# Derivations of two-step nilpotent algebras00footnotetext: Keywords: Leibniz
algebra, Nilpotent Leibniz algebra, Lie algebra, Derivation, Almost inner
derivation, Lie Theory. 00footnotetext: 2020 Mathematics Subject
Classification: 15B30, 16W25, 17A32, 17A36, 17B40.

###### Abstract

In this paper we study the Lie algebras of derivations of two-step nilpotent
algebras. We obtain a class of Lie algebras with trivial center and abelian
ideal of inner derivations. Among these, the relations between the complex and
the real case of the indecomposable _Heisenberg Leibniz algebras_ are
thoroughly described. Finally we show that every _almost inner derivation_ of
a complex nilpotent Leibniz algebra with one-dimensional commutator ideal,
with three exceptions, is an inner derivation.

Gianmarco La Rosa222Supported by University of Palermo and by the “National
Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA
– INdAM)., Manuel Mancini22footnotemark: 2
Dipartimento di Matematica e Informatica
Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
<EMAIL_ADDRESS>ORCID: 0000-0003-1047-5993
<EMAIL_ADDRESS>ORCID: 0000-0003-2142-6193

## Introduction

Leibniz algebras were first introduced by J.-L. Loday in [16] as a non-
antisymmetric version of Lie algebras, and many results of Lie algebras were
also established in the frame of Leibniz algebras. Earlier, such algebraic
structures were considered by A. Blokh, who called them D-algebras [6], for
their strict connection with the derivations. Leibniz algebras play a
significant role in different areas of mathematics and physics.
In [3] and [11] derivations of nilpotent Lie algebras with small dimensional
commutator ideal were studied; in [12] and [14] two-step nilpotent algebras
were classified. After a short preliminary section, in this paper we aim to
study the Lie algebras of derivations of two-step nilpotent algebras.
Independently of its intrinsic interest, derivations find concrete
applications in representation theory (cf. [15]), (sub-)Riemannian geometry
and control theory (see [5], and the bibliography therein), just to give two
instances.

The first section is devoted to some background material on Leibniz algebras
and derivations which will be useful for the rest of the paper. We address the
reader to [2] and [10] for more details. We recall that there are only three
classes of indecomposable nilpotent Leibniz algebras with one-dimensional
commutator ideal, namely the _Heisenberg Leibniz algebras_
$\mathfrak{l}_{2n+1}^{A}$, parametrized by their dimension $2n+1$ and a
$n\times n$ matrix $A$ in canonical form, the _Kronecker Leibniz algebras_
$\mathfrak{k}_{n}$ and the _Dieudonné Leibniz algebras_ $\mathfrak{d}_{n}$,
both parametrized by their dimension only. Recently noncommutative Heisenberg
algebras were also studied in [13].

In Sections 2, 3, 4 we find the derivation algebras of such nilpotent Leibniz
algebras. The properties of these Lie algebras are studied and, for the
Heisenberg algerbras, we analyze the relations between the complex and the
real case. We always obtain Lie algebras of derivations with trivial center
and abelian ideal of inner derivations. We use the _LieAlgebras_ package of
the software _Maple_ for finding the radical, the Levi decomposition and the
nilradical of such algebras.

In the last section we show that, for a complex nilpotent Leibniz algebra $L$
with one-dimensional commutator ideal, such that $L$ is not isomorphic to
$\mathfrak{l}_{2n+1}^{J_{\pm 1}}$ or $\mathfrak{d}_{n}$, every _almost inner
derivation_ is an inner derivation. This generalizes the result found by D.
Burde, K. Dekimpe and V. Verbeke in [7] for two-step nilpotent Lie algebras of
_genus_ $1$. We also find the Lie subalgebras
$\operatorname{AIDer}(\mathfrak{l}_{2n+1}^{J_{a}})$, when $a=\pm 1$, and
$\operatorname{AIDer}(\mathfrak{d}_{n})$, showing that in this case the
inclusions

$\operatorname{AIDer}(\mathfrak{l}_{2n+1}^{J_{\pm
1}})\supsetneq\operatorname{Inn}(\mathfrak{l}_{2n+1}^{J_{\pm 1}})\text{ and
}\operatorname{AIDer}(\mathfrak{d}_{n})\supsetneq\operatorname{Inn}(\mathfrak{d}_{n})$

are strict.

##  1 Preliminaries

We assume that $\mathbb{F}$ is a field with
$\operatorname{char}(\mathbb{F})\neq 2$. For the general theory we refer to
[2].

######  Definition 1.1.

A _left Leibniz algebra_ over $\mathbb{F}$ is a vector space $L$ over
$\mathbb{F}$ endowed with a bilinear map (called $commutator$ or $bracket$)
$\left[-,-\right]\colon L\times L\rightarrow L$ which satisfies the _left
Leibniz identity_

$\left[x,\left[y,z\right]\right]=\left[\left[x,y\right],z\right]+\left[y,\left[x,z\right]\right],\;\;\forall
x,y,z\in L.$

In the same way we can define a right Leibniz algebra, using the right Leibniz
identity

$\left[\left[x,y\right],z\right]=\left[[x,z],y\right]+\left[x,\left[y,z\right]\right],\;\;\forall
x,y,z\in L.$

A Leibniz algebra that is both left and right is called _symmetric Leibniz
algebra_. From now on we assume that $\dim_{\mathbb{F}}L<\infty$.

Clearly every Lie algebra is a Leibniz algebra and every Leibniz algebra with
skew-symmetric commutator is a Lie algebra. Thus it is defined an adjunction
(see [17]) between the category $\textbf{LieAlg}_{\mathbb{F}}$ of the Lie
algebras over $\mathbb{F}$ and the category $\textbf{LeibAlg}_{\mathbb{F}}$ of
the Leibniz algebras over $\mathbb{F}$. The left adjoint of the full inclusion
$i\colon\textbf{LieAlg}_{\mathbb{F}}\rightarrow\textbf{LeibAlg}_{\mathbb{F}}$
is the functor
$\pi\colon\textbf{LeibAlg}_{\mathbb{F}}\rightarrow\textbf{LieAlg}_{\mathbb{F}}$
that associates with every Leibniz algebra $L$ the quotient
$L/\operatorname{Leib}(L)$, where
$\operatorname{Leib}(L)=\langle\left[x,x\right]|\,x\in L\rangle$. We observe
that $\operatorname{Leib}(L)$ is the smallest bilateral ideal of $L$ such that
$L/\operatorname{Leib}(L)$ is a Lie algebra. Moreover $\operatorname{Leib}(L)$
is an abelian algebra.

The definition of _derivation_ is the same as in the case of Lie algebras.

######  Definition 1.2.

Let $L$ be a Leibniz algebra over $\mathbb{F}$. A _derivation_ of $L$ is a
linear map $d\colon L\rightarrow L$ such that

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

The left multiplications are particular derivations called _inner derivations_
and an equivalent way to define a left Leibniz algebra is to say that the
(left) adjoint map $\operatorname{ad}_{x}=\left[x,-\right]$ is a derivation,
for every $x\in L$. Meanwhile, for a left Leibniz algebra, the right adjoint
maps are not derivations in general.
With the usual bracket $\left[d_{1},d_{2}\right]=d_{1}\circ d_{2}-d_{2}\circ
d_{1}$, the set $\operatorname{Der}(L)$ is a Lie algebra and the set
$\operatorname{Inn}(L)$ of all inner derivations of $L$ is an ideal of
$\operatorname{Der}(L)$. Furthermore, $\operatorname{Aut}(L)$ is a Lie group
and the associated Lie algebra is $\operatorname{Der}(L)$.

######  Definition 1.3.

A derivation $d$ of a Leibniz algebra $L$ is called _almost inner derivation_
if $d(x)\in[L,x]$, for every $x\in L$.

In [1] Z. K. Adashev and T. K. Kurbanbaev proved that the space of all almost
inner derivations, denoted by $\operatorname{AIDer}(L)$, is a Lie subalgebra
of $\operatorname{Der}(L)$ and

$\operatorname{Inn}(L)\subseteq\operatorname{AIDer}(L).$

We define the left and the right center of a Leibniz algebra

$\operatorname{Z}_{l}(L)=\left\\{x\in
L\,|\,\left[x,L\right]=0\right\\},\,\,\,\operatorname{Z}_{r}(L)=\left\\{x\in
L\,|\,\left[L,x\right]=0\right\\},$

and we observe that they coincide when $L$ is a Lie algebra. The _center_ of
$L$ is
$\operatorname{Z}(L)=\operatorname{Z}_{l}(L)\cap\operatorname{Z}_{r}(L)$. In
the case of symmetric Leibniz algebras, the left center and the right center
are bilater ideals, but in general $\operatorname{Z}_{l}(L)$ is an ideal of
the left Leibniz algebra $L$, meanwhile the right center is not even a
subalgebra.

######  Definition 1.4.

Let $L$ be a left Leibniz algebra over $\mathbb{F}$ and let

$L^{(0)}=L,\,\,L^{(k+1)}=[L,L^{(k)}],\;\;\forall k\geq 0,$

be the _lower central series of $L$_. $L$ is _$n-$ step nilpotent_ if
$L^{(n-1)}\neq 0$ and $L^{(n)}=0$.

We observe that two-step nilpotent Leibniz algebras lie in different varieties
of non-associative algebras, such as associative, alternative and Zinbiel
algebras. For this reason we will refer at them as _two-step nilpotent
algebras_.

One can directly prove the following.

######  Proposition 1.5.

If $L$ is a left two-step nilpotent algebra, then
$L^{(1)}=[L,L]\subseteq\operatorname{Z}(L)$ and $L$ is symmetric.

