On n-Derivations and n-Homomorphisms in Perfect Lie Superalgebras

Abstract

Let $n\ge 2$ be a fixed integer. The aim of this paper is to investigate the properties of n-derivations within the framework of perfect Lie superalgebras over a commutative ring R. The main result shows that if the base ring contains ${\displaystyle \frac{1}{n-1}}$, and L is a perfect Lie superalgebra with a center equal to zero, then any n-derivation of L is necessarily a derivation. Additionally, every n-derivation of the derivation algebra $Der\left(L\right)$ is an inner derivation. Moreover, we extend the concept of n-homomorphisms to mappings between Lie superalgebras L and $L^{\prime }$ and prove that under specific assumptions, homomorphisms, anti-homomorphisms, and their combinations are all n-homomorphisms. Finally, we conclude our paper with some open problems.

1. Introduction

In the context of associative algebras [1,2,3], the concept of derivation naturally extends into several types of triple derivations, such as Jordan triple derivations and Lie triple derivations. Each type of derivation plays a role in understanding the broader mathematical structures, with Lie triple derivations being particularly significant. These derivations are not only important for associative algebras and rings but have broader applications in the study of Lie groups [4] and operator algebras [5,6,7]. The triple derivation in Lie algebras can be thought of as an extension of the more familiar concept of a derivation, and it serves as an analogy to the corresponding triple derivations in both associative and Jordan algebras. First introduced by Müller [4] as “prederivation,” this concept has gained importance in the field. A key property is that every derivation in a Lie algebra automatically qualifies as a Lie triple derivation, but the converse does not necessarily hold [8].

The relationships among homomorphisms and their variants such as anti-homomorphisms, Jordan homomorphisms, Lie homomorphisms, and Lie triple homomorphisms have long attracted interest. In particular, Bresar [9] characterized Lie triple isomorphisms on certain associative algebras. At the same time, Jacobson and Rickart’s study [10] established a theoretical structure showing that any Jordan homomorphism of a ring must necessarily act as either a standard homomorphism or an anti-homomorphism. These studies have inspired similar investigations in operator algebras [11,12], and a comparable result has been established for perfect Lie algebras [13] and the Cohomology theory of Lie groups and Lie algebras [14], broadening the applicability of these homomorphism relations.

Lie superalgebras, which are a natural extension of Lie algebras, have important applications across various fields, including both mathematics and physics. Beyond their applications, they also present intriguing mathematical properties [15,16,17]. We are motivated to generalize key results from [8,13,18] for Lie superalgebras, with a particular emphasis on triple derivations and triple homomorphisms. Our goal is to generalize results proved in [19] for triple derivations and triple homomorphisms of perfect Lie superalgebras in the setting of n-derivations and n-homomorphisms, thereby providing a broader framework for studying derivations of arbitrary order.

Recent developments in the study of higher-order algebraic structures have focused on generalized derivations and their applications in n-ary and Hom-type algebras. For instance, Mabrouk et al. [20] investigated generalized derivations and Rota–Baxter operators in n-ary Hom–Nambu superalgebras, while Zhou and Fan [21] studied generalized derivations on n-Hom–Lie superalgebras. The concept of double derivations in n-Lie algebras was explored by Bai et al. [22], and Arnlind et al. [23] provided constructions of n-Lie and n-ary Hom–Nambu–Lie algebras. These works provide a foundation for extending classical derivation theory to more general and higher-order algebraic settings.

By exploring these concepts, we seek to deepen our understanding of their properties and interactions in Lie superalgebras, building upon existing theoretical frameworks and expanding their applications. Our investigation will provide insights into the role of n-derivations and n-homomorphisms in the structure and behavior of Lie superalgebras, contributing to the broader field of algebraic research.

The notion of n-derivations has been studied in various algebraic settings, such as n-derivations for finitely generated graded Lie algebras [24] and n-derivations of Lie color algebras [25]. Our investigation builds upon this body of work, aiming to extend n-derivations, n-automorphisms, and Lie n-systems to the Lie algebra of a Lie group, thereby enhancing the understanding of their applications in Section 4.

2. Preliminaries

Let $n\ge 2$ be a fixed integer. We use L to refer to a Lie superalgebra over a commutative ring R with unity. A Lie superalgebra L is perfect if its derived subalgebra $[L,L]$ equals L. For any subset $S\subseteq L$, we denote ${\mathrm{C}}_L\left(S\right)$ the centralizer of S in L, while the center of L is represented by $\mathrm{Z}\left(L\right)$. A Lie superalgebra is called centerless if $\mathrm{Z}\left(L\right)=\left\{0\right\}$. The algebra of derivations of L is denoted by $Der\left(L\right)$ and the algebra of inner derivation of L by $\mathrm{ad}\left(L\right)$.

Now, we present some examples of centerless Lie superalgebras.

Example 1

([26]). Consider the Lie superalgebra $sl\left(2\right|1)$ over $\mathbb{C}$. A general element of $gl\left(2\right|1)$ can be written in block form as

$$X=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right),A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right),B=\left(\begin{array}{c}{b}_1\\ b_2\end{array}\right),C=\left(\begin{array}{cc}{c}_1& c_2\end{array}\right),D=\left(d\right).$$

Elements of $sl\left(2\right|1)$ satisfy the condition

$$str\left(X\right)=tr\left(A\right)-tr\left(D\right)=a_{11}+a_{22}-d=0.$$

Let $z\in sl\left(2\right|1)$ be of the same block form. Imposing the commutation relations $[z,E]=0$ with suitable elementary matrices E yields

$$A=\lambda I_2,D=\mu ,B=0,C=0.$$

Thus, any central element has the form

$$z=\left(\begin{array}{cc}\lambda I_2& 0\\ 0& \mu \end{array}\right),2\lambda -\mu =0.$$

Commutation with odd block units forces $\lambda =\mu$; hence, the supertrace condition becomes $\lambda =0$. Therefore

$$Z\left(sl\right(2\left|1\right))=\{0\}.$$

Hence, $sl\left(2\right|1)$ is centerless.

Next, we give an example of perfect Lie superalgebra with a trivial center.

Example 2

([15]). Consider the special linear Lie superalgebra $\mathfrak{sl}\left(m\right|n)$ over $\mathbb{C}$ with $m\ne n$, defined by

$$\mathfrak{sl}\left(m\right|n)=\{X\in \mathfrak{gl}\left(m\right|n)\mid str(X)=0\}.$$

This Lie superalgebra is perfect since

$$\left[\mathfrak{sl}\right(m\left|n\right),\mathfrak{sl}\left(m\right|n\left)\right]=\mathfrak{sl}\left(m\right|n).$$

Moreover, the supercenter of $\mathfrak{sl}\left(m\right|n)$ is trivial when $m\ne n$, namely,

$$Z\left(\mathfrak{sl}\right(m\left|n\right))=\{0\}.$$

Definition 1

([15]). A Lie superalgebra $L=L_{\overline{0}}\oplus L_{\overline{1}}$ is a ${\mathbf{Z}}_2$-graded algebra over a commutative ring R that contains a multiplicative identity. We say that L is a Lie superalgebra if the multiplication operation, denoted by [ , ], adheres to the following set of identities:

(i) 

For homogeneous $v_1,v_2\in L$,

$$[v_1,v_2]=-{(-1)}^{|v_1\left|\right|v_2|}[v_2,v_1]$$

(graded skew-symmetry).

(ii) 

For homogeneous $v_1,v_2,v_3\in L$,

$$[v_1,[v_2,v_3]]=[[v_1,v_2],v_3]+{(-1)}^{|v_1\left|\right|v_2|}[v_2,[v_1,v_3]]$$

(graded Jacobi identity).

Let $v_1,v_2,v_3\in hg\left(L\right)$, where $hg\left(L\right)$ represents the set of all ${\mathbf{Z}}_2$-homogeneous elements of L. Throughout this paper, whenever $\left|v\right|$ appears, we interpret v as a ${\mathbf{Z}}_2$-homogeneous element, and $\left|v\right|$ denotes the ${\mathbf{Z}}_2$-degree of v.

Definition 2

([19]). For a subset S of L, the enveloping Lie superalgebra of S is the Lie subalgebra of L generated by S. A Lie superalgebra is said to be indecomposable if it cannot be written as a direct sum of two nontrivial ideals.

Example 3

([27]). Consider the Lie superalgebra

$$L=\mathfrak{sl}\left(1\right|1)=\left\{\left(\begin{array}{cc}a& \alpha \\ \beta & -a\end{array}\right):a,\alpha ,\beta \in \mathbb{C}\right\},$$

with even and odd parts given by

$$L_{\overline{0}}=\left\{\left(\begin{array}{cc}a& 0\\ 0& -a\end{array}\right):a\in \mathbb{C}\right\},L_{\overline{1}}=\left\{\left(\begin{array}{cc}0& \alpha \\ \beta & 0\end{array}\right):\alpha ,\beta \in \mathbb{C}\right\}.$$

Define the standard homogeneous elements

$$e=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),f=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),h=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),$$

where $e,f\in L_{\overline{1}}$ and $h\in L_{\overline{0}}$.

Remark 1.

Let $S=\{e,f\}$. The smallest Lie subsuperalgebra of L containing S is the whole $\mathfrak{sl}\left(1\right|1)$. Indeed, the basic commutators are $[e,f]=h,[h,e]=2e,[h,f]=-2f.$ Thus, starting from e and f, one also obtains h, and the set $\{e,f,h\}$ spans L.

The Lie superalgebra $\mathfrak{sl}\left(1\right|1)$ is indecomposable, i.e., it cannot be expressed as a direct sum of two nontrivial ideals. Any nonzero ideal necessarily intersects the odd part $L_{\overline{1}}$; however, once it contains a nonzero odd element, the above commutator relations force it to contain $e,f$, and consequently h. Therefore, the only ideals are $\left\{0\right\}$ and L, and hence $\mathfrak{sl}\left(1\right|1)$ admits no nontrivial decomposition.

Definition 3.

