Local Superderivations on Solvable Lie and Leibniz Superalgebras

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.


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 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][15][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, L 2 := [L, L] can not be nilpotent, see [25]. Nevertheless, in [11] the authors proved that under the condition of being L 2 nilpotent, 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. 3 and 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).

Preliminary for Lie Superalgebras
A vector space V is said to be Z 2 -graded if it admits a decomposition into a direct sum, V = V0 ⊕ V1, where0,1 ∈ Z 2 . An element X ∈ V is called homogeneous of degree |x| if it is an element of V |x| , |x| ∈ Z 2 .
Thus, g0 is an ordinary Lie algebra, and g1 is a module over g0; the Lie superalgebra structure also contains the symmetric pairing S 2 g1 −→ g0.
Let us note that both the descending central sequence and the derived sequence of a Lie superalgebra g = g0 ⊕ g1 are defined in the same way as for Lie algebras: C 0 (g) := g, C k+1 (g) := [C k (g), g] and D 0 (g) := g, D k+1 (g) := [D k (g), D k (g)] respectively, for all k ≥ 0. Thus, if C k (g) = {0} (resp. D k (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 L is nilpotent if and only if ad L x is nilpotent for every homogeneous element x of L. Additionally, a Lie superalgebra L is solvable if and only if its even part L 0 (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 C k (g0) and C k (g1) which will play an important role in our study. They are defined as follows: Let us recall now, the definition of superderivations of superalgebras [21]. A superderivation of degree s of a superalgebra L, s ∈ Z 2 , is an endomorphism D ∈ End s L with the property denote Der s (L) ⊂ End s L the space of all superderivations of degree s. Then Der(L) = Der 0 (L) ⊕ Der 1 (L) is the Lie superalgebra of superderivations of L, with Der 0 (L) composed by even superderivations and Der 1 (L) by odd ones.
On the other hand, recall also that a homogeneous linear mapping Δ : L −→ L of degree s is called a local homogeneous superderivation of degree s if for any element x ∈ L, there exists a superderivation D x : L −→ L (depending on x) such that Δ(x) = D x (x). Then, the set of all local superderivations can be expressed with LocDer 0 (L) (resp. LocDer 1 (L)) composed by even (resp. odd) local superderivations. For more details it can be consulted [14].

Preliminaries for Leibniz Superalgebras
Let us note that many results and definitions of the above sub-section can be extended for Leibniz superalgebras. Note that if a Leibniz superalgebra L satisfies 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 Z 2 -graduation. Let us now denote by R x the right multiplication operator, i.e., R x : L → L given as R x (y) := [y, x] for y ∈ L, then the super Leibniz identity can be expressed as R [x,y] If we denote by R(L) the set of all right multiplication operators, then R(L) with respect to the following multiplication for R a ∈ R(L)ī, R b ∈ R(L)j, forms a Lie superalgebra. Note that R a is a derivation. In fact, the condition for being a derivation of a Leibniz superalgebra (for more details see [23] Since the degree of R z as homomorphism between Z 2 -graded vector spaces is the same as the degree of the homogeneous element z, that is |R z | = |z|, then the condition for R z to be a derivation is exactly Let V = V0 ⊕ V1 be the underlying vector space of L, L = L0 ⊕ L1 ∈ Leib n,m , being Leib n,m the variety of Leibniz superalgebras, and let G(V ) be the group of the invertible linear mappings of the form f = f0 + f1, such that f0 ∈ GL(n, C) and f1 ∈ GL(m, C) (then G(V ) = GL(n, C)⊕GL(m, C)). The action of G(V ) on Leib n,m induces an action on the Leibniz superalgebras variety: two laws λ 1 , λ 2 are isomorphic if there exists a linear mapping f = f0 + f1 ∈ G(V ), such that , fj(y))), for any x ∈ Vī, y ∈ Vj. 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. It is easy to see that Ann(L) is a two-sided ideal of L and [x, x] ∈ Ann(L) for any x ∈ L0. This notion is compatible with the right annihilator in Leibniz algebras. If we consider the ideal Let L = L0 ⊕ L1 be a nilpotent Leibniz superalgebra with dim L0 = n and dim L1 = m. From Equation (2.1) we have that R(L) is a Lie superalgebra, and in particular R(L0) is a Lie algebra. As L1 has L0-module structure we can consider R(L0) as a subset of GL(V1) , where V1 is the underlying vector space of L1. So, we have a Lie algebra formed by nilpotent endomorphisms of V1. Applying Engel's theorem we have the existence of a sequence of subspaces of V1, Then, it can be defined the descending sequences C k (L0) and C k (L1) and the super-nilindex in the same way as for Lie superalgebras. That is,