######  Proposition 1.6.

If $L$ is a left nilpotent Leibniz algebra with $\dim_{\mathbb{F}}L^{(1)}=1$,
then $L^{(1)}\subseteq\operatorname{Z}(L)$ and $L$ is symmetric.

Nilpotent Leibniz algebras with one-dimensional commutator ideal were
classified in [14]. It was proved that, up to isomorphism, there are only the
following three classes of indecomposable nilpotent Leibniz algebras with one-
dimensional commutator ideal.

######  Definition 1.7.

[14] Let $f(x)\in\mathbb{F}\left[x\right]$ be a monic irreducible polynomial.
Let $k\in\mathbb{N}$ and let $A=(a_{ij})_{i,j}$ be the companion matrix of
$f(x)^{k}$. The _Heisenberg_ Leibniz algebra $\mathfrak{l}_{2n+1}^{A}$ is the
$(2n+1)$-dimensional Leibniz algebra with basis
$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},z\right\\}$ and the Leibniz
brackets are given by

$[e_{i},f_{j}]=(\delta_{ij}+a_{ij})z,\;[f_{j},e_{i}]=(-\delta_{ij}+a_{ij})z,\;\;\forall
i,j=1,\ldots,n.$

Notice that, if $A$ is the zero matrix, then we obtain the
$(2n+1)-$dimensional Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$.

######  Definition 1.8.

[14] Let $n\in\mathbb{N}$. The _Kronecker_ Leibniz algebra $\mathfrak{k}_{n}$
is the $(2n+1)$-dimensional Leibniz algebra with basis
$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},z\right\\}$ and the Leibniz
brackets are given by

$\displaystyle[e_{i},f_{i}]=[f_{i},e_{i}]=z,\;\;\forall i=1,\ldots,n$
$\displaystyle[e_{i},f_{i-1}]=z,[f_{i-1},e_{i}]=-z,\;\;\forall i=2,\ldots,n.$

######  Definition 1.9.

[14] Let $n\in\mathbb{N}$. The _Dieudonné_ Leibniz algebra $\mathfrak{d}_{n}$
is the $(2n+2)$-dimensional Leibniz algebra with basis
$\left\\{e_{1},\ldots,e_{2n+1},z\right\\}$ and and the Leibniz brackets are
given by

$\displaystyle\left[e_{1},e_{n+2}\right]=z,$
$\displaystyle\left[e_{i},e_{n+i}\right]=\left[e_{i},e_{n+i+1}\right]=z,\;\;\forall
i=2,\ldots,n,$ $\displaystyle\left[e_{n+1},e_{2n+1}\right]=z,$
$\displaystyle\left[e_{i},e_{i-n}\right]=z,\;\;\left[e_{i},e_{i-n-1}\right]=-z,\;\;\forall
i=n+2,\ldots,2n+1.$

In the next sections we study the Lie algebra of derivations of the three
classes of indecomposable nilpotent Leibniz algebras with one-dimensional
commutator ideal. We observe that, given a derivation $d$ of a Leibniz algebra
$L$, we have

$d([L,L])\subseteq[L,L],$

thus, if $[L,L]=\mathbb{F}z$, it follows that $d(z)=\gamma z$, for some
$\gamma\in\mathbb{F}$.

##  2 Derivations of the Heisenberg Leibniz algebras $\mathfrak{l}_{2n+1}^{A}$

Now we want to study in details the Lie algebras of derivations of the
Heisenberg Leibniz algebras in the case the field $\mathbb{F}$ is $\mathbb{C}$
or $\mathbb{R}$.

###  2.1 The complex case

Let $n\in\mathbb{N}$ and let $f(x)=x-a\in\mathbb{C}[x]$. Then the companion
matrix $A$ of $f(x)^{n}$ is similar to the $n\times n$ Jordan block $J_{a}$ of
eigenvalue $a$. Thus $\mathfrak{l}_{2n+1}^{A}\cong\mathfrak{l}_{2n+1}^{J_{a}}$
and the Leibniz brackets are given by

$\displaystyle\left[e_{i},f_{i}\right]=(1+a)z,\;\;\left[f_{i},e_{i}\right]=(-1+a)z,\;\;\forall
i=1,\ldots,n,$
$\displaystyle\left[e_{i},f_{i-1}\right]=\left[f_{i-1},e_{i}\right]=z,\;\;\forall
i=2,\ldots,n,$

where $\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},z\right\\}$ is a basis of
$\mathfrak{l}_{2n+1}^{J_{a}}$. Moreover
$\mathfrak{l}_{2n+1}^{J_{a}}\cong\mathfrak{l}_{2n+1}^{J_{-a}}$.

Now let
$d\colon\mathfrak{l}_{2n+1}^{J_{a}}\rightarrow\mathfrak{l}_{2n+1}^{J_{a}}$ be
a linear endomorphism such that

$d(e_{i})=\sum_{j=1}^{n}a_{ji}e_{j}+\sum_{k=1}^{n}b_{ki}f_{k}+a_{i}z,\;\;\forall
i=1,\ldots,n,$
$d(f_{i})=\sum_{j=1}^{n}c_{ji}e_{j}+\sum_{k=1}^{n}d_{ki}f_{k}+b_{i}z,\;\;\forall
i=1,\ldots,n$

and

$d(z)=\gamma z.$

Then for $a\neq 0$ the linear endomorphism $d$ is a derivation of the
Heisenberg algebra $\mathfrak{l}_{2n+1}^{J_{a}}$ if and only if it has the
following form

$\begin{pmatrix}A&0&0\\\ 0&D&0\\\ \mu&\nu&\gamma\\\ \end{pmatrix}$

where

$A=\left(\begin{array}[]{*5{c}}\alpha_{1}&\alpha_{2}&\alpha_{3}&\ldots&\alpha_{n}\\\
0&\alpha_{1}&\alpha_{2}&\ldots&\alpha_{n-1}\\\
0&0&\alpha_{1}&\ldots&\alpha_{n-2}\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\ldots&\alpha_{1}\\\
\end{array}\right),\;\;D=\left(\begin{array}[]{*5{c}}\beta_{1}&0&0&\ldots&0\\\
-\alpha_{2}&\beta_{1}&0&\ldots&0\\\
-\alpha_{3}&-\alpha_{2}&\beta_{1}&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
-\alpha_{n}&-\alpha_{n-1}&-\alpha_{n-2}&\ldots&\beta_{1}\\\
\end{array}\right),$
$\mu=(\mu_{1},\mu_{2},\mu_{3},\ldots,\mu_{n}),\;\;\nu=(\nu_{1},\nu_{2},\nu_{3},\ldots,\nu_{n}),\;\;\gamma=\alpha_{1}+\beta_{1}.$

Thus, $\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}})$ is the Lie subalgebra
of $\operatorname{gl}(2n+1,\mathbb{C})$ of dimension $3n+1$ with basis

$\\{x,y,E_{1},\ldots,E_{n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\},$

where

$\displaystyle
x=\sum_{k=1}^{n}e_{k,k}+e_{2n+1,2n+1},\;\;y=\sum_{k=1}^{n}e_{n+k,n+k}+e_{2n+1,2n+1},$
$\displaystyle E_{i}=\sum_{k=1}^{n-i}(e_{k,k+i}-e_{n+i+k,n+k}),\;\;\forall
i=1,\ldots,n-1,$ $\displaystyle
A_{i}=e_{2n+1,i},\;\;B_{i}=e_{2n+1,n+i},\;\;\forall i=1,\ldots,n,$

and $e_{ij}$ are matrix units and non-trivial commutators given by

$\displaystyle[x,B_{i}]=B_{i},\;[y,A_{i}]=A_{i},\;\;\forall i=1,\ldots,n,$
$\displaystyle[E_{i},B_{k}]=B_{k-i},\;\;1\leq i<k\leq n,$
$\displaystyle[E_{i},A_{k}]=-A_{i+k},\;\;1\leq i\leq n-1,\;1\leq k\leq n-i.$

######  Remark 2.1.

With the change of basis

$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},z\right\\}\mapsto\left\\{e_{1},f_{1},\ldots,e_{n},f_{n},z\right\\},$

a derivation of $\mathfrak{l}_{2n+1}^{J_{a}}$ is represented by the
$(2n+1)\times(2n+1)$ matrix

$\begin{pmatrix}[cccccc|c]M_{1}&-\widetilde{M_{2}}&-\widetilde{M_{3}}&-\widetilde{M_{4}}&\cdots&-\widetilde{M}_{n}&0\\\
M_{2}&M_{1}&-\widetilde{M_{2}}&-\widetilde{M_{3}}&\cdots&-\widetilde{M}_{n-1}&0\\\
M_{3}&M_{2}&M_{1}&-\widetilde{M_{2}}&\cdots&-\widetilde{M}_{n-2}&0\\\
M_{4}&M_{3}&M_{2}&M_{1}&\cdots&-\widetilde{M}_{n-3}&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
M_{n}&M_{n-1}&M_{n-2}&M_{n-3}&\cdots&M_{1}&0\\\ \hline\cr
v_{1}&v_{2}&v_{3}&v_{4}&\cdots&v_{n}&\operatorname{tr}(M_{1})\end{pmatrix}$

where

$M_{1}=\begin{pmatrix}\alpha_{1}&0\\\ 0&\beta_{1}\\\
\end{pmatrix},\;\;M_{i}=\begin{pmatrix}0&0\\\ 0&\alpha_{i}\\\
\end{pmatrix},\;\;\widetilde{M}_{i}=\begin{pmatrix}\alpha_{i}&0\\\ 0&0\\\
\end{pmatrix},\;\;\forall i=2,\ldots,n,$

$v_{k}=(\mu_{k},\nu_{k})$, for any $k=1,\ldots,n$ and
$\operatorname{tr}(M_{1})=\alpha_{1}+\beta_{1}$.