An endomorphism D of an R-module L is called a triple derivation of L if for all $v_1,v_2,v_3\in L$, D satisfies the following condition:

$$\begin{array}{cc}\hfill D\left([[v_1,v_2],v_3]\right)& =[[D\left(v_1\right),v_2],v_3]+{(-1)}^{\left|D\right||v_1|}[[v_1,D\left(v_2\right)],v_3]\hfill \\ \hfill & {+(-1)}^{\left|D\right|\left(\right|v_1|+|v_2\left|\right)}[[v_1,v_2],D\left(v_3\right)].\hfill \end{array}$$

More generally, D is called an n-derivation of L if it satisfies the following identity.

$$\begin{array}{cc}\hfill D\left([\dots [[v_1,v_2],v_3],\dots ,v_n]\right)& =[\dots [[D\left(v_1\right),v_2],v_3],\dots ,v_n]\hfill \\ \hfill & +{(-1)}^{\left|D\right||v_1|}[\dots [[v_1,D\left(v_2\right)],v_3],\dots ,v_n]\hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(\right|v_1|+|v_2\left|\right)}[\dots [[v_1,v_2],D\left(v_3\right)],\dots ,v_n]\hfill \\ \hfill & +\dots \hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(\right|v_1|+|v_2|+|v_2|+\dots |v_{n-1}\left|\right)}[\dots [[v_1,v_2],v_3],\dots ,D\left(v_n\right)]\hfill \\ \hfill & \mathit{for}\mathit{all}{v}_1,v_2,v_3,\dots ,v_n\in L.\hfill \end{array}$$

Denote by $\mathrm{nDer}\left(L\right)$ the set of all n-derivations of a Lie algebra L. It is straightforward to verify that $\mathrm{nDer}\left(L\right)$ forms a Lie algebra under the usual commutator bracket of endomorphisms of the R-module. An n-derivation of a Lie algebra generalizes the usual derivation by satisfying a Leibniz rule for the n-fold Lie bracket. This concept parallels similar generalizations in associative and Jordan algebras. It was introduced independently in [4] by Müller, where it was referred to as a prederivation in the specific case of triple derivations. Müller [4] proved that if G is a Lie group equipped with a bi-invariant semi-Riemannian metric and $\mathfrak{g}$ its Lie algebra, then the Lie algebra of the group of isometries of G fixing the identity element is a subalgebra of $\mathrm{nDer}\left(\mathfrak{g}\right)$ when $n=3$. Therefore, the study of the Lie algebra of n-derivations is of interest not only from an algebraic perspective but also due to its relevance in the geometric theory of Lie groups.

Remark 2.

Let $L=L_{\overline{0}}\oplus L_{\overline{1}}$ be a Lie superalgebra over a commutative ring R as in Definition 1. For a subset $S\subseteq L$, the enveloping Lie superalgebra of S (Definition 2) is a Lie subalgebra of L, and the property of being indecomposable is determined by the ideal structure of L. On the other hand, an n-derivation $D\in En{d}_R\left(L\right)$ (Definition 3) is an endomorphism preserving the n-fold supercommutator structure of L. Hence, the study of n-derivations is intrinsically related to both the graded Lie structure of L and the way L decomposes into its ideals.

Definition 4.

Let $L\mathrm{and}{L}^{\prime }$ be two Lie superalgebras over R. An even R-linear mapping $f:L\to L^{\prime }$ is called the following:

(i) 

A homomorphism if it satisfies

$$f\left([v_1,v_2]\right)=[f\left(v_1\right),f\left(v_2\right)]\mathit{for}\mathit{all}{v}_1,v_2\in L.$$

(ii) 

An anti-homomorphism if it meets the condition

$$f\left([v_1,v_2]\right)={(-1)}^{|v_1\left|\right|v_2|}[f\left(v_2\right),f\left(v_1\right)]\mathit{for}\mathit{all}{v}_1,v_2\in hg\left(L\right).$$

(iii) 

A triple homomorphism if it meets the condition

$$f\left([v_1,[v_2,v_3]]\right)=[f\left(v_1\right),[f\left(v_2\right),f\left(v_3\right)]]\mathit{for}\mathit{all}{v}_1,v_2,v_3\in L.$$

(iv) 

A n-homomorphism if it meets the condition

$$\begin{array}{cc}\hfill f\left([v_1,v_2,v_3,\dots ,v_n]\right)& =[f\left(v_1\right),f\left(v_2\right),f\left(v_3\right),\dots ,f\left(v_n\right)]\hfill \\ \hfill & =[\dots [[f\left(v_1\right),f\left(v_2\right)],f\left(v_3\right)],\dots ,f\left(v_n\right)]\hfill \\ \hfill & \mathit{for}\mathit{all}{v}_1,v_2,v_3,\dots ,v_n\in L.\hfill \end{array}$$

Definition 5

([10]). Let L and $L^{\prime }$ be Lie superalgebras. A map $g:L\to L^{\prime }$ is said to be the direct sum of maps $g_1,g_2:L\to L^{\prime }$ if $g=g_1+g_2$, and there exist ideals $I_1$ and $I_2$ of the enveloping Lie superalgebra of $g\left(L\right)$ such that $I_1\cap I_2=\left\{0\right\}$ with $g_1\left(L\right)\subseteq I_1$ and $g_2\left(L\right)\subseteq I_2.$

Proposition 1.

If L is perfect, then $\mathrm{ad}\left(L\right)$ forms an ideal of the Lie superalgebra $nDer\left(L\right)$.

Proof. 

Let $D\in nDer\left(L\right),vs.\in hg\left(L\right)$. Then, for any $w\in L$, we have

$$\begin{array}{cc}\hfill [D,\mathrm{ad}v]\left(w\right)=& D\mathrm{ad}v\left(w\right)-{(-1)}^{\left|D\right|\left|v\right|}\mathrm{ad}v\left(D\left(w\right)\right)\hfill \\ \hfill & =D[v,w]-{(-1)}^{\left|D\right|\left|v\right|}[v,D\left(w\right)]\hfill \\ \hfill & =D\left([\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\right)\hfill \\ \hfill & -{(-1)}^{\left|D\right|\left|v\right|}[\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],D\left(w\right)]\hfill \\ \hfill & =\sum _{i\in I}D\left([[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\right)\hfill \\ \hfill & -\sum _{i\in I}{(-1)}^{\left|D\right|\left|v\right|}[[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],D\left(w\right)]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\sum _{i\in I}[[\dots [[D\left(v_{1i}\right),v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right||v_{1i}|}[[\dots [[v_{1i},D\left(v_{2i}\right)],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[[\dots [[v_{1i},v_{2i}],D\left(v_{3i}\right)],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\dots -\sum _{i\in I}{(-1)}^{\left|D\right|\left|v\right|}[[\dots [[v_{1i},v_{2i}],D\left(v_{3i}\right)],\dots ,v_{(n-1)i}],D\left(w\right)].\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill [D,\mathrm{ad}v]\left(w\right)& =\mathrm{ad}(\sum _{i\in I}[\dots [[D\left(v_{1i}\right),v_{2i}],v_{3i}],\dots ,v_{(n-1)i}]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right||v_{1i}|}[\dots [[v_{1i},D\left(v_{2i}\right)],v_{3i}],\dots ,v_{(n-1)i}]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[\dots [[v_{1i},v_{2i}],D\left(v_{3i}\right)],\dots ,v_{(n-1)i}]+\dots \hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-2)i}\left|\right)}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,D\left(v_{(n-1)i}\right)])\left(w\right).\hfill \end{array}$$

Since w is arbitrary, it follows that $[D,\mathrm{ad}v]$ is an inner derivation. Hence, $\mathrm{ad}\left(L\right)$ is an ideal of $nDer\left(L\right)$. □

3. The Proof of the Main Results

We state and prove our first main result of this paper.

Theorem 1.

Let $n\ge 2$ be a fixed integer and let L be a Lie superalgebra over a commutative ring R. If ${\displaystyle \frac{1}{n-1}}\in R$, and L is perfect with a trivial center, then the following hold:

(i) 

$nDer\left(L\right)=Der\left(L\right)$;

(ii) 

$nDer\left(Der\right(L\left)\right)=\mathrm{ad}\left(Der\right(L\left)\right)$.

We begin the proof of our main result through the following lemmas.

Lemma 1

([17]). For any Lie superalgebra L, if $v\in L$ and $D\in Der\left(L\right)$, then $[D,\mathrm{ad}v]=\mathrm{ad}\left(D\right(v\left)\right)$.

Lemma 2.

For any Lie superalgebra L, the set $nDer\left(L\right)$ is invariant with the standard Lie bracket operation.

Proof. 

Let $D_1,D_2\in nDer\left(L\right),v_1,v_2,v_2,\dots ,v_{n-1}\in hg\left(L\right),v_n\in L$. By the definition of n-derivation, we have

