The Structure of D-Derivations and Their Decomposition in Lie Algebras

Abstract

A D-derivation of a Lie algebra L is a linear map $\phi$ for which there exists a derivation D such that $\phi \left(\right[x,y\left]\right)=\left[\phi \right(x),y]+[x,D(y\left)\right]$ for all $x,y\in L$. This paper presents explicit structural results concerning D-derivations in Lie algebras over arbitrary fields. It is established that the set of D-derivations forms a Lie algebra, which decomposes as the sum of derivations and centroids, intersecting precisely at the space of central derivations. For centerless Lie algebras, the inclusion chain for D-derivations within existing derivation classes is completed, resulting in a refined hierarchy. It is proven that for both perfect and centerless Lie algebras, D-derivations decompose as a direct sum of derivations and centroids. In particular, for semisimple Lie algebras, it is shown that ${Der}_D\left(L\right)=ad\left(L\right)\oplus C\left(L\right)$, and for simple Lie algebras over an algebraically closed field of characteristic zero, ${Der}_D\left(L\right)=ad\left(L\right)\oplus \mathbb{F}{id}_L$. Furthermore, for any centerless Lie algebra, the Lie algebra of D-derivations is shown to be isomorphic to the semidirect product of the derivation and centroid algebras, with explicit descriptions provided for semisimple and solvable cases. Examples involving $\mathfrak{so}\left(3\right)$, $\mathfrak{so}(1,3)$, $\mathfrak{aff}\left(1\right)$, and ${\mathfrak{h}}_3$ confirm these decompositions and offer matrix realizations of their D-derivations, thereby supporting and illustrating the main theorems.

1. Introduction

Lie algebras and their derivations are central to many areas of mathematics and theoretical physics [1]. The study of derivations, defined as linear maps that satisfy the Leibniz rule with respect to the Lie bracket, has expanded to include generalized derivations, quasi-derivations, centroids, and central derivations [2]. These generalizations offer a deeper understanding of the symmetry and automorphism properties of Lie algebras.

Recent research has advanced the understanding of derivation-like structures under various algebraic constructions. Benkovič and Eremita [3] investigated conditions under which the decomposition $QDer\left(L\right)=Der\left(L\right)\oplus C\left(L\right)$ holds for quasi-derivations in current Lie algebras, particularly in the tensor product $L\otimes A$. Their findings extend previous work on functional identities and near-derivations [4,5]. Additionally, Chang, Chen, and Zhang [6] introduced $(\alpha ,\beta )$-derivations and G-derivations as broader generalizations and analyzed their algebraic structure using computational ideal theory.

This work examines a particular class of generalized derivations, D-derivations, which unify ordinary derivations and centroid maps. A D-derivation of a Lie algebra L is a linear map $\phi$ for which there exists a derivation D such that

$$\phi \left(\right[x,y\left]\right)=\left[\phi \right(x),y]+[x,D(y\left)\right]\forall x,y\in L.$$

This concept has appeared in earlier works such as [7,8], providing a unified framework that encompasses both derivations and centroids as special cases.

The investigation was prompted by computational experiments involving the rotation algebra $\mathfrak{so}\left(3\right)$ and the Lorentz algebra $\mathfrak{so}(1,3)$. In these instances, all D-derivations were found to be expressible as the sum of an inner derivation and a scalar multiple of the identity [9]. This observation raised the question of whether such a structural description applies to broader classes of Lie algebras. The study of D-derivations is both algebraically natural and physically significant. Within the context of symmetry analysis of differential equations, these derivations correspond to infinitesimal transformations that preserve the equation up to a source term represented by $D\left(y\right)$ (obtained by comparing the maps $T_y\left(x\right)=\phi \left([x,y]\right)-[\phi \left(x\right),y]$ with $T_y\left(x\right)=[x,D\left(y\right)]$). This framework unifies several generalizations of derivations encountered in deformation theory, integrable systems [10], and geometric mechanics, where concepts like centroids and generalized derivations are actively studied [11].

We demonstrate that the set ${Der}_D\left(L\right)$ of all D-derivations of a Lie algebra L forms a Lie sub-algebra of $\mathfrak{gl}\left(L\right)$. Furthermore, we establish the decomposition ${Der}_D\left(L\right)=Der\left(L\right)+C\left(L\right)$ for any Lie algebra L. We prove that $ZDer\left(L\right)=Der\left(L\right)\cap C\left(L\right)$, thereby identifying central derivations as those maps that are both derivations and centroids. For perfect Lie algebras L (i.e., $L=[L,L]$) and centerless Lie algebras L (i.e., $Z\left(L\right)=0$), we show that this sum is direct: ${Der}_D\left(L\right)=Der\left(L\right)\oplus C\left(L\right)$ as vector spaces. For centerless Lie algebras, we further refine the inclusion chain of derivation spaces. Specifically, we show that

$$ad\left(L\right)\subseteq Der\left(L\right)\subseteq {Der}_D\left(L\right)\subseteq QDer\left(L\right)\subseteq GenDer\left(L\right)\subseteq \mathfrak{gl}\left(L\right).$$

We establish a Lie algebra isomorphism ${Der}_D\left(L\right)\cong Der\left(L\right)⋉C\left(L\right)$ for any centerless Lie algebra L, where the centroid $C\left(L\right)$ forms an abelian ideal. In the semisimple case over an algebraically closed field, this semi-direct product becomes a direct sum: ${Der}_D\left(L\right)\cong L\oplus \mathbb{F}$. We present explicit matrix representations of D-derivations for specific examples, including $\mathfrak{so}\left(3\right)$, $\mathfrak{so}(1,3)$, $\mathfrak{aff}\left(1\right)$, and the Heisenberg algebra ${\mathfrak{h}}_3$. These examples validate the theoretical decompositions and illustrate computational techniques.

Our results generalize and refine several established statements in the literature, including those concerning $\delta$-derivations [12,13] and LR-algebra structures [14,15]. The semi-direct product structure demonstrates that the centroid $C\left(L\right)$ parameterizes scaling deformations of the Lie bracket, while $Der\left(L\right)$ captures infinitesimal automorphisms. This viewpoint is especially pertinent in deformation theory and the symmetry analysis of differential equations.