In this case $\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}})$ has basis

$\\{x,y,E_{1},\ldots,E_{n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\},$

where

$\displaystyle
x=\sum_{k=1}^{n}e_{2k-1,2k-1}+e_{2n+1,2n+1},\;\;y=\sum_{k=1}^{n}e_{2k,2k}+e_{2n+1,2n+1},$
$\displaystyle
E_{i}=\sum_{k=0}^{n-i+1}(e_{2(k+i+1),2(k+1)}-e_{2k+1,2(k+i)+1}),\;\;\forall
i=1,\ldots,n-1,$ $\displaystyle
A_{i}=e_{2n+1,2i-1},\;\;B_{i}=e_{2n+1,2i},\;\;\forall i=1,\ldots,n$

and the Lie brackets are given by

$\displaystyle[x,B_{i}]=B_{i},\;[y,A_{i}]=A_{i},\;\;\forall i=1,\ldots,n,$
$\displaystyle[E_{i},B_{k}]=-B_{k-2i},\;\;k>2i,$
$\displaystyle[E_{i},A_{k}]=A_{k+2i},\;\;k+2i\leq 2n.$

With this representation, one can check that

$\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}})\subseteq\operatorname{Der}(\mathfrak{h}_{2n+1}),$

where the derivations of the $(2n+1)-$dimensional Heisenberg Lie algebra
$\mathfrak{h}_{2n+1}$ were determined in [9], with respect to the symplectic
basis $\\{e_{1},f_{1},\ldots,e_{n},f_{n},z\\}$ of $\mathfrak{h}_{2n+1}$. Later
we present the derivations of the Heisenberg Leibniz algebra
$\mathfrak{l}_{2n+1}^{J_{0}}$ and the ones of the Kronecker Leibniz algebra
$\mathfrak{k}_{n}$ with respect to this basis, in order to compare them with
the corresponding ones of the Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$.

The commutator ideal of $\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}})$ is
the abelian algebra of dimension $2n$ with basis

$\\{A_{1},\ldots A_{n},B_{1},\ldots,B_{n}\\},$

thus $\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}})$ is a two-step solvable
Lie algebra. Moreover the lower central series is

$\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}})\supseteq\langle A_{1},\ldots
A_{n},B_{1},\ldots,B_{n}\rangle\supseteq\langle A_{1},\ldots
A_{n},B_{1},\ldots,B_{n}\rangle\supseteq....$

so $\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}})$ is not nilpotent and its
nilradical is the ideal

$N=\langle E_{1},\ldots,E_{n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\rangle.$

Finally the center
$\operatorname{Z}(\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}}))$ is trivial
and the algebra of inner derivations of $\mathfrak{l}_{2n+1}^{J_{a}}$ is

$\operatorname{Inn}(\mathfrak{l}_{2n+1}^{J_{a}})=\langle
A_{h},\ldots,A_{n},B_{1},\ldots,B_{k}\rangle,$

with $h=1$ and $k=n$ if $a\neq\pm 1$; $h=2$ and $k=n$ if $a=1$; and $h=1$ and
$k=n-1$ if $a=-1$. Indeed

$\operatorname{ad}_{e_{i}}=B_{i-1}+(1+a)B_{i},\;\;\forall i=2,\ldots,n,$
$\operatorname{ad}_{f_{j}}=A_{j+1}+(-1+a)A_{j},\;\;\forall j=1,\ldots,n-1$

and

$\operatorname{ad}_{e_{1}}=(1+a)B_{1},\;\operatorname{ad}_{f_{n}}=(-1+a)A_{n},$

thus for $a=1$ we have

$\operatorname{ad}_{f_{n}}=0,\;\operatorname{ad}_{f_{j}}=A_{j+1},\;\;\forall
j=1,\ldots,n-1$

and the matrix $A_{1}$ does not represent an inner derivations. In the same
way, if $a=-1$, then
$B_{n}\not\in\operatorname{Inn}(\mathfrak{l}_{2n+1}^{J_{1}})$. We will show
later that

$\operatorname{AIDer}(\mathfrak{l}_{2n+1}^{J_{a}})=\langle
A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\rangle$

for every $a\in\mathbb{C}$.

When $a=0$, a derivation of the Heisenberg Leibniz algebra
$\mathfrak{l}_{2n+1}^{J_{0}}$ has the form

$\begin{pmatrix}A&C&0\\\ B&D&0\\\ \mu&\nu&\gamma\\\ \end{pmatrix}$

where $A$, $D$, $\mu$, $\nu$ and $\gamma$ are as above,

$B=\left(\begin{array}[]{*9{c}}0&0&0&0&\ldots&0&0&0&0\\\
0&0&0&0&\ldots&0&0&0&-b_{n+2}\\\ 0&0&0&0&\ldots&0&0&b_{n+2}&0\\\
0&0&0&0&\ldots&0&-b_{n+2}&0&-b_{n+4}\\\
\vdots&\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&0&\ldots&b_{2n-6}&0&b_{2n-4}&0\\\
0&0&0&-b_{n+2}&\ldots&0&-b_{2n-4}&0&-b_{2n-2}\\\
0&0&b_{n+2}&0&\ldots&b_{2n-4}&0&b_{2n-2}&0\\\
0&-b_{n+2}&0&-b_{n+4}&\ldots&0&-b_{2n-2}&0&-b_{2n}\\\ \end{array}\right),\;\;$
$C=\left(\begin{array}[]{*9{c}}c_{2}&0&c_{4}&0&\ldots&c_{n-2}&0&c_{n}&0\\\
0&-c_{4}&0&-c_{6}&\ldots&0&-c_{n}&0&0\\\ c_{4}&0&c_{6}&0&\ldots&c_{n}&0&0&0\\\
0&-c_{6}&0&-c_{8}&\ldots&0&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots&\vdots\\\
c_{n-1}&0&c_{n}&0&\ldots&0&0&0&0\\\ 0&-c_{n}&0&0&\ldots&0&0&0&0\\\
c_{n}&0&0&0&\ldots&0&0&0&0\\\ 0&0&0&0&\ldots&0&0&0&0\\\
\end{array}\right),\;\;$

if $n$ is even,

$B=\left(\begin{array}[]{*9{c}}0&0&0&0&\ldots&0&0&0&b_{n+1}\\\
0&0&0&0&\ldots&0&0&-b_{n+1}&0\\\ 0&0&0&0&\ldots&0&b_{n+1}&0&b_{n+3}\\\
0&0&0&0&\ldots&-b_{n+1}&0&-b_{n+3}&0\\\
\vdots&\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&-b_{n+1}&\ldots&-b_{2n-3}&0&-b_{2n-2}&0\\\
0&0&b_{n+1}&0&\ldots&0&b_{2n-3}&0&b_{2n-2}\\\
0&-b_{n+1}&0&-b_{n+3}&\ldots&-b_{2n-2}&0&-b_{2n}&0\\\
b_{n+1}&0&b_{n+3}&0&\ldots&0&b_{2n-2}&0&b_{2n}\\\ \end{array}\right),\;\;$
$C=\left(\begin{array}[]{*9{c}}c_{2}&0&c_{4}&0&\ldots&0&c_{n-1}&0&c_{n+1}\\\
0&-c_{4}&0&-c_{6}&\ldots&-c_{n-1}&0&-c_{n+1}&0\\\
c_{4}&0&c_{6}&0&\ldots&0&c_{n+1}&0&0\\\
0&-c_{6}&0&-c_{8}&\ldots&-c_{n+1}&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots&\vdots\\\
0&-c_{n-1}&0&-c_{n+1}&\ldots&0&0&0&0\\\ c_{n-1}&0&c_{n+1}&0&\ldots&0&0&0&0\\\
0&-c_{n+1}&0&0&\ldots&0&0&0&0\\\ c_{n+1}&0&0&0&\ldots&0&0&0&0\\\
\end{array}\right),\;\;$

if $n$ is odd.