$$\begin{array}{cc}\hfill & D_1{D}_2\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right)\hfill \\ \hfill & =D_1([[\dots [[D_2\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_2\left|\right|v_1|}[[\dots [[v_1,D_2\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_2\left|\right(|v_1|+|v_2\left|\right)}[[\dots [[v_1,v_2],D_2\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +\dots +{(-1)}^{|D_2|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}[[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],D_2\left(v_n\right)])\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =[[\dots [[D_1{D}_2\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right(|D_2|+|v_1\left|\right)}[[\dots [[D_2\left(v_1\right),D_1\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right(|D_2|+|v_1|+|v_2\left|\right)}[[\dots [[D_2\left(v_1\right),v_2],D_1\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +\dots +{(-1)}^{|D_1|\left(|D_2|+|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}[[\dots [[D_2\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],D_1\left(v_n\right)]\hfill \\ \hfill & +{(-1)}^{|D_2\left|\right|v_1|}[[\dots [[D_1\left(v_1\right),D_2\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_2\left|\right|v_1|}{(-1)}^{|D_1\left|\right|v_1|}[[\dots [[v_1,D_1{D}_2\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_2\left|\right|v_1|}{(-1)}^{|D_1\left|\right(|D_2|+|v_1|+|v_2\left|\right)}[[\dots [[v_1,D_2\left(v_2\right)],D_1\left(v_3\right)],\dots ,v_{n-1}],v_n]+\dots \hfill \\ \hfill & +{(-1)}^{|D_2\left|\right|v_1|}{(-1)}^{|D_1|\left(|D_2|+|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}[[\dots [[v_1,D_2\left(v_2\right)],v_3],\dots ,v_{n-1}],D_1\left(v_n\right)]\hfill \\ \hfill & +{(-1)}^{|D_2\left|\right(|v_1|+|v_2\left|\right)}[[\dots [[D_1\left(v_1\right),v_2],D_2\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_2\left|\right(|v_1|+|v_2\left|\right)}{(-1)}^{|D_1\left|\right(|v_1\left|\right)}[[\dots [[v_1,D_1\left(v_2\right)],D_2\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_2\left|\right(|v_1|+|v_2\left|\right)}{(-1)}^{|D_1\left|\right(|v_1|+|v_2\left|\right)}[[\dots [[v_1,v_2],D_1{D}_2\left(v_3\right)],\dots ,v_{n-1}],v_n]+\dots \hfill \\ \hfill & +{(-1)}^{|D_2\left|\right(|v_1|+|v_2\left|\right)}{(-1)}^{|D_1|\left(|D_2|+|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}\hfill \\ \hfill & [[\dots [[v_1,v_2],D_2\left(v_3\right)],\dots ,v_{n-1}],D_1\left(v_n\right)]\hfill \\ \hfill & +\dots +{(-1)}^{|D_2|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}[[\dots [[D_1\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],D_2\left(v_n\right)]\hfill \\ \hfill & +{(-1)}^{|D_2|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}{(-1)}^{|D_1\left|\right|v_1|}[[\dots [[v_1,D_1\left(v_2\right)],v_3],\dots ,v_{n-1}],D_2\left(v_n\right)]\hfill \\ \hfill & +{(-1)}^{|D_2|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}{(-1)}^{|D_1\left|\right(|v_1|+|v_2\left|\right)}[[\dots [[v_1,v_2],D_1\left(v_3\right)],\dots ,v_{n-1}],D_2\left(v_n\right)]\hfill \\ \hfill & +\dots +{(-1)}^{|D_2|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}{(-1)}^{|D_1|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}\hfill \\ \hfill & [[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],D_1{D}_2\left(v_n\right)].\hfill \end{array}$$

Also, we have

$$\begin{array}{cc}\hfill & D_2{D}_1\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right)\hfill \\ \hfill & =D_2([[\dots [[D_1\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right|v_1|}[[\dots [[v_1,D_1\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right(|v_1|+|v_2\left|\right)}[[\dots [[v_1,v_2],D_1\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +\dots +{(-1)}^{|D_1|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}[[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],D_1\left(v_n\right)])\hfill \\ \hfill & =[[\dots [[D_2{D}_1\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],v_n]+{(-1)}^{|D_2\left|\right(|D_1|+|v_1\left|\right)}\hfill \\ \hfill & [[\dots [[D_1\left(v_1\right),D_2\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_2\left|\right(|D_1|+|v_1|+|v_2\left|\right)}[[\dots [[D_1\left(v_1\right),v_2],D_2\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +\dots +{(-1)}^{|D_2|\left(|D_1|+|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}[[\dots [[D_1\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],D_2\left(v_n\right)]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right|v_1|}[[\dots [[D_2\left(v_1\right),D_1\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right|v_1|}{(-1)}^{|D_2\left|\right|v_1|}[[\dots [[v_1,D_2{D}_1\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right|v_1|}{(-1)}^{|D_2\left|\right(|D_1|+|v_1|+|v_2\left|\right)}[[\dots [[v_1,D_2\left(v_2\right)],D_2\left(v_3\right)],\dots ,v_{n-1}],v_n]+\dots \hfill \\ \hfill & +{(-1)}^{|D_1\left|\right|v_1|}{(-1)}^{|D_2|\left(|D_1|+|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}[[\dots [[v_1,D_1\left(v_2\right)],v_3],\dots ,v_{n-1}],D_2\left(v_n\right)]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right(|v_1|+|v_2\left|\right)}[[\dots [[D_2\left(v_1\right),v_2],D_1\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right(|v_1|+|v_2\left|\right)}{(-1)}^{|D_2\left|\right(|v_1\left|\right)}[[\dots [[v_1,D_2\left(v_2\right)],D_1\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|D_1\left|\right(|v_1|+|v_2\left|\right)}{(-1)}^{|D_2\left|\right(|v_1|+|v_2\left|\right)}[[\dots [[v_1,v_2],D_2{D}_1\left(v_3\right)],\dots ,v_{n-1}],v_n]\dots \hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{(-1)}^{|D_1\left(\right|v_1|+|v_2\left|\right)}{(-1)}^{|D_2|\left(|D_1|+|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}[[\dots [[v_1,v_2],D_1\left(v_3\right)],\dots ,v_{n-1}],D_2\left(v_n\right)]\hfill \\ \hfill & +\dots +{(-1)}^{|D_1|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}[[\dots [[D_2\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],D_1\left(v_n\right)]\hfill \\ \hfill & +{(-1)}^{|D_1|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}{(-1)}^{|D_2\left|\right|v_1|}[[\dots [[v_1,D_2\left(v_2\right)],v_3],\dots ,v_{n-1}],D_1\left(v_n\right)]\hfill \\ \hfill & +{(-1)}^{|D_1|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}{(-1)}^{|D_2\left|\right(|v_1|+|v_2\left|\right)}[[\dots [[v_1,v_2],D_2\left(v_3\right)],\dots ,v_{n-1}],D_1\left(v_n\right)]\dots \hfill \\ \hfill & +{(-1)}^{|D_1|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}{(-1)}^{|D_2|\left(|v_1|+|v_2|+|v_2|+\dots |v_{n-1}|\right)}\hfill \\ \hfill & [[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],D_2{D}_1\left(v_n\right)].\hfill \end{array}$$

By simple calculation, we obtain

$$\begin{array}{cc}\hfill & [D_1,D_2]\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right)\hfill \\ \hfill & =D_1{D}_2\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right)\hfill \\ \hfill & -{(-1)}^{|D_1\left|\right|D_2|}{D}_2{D}_1\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right)\hfill \\ \hfill & =[[\dots [[[D_1,D_2]\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{|v_1\left(\right|D_2|+|D_1\left|\right)}[[\dots [[v_1,[D_1,D_2]\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{\left(\right|v_1|+|v_1\left|\right)\left(\right|D_2|+|D_1\left|\right)}[[\dots [[v_1,v_2],[D_1,D_2]\left(v_3\right)],\dots ,v_{n-1}],v_n]+\dots \hfill \\ & +{(-1)}^{\left(\right|D_1|+|D_2\left|\right)\left(\right|v_1|+|v_2|+|v_2|+\dots |v_{n-1}\left|\right)}[[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],[D_1,D_2]\left(v_n\right)].\hfill \end{array}$$

Hence, $[D_1,D_2]\in nDer\left(L\right)$, completing the proof of the lemma. □

It is evident that both $\mathrm{ad}\left(L\right)$ and $Der\left(L\right)$ are subalgebras of $nDer\left(L\right)$. Since L is perfect, every element $v\in (h\circ g)\left(L\right)$ can be written as a finite sum of Lie brackets, that is, there exists a finite index set I such that

$$\begin{array}{c}\hfill v=\sum _{\begin{array}{c}i\in I\\ |v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-1)i}|=|v|\end{array}}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],\end{array}$$

for some $v_{1i},v_{2i},v_{3i},\dots ,v_{(n-1)i}\in L$. In this article, we always put ∑ in place of

$$\begin{array}{c}\hfill \sum _{\begin{array}{c}i\in I\\ |v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-1)i}|=|v|\end{array}}\mathrm{for}\mathrm{convenience}.\end{array}$$

Lemma 3.

If L is a perfect Lie superalgebra with a trivial center, then there exists an R-module homomorphism $\delta :nDer\left(L\right)\to End\left(L\right)$, defined by $\delta \left(D\right)={\delta }_D$ such that for all $v\in L$ and $D\in nDer\left(L\right)$, the following holds:

$$[D,\mathrm{ad}v]=\mathrm{ad}{\delta }_D\left(v\right).$$

Proof. 

In view of Lemma 1, if L is perfect and L has zero center, $D\in nDer\left(L\right)$, then we can construct a module endomorphism ${\delta }_D$ on L such that for any $v=\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}]\in hg\left(L\right)$,

$$\begin{array}{cc}\hfill {\delta }_D\left(v\right)& =\sum _{i\in I}([\dots [[D\left(v_{1i}\right),v_{2i}],v_{3i}],\dots ,v_{(n-1)i}]\hfill \\ \hfill & +{(-1)}^{\left|D\right||v_{1i}|}[\dots [[v_{1i},D\left(v_{2i}\right)],v_{3i}],\dots ,v_{(n-1)i}]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[\dots [[v_{1i},v_{2i}],D\left(v_{3i}\right)],\dots ,v_{(n-1)i}]+\dots \hfill \\ \hfill & {+(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-2)i}\left|\right)}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,D\left(v_{(n-1)i}\right)]).\hfill \end{array}$$

In fact, the definition does not depend on the specific expression of v. To prove this, let

$$\begin{array}{cc}\hfill \mathbb{S}& =\sum _{i\in I}([\dots [[D\left(v_{1i}\right),v_{2i}],v_{3i}],\dots ,v_{(n-1)i}]+{(-1)}^{\left|D\right||v_{1i}|}[\dots [[v_{1i},D\left(v_{2i}\right)],v_{3i}],\dots ,v_{(n-1)i}]\hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[\dots [[v_{1i},v_{2i}],D\left(v_{3i}\right)],\dots ,v_{(n-1)i}]+\dots \hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-2)i}\left|\right)}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,D\left(v_{(n-1)i}\right)]).\hfill \end{array}$$

Next, let

$$\begin{array}{cc}\hfill \mathbb{M}& =\sum _{j\in I}([\dots [[D\left(u_{j1}\right),u_{j2}],u_{j3}],\dots ,u_{j(n-1)}]\hfill \\ \hfill & +{(-1)}^{\left|D\right||u_{J1}|}[\dots [[u_{j1},D\left(u_{j2}\right)],u_{j3}],\dots ,u_{j(n-1)}]\hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(\right|u_{j1}|+|u_{j2}\left|\right)}[\dots [[u_{j1},u_{j2}],D\left(u_{j3}\right)],\dots ,u_{j(n-1)}]+\dots \hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(\right|u_{j1}|+|u_{j2}|+|u_{j3}|+\dots +|u_{j(n-2)}\left|\right)}[\dots [[u_{j1},u_{j2}],u_{j3}],\dots ,D\left(u_{j(n-1)}\right)]).\hfill \end{array}$$