The structure of the paper is as follows. Section 1 introduces the background, motivation, and main contributions of this work. Section 2 reviews the necessary definitions and establishes the notation used throughout. Section 3 presents the main results, beginning with the decomposition of D-derivations into sums of derivations and centroids. A direct-sum decomposition for perfect and centerless Lie algebras is then established, along with a characterization and refinement of inclusion relations among derivations for centerless and semisimple Lie algebras. Special emphasis is given to semisimple and simple Lie algebras over an algebraically closed field. Section 4 provides concrete examples to validate and illustrate the theoretical findings. Section 5 concludes with a discussion and directions for future research. Section 6 concludes the paper.

We denote by $Der\left(L\right)$, $ad\left(L\right)$, $GenDer\left(L\right)$, $QDer\left(L\right)$, $C\left(L\right)$, $QC\left(L\right)$, ${Der}_D\left(L\right)$, and $ZDer\left(L\right)$ the sets of all derivations, inner derivations, generalized derivations, quasi-derivations, centroids, quasi-centroids, D-derivations, and central derivations of L, respectively.

2. Preliminary

Throughout the paper, suppose that $\mathbb{F}$ is an arbitrary field and L is a Lie algebra over $\mathbb{F}$. Denote by $Z\left(L\right)$ the center of L, $End\left(L\right)$ the set of all linear endomorphisms of L. Let $\mathfrak{gl}\left(L\right)$ be the general linear Lie algebra by defining the Lie bracket

$$[\phi ,\psi ]=\phi \circ \psi -\psi \circ \phi \forall \phi ,\psi \in \mathfrak{gl}\left(L\right),$$

where ∘ denotes the composition of linear endomorphisms.

Definition 1 

([1]). A linear endomorphism D of L is called a derivation if

$$D\left(\right[x,y\left]\right)=\left[D\right(x),y]+[x,D(y\left)\right]\forall x,y\in L.$$

For any $x\in L$, the linear endomorphism ${ad}_x:L\to L$ defined by ${ad}_x\left(y\right)=[x,y]$, $\forall y\in L$, is a derivation, which is called an inner derivation of L.

Definition 2 

([2]). Let φ be a linear endomorphism of L. Then,

1. 

φ is called a generalized derivation if there exist ${\phi }^{\prime },{\phi }^{\prime \prime }\in \mathfrak{gl}\left(L\right)$ such that

$$[\phi \left(x\right),y]+[x,{\phi }^{\prime }\left(y\right)]={\phi }^{\prime \prime }\left([x,y]\right)\forall x,y\in L.$$

2. 

φ is called a quasiderivation if there exists ${\phi }^{\prime }\in \mathfrak{gl}\left(L\right)$ such that

$$[\phi \left(x\right),y]+[x,\phi \left(y\right)]={\phi }^{\prime }\left([x,y]\right)\forall x,y\in L.$$

3. 

φ is called a centroid if

$$\left[\phi \right(x),y]=[x,\phi (y\left)\right]=\phi \left(\right[x,y\left]\right)\forall x,y\in L.$$

4. 

φ is called a quasicentroid if

$$\left[\phi \right(x),y]=[x,\phi (y\left)\right]\forall x,y\in L.$$

5. 

φ is called a central derivation if

$$\phi \left(L\right)\subseteq Z\left(L\right),\phi \left(\right[L,L\left]\right)=0.$$

Definition 3 

([8]). A linear endomorphism φ of L is called a D-derivation if there exists a derivation D of L such that

$$\phi \left(\right[x,y\left]\right)=\left[\phi \right(x),y]+[x,D(y\left)\right]\forall x,y\in L.$$

In [2], the authors prove that

$$ad\left(L\right)\subseteq Der\left(L\right)\subseteq QDer\left(L\right)\subseteq GenDer\left(L\right)\subseteq \mathfrak{gl}\left(L\right),$$

and $\mathrm{C}\left(L\right)\subseteq \mathrm{QC}\left(L\right)$ for centerless Lie algebras.

3. Main Results

In [2], the authors proved that $Der\left(L\right)+C\left(L\right)\subseteq QDer\left(L\right)$ for any Lie algebra L. In the following, we first establish that ${Der}_D\left(L\right)$ is a Lie algebra and then prove a stronger decomposition: ${Der}_D\left(L\right)=Der\left(L\right)+C\left(L\right)$. Moreover, we show that for perfect and centerless Lie algebras, this sum is direct.

3.1. Linear Space Decompositions

Theorem 1. 

${Der}_D\left(L\right)$ is a Lie subalgebra of $\mathfrak{gl}\left(L\right)$.

Proof. 

Let ${\phi }_1,{\phi }_2\in {Der}_D\left(L\right)$ be D-derivations with corresponding derivations $D_1,D_2\in Der\left(L\right)$, respectively, such that for all $x,y\in L$:

$$\begin{array}{cc}\hfill {\phi }_1\left([x,y]\right)& =[{\phi }_1\left(x\right),y]+[x,D_1\left(y\right)],\hfill \\ \hfill {\phi }_2\left([x,y]\right)& =[{\phi }_2\left(x\right),y]+[x,D_2\left(y\right)].\hfill \end{array}$$

Consider the Lie bracket $[{\phi }_1,{\phi }_2]={\phi }_1\circ {\phi }_2-{\phi }_2\circ {\phi }_1$. Then, for any $x,y\in L$:

$$\begin{array}{cc}\hfill [{\phi }_1,{\phi }_2]\left([x,y]\right)& ={\phi }_1\left({\phi }_2\left([x,y]\right)\right)-{\phi }_2\left({\phi }_1\left([x,y]\right)\right)\hfill \\ \hfill & ={\phi }_1([{\phi }_2\left(x\right),y]+[x,D_2\left(y\right)])-{\phi }_2([{\phi }_1\left(x\right),y]+[x,D_1\left(y\right)])\hfill \\ \hfill & =[{\phi }_1\left({\phi }_2\left(x\right)\right),y]+[{\phi }_2\left(x\right),D_1\left(y\right)]+[{\phi }_1\left(x\right),D_2\left(y\right)]+[x,D_1\left(D_2\left(y\right)\right)]\hfill \\ \hfill & -[{\phi }_2\left({\phi }_1\left(x\right)\right),y]-[{\phi }_1\left(x\right),D_2\left(y\right)]-[{\phi }_2\left(x\right),D_1\left(y\right)]-[x,D_2\left(D_1\left(y\right)\right)]\hfill \\ \hfill & =[[{\phi }_1,{\phi }_2]\left(x\right),y]+[x,[D_1,D_2]\left(y\right)].\hfill \end{array}$$