If we reorder the basis basis as in Remark 2.1, then a derivation of
$\mathfrak{l}_{2n+1}^{J_{0}}$ is represented by

$\setcounter{MaxMatrixCols}{12}\begin{pmatrix}[ccccccccccc|c]\alpha_{1}&c_{2}&-\alpha_{2}&0&-\alpha_{3}&c_{4}&\cdots&-\alpha_{n-1}&c_{n}&-\alpha_{n}&0&0\\\
0&\beta_{1}&0&0&0&0&\cdots&0&0&0&0&0\\\
0&0&\alpha_{1}&-c_{4}&-\alpha_{2}&0&\cdots&-\alpha_{n-2}&0&-\alpha_{n-1}&0&0\\\
0&\alpha_{2}&0&\beta_{1}&0&0&\cdots&0&0&-b_{n+2}&0&0\\\
0&c_{4}&0&0&\alpha_{1}&c_{6}&\cdots&-\alpha_{n-3}&0&-\alpha_{n-2}&0&0\\\
0&\alpha_{3}&0&\alpha_{2}&0&\beta_{1}&\cdots&b_{n+2}&0&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&c_{n}&0&0&0&0&\cdots&\alpha_{1}&0&-\alpha_{2}&0&0\\\
0&\alpha_{n-1}&0&\alpha_{n-2}&b_{n+2}&\alpha_{n-3}&\cdots&b_{2n-2}&\beta_{1}&0&0&0\\\
0&0&0&0&0&0&\cdots&0&0&\alpha_{1}&0&0\\\
0&\alpha_{n}&-b_{n+2}&\alpha_{n-1}&0&\alpha_{n-2}&\cdots&0&\alpha_{2}&b_{2n}&\beta_{1}&0\\\
\hline\cr\mu_{1}&\nu_{1}&\mu_{2}&\nu_{2}&\mu_{3}&\nu_{3}&\cdots&\mu_{n-1}&\nu_{n-1}&\mu_{n}&\nu_{n}&\alpha_{1}+\beta_{1}\end{pmatrix}$

if $n$ is even, and

$\setcounter{MaxMatrixCols}{12}\begin{pmatrix}[ccccccccccc|c]\alpha_{1}&c_{2}&-\alpha_{2}&0&-\alpha_{3}&c_{4}&\cdots&-\alpha_{n-1}&0&-\alpha_{n}&c_{n+1}&0\\\
0&\beta_{1}&0&0&0&0&\cdots&0&0&b_{n+1}&0&0\\\
0&0&\alpha_{1}&-c_{4}&-\alpha_{2}&0&\cdots&-\alpha_{n-2}&-c_{n+1}&-\alpha_{n-1}&0&0\\\
0&\alpha_{2}&0&\beta_{1}&0&0&\cdots&-b_{n+1}&0&0&0&0\\\
0&c_{4}&0&0&\alpha_{1}&c_{6}&\cdots&-\alpha_{n-3}&0&-\alpha_{n-2}&0&0\\\
0&\alpha_{3}&0&\alpha_{2}&0&\beta_{1}&\cdots&0&0&b_{n+3}&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&-c_{n+1}&0&0&\cdots&\alpha_{1}&0&-\alpha_{2}&0&0\\\
0&\alpha_{n-1}&-b_{n+1}&\alpha_{n-2}&0&\alpha_{n-3}&\cdots&-b_{2n-2}&\beta_{1}&0&0&0\\\
0&c_{n+1}&0&0&0&0&\cdots&0&0&\alpha_{1}&0&0\\\
b_{n+1}&\alpha_{n}&0&\alpha_{n-1}&b_{n+3}&\alpha_{n-2}&\cdots&0&\alpha_{1}&b_{2n}&\beta_{1}&0\\\
\hline\cr\mu_{1}&\nu_{1}&\mu_{2}&\nu_{2}&\mu_{3}&\nu_{3}&\cdots&\mu_{n-1}&\nu_{n-1}&\mu_{n}&\nu_{n}&\alpha_{1}+\beta_{1}\end{pmatrix}$

if $n$ is odd. We can now study in details these two cases.

If $n$ is even, then $\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})$ is a
Lie algebra of dimension $4n+1$ with basis

$\\{x,y,E_{1},\ldots,E_{n-1},c_{2},c_{4},\ldots,c_{n},b_{n+2},b_{n+4}\ldots,b_{2n},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\},$

where

$\displaystyle x=\sum_{k=1}^{n}e_{2k-1,2k-1}+e_{2k+1,2k+1},\;\;$
$\displaystyle y=\sum_{k=1}^{n}e_{2k,2k}+e_{2k+1,2k+1},$ $\displaystyle
E_{i}=\sum_{k=0}^{n-i+1}(e_{2(k+i+1),2(k+1)}-e_{2k+1,2(k+i)+1}),\;\;\forall
i=1,\ldots,n-1,$ $\displaystyle
A_{i}=e_{2n+1,2i-1},\;\;B_{i}=e_{2n+1,2i},\;\;\forall i=1,\ldots,n,$
$\displaystyle c_{h}=\sum_{i=0}^{h-2}(-1)^{i}e_{2(h-i-1)-1,2(1+i)},\;\;\forall
h=2,4,\ldots,n,$ $\displaystyle
b_{h}=\sum_{i=0}^{2n-h}(-1)^{i}e_{2(n-i),2(h-n+i)-1},\;\;\forall
h=n+2,n+4,\ldots,2n$

and commutators

$\displaystyle[x,B_{i}]=B_{i},\;[y,A_{i}]=A_{i},\;\;\forall i=1,\ldots,n,$
$\displaystyle[E_{i},B_{k}]=-B_{k-2i},\;\;k>2i,$
$\displaystyle[E_{i},A_{k}]=A_{k+2i},\;\;k+2i\leq 2n,$
$\displaystyle[x,c_{h}]=c_{h},\;[y,c_{h}]=-c_{h},\;\;\forall h=2,4,\ldots,n,$
$\displaystyle[x,b_{h}]=-b_{h},\;[y,b_{h}]=b_{h},\;\;\forall
h=n+2,n+4,\ldots,2n,$
$\displaystyle[A_{i},c_{k}]=(-1)^{i+1}B_{k-i},\;[B_{i},b_{k}]=(-1)^{i}A_{k-i},\;\;1\leq
k-i\leq n,$ $\displaystyle[c_{k},b_{h}]=E_{h-k},\;\;h-k\geq 1,$
$\displaystyle[\alpha_{2i},c_{h}]=-2c_{h-2i},\;\;h-2i>0,$
$\displaystyle[\alpha_{2i},b_{h}]=2b_{h+2i},\;\;h+2i\leq 2n.$

Then the commutator ideal of $\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})$
has basis

$\\{E_{2},E_{4},\ldots,E_{n-2},c_{2},c_{4},\ldots,c_{n},b_{n+2},b_{n+4}\ldots,b_{2n},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\}$

and we have a $(\frac{n}{2}+1)-$step solvable Lie algebra with derived series

$\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})\supseteq[\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}}),\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})]\supseteq$
$\supseteq\langle
E_{2},E_{4},\ldots,E_{n-2},c_{2},c_{4},\ldots,c_{n-2},b_{n+4},\ldots,b_{2n},A_{2},\ldots,A_{n},B_{1},\ldots,B_{n-1}\rangle\supseteq\ldots$
$\ldots\supseteq\langle
c_{2},b_{2n},A_{\frac{n}{2}+1},\ldots,A_{n},B_{1},\ldots,B_{\frac{n}{2}}\rangle\supseteq
0$

Moreover, $\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})$ is not nilpotent
and its nilradical is the ideal

$N=\langle
E_{1},\ldots,E_{n-1},c_{2},c_{4},\ldots,c_{n},b_{n+2},b_{n+4}\ldots,b_{2n},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\rangle.$

If $n$ is odd, then the algebra of derivations of
$\mathfrak{l}_{2n+1}^{J_{0}}$ has dimension $4n+2$ and it is generated by

$\\{x,y,E_{1},\ldots,E_{n-1},c_{2},c_{4},\ldots,c_{n+1},b_{n+1},b_{n+3}\ldots,b_{2n},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\}.$

The Lie brackets are the same of the ones listed for the case $n$ even, except
for the facts that $[B_{i},b_{k}]=(-1)^{i+1}A_{k-i}$, for any $1\leq k-i\leq
n$, and $[c_{n+1},b_{n+1}]=x-y$. Then the commutator ideal is the subspace
generated by

$\\{x-y,E_{2},E_{4},\ldots,E_{n-1},c_{2},c_{4},\ldots,c_{n+1},b_{n+1},b_{n+3}\ldots,b_{2n},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\}.$

In this case the Lie algebra of derivations is not solvable since

$[[\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}}),\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})],[\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}}),\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})]]=[\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}}),\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})]$

and the Levi decomposition is given by

$\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})=R\rtimes S,$

where the radical of the Lie algebra is

$R=\langle
x+y,E_{1},\ldots,E_{n-1},c_{2},c_{4},\ldots,c_{n-1},b_{n+3},b_{n+5}\ldots,b_{2n},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\rangle$

and the Levi complement is

$S=\langle x-y,c_{n+1},b_{n+1}\rangle.$

Finally the nilradical is the ideal

$N=\langle
E_{1},\ldots,E_{n-1},c_{2},c_{4},\ldots,c_{n-1},b_{n+3},b_{n+5}\ldots,b_{2n},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\rangle.$

In both cases $n$ is even or odd, we have that
$\operatorname{Z}(\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}}))=0$ and the
Lie algebra of inner derivations is represented by the matrices of the type

$\left(\begin{array}[]{c|c}&0\\\ \textbf{\Large 0}&\vdots\\\ &0\\\
\hline\cr\mu_{1}\;\nu_{1}\ldots\mu_{n}\;\nu_{n}&0\\\ \end{array}\right),$ (1)

thus $\operatorname{Inn}(\mathfrak{l}_{2n+1}^{J_{0}})$ is an abelian algebra
of dimension $2n$. Moreover, for every $n\in\mathbb{N}$ and for every
$a\in\mathbb{C}^{\ast}$, we observe that

$\operatorname{Der}(\mathfrak{h}_{2n+1})\supseteq\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})\supseteq\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}}).$

###  2.2 The real case

Irreducible polynomials in $\mathbb{R}[x]$ have degree one or two. Let
$f(x)\in\mathbb{R}[x]$ be an irreducible monic polynomial. If $f(x)=x-a$, then
we obtain the same results of the complex case. So we suppose that
$f(x)=x^{2}+Cx+D$, with $C^{2}-4D<0$.