Since $D\in nDer\left(L\right),\mathrm{for}\mathrm{all}w\in L$, we have

$$\begin{array}{cc}\hfill [\mathbb{S},w]& =\sum _{i\in I}([[\dots [[D\left(v_{1i}\right),v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +{(-1)}^{\left|D\right|\left|v_{1i}\right|}[[\dots [[v_{1i},D\left(v_{2i}\right)],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[[\dots [[v_{1i},v_{2i}],D\left(v_{3i}\right)],\dots ,v_{(n-1)i}],w]+\dots \hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-2)i}\left|\right)}[[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,D\left(v_{(n-1)i}\right)],w])\hfill \\ \hfill & =\sum _{i\in I}(D\left([[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\right)\hfill \\ \hfill & -{(-1)}^{\left|D\right||v_{1i}|}[[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],D\left(w\right)])\hfill \\ \hfill & =D\left([v,w]\right)-{(-1)}^{\left|D\right|\left|v\right|}[v,D\left(w\right)]\hfill \\ \hfill & =\sum _{j\in I}(D\left([[\dots [[u_{j1},u_{j2}],u_{j3}],\dots ,u_{j(n-1)}],w]\right)-{(-1)}^{\left|D\right|\left|u\right|}\hfill \\ \hfill & [[\dots [[u_{j1},u_{j2}],u_{j3}],\dots ,u_{j(n-1)}],D\left(w\right)])\hfill \\ \hfill & =\sum _{j\in I}([[\dots [[D\left(u_{j1}\right),u_{j2}],u_{j3}],\dots ,u_{j(n-1)}],w]\hfill \\ \hfill & +{(-1)}^{\left|D\right|\left|u_{j1}\right|}[[\dots [[u_{j1},D\left(u_{j2}\right)],u_{j3}],\dots ,u_{j(n-1)}],w]\hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(\right|u_{j1}|+|u_{j2}\left|\right)}[[\dots [[u_{j1},u_{j2}],D\left(u_{j3}\right)],\dots ,u_{j(n-1)}],w]\hfill \\ \hfill & +\dots +{(-1)}^{\left|D\right|\left(\right|u_{j1}|+|u_{j2}|+|u_{j3}|+\dots +|u_{j(n-2)}\left|\right)}[[\dots [[u_{j1},u_{j2}],u_{j3}],\dots ,D\left(u_{j(n-1)}\right)],w])\hfill \\ \hfill & =[\mathbb{M},w].\hfill \end{array}$$

Thus, $[\mathbb{S}-\mathbb{M},w]=0$, which implies that $\mathbb{S}-\mathbb{M}\in \mathrm{Z}\left(L\right)$. Since $\mathrm{Z}\left(L\right)=\left\{0\right\}$, it follows that $\mathbb{S}=\mathbb{M}$. Therefore, ${\delta }_D$ is well-defined. By Proposition 1, we will show that for each $D\in nDer\left(L\right)$ and $v\in L$, one has

$$[D,\mathrm{ad}v]\in \mathrm{ad}\left(L\right).$$

Hence, there exists ${\delta }_D\left(v\right)\in L$ such that

$$[D,\mathrm{ad}v]=\mathrm{ad}\left({\delta }_D\left(v\right)\right).$$

Since $Z\left(L\right)=\left\{0\right\}$, the element ${\delta }_D\left(v\right)$ is uniquely determined. Therefore, the map

$$\delta :nDer\left(L\right)⟶End\left(L\right),D\mapsto {\delta }_D,$$

is well defined, R-linear, and satisfies

$$[D,\mathrm{ad}v]=\mathrm{ad}\left({\delta }_D\left(v\right)\right)\mathrm{for}\mathrm{all}v\in L,D\in nDer\left(L\right).$$

This completes the proof. □

Lemma 4.

If L is a perfect Lie superalgebra with a trivial center, then for every $D\in nDer\left(L\right)$, ${\delta }_D$ belongs to $Der\left(L\right)$.

Proof. 

Let $D\in nDer\left(L\right)$ and from Proposition 1, we have

$$\begin{array}{c}\hfill v=\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}]\in hg\left(L\right),w\in L.\end{array}$$

Then

$$\begin{array}{cc}\hfill & [D,\mathrm{ad}\left([\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\right)]\hfill \\ \hfill & =\mathrm{ad}{\delta }_D\left([\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\right).\hfill \end{array}$$

Alternatively,

$$\begin{array}{cc}\hfill & [D,\mathrm{ad}\left([\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\right)]=[D,[\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]]\hfill \\ \hfill & =\sum _{i\in I}[[\dots [[[D,\mathrm{ad}\left(v_{1i}\right)],\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],\mathrm{ad}\left(w\right)]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right||v_{1i}|}[[\dots [\mathrm{ad}\left(v_{1i}\right),[D,\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],\mathrm{ad}\left(w\right)]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[[\dots [\mathrm{ad}\left(v_{1i}\right),[\mathrm{ad}\left(v_{2i}\right),[D,\mathrm{ad}\left(v_{3i}\right)]]],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],\mathrm{ad}\left(w\right)]\hfill \\ \hfill & +\dots +\sum _{i\in I}{(-1)}^{\left|D\right|\left|v\right|}[[\dots [[\mathrm{ad}\left(v_{1i}\right),\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,v_{(n-1)i}],[D,\mathrm{ad}\left(w\right)]]\hfill \\ \hfill & =\sum _{i\in I}[[\dots [[\mathrm{ad}{\delta }_D\left(v_{1i}\right),\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],\mathrm{ad}\left(w\right)]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right||v_{1i}|}[[\dots [[\mathrm{ad}\left(v_{1i}\right),\mathrm{ad}{\delta }_D\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],\mathrm{ad}\left(w\right)]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[[\dots [\mathrm{ad}\left(v_{1i}\right),[\mathrm{ad}\left(v_{2i}\right),\mathrm{ad}{\delta }_D\left(v_{3i}\right)]],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],\mathrm{ad}\left(w\right)]\hfill \\ \hfill & +\dots +\sum _{i\in I}{(-1)}^{\left|D\right|\left|v\right|}[\dots [[\mathrm{ad}\left(v_{1i}\right),\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,[v_{(n-1)i},\mathrm{ad}{\delta }_D\left(w\right)]].\hfill \\ \hfill & \mathrm{This}\mathrm{implies}\mathrm{that}\hfill \\ \hfill & =\mathrm{ad}(\sum _{i\in I}[[\dots [[{\delta }_D\left(v_{1i}\right),v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right||v_{1i}|}[[\dots [[v_{1i},{\delta }_D\left(v_{2i}\right)],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[[\dots [v_{1i},[v_{2i},{\delta }_D\left(v_{3i}\right)]],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\dots +\sum _{i\in I}{(-1)}^{\left|D\right|\left|v\right|}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,[v_{(n-1)i},{\delta }_D\left(w\right)]])\hfill \\ \hfill & =\mathrm{ad}{\delta }_D\left([\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\right)\hfill \\ \hfill & =\mathrm{ad}(\sum _{i\in I}[[\dots [[{\delta }_D\left(v_{1i}\right),v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right||v_{1i}|}[[\dots [[v_{1i},{\delta }_D\left(v_{2i}\right)],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[[\dots [v_{1i},[v_{2i},{\delta }_D\left(v_{3i}\right)]],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\dots +\sum _{i\in I}{(-1)}^{\left|D\right|\left|v\right|}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,[v_{(n-1)i},{\delta }_D\left(w\right)]]).\hfill \\ \hfill & \mathrm{Since}Z\left(L\right)=\left\{0\right\},\mathrm{it}\mathrm{follows}\mathrm{that}\hfill \\ \hfill & {\delta }_D\left([\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\right)\hfill \\ \hfill & =(\sum _{i\in I}[[\dots [[{\delta }_D\left(v_{1i}\right),v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right||v_{1i}|}[[\dots [[v_{1i},{\delta }_D\left(v_{2i}\right)],v_{3i}],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[[\dots [v_{1i},[v_{2i},{\delta }_D\left(v_{3i}\right)]],\dots ,v_{(n-1)i}],w]\hfill \\ \hfill & +\dots +\sum _{i\in I}{(-1)}^{\left|D\right|\left|v\right|}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,[v_{(n-1)i},{\delta }_D\left(w\right)]]).\hfill \end{array}$$

By the arbitrariness of $v={\sum }_{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],$ we conclude that ${\delta }_D\in Der\left(L\right)$. □

Lemma 5.

If the base ring R includes ${\displaystyle \frac{1}{n-1}}$ and L is perfect, then the centralizer of $\mathrm{ad}\left(L\right)$ in $nDer\left(L\right)$ is trivial, i.e., ${\mathrm{C}}_{nDer\left(L\right)}\left(\mathrm{ad}\left(L\right)\right)=\left\{0\right\}$. Consequently, the center of $nDer\left(L\right)$ is also trivial.

Proof. 