Since $[D_1,D_2]\in Der\left(L\right)$, we conclude that $[{\phi }_1,{\phi }_2]\in {Der}_D\left(L\right)$ with associated derivation $[D_1,D_2]$. Hence, ${Der}_D\left(L\right)$ is a Lie subalgebra of $\mathfrak{gl}\left(L\right)$. □

Since ${Der}_D\left(L\right)$ is a Lie algebra, it is clear that $Der\left(L\right)$ is a Lie subalgebra of ${Der}_D\left(L\right)$. We now establish a decomposition of D-derivations into ordinary derivations and centroids.

Theorem 2. 

Let L be a Lie algebra. Then,

$${Der}_D\left(L\right)=Der\left(L\right)+C\left(L\right).$$

Proof. 

We first prove the inclusion $Der\left(L\right)+C\left(L\right)\subseteq {Der}_D\left(L\right)$.

Let $D\in Der\left(L\right)$ and $\mathsf{\Gamma }\in C\left(L\right)$. Define $\phi =D+\mathsf{\Gamma }$. We verify that $\phi$ is a D-derivation for some D.

For all $x,y\in L$, we have

$$\begin{array}{cc}\hfill \phi \left(\right[x,y\left]\right)& =D\left(\right[x,y\left]\right)+\mathsf{\Gamma }\left(\right[x,y\left]\right)\hfill \\ & =\left[D\right(x),y]+[x,D(y\left)\right]+\left[\mathsf{\Gamma }\right(x),y](\mathrm{since}\mathsf{\Gamma }\in C(L\left)\right)\hfill \\ & =\left[D\right(x)+\mathsf{\Gamma }(x),y]+[x,D(y\left)\right]\hfill \\ & =\left[\phi \right(x),y]+[x,D(y\left)\right].\hfill \end{array}$$

Thus, $\phi \in {Der}_D\left(L\right)$. Hence, $Der\left(L\right)+C\left(L\right)\subseteq {Der}_D\left(L\right)$.

Now, we prove the reverse inclusion: ${Der}_D\left(L\right)\subseteq Der\left(L\right)+C\left(L\right)$.

Let $f\in {Der}_D\left(L\right)$. Then, there exists a derivation $D\in Der\left(L\right)$ such that for all $x,y\in L$:

$$f\left(\right[x,y\left]\right)=\left[f\right(x),y]+[x,D(y\left)\right].$$

Define the map $\mathsf{\Gamma }=f-D$. We claim that $\mathsf{\Gamma }\in C\left(L\right)$.

To verify this, we compute for all $x,y\in L$:

$$\begin{array}{cc}\hfill \mathsf{\Gamma }\left(\right[x,y\left]\right)& =f\left(\right[x,y\left]\right)-D\left(\right[x,y\left]\right)\hfill \\ & =\left(\left[f\right(x),y]+[x,D(y\left)\right]\right)-\left(\left[D\right(x),y]+[x,D(y\left)\right]\right)\hfill \\ & =\left[f\right(x),y]-\left[D\right(x),y]\hfill \\ & =\left[f\right(x)-D(x),y]=\left[\mathsf{\Gamma }\right(x),y].\hfill \end{array}$$

Similarly, using the skew-symmetry of the bracket and the definition of $\mathsf{\Gamma }$, we find

$$\begin{array}{cc}\hfill \mathsf{\Gamma }\left(\right[x,y\left]\right)& =-\mathsf{\Gamma }\left(\right[y,x\left]\right)=-\left[\mathsf{\Gamma }\right(y),x]=[x,\mathsf{\Gamma }(y\left)\right].\hfill \end{array}$$

Therefore, $\mathsf{\Gamma }\left(\right[x,y\left]\right)=\left[\mathsf{\Gamma }\right(x),y]=[x,\mathsf{\Gamma }(y\left)\right]$ for all $x,y\in L$, which implies $\mathsf{\Gamma }\in C\left(L\right)$.

Since $f=D+\mathsf{\Gamma }$ with $D\in Der\left(L\right)$ and $\mathsf{\Gamma }\in C\left(L\right)$, we conclude that

$$f\in Der\left(L\right)+C\left(L\right).$$

Hence, ${Der}_D\left(L\right)\subseteq Der\left(L\right)+C\left(L\right)$, and the equality follows. □

Theorem 3. 

$\mathrm{ZDer}\left(L\right)=\mathrm{Der}\left(L\right)\cap \mathrm{C}\left(L\right)$.

Proof. 

By definition, a central derivation satisfies

$$ZDer\left(L\right)\subseteq Der\left(L\right),ZDer\left(L\right)\subseteq C\left(L\right),$$

establishing the inclusion $ZDer\left(L\right)\subseteq Der\left(L\right)\cap C\left(L\right)$.

To prove the reverse inclusion, let $g\in Der\left(L\right)\cap C\left(L\right)$. We will show that g satisfies conditions of a central derivation.

Since $g\in C\left(L\right)$, for all $x,y\in L$, we have

$$g\left(\right[x,y\left]\right)=\left[g\right(x),y].$$

At the same time, since $g\in Der\left(L\right)$, the derivation property gives

$$g\left(\right[x,y\left]\right)=\left[g\right(x),y]+[x,g(y\left)\right].$$

Comparing these two expressions, we obtain

$$\left[g\right(x),y]=\left[g\right(x),y]+[x,g(y\left)\right].$$

Then, $[x,g(y\left)\right]=0$, $\forall x,y\in L$. This implies that $g\left(y\right)\in Z\left(L\right)$ for all $y\in L$, so $g\left(L\right)\subseteq Z\left(L\right)$.

Furthermore, using the definition of the centroid again, we have $g\left(\right[x,y\left]\right)=[x,g(y\left)\right]=0$, which shows that $g\left(\right[L,L\left]\right)=0$.