Let $z=a+ib\in\mathbb{C}\setminus\mathbb{R}$ be a root of $f(x)$. Then
$f(x)=(x-z)(x-\bar{z})$ and the companion matrix $A$ of $f(x)^{n}$ in similar
to the $2n\times 2n$ real block matrix

$J_{R}=\begin{pmatrix}R&0&\cdots&0\\\ I_{2}&R&\cdots&0\\\
\vdots&\ddots&\ddots&\vdots\\\ 0&\cdots&I_{2}&R\end{pmatrix},$

where

$R=\begin{pmatrix}a&b\\\ -b&a\end{pmatrix}$

is the realification of the complex number $z$. Thus
$\mathfrak{l}_{4n+1}^{A}\cong\mathfrak{l}_{4n+1}^{J_{R}}$ and
$\mathfrak{l}_{4n+1}^{J_{R}}$ is the realification of the complex algebra
$\mathfrak{l}_{2n+1}^{J_{z}}$. In [4] the derivations of the realification of
the $(2n+1)-$dimensional Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$ were
studied. We want to find the conditions such that the realification of a
derivation of the complex algebra $\mathfrak{l}_{2n+1}^{J_{z}}$, with
$z=a+ib\in\mathbb{C}\setminus\mathbb{R}$, is a derivation of the real algebra
$\mathfrak{l}_{4n+1}^{J_{R}}$. We will investigate the case $n=1$.

Let $\\{e_{1},f_{1},e_{2},f_{2},z\\}$ be a basis of the real algebra
$\mathfrak{l}_{5}^{R}$. The non-trivial commutators are

$\displaystyle[e_{i},f_{i}]=(1+a)z,\;[f_{i},e_{i}]=(-1+a)z,\;\;\forall i=1,2,$
$\displaystyle[e_{1},f_{2}]=[f_{2},e_{1}]=bz,\;[e_{2},f_{1}]=[f_{1},e_{2}]=-bz$

and it comes out that a general derivation of $\mathfrak{l}_{5}^{R}$ is
represented by the matrix

$\begin{pmatrix}[cccc|c]\alpha_{1}&0&\alpha_{2}&0&0\\\
0&\beta_{1}&0&\alpha_{2}&0\\\ -\alpha_{2}&0&\alpha_{1}&0&0\\\
0&-\alpha_{2}&0&\beta_{1}&0\\\
\hline\cr\mu_{1}&\nu_{1}&\mu_{2}&\nu_{2}&\alpha_{1}+\beta_{1}\\\
\end{pmatrix}$

if $a\neq 0$ and by

$\begin{pmatrix}[cccc|c]\alpha_{1}&\delta&\alpha_{2}&0&0\\\
\delta^{\prime}&\beta_{1}&0&\alpha_{2}&0\\\
-\alpha_{2}&0&\alpha_{1}&\delta&0\\\
0&-\alpha_{2}&\delta^{\prime}&\beta_{1}&0\\\
\hline\cr\mu_{1}&\nu_{1}&\mu_{2}&\nu_{2}&\alpha_{1}+\beta_{1}\\\
\end{pmatrix}$

if $a=0$. Then

  * •

if $a\neq 0$, $\operatorname{Der}(\mathfrak{l}_{5}^{R})$ is generated by the
set

$\\{x,y,E,A_{1},A_{2},B_{1},B_{2}\\},$

where $x=e_{11}+e_{33}+e_{55}$, $y=e_{22}+e_{44}+e_{55}$,
$E=e_{13}+e_{24}-e_{31}-e_{42}$, $A_{i}=e_{5,2i-1}$ and $B_{i}=e_{5,2i}$, for
every $i=1,2$, and the non-trivial Lie brackets are

$\displaystyle[x,B_{i}]=B_{i},\;[y,A_{i}]=A_{i},\;\;\forall i=1,2,$
$\displaystyle[E,A_{1}]=-A_{2},\;[E,A_{2}]=-A_{1},$
$\displaystyle[E,B_{1}]=-B_{2},\;[E,B_{2}]=-B_{1}.$

Then we have a solvable Lie algebra with abelian commutator ideal generated by

$\\{A_{1},A_{2},B_{1},B_{2}\\},$

which coincides with the ideal $\operatorname{Inn}(\mathfrak{l}_{5}^{R})$ and
with the nilradical of the Lie algebra itself. Moreover the center
$\operatorname{Z}(\operatorname{Der}(\mathfrak{l}_{5}^{R}))$ is trivial.

  * •

if $a=0$, a basis of $\operatorname{Der}(\mathfrak{l}_{5}^{R})$ is

$\\{x,y,E,F,G,A_{1},A_{2},B_{1},B_{2}\\},$

where $x,y,E,A_{1},A_{2},B_{1},B_{2}$ are defined as above, $F=e_{12}+e_{34}$,
$G=e_{21}+e_{43}$ and the non-trivial Lie brackets are given by the ones above
and by

$\displaystyle[x,F]=F,\;[x,G]=-G,\;[y,F]=-F,\;[y,G]=-G,$
$\displaystyle[F,G]=x-y,\;[F,A_{i}]=-B_{i},\;[F,B_{i}]=-A_{i},\;\,\forall
i=1,2.$

It follows that $\operatorname{Z}(\operatorname{Der}(\mathfrak{l}_{5}^{R}))=0$
and the Lie algebra is not solvable. Its radical is given by the ideal

$R=\langle x+y,E,A_{1},A_{2},B_{1},B_{2}\rangle,$

a Levi complement is the semisimple Lie algebra

$S=\langle x-y,F,G\rangle$

and the nilradical of $\operatorname{Der}(\mathfrak{l}_{5}^{R})$ is the
abelian four-dimensional algebra

$N=\langle A_{1},A_{2},B_{1},B_{2}\rangle\cong\mathbb{R}^{4}$

and again it coincides with the set of inner derivations of
$\mathfrak{l}_{5}^{R}$.

Now let

$\begin{pmatrix}\alpha&0&0\\\ 0&\beta&0\\\ \mu&\nu&\alpha+\beta\\\
\end{pmatrix}$

be a derivation of the complex Heinseberg algebra $\mathfrak{l}_{3}^{z}$,
where $z=a+ib$. Then its realification is represented by the matrix

$\begin{pmatrix}[cccc|c]\Re(\alpha)&\Im(\alpha)&0&0&0\\\
-\Im(\alpha)&\Re(\alpha)&0&0&0\\\ 0&0&\Re(\beta)&\Im(\beta)&0\\\
0&0&-\Im(\beta)&\Re(\beta)&0\\\
\hline\cr\mu_{1}&\nu_{1}&\mu_{2}&\nu_{2}&\gamma\\\ \end{pmatrix}$

and this is a derivation of the real Heisenberg Leibniz algebra
$\mathfrak{l}_{5}^{R}$ if and only if

$\alpha=\beta\in\mathbb{R}$

in both cases that $a\neq 0$ or $a=0$. Then the set of realifications of the
derivations of $\mathfrak{l}_{3}^{z}$ that are derivations of the real algebra
$\mathfrak{l}_{5}^{R}$ form the proper Lie subalgebra of the matrices of the
form

$\begin{pmatrix}[cccc|c]\alpha&0&0&0&0\\\ 0&\alpha&0&0&0\\\ 0&0&\alpha&0&0\\\
0&0&0&\alpha&0\\\ \hline\cr\mu_{1}&\nu_{1}&\mu_{2}&\nu_{2}&2\alpha\\\
\end{pmatrix}$

of $\operatorname{Der}(\mathfrak{l}_{5}^{R})$.

##  3 Derivations of the Kronecker Leibniz algebra $\mathfrak{k}_{n}$

Now we return to the case that $\mathbb{F}$ is a field with
$\operatorname{char}(\mathbb{F})\neq 2$. Let $n\in\mathbb{N}$ and let
$\mathfrak{k}_{n}$ be the _Kronecker Leibniz algebra_. We fix the basis
$\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},z\\}$ of $\mathfrak{k}_{n}$.

######  Theorem 3.1.

A linear endomorphism $d\colon\mathfrak{k}_{n}\rightarrow\mathfrak{k}_{n}$ is
a derivation of the Kronecker algebra $\mathfrak{k}_{n}$ if and only if it has
the form

$\begin{pmatrix}A&C&0\\\ B&D&0\\\ \mu&\nu&\gamma\\\ \end{pmatrix}$

where

$A=\left(\begin{array}[]{*5{c}}\alpha_{1}&\alpha_{2}&\alpha_{3}&\ldots&\alpha_{n}\\\
0&\alpha_{1}&\alpha_{2}&\ldots&\alpha_{n-1}\\\
0&0&\alpha_{1}&\ldots&\alpha_{n-2}\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\ldots&\alpha_{1}\\\
\end{array}\right),\;\;D=\left(\begin{array}[]{*5{c}}\beta_{1}&0&0&\ldots&0\\\
-\alpha_{2}&\beta_{1}&0&\ldots&0\\\
-\alpha_{3}&-\alpha_{2}&\beta_{1}&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
-\alpha_{n}&-\alpha_{n-1}&-\alpha_{n-2}&\ldots&\beta_{1}\\\
\end{array}\right),$
$B=\left(\begin{array}[]{*9{c}}0&0&0&0&\ldots&0&0&0&b_{n+1}\\\
0&0&0&0&\ldots&0&0&-b_{n+1}&0\\\ 0&0&0&0&\ldots&0&b_{n+1}&0&b_{n+3}\\\
0&0&0&0&\ldots&-b_{n+1}&0&-b_{n+3}&0\\\
\vdots&\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&b_{n+1}&\ldots&0&b_{2n-5}&0&b_{2n-3}\\\
0&0&-b_{n+1}&0&\ldots&-b_{2n-5}&0&-b_{2n-3}&0\\\
0&b_{n+1}&0&b_{n+3}&\ldots&0&b_{2n-3}&0&b_{2n-1}\\\
-b_{n+1}&0&-b_{n+3}&0&\ldots&-b_{2n-3}&0&-b_{2n-1}&0\\\
\end{array}\right),\;\;$
$C=\left(\begin{array}[]{*9{c}}0&c_{3}&0&c_{5}&\ldots&0&c_{n-1}&0&c_{n+1}\\\
-c_{3}&0&-c_{5}&0&\ldots&-c_{n-1}&0&-c_{n+1}&0\\\
0&c_{5}&0&c_{7}&\ldots&0&c_{n+1}&0&0\\\
-c_{5}&0&-c_{7}&0&\ldots&-c_{n+1}&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots&\vdots\\\
0&c_{n-1}&0&c_{n+1}&\ldots&0&0&0&0\\\ -c_{n-1}&0&-c_{n+1}&0&\ldots&0&0&0&0\\\
0&c_{n+1}&0&0&\ldots&0&0&0&0\\\ -c_{n+1}&0&0&0&\ldots&0&0&0&0\\\
\end{array}\right),\;\;$

