Derivation algebra of direct sum of lie algebras

Abstract

Let $L_1$ and $L_2$ be two finite dimensional Lie algebras on arbitrary field F with no common direct factor and $L=L_1\oplus L_2$. In this article, we express the structure and dimension of derivation algebra of $L$, $Der(L)$, and some of their subalgebras in terms of $Der(L_1)$, $Der(L_2)$, $Hom(L_1,Z(L_2))$, and $Hom(L_2,Z(L_1))$.

Keywords:

• Derivation
• Direct Sum
• Subalgebra

Maths Subject classifications:

• primary16w25
• secondary17b4017B99

PUBLIC INTEREST STATEMENT

There are many results on the automorphism of some of algebraic structures such as groups, Lie rings, and Lie algebras. The theory of Lie algebras is one of the most important parts of algebras. There are many other connections between the group theory and the theory of Lie algebras. Many papers make an attempt to generalize the results on finite p-groups to the theory of Lie algebras. Although there are some sporadic results for the Lie algebra that do not coincide with the results for groups. However, there are analogies between groups and Lie algebras, but these analogies are not completely identical and most of them should be checked carefully.

The study of the set of all commuting derivations, for this reason, is interesting that in spite of this in groups is not generally a subgroup from the automorphisms group but in Lie algebras, it is always a subalgebra of the derivations algebra.

1. Introduction

Lie algebras were first discovered by Sophus Lie (1842–1899) when he attempted to classify certain ”smooth” subgroups of general linear groups. The groups he considered are now called Lie groups. By taking the tangent space at the identity element of such a group, he obtained the Lie algebra and hence the problems on groups can be reduced to problems on Lie algebras so that it becomes more tractable. Lie algebra is applied in different domains of physics and mathematics, such as spectroscopy of molecules, atoms, nuclei, hadrons, hyperbolic, and stochastic differential equations. After the introduction of fuzzy sets by L. Zadeh (Zadeh, 1965), various notions of higher-order fuzzy sets have been proposed. Fuzzy and anti fuzzy Lie ideals in Lie algebras have been studied in (Akram, 2006; Davvaz, 2001; Keyun, Quanxi, & Chaoping, 2001; Kim & Lee, 1998).

Throughout this article, Lie algebras are considered finite dimensional.

Let $L$ be a Lie algebra over field $F$ with bracket [–,–]. A derivation of $L$ is an $F$-linear transformation $\alpha :L\to L$ such that $\alpha ([x,y])=[\alpha (x),y]+[x,\alpha (y)]$ for all $x,y\in L$. Set of all derivations L, by given bracket $[\alpha ,\beta ]=\alpha \beta -\beta \alpha$, where $\alpha$ and $\beta$ are derivations of $L$, forms a Lie algebra that we denote it by $Der(L)$.

Suppose that $L=L_1\oplus L_2$ is a direct sum of Lie algebras $L_1$ and $L_2$. First, we provide some of the symbols which we need. We denote inclusion maps by $i_{L_1}$ and $i_{L_2}$ from $L_1$ and $L_2$ into $L$, respectively, and projection maps by ${\pi }_{L_1}$ and ${\pi }_{L_2}$ from $L$ into $L_1$ and $L_2$, respectively.

For $\theta \in gl(L)=T(L,L)$ we put $\alpha ={\pi }_{L_1}\theta i_{L_1}$, $\beta ={\pi }_{L_1}\theta i_{L_2}$, $\gamma ={\pi }_{L_2}\theta i_{L_1}$, and $\delta ={\pi }_{L_2}\theta i_{L_2}$. Then $\alpha \in gl(L_1)$, $\beta \in T(L_2,L_1)$, $\gamma \in T(L_1,L_2)$, and $\delta \in gl(L_2)$, where $T(L_1,L_2)$ is the set of all $F$-linear transformations from $L_1$ to $L_2$.

Now assume that

$\text{{mathcal M} = left{ {left({matrix{ alpha \& beta cr gamma \& delta cr } } right)left|matrix{ {alpha in gl({L\_1})} \& {beta in T({L\_2},Z({L\_1}))} cr {gamma in T({L\_1},Z({L\_2}))} \& {delta in gl({L\_2})} cr } } right}.}$

It is easy to see that $\mathcal{M}$ forms a Lie algebra over field $F$ and $\left(\begin{array}{cc}{\pi }_{L_1}\theta i_{L_1}& {\pi }_{L_1}\theta i_{L_2}\\ {\pi }_{L_2}\theta i_{L_1}& {\pi }_{L_2}\theta i_{L_2}\end{array}\right)\in \mathcal{M}$. Conversely, suppose that $\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\in \mathcal{M}$. We define the map $\theta :L\to L$ by $\theta (x,y)=(\alpha (x)+\beta (y),\gamma (x)+\delta (y))$, where $x\in L_1$ and $y\in L_2$; then $\theta \in gl(L)$. It is easy to check that $f:gl(L)\to \mathcal{M}$, which defined by $f(\theta )=\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)$ is a Lie isomorphism. Thus, we have the following result.