Let $D\in {\mathrm{C}}_{\mathrm{nDer}L}\left(\mathrm{ad}\left(L\right)\right)$. Thus, for all $v\in L,[D,\mathrm{ad}v]=0$. Then, for all $v,u\in hg\left(L\right)$, $D\left([v,u]\right)-{(-1)}^{\left|D\right|\left|v\right|}[v,D\left(u\right)]=[D,\mathrm{ad}v]\left(u\right)=0$. Therefore, $D\left(\right[v,u\left]\right)=\left[D\right(v),u]=$${(-1)}^{\left|D\right|\left|v\right|}[v,D\left(u\right)]$. For $v_1,v_2,v_2,\dots ,v_{n-1},v_n\in hg\left(L\right)$, we have

$$\begin{array}{cc}\hfill & D\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right)\hfill \\ \hfill & ={(-1)}^{\left|D\right|\left(|v_1|+|v_2|+|v_3|+\dots |v_{n-1}|\right)}[[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],D\left(v_n\right)]\hfill \\ \hfill & ={(-1)}^{\left|D\right|\left(|v_1|+|v_2|+|v_3|+\dots |v_{n-2}|\right)}[[\dots [[v_1,v_2],v_3],\dots ,D\left(v_{n-1}\right)],v_n]\hfill \\ \hfill & ⋮\hfill \\ \hfill & ={(-1)}^{\left|D\right|\left(|v_1|+|v_2|\right)}[[\dots [[v_1,v_2],D\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & ={(-1)}^{\left|D\right||v_1|}[[\dots [[v_1,D\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & =[[\dots [[D\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],v_n].\hfill \end{array}$$

Thus, we arrive at

$$\begin{array}{cc}\hfill & D\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right)\hfill \\ \hfill & =[[\dots [[D\left(v_1\right),v_2],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{\left|D\right||v_1|}[[\dots [[v_1,D\left(v_2\right)],v_3],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(|v_1|+|v_2|\right)}[[\dots [[v_1,v_2],D\left(v_3\right)],\dots ,v_{n-1}],v_n]\hfill \\ \hfill & +\dots +{(-1)}^{\left|D\right|\left(|v_1|+|v_2|+|v_3|+\dots |v_{n-2}|\right)}[[\dots [[v_1,v_2],v_3],\dots ,D\left(v_{n-1}\right)],v_n]\hfill \\ \hfill & +{(-1)}^{\left|D\right|\left(|v_1|+|v_2|+|v_3|+\dots |v_{n-1}|\right)}[[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],D\left(v_n\right)]\hfill \\ \hfill & =nD\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right).\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill (n-1)D\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right)=0.\end{array}$$

This implies that

$$\begin{array}{c}\hfill D\left([[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]\right)=0.\end{array}$$

Since L is perfect, every element of L can be written as a linear combination of elements of the form $[[\dots [[v_1,v_2],v_3],\dots ,v_{n-1}],v_n]$. Therefore, we conclude that $D=0$, thus completing the proof. □

Lemma 6.

If the base ring R includes ${\displaystyle \frac{1}{n-1}}$, and L is a perfect Lie superalgebra with a trivial center, then $nDer\left(L\right)=Der\left(L\right)$.

Proof. 

Suppose $v\in L,D\in nDer\left(L\right)$. By Proposition 1, $[D,\mathrm{ad}v]=\mathrm{ad}{\delta }_D\left(v\right)$. By Lemmas 1 and 4, $\mathrm{ad}{\delta }_D\left(v\right)=[{\delta }_D,\mathrm{ad}v]$. Hence, $D-{\delta }_D\in {\mathrm{C}}_{\mathrm{nDer}\left(L\right)}\left(\mathrm{ad}\left(L\right)\right)$. By Lemma 5, $D-{\delta }_D=$ 0, i.e., $D={\delta }_D\in Der\left(L\right)$. Hence, $nDer\left(L\right)\subseteq Der\left(L\right)$. The lemma follows from Lemma 4. □

From the above discussion, we observe that Lemma 6 proves the first part of Theorem 1, i.e., $nDer\left(L\right)=Der\left(L\right)$. Next, we prove the second part of Theorem 1 through the following Lemma 9.

Lemma 7.

If L is a perfect Lie superalgebra and $D\in nDer\left(Der\right(L\left)\right)$, then $D\left(\mathrm{ad}\right(L\left)\right)$ is contained in $\mathrm{ad}\left(L\right)$.

Proof. 

Since L is perfect, we have

$$\begin{array}{cc}\hfill D\left(\mathrm{ad}v\right)& =\sum _{i\in I}D\left(\mathrm{ad}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{ni}]\right)\hfill \\ \hfill & =\sum _{i\in I}D\left([\dots [[\mathrm{ad}\left(v_{1i}\right),v_{2i}],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{ni}\right)]\right)\hfill \\ \hfill & =\sum _{i\in I}[\dots [[D\left(\mathrm{ad}\left(v_{1i}\right)\right),\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{ni}\right)]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right||v_{1i}|}[\dots [[\mathrm{ad}\left(v_{1i}\right),D\left(\mathrm{ad}\left(v_{2i}\right)\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{ni}\right)]\hfill \\ \hfill & +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[\dots [\mathrm{ad}\left(v_{1i}\right),[\mathrm{ad}\left(v_{2i}\right),D\left(\mathrm{ad}\left(v_{3i}\right)\right)]],\dots ,\mathrm{ad}\left(v_{ni}\right)]\hfill \\ \hfill & +\dots +\sum _{i\in I}{(-1)}^{\left|D\right|\left(\right|v_1|+|v_2|+|v_3|+\dots +|v_{n-1}\left|\right)}\hfill \\ \hfill & [\dots [[\mathrm{ad}\left(v_{1i}\right),\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,[\mathrm{ad}\left(v_{(n-1)i}\right),D\left(\mathrm{ad}\left(v_{ni}\right)\right)]].\hfill \end{array}$$

Hence, $D\left(\mathrm{ad}v\right)\in \mathrm{ad}\left(L\right)$ for all $u\in L$. The lemma holds thanks to Proposition 1. □

Lemma 8.

Assume that R is the base ring containing ${\displaystyle \frac{1}{n-1}}$, L is a perfect Lie superalgebra with a trivial center, and $D\in nDer\left(Der\right(L\left)\right)$. If $D\left(\mathrm{ad}\right(L\left)\right)=0$, then $D=0$.

Proof. 

For all $d\in Der\left(L\right),vs.\in hg\left(L\right)$, since L is perfect,

$$v=\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}].$$

We have that

$$\begin{array}{cc}\hfill & [\mathrm{ad}v,D(d\left)\right]\hfill \\ \hfill & =[\sum _{i\in I}[\dots [[\mathrm{ad}\left(v_{1i}\right),\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],D\left(d\right)]\hfill \\ \hfill & =\sum _{i\in I}({(-1)}^{\left|D\right|\left|v\right|}D\left([[\dots [[\mathrm{ad}\left(v_{1i}\right),\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],d]\right)\hfill \\ \hfill & -{(-1)}^{\left|D\right|\left|v\right|}[[\dots [[D\left(\mathrm{ad}\left(v_{1i}\right)\right),\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],d]\hfill \\ \hfill & -{(-1)}^{\left|D\right|\left|v\right|}{(-1)}^{\left|D\right|\left|v_{i1}\right|}[[\dots [[\mathrm{ad}\left(v_{1i}\right),D\left(\mathrm{ad}\left(v_{2i}\right)\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],d]\hfill \\ \hfill & -{(-1)}^{\left|D\right|\left|v\right|}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|}[[\dots [[\mathrm{ad}\left(v_{1i}\right),\mathrm{ad}\left(v_{2i}\right)],D\left(\mathrm{ad}\left(v_{3i}\right)\right)],\dots ,\mathrm{ad}\left(v_{(n-1)i}\right)],d]\hfill \\ \hfill & -\dots -{(-1)}^{\left|D\right|\left|v\right|}{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-2)i}\left|\right)}\hfill \\ & [[\dots [[\mathrm{ad}\left(v_{1i}\right),\mathrm{ad}\left(v_{2i}\right)],\mathrm{ad}\left(v_{3i}\right)],\dots ,D\left(\mathrm{ad}\left(v_{(n-1)i}\right)\right)],d]).\hfill \end{array}$$

By Proposition 1, $[\mathrm{ad}v,d]\in \mathrm{ad}\left(L\right)$, so $D\left(\right[\mathrm{ad}v,d\left]\right)=0$. Thus, $[\mathrm{ad}v,D(d\left)\right]=0$. This yields $D\left(d\right)\in {\mathrm{C}}_{\mathrm{nDer}\left(L\right)}\left(\mathrm{ad}\left(L\right)\right)$. Therefore, by Lemma 5, $D\left(d\right)=0$. Hence, $D=0$. □

Lemma 9.

Let L be a Lie superalgebra over a commutative ring R. Suppose that ${\displaystyle \frac{1}{n-1}}\in R$ and that L is perfect with a trivial center. If $D\in nDer\left(Der\right(L\left)\right)$, then there exists an element $d\in Der\left(L\right)$ such that $D\left(\mathrm{ad}v\right)=\mathrm{ad}d\left(v\right)$ for all $v\in L$.

Proof. 

For all $D\in nDer\left(Der\right(L\left)\right),vs.\in L$, by Lemma 7, $D\left(adv\right)\in \mathrm{ad}\left(L\right)$. Let $u\in L$ and $D\left(\mathrm{ad}v\right)=\mathrm{ad}u$. Since the center $\mathrm{Z}\left(L\right)$ is trivial, such a u is unique. Clearly, the map $d:L\to L$ given by $d\left(v\right)=u$ is an R-module endomorphism of L.