if $n$ is even,

$B=\left(\begin{array}[]{*9{c}}0&0&0&0&\ldots&0&0&0&0\\\
0&0&0&0&\ldots&0&0&0&-b_{n+2}\\\ 0&0&0&0&\ldots&0&0&b_{n+2}&0\\\
0&0&0&0&\ldots&0&-b_{n+2}&0&-b_{n+4}\\\
\vdots&\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&0&\ldots&0&-b_{2n-5}&0&-b_{2n-3}\\\
0&0&0&b_{n+2}&\ldots&b_{2n-5}&0&b_{2n-3}&0\\\
0&0&-b_{n+2}&0&\ldots&0&-b_{2n-3}&0&-b_{2n-1}\\\
0&b_{n+2}&0&b_{n+4}&\ldots&b_{2n-3}&0&b_{2n-1}&0\\\ \end{array}\right),\;\;$
$C=\left(\begin{array}[]{*9{c}}0&c_{3}&0&c_{5}&\ldots&c_{n-2}&0&c_{n}&0\\\
-c_{3}&0&-c_{5}&0&\ldots&0&-c_{n}&0&0\\\ 0&c_{5}&0&c_{7}&\ldots&c_{n}&0&0&0\\\
-c_{5}&0&-c_{7}&0&\ldots&0&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots&\vdots\\\
-c_{n-2}&0&-c_{n}&0&\ldots&0&0&0&0\\\ 0&c_{n}&0&0&\ldots&0&0&0&0\\\
-c_{n}&0&0&0&\ldots&0&0&0&0\\\ 0&0&0&0&\ldots&0&0&0&0\\\
\end{array}\right),\;\;$

if $n$ is odd and

$\mu=(\mu_{1},\mu_{2},\mu_{3},\ldots,\mu_{n}),\;\;\nu=(\nu_{1},\nu_{2},\nu_{3},\ldots,\nu_{n}),\;\;\gamma=\alpha_{1}+\beta_{1}.$

######  Remark 3.2.

Again, with the change of basis

$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},z\right\\}\mapsto\left\\{e_{1},f_{1},\ldots,e_{n},f_{n},z\right\\},$

a derivation of $\mathfrak{k}_{n}$ is represented by the $(2n+1)\times(2n+1)$
matrix

$\setcounter{MaxMatrixCols}{12}\begin{pmatrix}[ccccccccccc|c]\alpha_{1}&0&-\alpha_{2}&c_{3}&-\alpha_{3}&0&\cdots&-\alpha_{n-1}&0&-\alpha_{n}&c_{n+1}&0\\\
0&\beta_{1}&0&0&0&0&\cdots&0&0&b_{n+1}&0&0\\\
0&-c_{3}&\alpha_{1}&0&-\alpha_{2}&-c_{5}&\cdots&-\alpha_{n-2}&-c_{n+1}&-\alpha_{n-1}&0&0\\\
0&\alpha_{2}&0&\beta_{1}&0&0&\cdots&-b_{n+1}&0&0&0&0\\\
0&0&0&c_{5}&\alpha_{1}&0&\cdots&-\alpha_{n-3}&0&-\alpha_{n-2}&0&0\\\
0&\alpha_{1}&0&\alpha_{2}&0&\beta_{1}&\cdots&0&0&-b_{n+3}&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&c_{n+1}&0&0&\cdots&\alpha_{1}&0&-\alpha_{2}&0&0\\\
0&\alpha_{n-1}&b_{n+1}&\alpha_{n-2}&0&\alpha_{n-3}&\cdots&0&\beta_{1}&-b_{2n-1}&0&0\\\
0&-c_{n+1}&0&0&0&0&\cdots&0&0&\alpha_{1}&0&0\\\
-b_{n+1}&\alpha_{n}&0&\alpha_{n-1}&b_{n+3}&\alpha_{n-2}&\cdots&b_{2n-1}&\alpha_{2}&0&\beta_{1}&0\\\
\hline\cr\mu_{1}&\nu_{1}&\mu_{2}&\nu_{2}&\mu_{3}&\nu_{3}&\cdots&\mu_{n-1}&\nu_{n-1}&\mu_{n}&\nu_{n}&\alpha_{1}+\beta_{1}\end{pmatrix}$

if $n$ is even and

$\setcounter{MaxMatrixCols}{12}\begin{pmatrix}[ccccccccccc|c]\alpha_{1}&0&-\alpha_{2}&c_{3}&-\alpha_{3}&0&\cdots&-\alpha_{n-1}&c_{n}&-\alpha_{n}&0&0\\\
0&\beta_{1}&0&0&0&0&\cdots&0&0&0&0&0\\\
0&-c_{3}&\alpha_{1}&0&-\alpha_{2}&-c_{5}&\cdots&-\alpha_{n-2}&0&-\alpha_{n-1}&0&0\\\
0&\alpha_{2}&0&\beta_{1}&0&0&\cdots&0&0&-b_{n+2}&0&0\\\
0&0&0&-c_{5}&\alpha_{1}&0&\cdots&-\alpha_{n-3}&0&-\alpha_{n-2}&0&0\\\
0&\alpha_{3}&0&\alpha_{2}&0&\beta_{1}&\cdots&b_{n+2}&0&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&c_{n}&0&0&0&0&\cdots&\alpha_{1}&0&-\alpha_{2}&0&0\\\
0&\alpha_{n-1}&0&\alpha_{n-2}&-b_{n+2}&\alpha_{n-3}&\cdots&0&\beta_{1}&-b_{2n-1}&0&0\\\
0&0&0&0&0&0&\cdots&0&0&\alpha_{1}&0&0\\\
0&\alpha_{n}&b_{n+2}&\alpha_{n-1}&0&\alpha_{n-2}&\cdots&b_{2n-1}&\alpha_{2}&0&\beta_{1}&0\\\
\hline\cr\mu_{1}&\nu_{1}&\mu_{2}&\nu_{2}&\mu_{3}&\nu_{3}&\cdots&\mu_{n-1}&\nu_{n-1}&\mu_{n}&\nu_{n}&\alpha_{1}+\beta_{1}\end{pmatrix}$

if $n$ is odd.

We can now describe the main properties of the Lie algebra
$\operatorname{Der}(\mathfrak{k}_{n})$.

  * •

If $n$ is even, then $\operatorname{Der}(\mathfrak{k}_{n})$ has basis

$\\{x,y,E_{1},\ldots,E_{n-1},c_{3},c_{5},\ldots,c_{n+1},b_{n+1},b_{n+3}\ldots,b_{2n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\},$

where

$\displaystyle x=\sum_{k=1}^{n}e_{2k-1,2k-1}+e_{2k+1,2k+1},\;\;$
$\displaystyle y=\sum_{k=1}^{n}e_{2k,2k}+e_{2k+1,2k+1},$ $\displaystyle
E_{i}=\sum_{k=0}^{n-i+1}(e_{2(k+i+1),2(k+1)}-e_{2k+1,2(k+i)+1}),\;\;\forall
i=1,\ldots,n-1,$ $\displaystyle
A_{i}=e_{2n+1,2i-1},\;\;B_{i}=e_{2n+1,2i},\;\;\forall i=1,\ldots,n,$
$\displaystyle
c_{h}=\sum_{i=0}^{h-2}(-1)^{i+1}e_{2(h-i-1)-1,2(1+i)},\;\;\forall
h=3,5,\ldots,n+1,$ $\displaystyle
b_{h}=\sum_{i=0}^{2n-h}(-1)^{i+1}e_{2(n-i),2(h-n+i)-1},\;\;\forall
h=n+1,n+3,\ldots,2n-1,$

and

$\displaystyle[x,B_{i}]=B_{i},\;[y,A_{i}]=A_{i},\;\;\forall i=1,\ldots,n,$
$\displaystyle[E_{i},B_{k}]=-B_{k-2i},\;\;k>2i,$
$\displaystyle[E_{i},A_{k}]=A_{k+2i},\;\;k+2i\leq 2n,$
$\displaystyle[x,c_{h}]=c_{h},\;[y,c_{h}]=-c_{h},\;\;\forall
h=3,5,\ldots,n+1,$
$\displaystyle[x,b_{h}]=-b_{h},\;[y,b_{h}]=b_{h},\;\;\forall
h=n+1,n+3,\ldots,2n-1,$
$\displaystyle[A_{i},c_{k}]=(-1)^{i+1}B_{k-i},\;[B_{i},b_{k}]=(-1)^{i+1}A_{k-i},\;\;1\leq
k-i\leq n,$ $\displaystyle[c_{k},b_{h}]=E_{h-k},\;\;h-k\geq 1,$
$\displaystyle[c_{n+1},b_{n+1}]=-x+y,$
$\displaystyle[E_{2i},c_{h}]=-2c_{h-2i},\;\;h-2i>0,$
$\displaystyle[E_{2i},b_{h}]=2b_{h+2i},\;\;h+2i\leq 2n.$