Proposition 1.1. If $L=L_1\oplus L_2$, then $gl(L)\cong \mathcal{M}$.

Consider an arbitrary Lie algebra $L$ and an arbitrary abelian Lie algebra $A$. Notations $Z(L)$, $L^2$, $Hom(L,A)$, and $T(L,A)$ denote the center $L$, derived subalgebra $L$, set of all lie homomorphism from $L$ into $A$, and set of all linear transformations from $L$ into $A$, respectively.

Let $L_1$ and $L_2$ be two finite dimensional Lie algebras with no common direct factor, and let $L=L_1\oplus L_2$. The main aim of this article is to express the structure and dimensional $Der(L)$ in terms of $Der(L_1)$, $Der(L_2)$, and Lie algebras of central homomorphisms $Hom(L_1,Z(L_2))$ and $Hom(L_2,Z(L_1))$. Note that, before in groups theory, the structure and order automorphisms group of $G$ had been investigated, which $G$ is a group in the form of direct product of two finite groups; see (Bidwell, Curran, & McCaughan, 2006). Therefore, this is our main theorem.

Theorem 1.2. Let $L=L_1\oplus L_2$ be direct sum of two Lie algebras of finite dimensional that $L_1$ and $L_2$ do not have nontrivial common direct factor, and let

$\text{{mathcal A} = left{ {left({matrix{ alpha \& beta cr gamma \& delta cr } } right)left|matrix{ {alpha in Der({L\_1})} \& {beta in Hom({L\_2},Z({L\_1}))} cr {gamma in Hom({L\_1},Z({L\_2}))} \& {delta in Der({L\_2})} cr } } right}.}$

Then $Der(L)\cong \mathcal{A}$; furthermore

$dimDer(L)=dimDer(L_1)+dimDer(L_2)+dimHom(L_1,Z(L_2))+dimHom(L_2,Z(L_1)).$

In Section 2, among the proof of Theorem 1.2, we get the structure and dimension of some subalgebras $Der(L)$. In the last section, we create conditions that under it $Der(L_1\oplus L_2)\cong Der(L_1)\oplus Der(L_2).$

2. Main result

Let $L$ be a Lie algebra, and let $De{r}_c(L)$ be the set all of derivations $\alpha$ from $Der(L)$ that $\alpha (x)\in \left[x,L\right]$ for all $x\in L$, which is subalgebra of derivation algebra $L.$ A derivation of $\alpha$ from $L$ is called ID-derivation when $\alpha (x)\in L^2$ for all $x\in L$. Set all of ID-derivations are denoted by ID(L). Also the set all of $\alpha \in ID(L)$ that sends elements of $Z(L)$ to zero is denoted by ID${ }^{\ast }(L)$.

A derivation $\alpha$ from $L$ is central if $\alpha$ commutes with all of inner derivations from $L$, or equivalently if $\alpha (x)$ lies in center of $L$ for all $x\in L$. Central derivations of $L$ are ideals of $Der(L)$that we denote it by Der${ }_z(L)$.

It is obviously that always

(1) $ad(L)\subseteq De{r}_c(L)\subseteq I{D}^{\ast }(L)\subseteq ID(L)\subseteq Der(L)\subseteq gl(L)$(1)

wherein $ad(L)$ the ideal of $Der(L)$ consists the set all of inner derivations.

Many authors investigated characterizing Lie algebras with the help of the above subalgebras of $Der(L)$. For example, Tôgô (Tôgô, 1964) characterized Lie algebras of $L$ over fields with $char(L)=0$ and $Z(L)\ne \{0\}$ that $ad(L)=Der(L)$; he also proved $De{r}_z(L)=Der(L)$ if only if $L$ is abelian. Sheikh-Mohseni et al. (Saeedi & Sheikh-Mohseni, 2015; Sheikh-Mohseni, Saeedi, & Badrkhani Asl, 2015) found conditions for Lie algebra of $L$ that $ad(L)=De{r}_c(L)$ or $De{r}_c(L)=ID(L)$. For more information about this subalgebras, refer to (Saeedi & Sheikh-Mohseni, 2018; Tôgô, 1955, 1961, 1964, 1967). Following lemmas are helpful to prove Theorem 1.2. The proof of the following lemma is straightforward.