Therefore, g satisfies $g\left(L\right)\subseteq Z\left(L\right)$ and $g\left(\right[L,L\left]\right)=0$, proving that $g\in ZDer\left(L\right)$.

Hence, $Der\left(L\right)\cap C\left(L\right)\subseteq ZDer\left(L\right)$, and the equality $ZDer\left(L\right)=Der\left(L\right)\cap C\left(L\right)$ is established. □

Remark 1. 

For any centerless Lie algebra L, we have the following chain of inclusions:

$$ad\left(L\right)\subseteq Der\left(L\right)\subseteq {Der}_D\left(L\right)\subseteq QDer\left(L\right)\subseteq GenDer\left(L\right)\subseteq \mathfrak{gl}\left(L\right).$$

The inclusion $Der\left(L\right)\subseteq {Der}_D\left(L\right)$ follows from Theorem 2 and the fact that $Der\left(L\right)\subseteq Der\left(L\right)+C\left(L\right)={Der}_D\left(L\right)$. The inclusion ${Der}_D\left(L\right)\subseteq QDer\left(L\right)$ is obtained by setting ${\phi }^{\prime }=D$ in the definition of quasiderivations. The remaining inclusions are clear.

Theorem 4. 

For any perfect or centerless Lie algebra L, the space of D-derivations admits a direct sum decomposition

$${Der}_D\left(L\right)=Der\left(L\right)\oplus C\left(L\right)(\mathit{direct}\mathit{sum}\mathit{of}\mathit{vector}\mathit{spaces}).$$

Proof. 

Theorem 2 establishes that ${Der}_D\left(L\right)=Der\left(L\right)+C\left(L\right)$. To demonstrate that this sum is direct, it is sufficient to show that $Der\left(L\right)\cap C\left(L\right)=\left\{0\right\}$.

Let f be an arbitrary element in $Der\left(L\right)\cap C\left(L\right)$. The centroid property implies

$$f\left(\right[x,y\left]\right)=\left[f\right(x),y],\forall x,y\in L,$$

while the derivation property yields:

$$f\left(\right[x,y\left]\right)=\left[f\right(x),y]+[x,f(y\left)\right],\forall x,y\in L.$$

Comparing these identities yields $[x,f(y\left)\right]=0$ for all $x,y\in L$. This result implies that $f\left(y\right)$ lies in the center $Z\left(L\right)$ for every $y\in L$, so $f\left(L\right)\subseteq Z\left(L\right)$.

The proof proceeds by considering two cases:

• If L is perfect, then $L=[L,L]$, and we have

  $$f\left(L\right)=f\left(\right[L,L\left]\right)=\left[f\right(L),L]\subseteq \left[Z\right(L),L]=\left\{0\right\},$$

  where the last equality follows from the definition of the center. Therefore, $f=0$.
• If L is centerless, meaning $Z\left(L\right)=0$, then since $f\left(L\right)\subseteq Z\left(L\right)=0$, it follows that $f=0$.
  In both cases, it follows that $f=0$, confirming that $Der\left(L\right)\cap C\left(L\right)=\left\{0\right\}$.
  Therefore, ${Der}_D\left(L\right)$ decomposes as the direct sum $Der\left(L\right)\oplus C\left(L\right)$. □

Let L be a semisimple Lie algebra. Then, $Der\left(L\right)=ad\left(L\right)$ and $C\left(L\right)$ is the set of scalar multiples of the identity map if L is simple over an algebraically closed field of characteristic zero.

Corollary 1. 

For any semisimple Lie algebra L, the space of D-derivations decomposes as follows:

$${Der}_D\left(L\right)=ad\left(L\right)\oplus C\left(L\right).$$

Corollary 2. 

Let L be a simple Lie algebra over an algebraically closed field $\mathbb{F}$ of characteristic zero. Then, the space of D-derivations admits the decomposition:

$${Der}_D\left(L\right)=ad\left(L\right)\oplus \mathbb{F}{id}_L.$$

3.2. Inclusion Chains

In [2], the authors give that $Der\left(L\right)+C\left(L\right)\subseteq QDer\left(L\right)$ and the tower

$$L=ad\left(L\right)\subseteq Der\left(L\right)\subseteq QDer\left(L\right)\subseteq GenDer\left(L\right)\subseteq \mathfrak{gl}\left(L\right).$$

Theorem 5. 

For any semisimple Lie algebra L, the derivation spaces satisfy the following chain:

$$L\cong ad\left(L\right)=Der\left(L\right)⊊{Der}_D\left(L\right)\subseteq QDer\left(L\right)=GenDer\left(L\right)⊊\mathfrak{gl}\left(L\right).$$

Proof. 

Let L be a semisimple Lie algebra. It follows that $L\cong ad\left(L\right)$ from the adjoint representation, while the equality $Der\left(L\right)=ad\left(L\right)$ is a classical result for semisimple Lie algebras.

The strict inclusion $Der\left(L\right)⊊{Der}_D\left(L\right)$ is established by considering the decomposition ${Der}_D\left(L\right)=Der\left(L\right)+C\left(L\right)$ from Theorem 2. Since L is semisimple, its centroid $C\left(L\right)$ consists of scalar multiples of the identity map. The identity map lies in $C\left(L\right)$ but not in $Der\left(L\right)$, confirming the proper inclusion.

The relation ${Der}_D\left(L\right)\subseteq QDer\left(L\right)$ follows from Remark 1, utilizing the fact that semisimple Lie algebras are centerless. Furthermore, the equality $QDer\left(L\right)=GenDer\left(L\right)$ is a direct consequence of Theorem 5.4 in [2], which applies to perfect centerless Lie algebras.

To establish the final strict inclusion $GenDer\left(L\right)⊊\mathfrak{gl}\left(L\right)$, observe that for semisimple Lie algebras of rank at least 2, we have the decomposition $QDer\left(L\right)=ad\left(L\right)\oplus \mathbb{F}{id}_L$. This forms a proper subspace of $\mathfrak{gl}\left(L\right)$, as there exist linear endomorphisms that are neither derivations nor scalar multiples of the identity. Although the special case ${\mathrm{sl}}_2\left(\mathbb{F}\right)$ satisfies $QDer\left(L\right)=\mathfrak{gl}\left(L\right)$, this equality fails for higher rank semisimple Lie algebras, thereby confirming the strict inclusion in general. □

3.3. The Lie Algebra Structure of D-Derivations

Theorem 6. 