The commutator ideal of $\operatorname{Der}(\mathfrak{k}_{n})$ has basis

$\\{x-y,E_{2},E_{4},\ldots,E_{n-2},c_{3},c_{5},\ldots,c_{n+1},b_{n+1},b_{n+3}\ldots,b_{2n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\}$

and, as in the case of the Heisenberg Leibniz algebra
$\mathfrak{l}_{2n+1}^{J_{0}}$ with $n$ odd, we have that the Lie algebra of
derivations is not solvable. The Levi decomposition is

$\operatorname{Der}(\mathfrak{k}_{n})=R\rtimes S,$

where

$R=\langle
x+y,E_{1},\ldots,E_{n-1},c_{3},c_{5},\ldots,c_{n-1},b_{n+3},b_{n+5}\ldots,b_{2n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\rangle$

is the radical and

$S=\langle x-y,c_{n+1},b_{n+1}\rangle$

is a Levi complement. Moreover the nilradical of
$\operatorname{Der}(\mathfrak{k}_{n})$ is

$N=\langle
E_{1},\ldots,E_{n-1},c_{3},c_{5},\ldots,c_{n-1},b_{n+3},b_{n+5}\ldots,b_{2n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\rangle.$

  * •

If $n$ is odd, then the algebra of derivations of $\mathfrak{k}_{n}$ has
dimension $4n$ and it is generated by

$\\{x,y,E_{1},\ldots,E_{n-1},c_{3},c_{5},\ldots,c_{n},b_{n+2},b_{n+4}\ldots,b_{2n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\}.$

The Lie brackets are the same of the ones listed before when $n$ is even,
except for the facts that $x-y$ does not belong to the commutator ideal and
$[B_{i},b_{k}]=(-1)^{i}A_{k-i}$, for any $1\leq k-i\leq n$. In this case the
commutator ideal is the subspace generated by

$\\{E_{2},E_{4},\ldots,E_{n-1},c_{3},c_{5},\ldots,c_{n},b_{n+2},b_{n+4}\ldots,b_{2n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\\}$

and we have a $(\frac{n+1}{2}+1)-$step solvable Lie algebra with nilradical

$N=\langle
E_{1},\ldots,E_{n-1},c_{3},c_{5},\ldots,c_{n},b_{n+2},b_{n+4}\ldots,b_{2n-1},A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\rangle.$

In both cases $n$ is odd or even, the center
$\operatorname{Z}(\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}}))$ is
trivial, the Lie algebra of inner derivations is

$\operatorname{Inn}(\mathfrak{k}_{n})=\langle
A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\rangle\cong\mathbb{F}^{2n},$

since

$\operatorname{ad}_{e_{i}}=B_{i-1}+B_{i},\;\operatorname{ad}_{f_{i}}=A_{i}+A_{i+1},\;\;\forall
i=1,\ldots,n$

and

$\operatorname{Der}(\mathfrak{h}_{2n+1})\supseteq\operatorname{Der}(\mathfrak{k}_{n})\supseteq\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}}),$

for any $a\neq 0$. More precisely

$\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{0}})\cap\operatorname{Der}(\mathfrak{k}_{n})=\operatorname{Der}(\mathfrak{l}_{2n+1}^{J_{a}}).$

##  4 Derivations of the Dieudonné Leibniz algebra $\mathfrak{d}_{n}$

Finally we study the derivations of the _Dieudonné Leibniz algebra_
$\mathfrak{d}_{n}$. We fix the basis $\\{e_{1},\ldots,e_{2n+1},z\\}$ of
$\mathfrak{d}_{n}$.

######  Theorem 4.1.

A derivation of $\mathfrak{d}_{n}$ is represented by a $(2n+2)\times(2n+2)$
matrix

$\left(\begin{array}[]{c|c|c}&&0\\\ \alpha I_{n+1}&C&\vdots\\\ &&0\\\
\hline\cr&&0\\\ 0&\beta I_{n}&\vdots\\\ &&0\\\
\hline\cr\mu_{1}\ldots\mu_{n+1}&\nu_{1}\ldots\nu_{n}&\alpha+\beta\end{array}\right)$
(2)

where the $(n+1)\times n$ matrix $C$ is

$\begin{pmatrix}\alpha_{1}&0&\alpha_{2}&0&\alpha_{3}&0&\cdots&\alpha_{\frac{n}{2}}&0\\\
0&-\alpha_{2}&0&-\alpha_{3}&0&-\alpha_{4}&\cdots&0&-\alpha_{\frac{n}{2}+1}\\\
\alpha_{2}&0&\alpha_{3}&0&\alpha_{4}&0&\cdots&\alpha_{\frac{n}{2}+1}&0\\\
0&-\alpha_{3}&0&-\alpha_{4}&0&-\alpha_{5}&\cdots&0&-\alpha_{\frac{n}{2}+2}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&-\alpha_{\frac{n}{2}+1}&0&-\alpha_{\frac{n}{2}+2}&0&-\alpha_{\frac{n}{2}+3}&\cdots&0&-\alpha_{n}\\\
\alpha_{\frac{n}{2}+1}&0&\alpha_{\frac{n}{2}+2}&0&\alpha_{\frac{n}{2}+3}&0&\cdots&\alpha_{n}&0\\\
\end{pmatrix}$

if $n$ is even and

$\begin{pmatrix}\alpha_{1}&0&\alpha_{2}&0&\alpha_{3}&0&\cdots&0&\alpha_{\frac{n+1}{2}}\\\
0&-\alpha_{2}&0&-\alpha_{3}&0&-\alpha_{4}&\cdots&-\alpha_{\frac{n+1}{2}}&0\\\
\alpha_{2}&0&\alpha_{3}&0&\alpha_{4}&0&\cdots&0&\alpha_{\frac{n+1}{2}+1}\\\
0&-\alpha_{3}&0&-\alpha_{4}&0&-\alpha_{5}&\cdots&-\alpha_{\frac{n+1}{2}+1}&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
\alpha_{\frac{n+1}{2}}&0&\alpha_{\frac{n+1}{2}+1}&0&\alpha_{\frac{n+1}{2}+2}&0&\cdots&0&\alpha_{n}\\\
0&-\alpha_{\frac{n+1}{2}+1}&0&-\alpha_{\frac{n+1}{2}+2}&0&-\alpha_{\frac{n+1}{2}+3}&\cdots&-\alpha_{n}&0\\\
\end{pmatrix}$

if $n$ is odd.

The Lie algebra $\operatorname{Der}(\mathfrak{d}_{n})$ has dimension $3n+3$
and basis

$\\{x,y,E_{1},\ldots,E_{n},A_{1},\ldots,A_{2n+1}\\},$

where

$\displaystyle x=\sum_{i=1}^{n+1}e_{ii}+e_{2n+1,2n+1},\;$ $\displaystyle
y=\sum_{i=n+2}^{2n+2}e_{ii},\;A_{i}=e_{2n+1,i},\;\;\forall i=1,\ldots,2n+1,$
$\displaystyle E_{i}=\sum_{k=1}^{2i-1}(-1)^{k+1}e_{k,n+2i+1-k},\;\;\forall
i=1,\ldots,\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor},$ $\displaystyle
E_{\frac{n}{2}+j}=\sum_{k=1}^{n+2-2j}(-1)^{k+1}e_{n+2-k,n+2j-1+k},\;\;\forall
j=1,\ldots,\frac{n}{2},$

if $n$ even, and

$E_{\frac{n+1}{2}+j}=\sum_{k=1}^{n+1-2j}(-1)^{k}e_{n+2-k,n+2j+k},\;\;\forall
j=1,\ldots,\frac{n-1}{2},$

if $n$ is odd. The non-zero Lie brackets are

$\displaystyle[x,E_{i}]=[E_{i},y]=\alpha_{i},\;\;\forall i=1,\ldots,n,$
$\displaystyle[y,A_{h}]=A_{h},\;\;\forall h=1,\ldots,n+1,$
$\displaystyle[x,A_{k}]=A_{k},\;\;\forall k=n+2,\ldots,2n+1,$
$\displaystyle[A_{i},E_{k}]=\varepsilon_{j}A_{j},\;\;\forall i=1,\ldots,n+1,$

where $\varepsilon_{j}=\pm 1$ is the only entry different than zero in the
$i^{th}$ row of the matrix $\alpha_{k}$ and $j\in\\{n+2,\ldots,2n+1\\}$ is its
column. Thus $\operatorname{Der}(\mathfrak{d}_{n})$ is a $3-$step solvable Lie
algebra with commutator ideal consisting of the matrices

$\left(\begin{array}[]{c|c|c}&&0\\\ 0&C&\vdots\\\ &&0\\\ \hline\cr&&0\\\
0&0&\vdots\\\ &&0\\\
\hline\cr\mu_{1}\ldots\mu_{n+1}&\nu_{1}\ldots\nu_{n}&0\end{array}\right)$ (3)