Lemma 2.1 Let $L$ be a Lie algebra, and let $A$ be an abelian Lie algebra . Then

(i) $Hom(L,A)$ is an abelian Lie algebra with the following bracket:

$\left[\alpha ,\beta \right](x)=\left[\alpha (x),\beta (x)\right]\mathrm{\forall }\alpha ,\beta \in Hom(L,A),x\in L.$

(ii) $T(L/L^2,A)=Hom(L/L^2,A)\cong Hom(L,A).$

Lemma 2.2. Let $L=L_1\oplus L_2$ be such that L₁ and L₂ with no nontrivial common direct factor. Then

(i) If $\theta$ belongs to $Der(L)$, then $\alpha ={\pi }_{L_1}\theta i_{L_1}\in Der(L_1)$, $\beta ={\pi }_{L_1}\theta i_{L_2}\in Hom(L_2,Z(L_1))$, $\gamma ={\pi }_{L_2}\theta i_{L_1}\in Hom(L_1,Z(L_2))$, and $\delta ={\pi }_{L_2}\theta i_{L_2}\in Der(L_2)$.

(ii) If $\theta$ belongs to $De{r}_c(L)$, then $\alpha \in De{r}_c(L_1)$, $\delta \in De{r}_c(L_2)$ and $\beta =\gamma =0$.

(iii) If $\theta$ belongs to $ID(L)$, then $\alpha \in ID(L_1)$, $\beta \in Hom(L_2,Z(L_1)\cap L_1^2)$, $\gamma \in Hom(L_1,Z(L_2)\cap L_2^2)$, and $\delta \in ID(L_2)$.

(iv) If $\theta$ belongs to $I{D}^{\ast }(L)$, then $\alpha \in I{D}^{\ast }(L_1)$, $\beta \in Hom(L_2,Z(L_1)\cap L_1^2)$, $\gamma \in Hom(L_1,Z(L_2)\cap L_2^2)$, and $\delta \in I{D}^{\ast }(L_2)$.

(v) If $\theta$ belongs to $De{r}_z(L)$, then $\alpha \in De{r}_z(L_1)$, $\beta \in Hom(L_2,Z(L_1))$, $\gamma \in Hom(L_1,Z(L_2))$, and $\delta \in De{r}_z(L_2)$.

Proof. Since the proofs are similar, only we give the proof of (i)

(i) Let $x_1,x_2\in L_1$; then

$\alpha ([x_1,x_2])={\pi }_{L_1}\theta [(x_1,0),(x_2,0)]={\pi }_{L_1}[\theta (x_1,0),(x_2,0)]+{\pi }_{L_1}[(x_1,0),\theta (x_2,0)].$

Now suppose that $\theta (x_1,0)=(x_1^{\prime },y_1^{\prime })$ and $\theta (x_2,0)=(x_2^{\prime },y_2^{\prime })$; then

${\pi }_{L_1}[(x_1^{\prime },y_1^{\prime }),(x_2,0)]+{\pi }_{L_1}[(x_1,0),(x_2^{\prime },y_2^{\prime })]=[x_1^{\prime },x_2]+[x_1,x_2^{\prime }].$

On the other hand,

$[\alpha (x_1),x_2]=[{\pi }_{L_1}\theta (x_1,0),x_2]=[{\pi }_{L_1}(x_1^{\prime },y_1^{\prime }),x_2]=[x_1^{\prime },x_2],$

$[x_1,\alpha (x_2)]=[x_1,{\pi }_{L_2}\theta (x_2,0)]=[x_1,{\pi }_{L_1}(x_2^{\prime },y_2^{\prime })]=[x_1,x_2^{\prime }].$

Therefore $\alpha ={\pi }_{L_1}\theta i_{L_1}\in Der(L_1)$. Similarly $\delta ={\pi }_{L_2}\theta i_{L_2}\in Der(L_2)$.

Now, Let $(x,0),(0,y)\in L_1\oplus L_2$; we have

$\theta ([(x,0),(0,y)])=[\theta (x,0),(0,y)]+[(x,0)+\theta (0,y)].$

The left side of equality is $(0,0)$, which implies $[(x,0),\theta (0,y)]=-[\theta (x,0),(0,y)]$. Notice that projection mappings ${\pi }_{L_1}$ and ${\pi }_{L_2}$ are lie epimorphism. First suppose $y\in L_2^2$. Then, there exist $y_1,y_2\in L_2$ such that $y=[y_1,y_2]$. we have

$\begin{array}{cc}\beta (y)=\beta ([y_1,y_2])& ={\pi }_{L_1}\theta (0,[y_1,y_2])={\pi }_{L_1}\theta ([(0,y_1),(0,y_2)])\\ \\ \hspace{0.17em}& ={\pi }_{L_1}[\theta (0,y_1),(0,y_2)]+{\pi }_{L_1}[(0,y_1),\theta (0,y_2)]\end{array}$

Now, if $\theta (0,y_1)=(x_1^{\prime },y_1^{\prime })$ and $\theta (0,y_2)=(x_2^{\prime },y_2^{\prime })$, Then

${\pi }_{L_1}[(x_1^{\prime },y_1^{\prime }),(0,y_2)]+{\pi }_{L_1}[(0,y_1),(x_2^{\prime },y_2^{\prime })]={\pi }_{L_1}(0,[y_1^{\prime },y_2])+{\pi }_{L_1}(0,[y_1,y_2^{\prime }])=0.$

That is, $\beta$ maps $L_2^2$ to zero.