Let L be a Lie algebra with trivial center $Z\left(L\right)=0$. Then, there exists a Lie algebra isomorphism

$${Der}_D\left(L\right)\cong Der\left(L\right)⋉C\left(L\right),$$

The Lie bracket for $D_1,D_2\in Der\left(L\right)$ and ${\mathsf{\Gamma }}_1,{\mathsf{\Gamma }}_2\in C\left(L\right)$ is given by

$$\left[(D_1,{\mathsf{\Gamma }}_1),(D_2,{\mathsf{\Gamma }}_2)\right]=\left([D_1,D_2],D_1·{\mathsf{\Gamma }}_2-D_2·{\mathsf{\Gamma }}_1\right),$$

where the action of $Der\left(L\right)$ on $C\left(L\right)$ is given by the adjoint action $D·\mathsf{\Gamma }=[D,\mathsf{\Gamma }]=D\circ \mathsf{\Gamma }-\mathsf{\Gamma }\circ D$. Moreover, $C\left(L\right)$ is an abelian ideal of ${Der}_D\left(L\right)$.

Proof. 

Since $Z\left(L\right)=0$, for any $\phi \in {\mathrm{Der}}_D\left(L\right)$, the derivation D appearing in the defining equation $\phi \left(\right[x,y\left]\right)=\left[\phi \right(x),y]+[x,D(y\left)\right]$ is uniquely determined. This allows us to define a map $\pi :{\mathrm{Der}}_D\left(L\right)\to \mathrm{Der}\left(L\right)$ by $\pi \left(\phi \right)=D$, which is a well-defined Lie algebra homomorphism. Its kernel is precisely

$$ker\pi =\{\phi \in {Der}_D\left(L\right)\mid D=0\}=C\left(L\right).$$

Indeed, if $D=0$, then $\phi \left(\right[x,y\left]\right)=\left[\phi \right(x),y]$ for all $x,y\in L$, which is exactly the condition for $\phi$ to be a centroid. Thus, we have a short exact sequence of Lie algebras:

$$0⟶C\left(L\right)⟶{Der}_D\left(L\right)\stackrel{\pi }{⟶}Der\left(L\right)⟶0.$$

The sequence splits via the inclusion $\iota :Der\left(L\right)\hookrightarrow {Der}_D\left(L\right)$ given by $\iota \left(D\right)=D$. Thus, ${Der}_D\left(L\right)$ is isomorphic as a Lie algebra to the semi-direct product $Der\left(L\right)⋉C\left(L\right)$. Since $C\left(L\right)$ is abelian, its internal Lie bracket vanishes; the bracket formula follows.

Now we show that $C\left(L\right)$ is abelian. For any ${\mathsf{\Gamma }}_1,{\mathsf{\Gamma }}_2\in C\left(L\right)$ and $x,y\in L$, we have $[{\mathsf{\Gamma }}_1{\mathsf{\Gamma }}_2\left(x\right)-{\mathsf{\Gamma }}_2{\mathsf{\Gamma }}_1\left(x\right),y]=0$ for all $x,y\in L$.

$$[{\mathsf{\Gamma }}_1\left({\mathsf{\Gamma }}_2\left(x\right)\right)-{\mathsf{\Gamma }}_2\left({\mathsf{\Gamma }}_1\left(x\right)\right),y]=0\forall x,y\in L.$$

Since $Z\left(L\right)=0$, it follows that ${\mathsf{\Gamma }}_1\circ {\mathsf{\Gamma }}_2={\mathsf{\Gamma }}_2\circ {\mathsf{\Gamma }}_1$, so $C\left(L\right)$ is commutative as an associative algebra and is abelian as a Lie algebra. Therefore, $C\left(L\right)$ is an abelian ideal of ${Der}_D\left(L\right)$. □

Corollary 3. 

If L is a finite-dimensional semisimple Lie algebra over an algebraically closed field $\mathbb{F}$ of characteristic zero, then

$${Der}_D\left(L\right)\cong L\oplus \mathbb{F},$$

where $\mathbb{F}$ is a one-dimensional central ideal spanned by the identity map.

Proof. 

For a semisimple Lie algebra over an algebraically closed field of characteristic zero, we have $Der\left(L\right)=ad\left(L\right)\cong L$ and $C\left(L\right)=\mathbb{F}·{id}_L$. Moreover, the action of $Der\left(L\right)$ on $C\left(L\right)$ is trivial because

$$[{ad}_x,\lambda id]=0\forall x\in L,\lambda \in \mathbb{F}.$$

Hence, the semi-direct product reduces to a direct sum:

$${Der}_D\left(L\right)\cong L\oplus \mathbb{F}.$$

The one-dimensional factor $\mathbb{F}·{id}_L$ is central in ${Der}_D\left(L\right)$. □

4. Examples of $\mathit{D}$-Derivations

4.1. Example 1: D-Derivations on the Rotation Algebra

We begin by examining the real special orthogonal Lie algebra $\mathfrak{so}\left(3\right)$, which is the Lie algebra of the three-dimensional rotation group. Let $\mathcal{A}=\{e_1,e_2,e_3\}$ denote its standard basis, with the Lie brackets:

$$[e_1,e_2]=e_3,[e_2,e_3]=e_1,[e_3,e_1]=e_2.$$

Since $\mathfrak{so}\left(3\right)$ is a simple Lie algebra, it has trivial center $Z\left(\mathfrak{so}\right(3\left)\right)=0$ and satisfies $Der\left(\mathfrak{so}\right(3\left)\right)=ad\left(\mathfrak{so}\right(3\left)\right)\cong \mathfrak{so}\left(3\right)$. The derivation algebra has dimension $dimDer\left(\mathfrak{so}\right(3\left)\right)=3$. The centroid is one-dimensional: $C\left(\mathfrak{so}\right(3\left)\right)=\mathbb{R}id$, so $dimC\left(\mathfrak{so}\right(3\left)\right)=1$.

