +(βe 1+γe 1+γe 2)e3+βe 1e4. On account of the coefficients of e1,...,e 4we have (1) γ1−β1=γe 1−βe 1, (2) γ2=γe 2, (3) γ3,1+γ3,2+β1=γe 1+γe 2+βe 1, (4) β1=βe 1. From (1) + (2) + (4) we have that γ1+γ2=γe 1+γe 2and from (3) −(4) we get γ3,1+γ3,2=γe 1+γe 2. Then, γ3,1+γ3,2=γ1+γ2. Now, we study Δ(t1+e1+e2+ej)for4≤j≤nand we obtain the following equations: (1) γ1−β1=γe 1−βe 1, (2) γ2=γe 2, (3) β1=βe 1, (4) (j−2)γj,1+γj,2=(j−2)γe 1+γe 2, From (j−2)×(1)+(2)+(j−2)×(3) we get (j−2)γe 1+γe 2=(j−2)γ1+γ2 and from (4) we leads to (j−2)γj,1+γj,2=(j−2)γ1+γ2. We now consider the vectors t1+e1+en+1 +en+jfor a fixed jverifying 2≤j≤m. For j=2,we have that Δ(t1)+Δ(e1)+Δ(en+1)+Δ(en+2)=(γ1−β1)e1+γ3en+1 + (γn+2,1+γn+2,3+β1)en+2 +β1en+3 on the other hand, we get Re(t1)+Re(e1)+Re(en+1)+Re(en+2)=(γe 1−βe 1)e1+γe 3en+1 +(γe 1+γe 3+βe 1)en+2 +βe 1en+3, MJOM Local Superderivations on Solvable Page 19 of 22 76 which leads to (1) γ1−β1=γe 1−βe 1, (2) γ3=γe 3, (3) γn+2,1+γn+2,2+β1=γe 1+γe 2+βe 1, (4) β1=βe 1. From (1) + (2) + (3) we obtain γ1+γ3=γe 1+γe 3and from (3) −(4) we get γn+2,1+γn+2,3=γ1+γ3. We repeat the calculations for the vectors t1+e1+en+1 +en+jwith 3≤j≤mand we derive the following equations: (1) γ1−β1=γe 1−βe 1, (2) γ3=γe 3, (3) β1=βe 1, (4) (j−1)γn+j,1+γn+j,3=(j−1)γe 1+γe 3, From (j−1)×(1)+(2)+(j−1)×(3) we get (j−1)γe 1+γe 3=(j−1)γ1+γ3 and from (4) we derive (j−1)γn+j,1+γn+j,3=(j−1)γ1+γ3with 3 ≤j≤m which completes the proof of the theorem.  Let us consider now the complex maximal-dimensional solvable Leibniz superalgebra with model nilpotent non-Lie nilradical [12]. We denote this superalgebra by SNP(n1,··· ,n k,1|m1,...,m p) given by: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [xj,x 1]=xj+1,2≤j≤n1; [xn1+···+nj+i,x 1]=xn1+···+nj+i+1,1≤j≤k−1,2≤i≤nj+1; [yj,x 1]=yj+1,1≤j≤m1−1; [ym1+···+mj+i,x 1]=ym1+···+mj+i+1,1≤j≤p−1,1≤i≤mj+1 −1; [t1,x 1]=−x1, [x1,t 1]=x1, [xi,t 1]=(i−2)xi,3≤i≤n1+1; [xn1+···+nj+i,t 1]=(i−2)xn1+···+nj+i,1≤j≤k−1,3≤i≤nj+1 +1; [yj,t 1]=(i−1)yj,2≤j≤m1; [ym1+···+mj+i,t 1]=(i−1)ym1+···+mj+i,1≤j≤p−1,2≤i≤mj+1, [xi,t 2]=xi,2≤i≤n1+1; [xn1+···+nj+i,t j+2]=xn1+···+nj+i,1≤j≤k−1,2≤i≤nj+1 +1; [yj,t  1]=yj,1≤j≤m1; [ym1+···+mj+i,t  j+1]=ym1+···+mj+i,1≤j≤p−1,1≤i≤mj+1; with {x1,...,x n1+···nk+1,t 1,...,t k+1,t  1,...,t  p}even basis vectors and {y1, ..., y m1+···+mp}odd basis vectors. In [12] it is proved that all superderivation are inner. Analogously to Theorem 6.1, we have the following result. Theorem 6.2. On the maximal-dimensional solvable Leibniz superalgebra with model nilpotent non-Lie nilradical, SNP(n1,...,n k,1|m1,...,m p)every local superderivation is a superderivation. 76 Page 20 of 22 L. M. Camacho et al. MJOM Proof. The proof is carrying out by arguments that used in the previous theorem. We omit the proof of this theorem because the computations are rather cumbersome and do not contain any new idea.  Funding Funding for open access publishing: Universidad de Sevilla/CBUA. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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MJOM Rosa Mar´ıa Navarro Dpto. de Matem´aticas Universidad de Extremadura C´aceres Spain e-mail: [email protected] Bakhrom Omirov National University of Uzbekistan University street 4 100174 Tashkent Uzbekistan e-mail: [email protected] and Akfa University Milliy Bog street 264 111221 Tashkent Uzbekistan Received: March 25, 2021. Revised: August 4, 2022. Accepted: January 3, 2023.