Now suppose $x\in L$. Then, we have

$\begin{array}{cc}[x,\beta (y)]& =[x,{\pi }_{L_1}\theta i_{L_2}(y)]=[{\pi }_{L_1}(x,0),{\pi }_{L_1}\theta (0,y)]={\pi }_{L_1}[(x,0),\theta (0,y)]\\ \\ \hspace{0.17em}& =-{\pi }_{L_1}[\theta (x,0),(0,y)]\end{array}$

Suppose $\theta (x,0)=(x^{\prime },y^{\prime })$. Then, we have $-{\pi }_{L_1}[(x^{\prime },y^{\prime }),(0,y)]=-{\pi }_{L_1}(0,[y^{\prime },y])=0$, which implies $\beta (y)\in Z(L_1)$. Thus $\beta ={\pi }_{L_1}\theta i_{L_2}\in Hom(L_2,Z(L_1))$. Similarly $\gamma ={\pi }_{L_2}\theta i_{L_1}\in Hom(L_1,Z(L_1))$.

Now we are ready to prove Theorem 1.2. ⁏

Proof of Theorem 1.2. Let $g=f_{|Der(L)}$, where $f$ is the same of lie homomorphism of Proposition 1.1; thus, by Lemma 2.2(i), $Im(g)\subseteq \mathcal{A}$.

Now let $\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\in \mathcal{A}$. So $f$ is onto; then there exists $\theta \in gl(L)$ such that $g(\theta )=\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)$. We prove $\theta \in Der(L)$. Let $(x_1,y_1),(x_2,y_2)\in L_1\oplus L_2$; then

$\begin{array}{c}\hspace{0.17em}\theta ([(x_1,y_1),(x_2,y_2)])=\left[\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\left(\begin{array}{c}{x}_1\\ y_1\end{array}\right),\left(\begin{array}{c}{x}_2\\ y_2\end{array}\right)\right]=\left[\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\left(\begin{array}{c}[x_1,x_2]\\ [y_1,y_2]\end{array}\right)\right]\\ =\left(\begin{array}{c}\alpha ([x_1,x_2])+\beta ([y_1,y_2])\\ \gamma ([x_1,x_2])+\delta ([y_1,y_2])\end{array}\right)=\left(\begin{array}{c}\alpha ([x_1,x_2])\\ \delta ([y_1,y_2])\end{array}\right).\end{array}$

On the other hand,

$\begin{array}{c}[\theta (x_1,y_1),(x_2,y_2)]=\left[\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\left(\begin{array}{c}{x}_1\\ y_1\end{array}\right),\left(\begin{array}{c}{x}_2\\ y_2\end{array}\right)\right]=\left[\left(\begin{array}{c}\alpha (x_1)+\beta (y_1)\\ \gamma (x_1)+\delta (y_1)\end{array}\right),\left(\begin{array}{c}{x}_2\\ y_2\end{array}\right)\right]\\ =\left(\begin{array}{c}[\alpha (x_1)+\beta (y_1),x_2]\\ [\gamma (x_2)+\delta (y_1),y_2]\end{array}\right)=\left(\begin{array}{c}[\alpha (x_1),x_2]\\ [\delta (y_1),y_2]\end{array}\right).\end{array}$

Similarly,

$[(x_1,y_1),\theta (x_2,y_2)]=\left(\begin{array}{c}[x_1,\alpha (x_2)]\\ \left[y_1,\delta (y_2)\right]\end{array}\right).$

Therefore, the favorable result obtains. ⁏

By the above theorem, we can obtain the structure and dimension subalgebras of $Der(L)$ that is in sequence (1).

Consider the following sets:

$\text{{{mathbb J}\_z} = left{ {left({matrix{ alpha \& beta cr gamma \& delta cr } } right)left|matrix{ {alpha in ID({L\_1})} \& {beta in Hom({L\_2},Z({L\_1})cap L^2\_1)} cr {gamma in Hom({L\_1},Z({L\_2})cap L^2\_2)} \& {delta in ID{}({L\_2})} cr } } right} underline{underline subset } = {mathcal A}.}$

$\text{{{mathbb J}^*} = left{ {left({matrix{ alpha \& beta cr gamma \& delta cr } } right)left|matrix{ {alpha in ID{^*}({L\_1})} \& {beta in Hom({L\_2},Z({L\_1})cap L^2\_1)} cr {gamma in Hom({L\_1},Z({L\_2})cap L^2\_2)} \& {delta in ID{^*}({L\_2})} cr } } right} underline{underline subset } = {mathcal J}.}$

$\text{{{mathcal A}\_c} = left{ {left({matrix{ alpha \& 0 cr 0 \& delta cr } } right)left|matrix{ {alpha in De{r\_c}({L\_1})} cr {delta in De{r\_c}({L\_2})} cr } } right} underline{underline subset } = {{mathbb J}^ * },}$