The real matrix representation of $D\in Der\left(\mathfrak{so}\right(3\left)\right)$ with respect to the basis $\mathcal{A}$ takes the form:

$${\left[D\right]}_{\mathcal{A}}=\left(\begin{array}{ccc}0& -c& b\\ c& 0& -a\\ -b& a& 0\end{array}\right),$$

where $a,b,c\in \mathbb{R}$.

By explicit computation, we find that the corresponding D-derivation $\phi \in {Der}_D\left(\mathfrak{so}\left(3\right)\right)$ admits the matrix representation:

$${\left[\phi \right]}_{\mathcal{A}}=\lambda I_3+{\left[D\right]}_{\mathcal{A}}=\left(\begin{array}{ccc}\lambda & -c& b\\ c& \lambda & -a\\ -b& a& \lambda \end{array}\right),$$

where $I_3$ is the $3\times 3$ identity matrix and $\lambda \in \mathbb{R}$.

This matrix representation illustrates the direct sum decomposition:

$${Der}_D\left(\mathfrak{so}\left(3\right)\right)=ad\left(\mathfrak{so}\left(3\right)\right)\oplus \mathbb{R}id=Der\left(\mathfrak{so}\left(3\right)\right)\oplus C\left(\mathfrak{so}\left(3\right)\right).$$

Since $\mathfrak{so}\left(3\right)$ is semisimple, the centroid $C\left(\mathfrak{so}\right(3\left)\right)=\mathbb{R}id$ is central in ${Der}_D\left(\mathfrak{so}\left(3\right)\right)$. Therefore, the semi-direct product of Theorem 6 reduces to a direct sum:

$${Der}_D\left(\mathfrak{so}\left(3\right)\right)\cong \mathfrak{so}\left(3\right)\oplus \mathbb{R},$$

with total dimension $dim{Der}_D\left(\mathfrak{so}\left(3\right)\right)=3+1=4$.

4.2. Example 2: D-Derivations on the Lorentz Algebra $\mathfrak{so}(1,3)$

We now investigate the real Lorentz Lie algebra $\mathfrak{so}(1,3)$, which describes the symmetries of spacetime in special relativity. Let $\mathcal{B}=\{e_1,\dots ,e_6\}$ be its standard basis, where $\{e_1,e_2,e_3\}$ generate spatial rotations and $\{e_4,e_5,e_6\}$ generate Lorentz boosts. Please refer to Table 1 for Lie brackets of $\mathcal{B}$.

Table 1. Lie brackets of $\mathcal{B}$.

$[{\mathit{e}}_{\mathit{i}},{\mathit{e}}_{\mathit{j}}]$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$	${\mathit{e}}_6$
$e_1$	0	$e_3$	$-e_2$	0	$e_6$	$-e_5$
$e_2$	$-e_3$	0	$e_1$	$-e_6$	0	$e_4$
$e_3$	$e_2$	$-e_1$	0	$e_5$	$-e_4$	0
$e_4$	0	$e_6$	$-e_5$	0	$-e_3$	$e_2$
$e_5$	$-e_6$	0	$e_4$	$e_3$	0	$-e_1$
$e_6$	$e_5$	$-e_4$	0	$-e_2$	$e_1$	0

The Lorentz algebra $\mathfrak{so}(1,3)$ is semisimple with trivial center, satisfying $Z\left(\mathfrak{so}\right(1,3\left)\right)=0$ and $Der\left(\mathfrak{so}\right(1,3\left)\right)=ad\left(\mathfrak{so}\right(1,3\left)\right)\cong \mathfrak{so}(1,3)$. Thus $dimDer\left(\mathfrak{so}\right(1,3\left)\right)=6$. A concrete description can be given by noting that the complexification satisfies $\mathfrak{so}{(1,3)}_{\mathbb{C}}\cong {\mathfrak{sl}}_2\left(\mathbb{C}\right)\oplus {\mathfrak{sl}}_2\left(\mathbb{C}\right)$. Over $\mathbb{R}$, the centroid takes the form $\mathrm{C}\left(\mathfrak{so}\right(1,3\left)\right)=\lambda id+kJ\mid \lambda ,k\in \mathbb{R}$, where J is a linear map corresponding to a complex structure on the underlying vector space, satisfying $J^2=-id$. Therefore, $dimC\left(\mathfrak{so}\right(1,3\left)\right)=2$.

Let $D\in Der\left(\mathfrak{so}\right(1,3\left)\right)$. A straightforward calculation reveals that real matrix representations of derivation D, centroid $\mathsf{\Gamma }$, and D-derivation $\phi$ of $\mathfrak{so}(1,3)$ with respect to the basis $\mathcal{B}$ have the following forms:

$${\left[D\right]}_{\mathcal{B}}=\left(\begin{array}{cc}A& B\\ -B& A\end{array}\right),{\left[\mathsf{\Gamma }\right]}_{\mathcal{B}}=\left(\begin{array}{cc}\lambda I_3& k{I}_3\\ -k{I}_3& \lambda I_3\end{array}\right),{\left[\phi \right]}_{\mathcal{B}}={\left[D\right]}_{\mathcal{B}}+{\left[\mathsf{\Gamma }\right]}_{\mathcal{B}},$$

where A, B are $3\times 3$ skew-symmetric matrices and $I_3$ is the $3\times 3$ identity matrix.

This exemplifies the decomposition ${Der}_D\left(\mathfrak{so}(1,3)\right)=Der\left(\mathfrak{so}(1,3)\right)+C\left(\mathfrak{so}(1,3)\right)$. Since $\mathfrak{so}(1,3)$ is perfect and centerless, this sum is direct. Moreover, the centroid is central in ${Der}_D\left(\mathfrak{so}(1,3)\right)$, so the semi-direct product of Theorem 6 reduces to a direct sum:

$${Der}_D\left(\mathfrak{so}(1,3)\right)\cong \mathfrak{so}(1,3)\oplus {\mathbb{R}}^2,$$

with total dimension $dim{Der}_D\left(\mathfrak{so}(1,3)\right)=6+2=8$.