Let $v_1,v_2,v_2,\dots ,v_{n-1}\in hg\left(L\right),v_n\in L$, we have

$$\begin{array}{c}\mathrm{ad}d\left([\dots [[v_1,v_2],v_3],\dots ,v_n]\right)\hfill \\ =D\left(\mathrm{ad}\left([\dots [[v_1,v_2],v_3],\dots ,v_n]\right)\right)\hfill \\ =D\left([\dots [[\mathrm{ad}\left(v_1\right),\mathrm{ad}\left(v_2\right)],\mathrm{ad}\left(v_3\right)],\dots ,\mathrm{ad}\left(v_n\right)]\right)\hfill \\ =[\dots [[D\left(\mathrm{ad}\left(v_1\right)\right),\mathrm{ad}\left(v_2\right)],\mathrm{ad}\left(v_3\right)],\dots ,\mathrm{ad}\left(v_n\right)]\hfill \\ +{(-1)}^{\left|D\right|\left|v_{i1}\right|}[\dots [[\mathrm{ad}\left(v_1\right),D\left(\mathrm{ad}\left(v_2\right)\right)],\mathrm{ad}\left(v_3\right)],\dots ,\mathrm{ad}\left(v_n\right)]\hfill \\ +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[\dots [[\mathrm{ad}\left(v_1\right),\mathrm{ad}\left(v_2\right)],D\left(\mathrm{ad}\left(v_3\right)\right)],\dots ,\mathrm{ad}\left(v_n\right)]\hfill \\ +\dots +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|+|v_{i3}|+\dots +|v_{(n-1)i}\left|\right)}[\dots [[\mathrm{ad}\left(v_1\right),\mathrm{ad}\left(v_2\right)],\mathrm{ad}\left(v_3\right)],\dots ,D\left(\mathrm{ad}\left(v_n\right)\right)]\hfill \\ =[\dots [[\mathrm{ad}d\left(v_1\right),\mathrm{ad}{v}_2],\mathrm{ad}{v}_3],\dots ,\mathrm{ad}{v}_n]\hfill \\ +{(-1)}^{\left|D\right|\left|v_{i1}\right|}[\dots [[\mathrm{ad}{v}_1,{\mathrm{ad}}_{d\left(v_2\right)}],\mathrm{ad}{v}_3],\dots ,\mathrm{ad}{v}_n]\hfill \\ +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[\dots [[\mathrm{ad}{v}_1,\mathrm{ad}{v}_2],{\mathrm{ad}}_{d\left(v_3\right)}],\dots ,\mathrm{ad}{v}_n]\hfill \\ +\dots +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|+|v_{i3}|+\dots +|v_{(n-1)i}\left|\right)}[\dots [[\mathrm{ad}{v}_1,\mathrm{ad}{v}_2],\mathrm{ad}{v}_3],\dots ,{\mathrm{ad}}_{d\left(v_n\right)}]\hfill \\ =\mathrm{ad}([\dots [[d\left(v_1\right),v_2],v_3],\dots ,v_n]+{(-1)}^{\left|D\right|\left|v_{i1}\right|}[\dots [[v_1,d\left(v_2\right)],v_3],\dots ,v_n]\hfill \\ +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}\left|\right)}[\dots [[v_1,v_2],d\left(v_3\right)],\dots ,v_n]\hfill \\ +\dots +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|+|v_{i3}|+\dots +|v_{(n-1)i}\left|\right)}[\dots [[v_1,v_2],v_3],\dots ,d\left(v_n\right)]).\hfill \end{array}$$

Since $Z\left(L\right)=\left\{0\right\}$,

$$\begin{array}{cc}\hfill d\left([\dots [[v_1,v_2],v_3],\dots ,v_n]\right)& =[\dots [[d\left(v_1\right),v_2],v_3],\dots ,v_n]\hfill \\ & +{(-1)}^{\left|D\right|\left|v_{i1}\right|}[\dots [[v_1,d\left(v_2\right)],v_3],\dots ,v_n]\hfill \\ & +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|}[\dots [[v_1,v_2],d\left(v_3\right)],\dots ,v_n]+\dots \hfill \\ & +{(-1)}^{\left|D\right|\left(\right|v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-1)i}\left|\right)}[\dots [[v_{1i},v_2],v_3],\dots ,d\left(v_n\right)].\hfill \end{array}$$

The above relation gives $d\in nDer\left(L\right)$. By Lemma 6, $d\in Der\left(L\right)$. □

Proof of Theorem 1.

In view of Lemma 6, it remains only to prove the second assertion. By Lemma 9, for $\mathrm{all}D\in \left(Der\right(L\left)\right),vs.\in L$, there exists $d\in Der\left(L\right)$ such that for all $v\in L,D\left(\mathrm{ad}v\right)=\mathrm{ad}d\left(v\right)$. Application of Lemma 1 yields that, $\mathrm{ad}d\left(v\right)=[d,\mathrm{ad}v]$ for all $v\in L$.

Therefore, we obtain

$$D\left(\mathrm{ad}v\right)=\mathrm{ad}d\left(v\right)=[d,\mathrm{ad}v]=\mathrm{ad}d\left(\mathrm{ad}v\right)\mathrm{for}\mathrm{all}vs.\in L.$$

This gives

$$(D-\mathrm{ad}d)\left(\mathrm{ad}v\right)=0\mathrm{for}\mathrm{all}vs.\in L.$$

By Lemma 8, $D=\mathrm{ad}d$. Hence, $nDer\left(Der\right(L\left)\right)=\mathrm{ad}\left(Der\right(L\left)\right)$. The theorem holds. □

As an immediate consequence of Theorem 1, we have the following result.

Corollary 1.

If L is a simple Lie superalgebra over a commutative ring R with ${\displaystyle \frac{1}{n-1}}\in R$, then

$$nDer\left(L\right)=Der\left(L\right),nDer\left(Der\right(L\left)\right)=\mathrm{ad}\left(Der\right(L\left)\right).$$

Remark 3.

It is important to distinguish between simple and perfect Lie superalgebras. Every non-abelian simple Lie superalgebra is perfect since the derived superalgebra $[L,L]$ forms a nontrivial ideal, which must coincide with L itself. However, the converse meed not be holds in general. A perfect Lie superalgebra may possess nontrivial ideals and hence fail to be simple. For instance, $\mathfrak{sl}\left(1\right|1)$ is perfect, as $\left[\mathfrak{sl}\right(1\left|1\right),\mathfrak{sl}\left(1\right|1\left)\right]=\mathfrak{sl}\left(1\right|1)$, but it is not simple due to the existence of proper nontrivial ideals contained in its odd part.

Consider the Lie superalgebras L and $L^{\prime }$ over the commutative ring R. Assume that M is the enveloping Lie superalgebra of $f\left(L\right)$ and that f is a n-homomorphism from L to $L^{\prime }$. It may be represented as a direct sum of indecomposable ideals (cf; [10]), and it is assumed that L is perfect and M is centerless. The second main result of this paper is the following theorem.

Theorem 2.

Let R be a commutative ring with unity, and assume that $(n-1)$ be invertible in R. Let L and $L^{\prime }$ be Lie superalgebras over R, with f being an n-homomorphism from L to $L^{\prime }$, and let M represent the enveloping Lie superalgebra of $f\left(L\right)$. Then, the following hold:

(i) 

L is perfect;

(ii) 

M is centerless and can be decomposed into a direct sum of indecomposable ideals. In this case, f is either a homomorphism, an anti-homomorphism, or a direct sum of both a homomorphism and an anti-homomorphism.

We prove the above mentioned result through the sequence of the following lemmas.

Lemma 10.

Let L and $L^{\prime }$ be Lie superalgebras over a commutative ring with unity. There exists an even R-linear mapping ${\delta }_f:L\to L^{\prime }$ such that

$${\delta }_f\left(v\right)=\sum _{\begin{array}{c}i\in I\\ |v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-1)i}|=|v|\end{array}}[\dots [[f\left(v_{1i}\right),f\left(v_{2i}\right)],f\left(v_{3i}\right)],\dots ,f\left(v_{(n-1)i}\right)]$$

for all $v\in hg\left(L\right)$, where,

$$\begin{array}{cc}\hfill v& =\sum _{\begin{array}{c}i\in I\\ |v_{1i}|+|v_{2i}|+|v_{3i}|+\dots +|v_{(n-1)i}|=|v|\end{array}}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],\left(v_{1i},v_{2i},v_{3i},\dots ,v_{(n-1)i}\in L\right).\hfill \end{array}$$

Proof. 

It is sufficient to prove that $\sum [\dots [f\left(v_{i1}\right),f\left(v_{i2}\right)],\dots ,f\left(v_{(n-1)i}\right)]$ is independent of the expression of v. Suppose that $[\dots [f\left(v_{1i}\right),f\left(v_{2i}\right)],\dots ,f\left(v_{(n-1)i}\right)]$.

Assume that

$$v=\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}]\in hg\left(L\right),$$

and

$${\delta }_D\left(v\right)=\sum _{i\in I}\left([\dots [[f\left(v_{1i}\right),f\left(v_{2i}\right)],f\left(v_{3i}\right)],\dots ,f\left(v_{(n-1)i}\right)]\right).$$

In actuality, the definition is unaffected by how v is expressed. To demonstrate it, let

$$\mathbb{S}=\sum _{i\in I}\left([\dots [[f\left(v_{1i}\right),f\left(v_{2i}\right)],f\left(v_{3i}\right)],\dots ,f\left(v_{(n-1)i}\right)]\right).$$

Let $v={\sum }_{i\in I}\left([\dots [[u_{i1},u_{i2}],u_{i3}],\dots ,u_{i(n-1)}]\right)\mathrm{and}$ let

$$\mathbb{M}=\sum _{i\in I}\left([\dots [[f\left(u_{i1}\right),f\left(u_{i2}\right)],f\left(u_{i3}\right)],\dots ,f\left(u_{i(n-1)}\right)]\right).$$

Then, for all $w\in L$, we have

$$\begin{array}{cc}\hfill \left[f\right(w),\mathbb{S}-\mathbb{M}]& =[f\left(w\right),\sum _{i\in I}[\dots [[f\left(v_{1i}\right),f\left(v_{2i}\right)],f\left(v_{3i}\right)],\dots ,f\left(v_{(n-1)i}\right)]\hfill \\ \hfill & -\sum _{i\in I}[f\left(w\right),[\dots [[f\left(u_{i1}\right),f\left(u_{i2}\right)],f\left(u_{i3}\right)],\dots ,f\left(u_{i(n-1)}\right)]]]\hfill \\ \hfill & =\sum _{i\in I}[f\left(w\right),[\dots [[f\left(v_{1i}\right),f\left(v_{2i}\right)],f\left(v_{3i}\right)],\dots ,f\left(v_{(n-1)i}\right)]]\hfill \\ \hfill & -\sum _{i\in I}[f\left(w\right),[\dots [[f\left(u_{i1}\right),f\left(u_{i2}\right)],f\left(u_{i3}\right)],\dots ,f\left(u_{i(n-1)}\right)]]\hfill \\ \hfill & =f([w,\sum _{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}]]\hfill \\ \hfill & -[w,\sum _{i\in I}[\dots [[u_{i1},u_{i2}],u_{i3}],\dots ,u_{i(n-1)}]])\hfill \\ \hfill & =f\left(\right[w,v]-[w,v\left]\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

It follows that $\mathbb{S}-\mathbb{M}\in \mathrm{Z}\left(M\right)$, and hence $\mathbb{S}=\mathbb{M}$ since M is centerless. This completes the proof. □

Lemma 11.

Let ${\delta }_f$ be the mapping in Lemma 10. Then, $f\mathrm{ad}v=\mathrm{ad}{\delta }_f\left(v\right)f$.

Proof. 