$\text{{{mathcal A}\_z} = left{ {left({matrix{ alpha \& beta cr gamma \& delta cr } } right)left|matrix{ {alpha in De{r\_z}({L\_1})} \& {beta in Hom({L\_2},Z({L\_1}))} cr {gamma in Hom({L\_1},Z({L\_2}))} \& {delta in De{r\_z}({L\_2})} cr } } right} underline{underline subset } = {mathcal A}.}$

Corollary 2.3. Let $L_1$ and $L_2$ be two Lie algebras finite dimension over field $F$ with no nontrivial common direct factor, and let $L=L_1\oplus L_2$. Then

(i) ID(L) $\cong \mathcal{I}$ in Lie algebra. Furthermore,

$dimID(L)=dimID(L_1)+dimID(L_2)+dimHom(L_1,Z(L_2)\cap L_2^2)+dimHom(L_2,Z(L_1)\cap L_1^2).$

(ii) $I{D}^{\ast }(L)\cong {\mathcal{I}}^{\ast }$ in Lie algebra. Moreover,

$dimI{D}^{\ast }(L)=dimI{D}^{\ast }(L_1)+dimI{D}^{\ast }(L_2)+dimHom(L_1,Z(L_2)\cap L_2^2)+dimHom(L_2,Z(L_1)\cap L_1^2).$

(iii) $De{r}_c(L)\cong {\mathcal{A}}_c$ in Lie algebra. Hence,

$dimDe{r}_c(L)=dimDe{r}_c(L_1)+dimDe{r}_c(L_2).$

(iv) $De{r}_z(L)\cong {\mathcal{A}}_z$ in Lie algebra. Hence,

$dimDe{r}_z(L)=dimDe{r}_z(L_1)+dimDe{r}_z(L_2)+dimHom(L_1,Z(L_2))+dimHom(L_2,Z(L_1)).$

Proof. (i) We have $ID(L)\le Der(L)$; suppose that $h=g_{|ID(L)}$, where $g$ is the same of lie homomorphism of Theorem 1.2. Thus, by Lemma 2.2(iii), $Im(h)\subseteq \mathbb{J}$. Now suppose that $\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\in \mathbb{J}\subseteq \mathcal{A}$. By Theorem 1.2, there exists $\theta \in Der(L)$ such that $h(\theta )=\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)$. Now let $(x,y)\in L_1\oplus L_2$; then we have

$\theta (x,y)=(\alpha (x)+\beta (y),\gamma (x)+\delta (y))\in L_1^2\oplus L_2^2=(L_1\oplus L_2{)}^2$

Therefore $\theta \in ID(L)$ as required.

Next, parts are similar to part (i). ⁏

Let $L_1$ and $L_2$ be two Lie algebras over an arbitrary field $F$, and let $\beta =\{x_1,\dots ,x_n\}$ and ${\beta }^{\prime }=\{y_1,\dots ,y_m\}$ be, respectively, two ordered basis of $L_1$ and $L_2$. Suppose that $\gamma \in T(L_1,L_2)$. Using this two basis, we may define scalars ${\gamma }_{i,j}$ by $\gamma (x_j)={\sum }_{i=1}^m{\gamma }_{i,j}{y}_i$; then the matrix of $\gamma$ is such as follows:

$\left[\begin{array}{cccc}{\gamma }_{1,1}& {\gamma }_{1,2}& \cdots & {\gamma }_{1,n}\\ {\gamma }_{2,1}& {\gamma }_{2,2}& \cdots & {\gamma }_{2,n}\\ ⋮& ⋮& \hspace{0.17em}& ⋮\\ {\gamma }_{m,1}& {\gamma }_{m,2}& \cdots & {\gamma }_{m,n}\end{array}\right]$

Also, we recall that a Lie algebra $L$ is called Heisenberg if $L^2=Z(L)$ and dim$L^2$= 1. Heisenberg Lie algebras of finite dimension are from odd dimension with basis of $\{x_1,x_2,\dots ,x_{2k},x_{2k+1}\}$, that is the nonzero bracket between the basis elements in the form $\left[x_{2i-1},x_{2i}\right]=x_{2k+1}$ for all $i=1,2,\dots ,k$. Symbol of $H(k)$ is a Heisenberg Lie algebra of dimension 2k+1. A Lie algebra $L$ is called purely nonabelian, when $L$ has no nontrivial abelian direct factor.

The following example illustrates dimension $Der(A(m)\oplus H(1))$ and some subalgebras of derivation algebra it, by using Theorem1.2 and Corollary 2.3.