For the real simple Lie algebra $\mathfrak{so}\left(3\right)$, although the base field $\mathbb{R}$ is not algebraically closed, one still has $C\left(\mathfrak{so}\right(3\left)\right)=\mathbb{R}·id$. This is a special property of this particular algebra. In contrast, for $\mathfrak{so}(1,3)$, the centroid is two-dimensional, reflecting the fact that $\mathfrak{so}{(1,3)}_{\mathbb{C}}\cong {\mathfrak{sl}}_2\left(\mathbb{C}\right)\oplus {\mathfrak{sl}}_2\left(\mathbb{C}\right)$ as complex abstract algebraic structures.

4.3. Example 3: D-Derivations on the Two-Dimensional Non-Abelian Lie Algebra

Consider the two-dimensional non-abelian Lie algebra $\mathfrak{aff}\left(1\right)$ with basis $\{e_1,e_2\}$ and the Lie bracket

$$[e_1,e_2]=e_1.$$

This Lie algebra is solvable and has trivial center $Z\left(\mathfrak{aff}\right(1\left)\right)=0$.

We first compute its derivation algebra $Der\left(\mathfrak{aff}\right(1\left)\right)$. Let $D\in Der\left(\mathfrak{aff}\right(1\left)\right)$ be represented by

$$D\left(e_1\right)=a{e}_1+b{e}_2,D\left(e_2\right)=c{e}_1+d{e}_2.$$

Applying the derivation property to the bracket $[e_1,e_2]=e_1$, we obtain

$$D\left(e_1\right)=[D\left(e_1\right),e_2]+[e_1,D\left(e_2\right)]=a{e}_1+d{e}_1=(a+d)e_1.$$

Hence, $b=0$ and $a=a+d$, i.e., $d=0$. Therefore, every derivation takes the form

$$D\left(e_1\right)=a{e}_1,D\left(e_2\right)=c{e}_1,$$

with $a,c\in \mathbb{F}$. Consequently, $dimDer\left(\mathfrak{aff}\right(1\left)\right)=2$. Moreover, since ${ad}_{e_1}\left(e_1\right)=0$, ${ad}_{e_1}\left(e_2\right)=e_1$, and ${ad}_{e_2}\left(e_1\right)=-e_1$, ${ad}_{e_2}\left(e_2\right)=0$, we see that $Der\left(\mathfrak{aff}\right(1\left)\right)=ad\left(\mathfrak{aff}\right(1\left)\right)$.

Next, we determine the centroid $C\left(\mathfrak{aff}\right(1\left)\right)$. For $\mathsf{\Gamma }\in C\left(\mathfrak{aff}\right(1\left)\right)$, write

$$\mathsf{\Gamma }\left(e_1\right)=\alpha e_1+\beta e_2,\mathsf{\Gamma }\left(e_2\right)=\gamma e_1+\delta e_2.$$

Using the centroid condition $\mathsf{\Gamma }\left([e_1,e_2]\right)=[\mathsf{\Gamma }\left(e_1\right),e_2]=[e_1,\mathsf{\Gamma }\left(e_2\right)]$, we obtain

$$\begin{array}{cc}\hfill \mathsf{\Gamma }\left(e_1\right)& =[\alpha e_1+\beta e_2,e_2]=\alpha e_1,\hfill \\ \hfill \mathsf{\Gamma }\left(e_1\right)& =[e_1,\gamma e_1+\delta e_2]=\delta e_1.\hfill \end{array}$$

Thus, $\beta =0$ and $\alpha =\delta$. Hence, every centroid is of the form

$$\mathsf{\Gamma }\left(e_1\right)=\alpha e_1,\mathsf{\Gamma }\left(e_2\right)=\gamma e_1+\alpha e_2,$$

with $\alpha ,\gamma \in \mathbb{F}$, so $dimC\left(\mathfrak{aff}\right(1\left)\right)=2$.

Now let $\phi \in {Der}_D\left(\mathfrak{aff}\left(1\right)\right)$ be a D-derivation. Write

$$\phi \left(e_1\right)=a_{\phi }{e}_1+b_{\phi }{e}_2,\phi \left(e_2\right)=c_{\phi }{e}_1+d_{\phi }{e}_2.$$

By definition, there exists a derivation D with $D\left(e_1\right)=a{e}_1$, $D\left(e_2\right)=c{e}_1$ such that for all $x,y\in \mathfrak{aff}\left(1\right)$,

$$\phi \left(\right[x,y\left]\right)=\left[\phi \right(x),y]+[x,D(y\left)\right].$$

Applying this condition to the pairs $(x,y)=(e_1,e_2)$ and $(x,y)=(e_2,e_1)$ yields

$$\begin{array}{cc}\hfill \phi \left(e_1\right)& =[\phi \left(e_1\right),e_2]+[e_1,D\left(e_2\right)],\hfill \\ \hfill \phi (-e_1)& =[\phi \left(e_2\right),e_1]+[e_2,D\left(e_1\right)].\hfill \end{array}$$

Explicit computation gives

$$a_{\phi }{e}_1+b_{\phi }{e}_2=a_{\phi }{e}_1,-a_{\phi }{e}_1=-d_{\phi }{e}_1+a{e}_1.$$

Hence, $b_{\phi }=0$ and $d_{\phi }=a_{\phi }+a$. Moreover, the derivation D associated to $\phi$ is uniquely determined by the formulas (obtained by comparing the maps $T_y\left(x\right)=\phi \left([x,y]\right)-[\phi \left(x\right),y]$ with $T_y\left(x\right)=[x,D\left(y\right)]$)

$$a=a_{\phi }-d_{\phi },c=c_{\phi }.$$

Thus, $D\left(e_1\right)=(a_{\phi }-d_{\phi })e_1$, $D\left(e_2\right)=c_{\phi }{e}_1$.