Let $v={\sum }_{i\in I}[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}]\in L$ for any $w\in L$, and we have

$$\begin{array}{cc}\hfill f\mathrm{ad}v\left(w\right)=f\left(\right[v,w\left]\right)& =\sum f\left([[\dots [[v_{1i},v_{2i}],v_{3i}],\dots ,v_{(n-1)i}],w]\right)\hfill \\ & =\sum [[\dots [[f\left(v_{1i}\right),f\left(v_{2i}\right)],f\left(v_{3i}\right)],\dots ,f\left(v_{(n-1)i}\right)],f\left(w\right)]\hfill \\ & =[\sum [\dots [[f\left(v_{1i}\right),f\left(v_{2i}\right)],f\left(v_{3i}\right)],\dots ,f\left(v_{(n-1)i}\right)],f\left(w\right)]\hfill \\ & =[{\delta }_f\left(v\right),f\left(w\right)]\hfill \\ & =\mathrm{ad}{\delta }_f\left(v\right)f\left(w\right).\hfill \end{array}$$

Hence $f\mathrm{ad}v=\mathrm{ad}{\delta }_f\left(v\right)f$ for all $v\in L$. □

Lemma 12.

The mapping ${\delta }_f$ in Lemma 10 is a homomorphism of Lie superalgebras.

Proof. 

For $\mathrm{all}v,u,w\in L$, it follows from Lemmas 10 and 11 that

$$\begin{array}{cc}\hfill & [{\delta }_f\left([v,u]\right)-[{\delta }_f\left(v\right),{\delta }_f\left(u\right)],f\left(w\right)]\hfill \\ \hfill & =[{\delta }_f\left([v,u]\right),f\left(w\right)]-[{\delta }_f\left(v\right),[{\delta }_f\left(u\right),f\left(w\right)]]+{(-1)}^{\left|v\right|\left|u\right|}[{\delta }_f\left(u\right),[{\delta }_f\left(v\right),f\left(w\right)]]\hfill \\ \hfill & =[[f\left(v\right),f\left(u\right)],f\left(w\right)]-[{\delta }_f\left(v\right),\mathrm{ad}{\delta }_f\left(u\right)f\left(w\right)]+{(-1)}^{\left|v\right|\left|u\right|}[{\delta }_f\left(u\right),\mathrm{ad}{\delta }_f\left(v\right)f\left(w\right)]\hfill \\ \hfill & =[[f\left(v\right),f\left(u\right)],f\left(w\right)]-\left[{\delta }_f\left(v\right)f\left([y,w]\right)\right]+{(-1)}^{\left|v\right|\left|u\right|}[{\delta }_f\left(u\right),f\left([v,w]\right)]\hfill \\ \hfill & =[[f\left(v\right),f\left(u\right)],f\left(w\right)]-\mathrm{ad}{\delta }_f\left(v\right)f\left([y,w]\right)+{(-1)}^{\left|v\right|\left|u\right|}\mathrm{ad}{\delta }_f\left(u\right)f\left([v,w]\right)\hfill \\ \hfill & =f\left([[v,u],w]\right)-f\left([v,[y,w]]\right)+{(-1)}^{\left|v\right|\left|u\right|}f\left([y,[v,w]]\right)\hfill \\ \hfill & =f\left([[v,u],w]-[v,[y,w]]+{(-1)}^{\left|v\right|\left|u\right|}[y,[v,w]]\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

By the Jacobi identity, we have $\left[{\delta }_f\left([v,u]\right)-[{\delta }_f\left(v\right),{\delta }_f\left(u\right)],f\left(w\right)\right]=0$ for arbitrary $w\in L$. Since M is the enveloping Lie superalgebra of $f\left(L\right)$, it follows that ${\delta }_f\left([v,u]\right)-[{\delta }_f\left(v\right),{\delta }_f\left(u\right)]\in \mathrm{Z}\left(M\right).$ Because M is centerless, ${\delta }_f\left([v,u]\right)=\left[{\delta }_f\left(v\right),{\delta }_f\left(u\right)\right]$ for all $v,u\in L$. □

Lemma 13.

Let ${\mathbb{Z}}^{+}=Im(f+{\delta }_f)$ and ${\mathbb{Z}}^{-}=Imt(f-{\delta }_f)$. Then, ${\mathbb{Z}}^{+}$ and ${\mathbb{Z}}^{-}$ are both ideals of M.

Proof. 

It is clear that ${\mathbb{Z}}^{+},{\mathbb{Z}}^{-}\subseteq M$. For any $v,u\in L$, by Lemma 11, we have

$$\begin{array}{cc}\hfill [f\left(v\right)-{\delta }_f\left(v\right),f\left(u\right)]& =[f\left(v\right),f\left(u\right)]-[{\delta }_f\left(v\right),f\left(u\right)]\hfill \\ & =[f\left(v\right),f\left(u\right)]-{\mathrm{ad}}_{{\delta }_f\left(v\right)}f\left(u\right)\hfill \\ \hfill & ={\delta }_f\left([v,u]\right)-f{\mathrm{ad}}_v\left(u\right)\hfill \\ \hfill & ={\delta }_f\left([v,u]\right)-f\left([v,u]\right)\hfill \\ \hfill & =\left(f-{\delta }_f\right)\left([v,u]\right)\in {\mathbb{Z}}^{-}.\hfill \end{array}$$

Hence, ${\mathbb{Z}}^{-}$ is an ideal of M.

Similarly, we can prove that ${\mathbb{Z}}^{+}$ is an ideal of M. □

Lemma 14.

Prove that $[{\mathbb{Z}}^{+},{\mathbb{Z}}^{-}]=\left\{0\right\}$.

Proof. 

Take $v,u,w\in L$; by Lemmas 10–13, we have that

$$\begin{array}{cc}\hfill & [[f\left(v\right)+{\delta }_f\left(v\right),f\left(u\right)-{\delta }_f\left(u\right)],f\left(w\right)]\hfill \\ \hfill & =[[f\left(v\right),f\left(u\right)],f\left(w\right)]-[[f\left(v\right),{\delta }_f\left(u\right)],f\left(w\right)]\hfill \\ \hfill & +[[{\delta }_f\left(v\right),f\left(u\right)],f\left(w\right)]-[[{\delta }_f\left(v\right),{\delta }_f\left(u\right)],f\left(w\right)]\hfill \\ \hfill & =f\left([[v,u],w]\right)+{(-1)}^{\left|v\right|\left|u\right|}[\mathrm{ad}{\delta }_f\left(u\right)f\left(v\right),f\left(w\right)]+[\mathrm{ad}{\delta }_f\left(v\right)f\left(u\right),f\left(w\right)]\hfill \\ \hfill & -[{\delta }_f\left(v\right),[{\delta }_f\left(u\right),f\left(w\right)]]+{(-1)}^{\left|v\right|\left|u\right|}[{\delta }_f\left(u\right),\mathrm{ad}{\delta }_f\left(v\right)f\left(w\right)]\hfill \\ \hfill & =f\left([[v,u],w]\right)+{(-1)}^{\left|v\right|\left|u\right|}[f\left([y,v]\right),f\left(w\right)]+[f\left([v,u]\right),f\left(w\right)]\hfill \\ \hfill & -[{\delta }_f\left(v\right),\mathrm{ad}{\delta }_f\left(u\right)f\left(w\right)]+{(-1)}^{\left|v\right|\left|u\right|}[{\delta }_f\left(u\right),f\left([v,w]\right)]\hfill \\ \hfill & =f\left([[v,u],w]\right)+{(-1)}^{\left|v\right|\left|u\right|}[f\left([y,v]\right),f\left(w\right)]+[f\left([v,u]\right),f\left(w\right)]\hfill \\ \hfill & -[{\delta }_f\left(v\right),f\left([y,w]\right)]-{(-1)}^{\left|v\right|\left|u\right|}{(-1)}^{\left|u\right|\left(\right|v|+|w|}[f\left([v,w]\right),{\delta }_f\left(u\right)]\hfill \\ \hfill & =f\left([[v,u],w]\right)-[{\delta }_f\left(v\right),f\left([y,w]\right)]-{(-1)}^{\left|u\right|\left|w\right|}[f\left([v,w]\right),{\delta }_f\left(u\right)]\hfill \\ \hfill & =f\left([[v,u],w]\right)-f\left([v,[y,w]]\right)+{(-1)}^{\left|u\right|\left|w\right|}{(-1)}^{\left|u\right|\left(\right|v|+|w\left|\right)}f\left([y,[v,w]]\right)\hfill \\ \hfill & =f\left([[v,u],w]-[v,[y,w]]+{(-1)}^{\left|v\right|\left|u\right|}[y,[v,w]]\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

This implies that $[f\left(v\right)+{\delta }_f\left(v\right),f\left(u\right)-{\delta }_f\left(u\right)]\in Z\left(M\right)$. Since $Z\left(M\right)=\left\{0\right\}$, we conclude that $[f\left(v\right)+{\delta }_f\left(v\right),f\left(u\right)-{\delta }_f\left(u\right)]=0$. Hence the lemma follows. □

Lemma 15.

Prove that ${\mathbb{Z}}^{+}\cap {\mathbb{Z}}^{-}=\left\{0\right\}$.

Proof. 

Let $z\in {\mathbb{Z}}^{+}\cap {\mathbb{Z}}^{-}$. By Lemma 14, $[{\mathbb{Z}}^{+},w]=[{\mathbb{Z}}^{-},w]=0$. Thus, for any $w\in L$, we have

$$[f\left(v\right)+{\delta }_f\left(v\right),w]=[f\left(v\right)-{\delta }_f\left(v\right),w]=0.$$

Under the assumption ${\displaystyle \frac{1}{n-1}}\in R$, we have $\left[f\right(v),w]=0.$ Since M is the enveloping Lie algebra of $f\left(L\right)$ and v is arbitrary, it follows that $w\in Z\left(M\right)$. As $Z\left(M\right)=\left\{0\right\}$, we conclude that $w=0$. Hence, the lemma follows. □

Lemma 16.

If M cannot be decomposed into a direct sum of two nontrivial ideals, then f is either a homomorphism or an anti-homomorphism of Lie superalgebras.

Proof. 