Example 1. Let $L=A(m)\oplus H(1)$ be direct sum of abelian Lie algebra of dimension $m$, $A(m)$, and Heisenberg Lie algebra of dimension three, $H(1)$. It is easy to see that the matrix representation of each element in $Der(H(1))$ is such as follows:

$\left[\begin{array}{ccc}{\gamma }_{1,1}& {\gamma }_{1,2}& 0\\ {\gamma }_{2,1}& {\gamma }_{2,2}& 0\\ {\gamma }_{3,1}& {\gamma }_{3,2}& {\gamma }_{1,1}+{\gamma }_{2,2}\end{array}\right]$

Because $H(1)$ is purely nonabelian, thus, by Theorem1.2 and Corollary 2.2, we have

$\begin{array}{rl}& dimDer(L)=dimDer(A(m))+dimDer(H(1))+dimHom(A(m),Z(H(1)))\\ & +dimHom(H(1),Z(A(m)))=m^2+6+m+2m=m^2+3m+6\end{array}$

$\begin{array}{c}dimID(L)=dimID(A(m))+dimID(H(1))+dimHom(A(m),Z(H(1))\cap H{(1)}^2)\\ +dimHom(H(1),Z(A(m))\cap A{(m)}^2)=m+2\end{array}$

$\begin{array}{c}dimI{D}^{\ast }(L)=dimI{D}^{\ast }(A(m))+dimI{D}^{\ast }(H(1))+dimHom(A(m),Z(H(1))\cap H{(1)}^2)\\ +dimHom(H(1),Z(A(m))\cap A{(m)}^2)=m+2\end{array}$

$dimDe{r}_c(L)=dimDe{r}_c(A(m))+dimDe{r}_c(H(1))=2$

$\begin{array}{rl}& dimDe{r}_z(L)=dimDe{r}_z(A(m))+dimDe{r}_z(H(1))+dimHom(A(m),Z(H(1)))\\ & +dimHom(H(1),Z(A(m)))=m^2+2+m+2m=m^2+3m+2\end{array}$

In the following example, by using Theorem 1.2, Corollary 2.3, and the induction, we obtain dimension of derivation n-copy of direct sum of Lie algebras $H(1)$.

Example 2. Let $L=H(1)\oplus H(1)\oplus \cdots \oplus H(1)$ be direct sum n-copy of Lie algebras $H(1)$. Then using induction on n, for n $\ge$ 1, we have

$dimDer(L)=2n(n+2).$

In special case, if n = 2, then dimDer(L) = 16.

In Theorem 1.2, it is necessary that $L_1$ and $L_2$ has no nontrivial common direct factor. Consider the following example.

Example 3. Let $L_1=A(1)\oplus ⟨x_1,x_2,x_3|\left[x_1,x_2\right]=\left[x_1,x_3\right]=x_2⟩$ and $L_2=A(1)\oplus ⟨y_1,y_2|\left[y_1,y_2\right]=y_1⟩$. Put $L=L_1\oplus L_2$; then

$\begin{array}{rl}& dimDer(L_1)+dimDer(L_2)+dimHom(L_1,Z(L_2))+dimHom(L_2,Z(L_1))\\ & =9+5+3+2=19.\end{array}$

While $Der(L_1\oplus L_2)$ is the form of the following matrix:

$\left[\begin{array}{ccccccc}{\alpha }_{11}& 0& 0& 0& 0& 0& 0\\ {\alpha }_{21}& {\alpha }_{11}+{\alpha }_{22}& {\alpha }_{23}& {\alpha }_{24}& 0& {\alpha }_{26}& {\alpha }_{27}\\ {\alpha }_{31}& 0& {\alpha }_{22}-{\alpha }_{23}& -{\alpha }_{24}& 0& -{\alpha }_{26}& -{\alpha }_{27}\\ {\alpha }_{41}& 0& {\alpha }_{43}& {\alpha }_{44}& 0& {\alpha }_{46}& {\alpha }_{47}\\ 0& 0& 0& 0& {\alpha }_{55}+{\alpha }_{66}& {\alpha }_{56}& 0\\ 0& 0& 0& 0& 0& {\alpha }_{66}& 0\\ {\alpha }_{71}& 0& {\alpha }_{73}& {\alpha }_{74}& 0& {\alpha }_{76}& {\alpha }_{77}\end{array}\right]$

Therefore dimDer($L_1\oplus L_2$) = 21; so the equality is not established.

Conversely, Theorem 1.2 is not correct generally. See the following example.