Consequently, $dim{Der}_D\left(\mathfrak{aff}\left(1\right)\right)=4$ (parameters $a_{\phi },c_{\phi },d_{\phi }$ and, for instance, a or c, but note that $d_{\phi }$ is already a parameter and a is expressed in terms of $a_{\phi }$ and $d_{\phi }$). Moreover, the decomposition $\phi =D+\mathsf{\Gamma }$ with D as above and $\mathsf{\Gamma }=\phi -D$ yields a centroid:

$$\mathsf{\Gamma }\left(e_1\right)=d_{\phi }{e}_1,\mathsf{\Gamma }\left(e_2\right)=d_{\phi }{e}_2,$$

i.e., $\mathsf{\Gamma }=d_{\phi }·id$. This illustrates the semi-direct product structure of Theorem 6:

$${Der}_D\left(\mathfrak{aff}\left(1\right)\right)\cong Der\left(\mathfrak{aff}\left(1\right)\right)⋉C\left(\mathfrak{aff}\left(1\right)\right).$$

Notice that in this example the centroid is not central in ${Der}_D\left(\mathfrak{aff}\left(1\right)\right)$; indeed, for a non-zero $\mathsf{\Gamma }$ in the general centroid, the commutator $[D,\mathsf{\Gamma }]$ need not vanish. Hence, the semi-direct product does not reduce to a direct sum in contrast to the semisimple cases of Examples 1 and 2.

4.4. Example 4: D-Derivations on the Heisenberg Algebra

Consider the three-dimensional Heisenberg Lie algebra ${\mathfrak{h}}_3$ with basis $\{e_1,e_2,e_3\}$ and bracket

$$[e_1,e_2]=e_3,[e_1,e_3]=[e_2,e_3]=0.$$

The center is $Z\left({\mathfrak{h}}_3\right)=\mathbb{R}{e}_3$.

It is known that $Der\left({\mathfrak{h}}_3\right)$ is six-dimensional and solvable, while $C\left({\mathfrak{h}}_3\right)$ consists of maps of the form

$$\mathsf{\Gamma }\left(e_1\right)=\alpha e_1+\beta e_2,\mathsf{\Gamma }\left(e_2\right)=\gamma e_1+\delta e_2,\mathsf{\Gamma }\left(e_3\right)=(\alpha +\delta )e_3,$$

where $\alpha ,\beta ,\gamma ,\delta \in \mathbb{R}$. Thus, $dimC\left({\mathfrak{h}}_3\right)=4$.

By Theorem 6, we have

$${Der}_D\left({\mathfrak{h}}_3\right)\cong Der\left({\mathfrak{h}}_3\right)⋉C\left({\mathfrak{h}}_3\right),$$

with $dim{Der}_D\left({\mathfrak{h}}_3\right)=6+4=10$. This illustrates that, even for a non-semisimple Lie algebra, the space of D-derivations inherits a rich structure combining derivations and centroids into a semidirect product.

5. Discussion

This work shows that D-derivations of a Lie algebra form a Lie algebra, given by the sum of the spaces of derivations and centroids. Central derivations are the intersection of these spaces, unifying several previously studied derivation types.

For any Lie algebra, the decomposition ${Der}_D\left(L\right)=Der\left(L\right)+C\left(L\right)$ holds, while the direct sum decomposition ${Der}_D\left(L\right)=Der\left(L\right)\oplus C\left(L\right)$ applies only to perfect and centerless Lie algebras. This result clarifies and expands previous research on the hierarchy of derivation types, such as Leger and Luks’s analysis [2] of generalized derivations and quasi-derivations. The placement of D-derivations within this hierarchy emerges clearly, illuminating a more general structural pattern.

For semisimple Lie algebras, a concise decomposition ${Der}_D\left(L\right)=ad\left(L\right)\oplus C\left(L\right)$ is derived. For simple Lie algebras over algebraically closed fields, this further reduces to ${Der}_D\left(L\right)=ad\left(L\right)\oplus \mathbb{F}{id}_L$. These results demonstrate that D-derivations serve as a natural bridge between classical derivation theory and centroid maps.

The numerical examples of $\mathfrak{so}\left(3\right)$ and $\mathfrak{so}(1,3)$ concretely illustrate the computation of D-derivations. The matrix representations reveal how D-derivations split into antisymmetric parts (derivations) and symmetric parts (centroids), thus providing tangible insight into the underlying algebraic structure.

The semi-direct product structure ${Der}_D\left(L\right)\cong Der\left(L\right)⋉C\left(L\right)$ for centerless Lie algebras yields a deeper algebraic understanding. The centroid $C\left(L\right)$ forms an abelian ideal that quantifies scaling deformations of the Lie bracket, while $Der\left(L\right)$ encodes the infinitesimal automorphisms. This structure is particularly significant in deformation theory and the study of symmetries in differential equations, where extensions of symmetry algebras by central or scaling terms frequently arise in the context of conformal transformations or mass-like perturbations. For instance, the decomposition ${Der}_D\left(\mathfrak{so}(1,3)\right)\cong \mathfrak{so}(1,3)\oplus {\mathbb{R}}^2$ corresponds to enlarging the Lorentz algebra by a two-dimensional center, which may be linked to conformal symmetries or scale invariance in relativistic field equations.

Future research will generalize these findings to other non-associative abstract algebraic structures and examine connections with various extensions of the notion of a derivation. Specifically, the following avenues are identified:

• Investigate D-derivations in Jacobi–Jordan algebras, commutative algebras satisfying the Jordan identity that emerge in differential geometry and mathematical physics. Recent work by Baklouti and Benayadi [16] on symplectic Jacobi–Jordan algebras reveals rich structures that may permit comparable decomposition theorems for their derivation spaces.
• Explore how the framework of D-derivations can unify and extend various existing generalizations of derivations. In particular, the $\delta$-derivations introduced by Filippov [12,13] and the LR-algebra structures examined by Burde et al. [14,15] may be interpreted within the D-derivation framework. Similarly, the $(\alpha ,\beta ,\gamma )$-derivations of Hirvnak and Novotný [17] could be naturally understood through the decomposition theorems, offering a more coherent perspective on these diverse concepts.
• Study D-derivations in infinite-dimensional Lie algebras and Leibniz algebras, as well as their relevance to representation theory and mathematical physics, especially regarding symmetry algebras in field theories and integrable systems.

6. Conclusions

This study investigates the structure of D-derivations in Lie algebras and establishes decomposition theorems that relate them to ordinary derivations and centroids. The results provide a unified framework for several known classes of derivations and offer new insights into the abstract algebraic structure of derivation spaces. The presented examples demonstrate the practical applicability of these theoretical findings. Future research will aim to extend these results to infinite-dimensional Lie algebras and to explore their connections with representation theory and mathematical physics.