For every $v\in L$, define ${\mathbb{Z}}^{+}={\displaystyle \frac{1}{n-1}}(f\left(v\right)+{\delta }_f\left(v\right))$ and ${\mathbb{Z}}^{-}={\displaystyle \frac{1}{n-1}}(f\left(v\right)-{\delta }_f\left(v\right))$. It follows that ${\mathbb{Z}}^{+}\in {\mathbb{Z}}^{+}$ and ${\mathbb{Z}}^{-}\in {\mathbb{Z}}^{-}$ and that $f\left(v\right)={\mathbb{Z}}^{+}+{\mathbb{Z}}^{-}$. This implies that $f\left(L\right)\subseteq {\mathbb{Z}}^{+}+{\mathbb{Z}}^{-}$, and consequently, $M\subseteq {\mathbb{Z}}^{+}+{\mathbb{Z}}^{-}$. By Lemma 15, we know that $M={\mathbb{Z}}^{+}\oplus {\mathbb{Z}}^{-}$. Since M cannot be written as a direct sum of two nontrivial ideals, exactly one of ${\mathbb{Z}}^{+}$ or ${\mathbb{Z}}^{-}$ must be trivial. If ${\mathbb{Z}}^{+}$ is trivial (that is, $(f+{\delta }_f)\left([v,u]\right)=0$), then $f\left([v,u]\right)=-{\delta }_f\left([v,u]\right)=-[f\left(v\right),f\left(u\right)]={(-1)}^{\left|v\right|\left|u\right|}[f\left(u\right),f\left(v\right)],$ so f is an anti-homomorphism. Conversely, if ${\mathbb{Z}}^{-}$ is trivial (that is, $(f-{\delta }_f)\left([v,u]\right)=0$), then $f\left([v,u]\right)={\delta }_f\left([v,u]\right)=[f\left(v\right),f\left(u\right)],$ and thus f is a homomorphism. This completes the proof of the lemma. □

Proof of Theorem 2.

According to Lemma 16, it is sufficient to prove the theorem when M is decomposable. Given the assumptions, we can express M as

$$M=M_1\oplus M_2\oplus \dots \oplus M_{i^{\prime s}}.$$

Each $M_i$ is an indecomposable ideal of M. Since $Z\left(M\right)=\left\{0\right\}$, Lemma 10 in [18] implies that every $M_i$ is also centerless. Let $p_i:M\to M_i$ denote the canonical projection. Then, $f={\sum }_{i=1}^s{p}_{i}f,$ and each $p_{i}f:L\to M_i$ is a triple homomorphism, with $M_i$ the enveloping Lie superalgebra of $p_{i}f\left(L\right)$ for $i=1,2,\dots ,s$. Because each $M_i$ is indecomposable, Lemma 16 yields that $p_{i}f$ is either a homomorphism or an anti-homomorphism from L to $M_i$. Let $P=\{i\mid p_{i}f\mathrm{is}\mathrm{a}\mathrm{homomorphism}\},$ and let Q be the complementary set of P within $\{1,2,\dots ,s\}$. Define $M_1={\sum }_{i\in P}{M}_i,M_2={\sum }_{i\in Q}{M}_i.$ Let $f_1={\sum }_{i\in P}{p}_{i}f$ and $f_2={\sum }_{i\in Q}{p}_{i}f$. By direct verification, we can check that $M=M_1\oplus M_2$ and that $[M_1,M_2]=0$. Moreover, $f=f_1+f_2$, where $f_1$ is a homomorphism and $f_2$ is an anti-homomorphism of Lie superalgebras. This completes the proof. □

Every finite-dimensional semisimple Lie superalgebra over an algebraically closed field of characteristic 0 is perfect, centerless, and decomposes into a direct sum of simple ideals; hence, we have the following corollary.

Corollary 2.

Let L and $L^{\prime }$ be finite-dimensional semisimple Lie superalgebras over an algebraically closed field of characteristic 0. Then every n-homomorphism $f:L\to L^{\prime }$ is of exactly one of the following types:

(i) 

A homomorphism;

(ii) 

An anti-homomorphism;

(iii) 

A direct sum of a homomorphism and an anti-homomorphism.

4. Applications and Future Research Problems

The results of this study are established under specific structural and algebraic assumptions—namely, the Lie superalgebra L is perfect with a trivial center, and the base ring R contains ${\displaystyle \frac{1}{n-1}}$. A natural future direction is to extend the analysis of n-derivations to more general classes of Lie superalgebras, such as those are not perfect or have a nontrivial center. Additionally, investigating the behavior of n-derivations over rings that do not contain ${\displaystyle \frac{1}{n-1}}$ could reveal whether this condition is essential. Another promising direction lies in studying n-homomorphisms between non-isomorphic or structurally different Lie superalgebras or in exploring their categorical or cohomological interpretations. Expanding the framework to include color Lie superalgebras, graded Lie algebras, or infinite-dimensional cases may also offer deeper insights and broader applicability. Future directions include relaxing these conditions to consider the following:

(i)

Structure Theory of Lie Superalgebras: The characterization of n-derivations as derivations strengthens the structural understanding of perfect Lie superalgebras. It provides criteria for decomposing L into subalgebras that are invariant with higher-order derivations, aiding classification and structural analysis.

(ii)

Representation Theory: The correspondence between n-derivations and ordinary derivations has implications for representations of Lie superalgebras. Fixed-point subalgebras with n-homomorphisms identify invariant modules and constrain the action of symmetries in representation spaces.

(iii)

Lie and Jordan Structures: Since n-derivations generalize the classical derivation concept, the results provide a bridge between Lie superalgebras theory and Jordan-type structures. The inner derivation property of n-derivations of $\mathrm{Der}\left(L\right)$ facilitates the study of generalized Jordan derivations and commutator identities.

(iv)

Algebraic Stability and Perturbations: Understanding that n-derivations reduce to derivations ensures stability of the algebraic structure with higher-order transformations. This has potential implications in perturbation theory where maintaining the integrity of algebraic operations under deformation conditions is crucial.

(v)

Symmetry Analysis in Quantum and Mathematical Physics: Perfect Lie superalgebras frequently model symmetries in supersymmetric and quantum systems. The results on n-homomorphisms can be used to identify higher-order symmetries and invariant subspaces, contributing to the analysis of quantum observables and supersymmetric structures.

(vi)

Symbolic and Computational Algebra: The explicit structural identities of n-derivations and n-homomorphisms can be implemented in computer algebra systems, enabling automated verification of algebraic properties and the study of perfect Lie superalgebras algorithmically.

We conclude the paper with the following interesting open problems related to Theorems 1 and 2.

Open Problem 1.

Investigate whether Theorem 1 remains valid under weaker assumptions. In particular:

(i) 

What is the structure of $nDer\left(L\right)$ if $\frac{1}{n-1}\notin R$?

(ii) 

Does the equality $nDer\left(L\right)=Der\left(L\right)$ hold when L is not perfect or when $Z\left(L\right)\ne \left\{0\right\}$?

(iii) 

Can analogous results be established for other algebraic structures such as Hom–Lie superalgebras (or an Leibniz superalgebras)?

Open Problem 2.

Let $L={⨁}_{\gamma \in G}{L}_{\gamma }$ be a Lie (super) algebra over a commutative ring R, where G is an abelian grading group (for the ordinary Lie superalgebra case take $G={\mathbb{Z}}_2$). This includes

(i) 

Color Lie (super) algebras (with a bicharacter $\epsilon :G\times G\to R^{\times }$);

(ii) 

More general G-graded Lie algebras;

(iii) 

Infinite-dimensional Lie (super)algebras.

For a fixed integer $n\ge 2$, define

$$nDer\left(L\right)=\{D:L\to L\mid D\mathrm{is}\mathrm{an}n-\mathrm{derivation}\mathrm{of}L\}.$$

Here, an R-linear map $D:L\to L$ is called an n-derivation if for all homogeneous elements $v_1,\dots ,v_n\in L$ the following holds:

(a) 

Graded/super case ($G={\mathbb{Z}}_2$). If $|·|$ denotes degree, then

$$D\left([v_1,\dots ,v_n]\right)=\sum _{i=1}^n{(-1)}^{\left|D\right|\left(\right|v_1|+\dots +|v_{i-1}\left|\right)}[v_1,\dots ,D\left(v_i\right),\dots ,v_n],$$

where $[·,\dots ,·]$ denotes the iterated (or n-ary) Lie bracket and $\left|D\right|$ is the degree of D (for even D the signs are all $+1$).

(b) 

Color/general graded case. If L is G-graded with bicharacter ε, then for homogeneous $v_j$ one requires

$$D\left([v_1,\dots ,v_n]\right)=\sum _{i=1}^n\epsilon \left(\left|D\right|,|v_1|+\dots +|v_{i-1}|\right)[v_1,\dots ,D\left(v_i\right),\dots ,v_n],$$

where $\left|D\right|\in G$ is the degree of D (take $\epsilon (v_1,v_2)={(-1)}^{v_1{v}_2}$ in the super case).

What information can be deduced regarding the structure of $nDer\left(L\right)$?

Open Problem 3.

The following questions remain open concerning Theorem 2:

(i) 

Does the conclusion hold if $(n-1)$ is not invertible in R?

(ii) 

Can similar results be obtained for non-perfect Lie superalgebras?

(iii) 

Does an analogue of the theorem exists for Hom–Lie (or an Leibniz superalgebras)?

(iv) 

Is the decomposition of M into indecomposable ideals unique up to isomorphism?

5. Conclusions

In this paper, we studied n-derivations and n-homomorphisms of perfect Lie superalgebras. Our investigation provided insights into the role of n-derivations and n-homomorphisms in the structure and behavior of Lie superalgebras, contributing to the broader field of algebraic research. These findings significantly enhanced our understanding of n-derivations and n-homomorphisms on perfect Lie superalgebras over the commutative rings. The results demonstrated that under specific conditions, such as the base ring containing ${\displaystyle \frac{1}{n-1}}$ and the center of L being zero, n-derivations of L coincide with standard derivations, and every n-derivation of the derivation algebra $Der\left(L\right)$ is shown to be inner. Furthermore, the extension of the concept of n-homomorphisms to mappings between Lie superalgebras L and $L^{\prime }$ reveals that, under certain assumptions, homomorphisms, anti-homomorphisms, and their combinations all qualify as n-homomorphisms. These results contributed to the advancement of the theory of derivations and homomorphisms in non-associative algebras, paving the way for further exploration of their applications in broader algebraic structures.