Example 4. Let $L_1=⟨x_1,\cdots ,x_6|\left[x_1,x_2\right]=x_3,\left[x_4,x_5\right]=x_6⟩\cong H(1)\oplus H(1)$ and $L_2=⟨y_1,y_2,y_3,y_4|$ $\left[y_1,y_2\right]=y_3⟩\cong H(1)\oplus A(1)$. Put $L=L_1\oplus L_2$; then representation of matrix each element of $Der(L)$ is as follows:

$\left[\begin{array}{cccccccccc}{\alpha }_{1,1}& {\alpha }_{1,2}& 0& 0& 0& 0& 0& 0& 0& 0\\ {\alpha }_{2,1}& {\alpha }_{2,2}& 0& 0& 0& 0& 0& 0& 0& 0\\ {\alpha }_{3,1}& {\alpha }_{3,2}& {\alpha }_{1,1}+{\alpha }_{2,2}& {\alpha }_{3,4}& {\alpha }_{3,5}& 0& {\alpha }_{3,7}& {\alpha }_{3,8}& 0& {\alpha }_{3,10}\\ 0& 0& 0& {\alpha }_{4,4}& {\alpha }_{4,5}& 0& 0& 0& 0& 0\\ 0& 0& 0& {\alpha }_{5,4}& {\alpha }_{5,5}& 0& 0& 0& 0& 0\\ {\alpha }_{6,1}& {\alpha }_{6,2}& 0& {\alpha }_{6,4}& {\alpha }_{6,5}& {\alpha }_{4,4}+{\alpha }_{5,5}& {\alpha }_{6,7}& {\alpha }_{6,8}& 0& {\alpha }_{6,10}\\ 0& 0& 0& 0& 0& 0& {\alpha }_{7,7}& {\alpha }_{7,8}& 0& 0\\ 0& 0& 0& 0& 0& 0& {\alpha }_{8,7}& {\alpha }_{8,8}& 0& 0\\ {\alpha }_{9,1}& {\alpha }_{9,2}& 0& {\alpha }_{9,4}& {\alpha }_{9,5}& 0& {\alpha }_{9,7}& {\alpha }_{9,8}& {\alpha }_{7,7}+{\alpha }_{8,8}& {\alpha }_{9,10}\\ {\alpha }_{10,1}& {\alpha }_{10,2}& 0& {\alpha }_{10,4}& {\alpha }_{10,5}& 0& {\alpha }_{10,7}& {\alpha }_{10,8}& 0& {\alpha }_{10,10}\end{array}\right]$

Therefore dimDer(L) = 40, also

$\begin{array}{rl}& dimDer(L_1)+dimDer(L_2)+dimHom(L_1,Z(L_2))+dimHom(L_2,Z(L_1))\\ & =16+10+8+6=40.\end{array}$

As seen on, equality established while $L_1$ and $L_2$ have nontrivial common direct factor.

3. Relation between derivation of direct sum and direct sum of derivations

First, we recall that Lie algebra $L$ is called $stem$, when $Z(L)\subseteq L^2$, and it is called perfect if $L=L^2$. Suppose that $L$ is a Lie algebra and that $H$ is a subalgebra of $L$ and $K$ is an ideal of it such that $L=H+K$ and $H\cap K=\{0\}$; then we say $L$ is the semidirect sum of $H$ with $K$ and we write $L=HK$.

Now we turn to the structure $Der(L)$, where $L=L_1\oplus L_2$. As we saw in Example 2, if $L=L_1\oplus L_2$ is the direct sum of two Lie algebras with no nontrivial common direct factor, then it could $Der(L)\mathit{/}\cong Der(L_1)\oplus Der(L_2)$. In this section, we give conditions that create this Isomorphism.

Let

$\text{A = left{ {left({matrix{ alpha \& 0 cr 0 \& delta cr } } right)left|matrix{ {alpha in Der({L\_1})} cr {delta in Der({L\_2})} cr } } right}quadquadquadquadquadquad B = left{ {left({matrix{ 0 \& beta cr gamma \& 0 cr } } right)Big|matrix{ {beta in Hom({L\_2},Z({L\_1}))} cr {gamma in Hom({L\_1},Z({L\_2}))} cr } } right}}$

It is obvious that $A$ is a subalgebra of $\mathcal{A}$, and $A\cong Der(L_1)\oplus Der(L_2)$. Now if $L_1$ and $L_2$ be stem, then $B$ is not only a subalgebra of $\mathcal{A}$ but also it is an ideal of it; thus, by Theorem 1.2, we have

Corollary 3.1. Let $L=L_1\oplus L_2$ be such that $L_1$ and $L_2$ are stem Lie algebras with no nontrivial common direct factor; then

$Der(L)\cong (Der(L_1)\oplus Der(L_2))\left(Hom(L_1,Z(L_2))\oplus Hom(L_2,Z(L_1))\right).$

Proof. It is easy to check that $B\cong \left(Hom(L_1,Z(L_2))\oplus Hom(L_2,Z(L_1))\right)$ and $\mathcal{A}\cong A\times B$; then, by Theorem 1.2, the assert is valid.