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 Z 2 0 (L n,m , L n,m ). We consider then, the model filiform Lie superalgebra L n,m , that is, the simplest filiform Lie superalgebra which is defined by the only non-zero bracket products that follow Applying induction and the even superderivation condition for the prod- Similarly, from the products [x 1 , y j ] we get Let now D be an odd superderivation of L n,m . Then we have According to the odd superderivation condition on the products of L n,m and induction, similar to even superderivation case, we obtain Considering superderivation property for the products [x 2 , y 1 ] and [x 1 , Proof. Consider the homogeneous linear mappings Δ t : L n,m −→ L n,m , t = 2 of degree 0 defined on the basis vectors of L n,m by Clearly, Δ t is not an even superderivation (because its matrix does not fit with the general matrix of even superderivations).
Consider the following superderivations: • d 1 is the resultant even superderivation after replacing a 1 by 1 and all the rest of parameters by 0 on the general matrix of Der 0 (L n,m ), • d 2 is the resultant even superderivation after replacing b 2 by 1 and all the rest of parameters by 0 on the general matrix of Der 0 (L n,m ), Clearly, For an arbitrary element e = α 1 Define superderivation d e as follows d e (x i ) = d e (y j ) = 0 with 4 ≤ i ≤ n and 1 ≤ j ≤ m and where β 1 , β 2 , β 3 , γ 2 , γ 3 are some unknowns parameters. From Δ t (e) = d e (e) we derive β 1 = 1 and the following linear system of equations , which always has a solution with respect to unknowns β 2 , β 3 , γ 2 , γ 3 . Thus, we obtain the existence a superderivation d e such that d e (e) = Δ t (e). The proof is complete. Along the next sections, we consider non-nilpotent solvable Lie and Leibniz superalgebras with different types of nilradical, starting with abelian nilradical.

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: with {x 1 , . . . , x n , t 1 , t 2 , t 3 } a basis of (SL n,m )0 and {y 1 , . . . , y m } a basis of (SL n,m )1. Its superalgebra of superderivations was obtained in [12]. Next, we prove the following result.
In [12] the authors proved every superderivation is exactly the adjoint operator of an element of SL n,m . Let us fix an arbitrary element z = γ 1 t 1 + γ 2 t 2 + γ 3 t 3 + n+m p=1 β p e p of the superalgebra SL n,m , then for its adjoint operator ad z we obtain 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 : SL n,m −→ SL n,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: The goal now is to show that the expressions for ad z and Δ coincide. Firstly, we will show this coincidence on the generators of the basis vectors, i.e. t 1 , t 2 , t 3 , e 1 , e 2 and e n+1 . Let us consider Δ(st 2 −t 1 ) with a fixed s verifying 2 ≤ s ≤ n. Thus as Δ is linear we obtain (3β e p − pβ e p + β e p−1 ) considering the coefficients of e 2 and e 3 we obtain δ 1,2 = β 2 . On account of Δ(t 1 − (i + 1)t 2 − e 1 ) inductively we get δ 1,i = β i for i verifying 2 ≤ i ≤ n − 1.
In [12] it is proved that all the superderivations are inner, then and following the spirit of the proof of the theorem for model filiform nilradical we have the next result. We omit the computations because they are rather cumbersome and do not contain any new idea.

Local Superderivations of Solvable Lie Superalgebras with Non-model Nilradical
Along this section, we use an example of solvable Lie superalgebra whose nilradical is a non-model one, in particular the nilradical is the only one Lie superalgebra of maximal nilindex K 2,m (for more details regarding K 2,m see Theorem 4.17 of [17]). We build over this solvable Lie superalgebra infinitely many local superderivations which are not superderivations. Thus, consider for any m odd positive integer m ≥ 3, the (m + 3)dimensional solvable Lie superalgebra L m+3 (named L m+3 1, 2−m 2 ,1,0,...,0 in [11]). For that Lie superalgebra there exists a basis, namely {z, x 1 , x 2 , y 1 , . . . , y m } with {z, x 1 , x 2 } even basis vectors and {y 1 , . . . , y m } odd basis vectors, in which L m+3 can be expressed by the only non-null bracket products that follow: being its nilradical: . Note that [·, ·] is the standard skew-symmetric bracket product whereas (·, ·) denotes the symmetrical ones, recall that a Lie superalgebra structure g = g0 ⊕ g1 contains in particular the symmetric pairing S 2 g1 −→ g0. Moreover, because of these symmetric products K 2,m is not a model nilpotent Lie superalgebra.
It can be easily checked that Δ is not an even superderivation because its matrix does not fit with the general matrix of even superderivations. Nevertheless, for any t the map Δ t is an even local superderivation. Indeed, we have Δ t (z) = d 0 (z), Δ t (x 1 ) = d 1 (x 1 ), Δ t (x 2 ) = td 1 (x 2 ), Δ t (y j ) = d 0 (y j ), 1 ≤ j ≤ m, where d 1 is the resultant even superderivation after replacing α 11 by 1 and all the rest of parameters by 0 on the general matrix of Der 0 (L m+3 ) and d 0 is the null superderivation. Analogously to proof of Theorem 3.1 for a fixed element e = α 0 z + α 1 x 1 + α 2 x 2 + β 1 y 1 + · · · + β m y m we have Δ t (e) = α 1 x 1 + α 2 tx 2 . Now our goal is to prove the existence of a derivation d e such that Δ t (e) = d e (e).
Let d e (z) = d e (y j ) = 0 be with 1 ≤ i ≤ m and d e (x 1 ) = a 1 x 1 + a 2 x 2 , d e (x 2 ) = a 3 x 2 where a 1 , a 2 , a 3 are unknowns. From the constraint Δ t (e) = d e (e) we have that a 1 = 1 and the following equation α 1 a 2 + α 2 (a 3 − t) = 0. This equation always has solution with respect to unknowns a 2 , a 3 . Replacing one of theses solutions of d e we get that Δ t is 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 L m+3 which are not superderivations. Note also, that analogously it can be found infinitely many odd local superderivations which are not odd superderivations.

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: SLP n,m : with {x 1 , x 2 , . . . , x n , t 1 , t 2 , t 3 } a basis of (SLP n,m ) 0 and {y 1 , y 2 , . . . , y m } a basis of (SLP n,m ) 1 . Its superalgebra of superderivations was obtained in [12]. Next, we prove the following result.