⁏

Corollary 3.2. Let $L_1$ and $L_2$ be two Lie algebras with no nontrivial common direct factor and $L=L_1\oplus L_2$; then

$\text{ID(L) cong (ID({L\_1}) oplus ID({L\_2}))left({Hom({L\_1},Z({L\_2})mathop cap nolimits^ L\_2^2) oplus Hom({L\_2},Z({L\_1})mathop cap nolimits^ L\_1^2)} right).}$

Proof. By Theorem 2.3(i), we have

$A\cong (ID(L_1)\oplus ID(L_2))$

and

$B\cong \left(Hom(L_1,Z(L_2)\cap L_2^2)\oplus Hom(L_2,Z(L_1)\cap L_1^2)\right)$

Also B is an ideal of $\mathcal{I}$; thus, $\mathcal{I}\cong A\times B$ .⁏

Corollary 3.3 Let $L_1$ and $L_2$ be two Lie algebras with no nontrivial common direct factor and $L=L_1\oplus L_2$; then

$\text{I{D^*}(L) cong (I{D^*}({L\_1}) oplus I{D^*}({L\_2}))left({Hom({L\_1},Z({L\_2})mathop cap nolimits^ L\_2^2) oplus Hom({L\_2},Z({L\_1})mathop cap nolimits^ L\_1^2)} right).}$

Proof. It is similar to previous corollary.

Corollary 3.4. Let $L=L_1\oplus L_2$ be such that $L_1$ and $L_2$ with no nontrivial common direct factor and $Z(L_2)=\{0\}$. Then

$Der(L)\cong (Der(L_1)\oplus Der(L_2))Hom\left(L_2,Z(L_1)\right).$

Proof. Since $Z(L_2)=\{0\}$, then $B$ is an ideal of $\mathcal{A}$ and $B\cong Hom\left(L_2,Z(L_1)\right)$. Therefore $\mathcal{A}\cong A⋉B$, and, by Theorem 1.2, the assert is valid.

Corollary 3.5. Let $L_1$ and $L_2$ with no common direct factor, and let $L=L_1\oplus L_2$. Then if $L_1$ and $L_2$ are perfect Lie algebras or $Z(L_1)$ and $Z(L_2)$ are trivial, then we have

$Der(L)\cong Der(L_1)\oplus Der(L_2).$

Proof. Since $Hom\left(L_1,Z(L_2)\right)=Hom\left(L_2,Z(L_1)\right)=\{0\}$, thus, $B=\{0\}$ and $\mathcal{A}\cong A$.

Example 5 Let $L=⟨x_1,x_2,x_3|[x_1,x_2]=x_3,[x_2,x_3]=x_1,[x_3,x_1]=x_2⟩$ be a three-dimensional Lie algebra such that $L^2=L$. Then, $Der(L)$ and $Der(L\oplus L)$ have the following matrix form, respectively,

$\left[\begin{array}{ccc}{\alpha }_{22}+{\alpha }_{33}& -{\alpha }_{21}& -{\alpha }_{31}\\ -{\alpha }_{12}& {\alpha }_{11}+{\alpha }_{22}& -{\alpha }_{32}\\ -{\alpha }_{13}& -{\alpha }_{23}& {\alpha }_{11}+{\alpha }_{33}\end{array}\right]$

$\left[\begin{array}{cccccc}{\alpha }_{22}+{\alpha }_{33}& -{\alpha }_{21}& -{\alpha }_{31}& 0& 0& 0\\ -{\alpha }_{12}& {\alpha }_{11}+{\alpha }_{22}& -{\alpha }_{32}& 0& 0& 0\\ -{\alpha }_{13}& -{\alpha }_{23}& {\alpha }_{11}+{\alpha }_{33}& 0& 0& 0\\ 0& 0& 0& {\alpha }_{55}+{\alpha }_{66}& -{\alpha }_{54}& -{\alpha }_{64}\\ 0& 0& 0& -{\alpha }_{45}& {\alpha }_{44}+{\alpha }_{66}& -{\alpha }_{65}\\ 0& 0& 0& -{\alpha }_{46}& -{\alpha }_{56}& {\alpha }_{44}+{\alpha }_{55}\end{array}\right]$

It is obviously $Der(L\oplus L)\cong Der(L)\oplus Der(L)$.

Corollary 3.6 Let $L_1$ and $L_2$ with no common direct factor, and let $L=L_1\oplus L_2$. Then

$De{r}_c(L)\cong De{r}_c(L_1)\oplus De{r}_c(L_2).$

Proof. By lemma 2.3(iii), the assert is valid.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Mohammad Reza Alemi

Mohammad Reza Alemi is Ph.D. student at Islamic Azad University, Mashhad Branch.

Farshid Saeedi

Dr. Farshid Saeedi is a Mathematician and a faculty in the Department of Mathematics, Mashhd Branch, Islamic Azad University, Iran. He has Bachelor of Science (B.Sc.) and Master of Science (M.Sc.) degrees in pure Mathematics in 1992 and 1996 from University of Birjand and University of Amir Kabir, respectively. He earned his Ph.D. degree in pure Mathematics (group theory) in 2006 from Islamic Azad University, Iran. His area of specialization are group theory and Lie algebra.