The derived series is

$\operatorname{Der}(\mathfrak{d}_{n})\supseteq\langle
E_{1},\ldots,E_{n},A_{1},\ldots,A_{2n+1}\rangle\supseteq\langle
A_{n+2},\ldots,A_{2n+1}\rangle\supseteq 0$

and the nilradical coincides with the commutator ideal (which is a two-step
nilpotent Lie algebra). Finally
$\operatorname{Z}(\operatorname{Der}(\mathfrak{d}_{n}))$ is trivial and the
left adjoint maps are

$\operatorname{ad}_{e_{1}}=A_{n+2},\;\operatorname{ad}_{e_{i}}=A_{n+i}+A_{n+i+1},\;\;\forall
i=2,\ldots,n,$
$\operatorname{ad}_{e_{n+1}}=A_{2n+1},\;\operatorname{ad}_{e_{j}}=A_{j-n}-A_{j-n-1},\;\;\forall
j=n+2,\ldots,2n+1,$

thus the inner derivations of the Dieudonné algebra $\mathfrak{d}_{n}$ are
represented by the matrices of the form

$\left(\begin{array}[]{c|c}&0\\\ \textbf{\Large 0}&\vdots\\\ &0\\\
\hline\cr\mu_{1}\;\mu_{2}\ldots\mu_{n}\;\mu\;\nu_{1}\ldots\nu_{n}&0\end{array}\right)$
(4)

where $\mu=-\displaystyle\sum_{k=1}^{n}\mu_{k}$. More precisely, the matrix

$\left(\begin{array}[]{c|c}&0\\\ \textbf{\Large 0}&\vdots\\\ &0\\\ \hline\cr
0\ldots 0\;\mu_{n+1}\;0\ldots 0&0\end{array}\right)$ (5)

does not represent an inner derivation, for every $a_{n+1}\neq 0$.

For example, we study $\operatorname{Der}(\mathfrak{d}_{n})$ in the case that
$n\leq 3$.

######  Example 4.2.

If $n=1$, then

$D=\begin{pmatrix}[ccc|c]\alpha&0&\alpha_{1}&0\\\ 0&\alpha&0&0\\\
0&0&\beta&0\\\ \hline\cr\mu_{1}&\mu_{2}&\nu_{1}&\alpha+\beta\\\ \end{pmatrix}$

thus $\operatorname{Der}(\mathfrak{d}_{1})$ is the six-dimensional solvable
Lie algebra with basis

$\\{x,y,E,A_{1},A_{2},A_{3}\\},$

where $x=e_{11}+e_{22}+e_{44},y=e_{33}+e_{44},E=e_{1,3}$ and $A_{i}=e_{4,i}$,
for any $i=1,2,3$, and with non-trivial commutators

$[x,E]=[E,y]=E,\;[x,A_{3}]=A_{3},\;[y,A_{1}]=A_{1},\;[y,A_{2}]=A_{2},\;[A_{1},E]=A_{3}.$

######  Example 4.3.

If $n=2$, the derivations of $\mathfrak{d}_{2}$ are of the form

$\begin{pmatrix}[ccccc|c]\alpha&0&0&\alpha_{1}&0&0\\\
0&\alpha&0&0&-\alpha_{2}&0\\\ 0&0&\alpha&\alpha_{2}&0&0\\\ 0&0&0&\beta&0&0\\\
0&0&0&0&\beta&0\\\
\hline\cr\mu_{1}&\mu_{2}&\mu_{3}&\nu_{1}&\nu_{2}&\alpha+\beta\\\
\end{pmatrix}$

and $\operatorname{Der}(\mathfrak{d}_{2})$ is a nine-dimensional Lie algebra
with commutator ideal consisting of the matrices

$\begin{pmatrix}[ccccc|c]0&0&0&\alpha_{1}&0&0\\\ 0&0&0&0&-\alpha_{2}&0\\\
0&0&0&\alpha_{2}&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
\hline\cr\mu_{1}&\mu_{2}&\mu_{3}&\nu_{1}&\nu_{2}&0\\\ \end{pmatrix}$

######  Example 4.4.

If $n=3$, then

$D=\left(\begin{array}[]{c|c|c}&&0\\\ \alpha I_{4}&C&\vdots\\\ &&0\\\
\hline\cr&&0\\\ 0&\beta I_{3}&\vdots\\\ &&0\\\
\hline\cr\mu_{1}\;\mu_{2}\;\mu_{3}\;\mu_{4}&\nu_{1}\;\nu_{2}\;\nu_{3}&\alpha+\beta\end{array}\right)$
(6)

where

$C=\begin{pmatrix}\alpha_{1}&0&\alpha_{2}\\\ 0&-\alpha_{2}&0\\\
\alpha_{2}&0&\alpha_{3}\\\ 0&-\alpha_{3}&0\\\ \end{pmatrix}$

and the Lie algebra $\operatorname{Der}(\mathfrak{d}_{3})$ has dimension $9$
with generators

$\\{x,y,E_{1},E_{2},E_{3},A_{1},\ldots,A_{8}\\},$

where

$\displaystyle
x=e_{11}+e_{22}+e_{33}+e_{44}+e_{88},\;y=e_{55}+e_{66}+e_{77}+e_{88},$
$\displaystyle
E_{1}=e_{1,5},\;E_{2}=e_{1,7}-e_{2,6}+e_{3,5},E_{3}=e_{3,7}-e_{2,8},$
$\displaystyle A_{i}=e_{8,i},\;\;\forall i=1,\ldots,7$

and with Lie brackets

$\displaystyle[x,E_{i}]=[E_{i},y]=E_{i},\;\;\forall i=1,2,3,$
$\displaystyle[y,A_{h}]=A_{h},\;[x,A_{k}]=A_{k}\;\;\forall h=1,2,3,4,\;\forall
k=5,6,7,$
$\displaystyle[A_{1},E_{1}]=A_{5},\;[A_{1},E_{2}]=A_{7},\;[A_{2},E_{2}]=-A_{6},$
$\displaystyle[A_{3},E_{2}]=A_{5},\;[A_{3},E_{3}]=A_{7},\;[A_{4},E_{3}]=-A_{6}.$

##  5 Almost inner derivations of nilpotent Leibniz algebras with one-
dimensional commutator ideal

We recall that a derivation $d$ of a left Leibniz algebra $L$ is an _almost
inner derivation_ if $d(x)\in[L,x]$, for every $x\in L$. The set of all almost
inner derivations of $L$ forms a Lie subalgebra of $\operatorname{Der}(L)$,
denoted by $\operatorname{AIDer}(L)$, containing the ideal
$\operatorname{Inn}(L)$ of inner derivations of $L$.

Derivations of two-step nilpotent Lie algebras were studied in [7] and [8] by
D. Burde, K. Dekimpe and B. Verbeke. They proved that every almost inner
derivations of a Lie algebra of _genus_ 1 (i.e. with one-dimensional
commutator ideal) is an inner derivation. We want to generalize this result in
the frame of Leibniz algebras.

######  Proposition 5.1.

Let $L$ be a complex nilpotent Leibniz algebra with $[L,L]=\mathbb{C}z$, such
that $L\not\cong\mathfrak{l}_{2n+1}^{J_{\pm 1}}$ and
$L\not\cong\mathfrak{d}_{n}$. Then every almost inner derivation
$d\in\operatorname{AIDer}(L)$ is an inner derivation.

###### Proof.

Let $d\in\operatorname{AIDer}(L)$. Then
$d(y)\in[L,y]\subseteq[L,L]=\mathbb{C}z$, for any $y\in L$ and $d([L,L])=0$.
Fixed a basis $\\{e_{1},\ldots,e_{t-1},z\\}$ of $L$, where
$t=\dim_{\mathbb{C}}L$, we have that

$d(e_{i})=a_{i}z,\;\;\forall i=1,\ldots,t-1,$

with $a_{i}\in\mathbb{F}$ and $d(z)=0$, thus $d\in\operatorname{Inn}(L)$. ∎

For the Heisenberg Leibniz algebras $\mathfrak{l}_{2n+1}^{J_{a}}$ with $a=\pm
1$ (in [14] it was proved that these two algebras are isomorphic) and for the
Dieudonné Leibniz algebra $\mathfrak{d}_{n}$, it is possible to define an
almost inner derivation $d$ which is not inner. For instance, if $a=1$ and we
fixed the basis $\\{e_{1},f_{1},\ldots,e_{n},f_{n},z\\}$ of
$\mathfrak{l}_{2n+1}^{J_{1}}$, then the matrix

$\left(\begin{array}[]{c|c}&0\\\ \textbf{\Large 0}&\vdots\\\ &0\\\ \hline\cr
1\;0\ldots 0\;0&0\\\ \end{array}\right)$ (7)

defines a derivation
$d\in\operatorname{AIDer}(\mathfrak{l}_{2n+1}^{J_{1}})\setminus\operatorname{Inn}(\mathfrak{l}_{2n+1}^{J_{1}})$.
In the same way

$\left(\begin{array}[]{c|c}&0\\\ \textbf{\Large 0}&\vdots\\\ &0\\\ \hline\cr
0\;0\ldots 0\;1&0\\\ \end{array}\right)$ (8)

is an almost inner but non-inner derivation of $\mathfrak{l}_{2n+1}^{J_{-1}}$.
More precisely every almost inner derivation of $\mathfrak{l}_{2n+1}^{J_{\pm
1}}$ is of the form

$\left(\begin{array}[]{c|c}&0\\\ \textbf{\Large 0}&\vdots\\\ &0\\\
\hline\cr\mu_{1}\;\nu_{1}\ldots\mu_{n}\;\nu_{n}&0\\\ \end{array}\right)$ (9)

with $a_{1},b_{1},\ldots,a_{n},b_{n}\in\mathbb{C}$, meanwhile the inner
derivations are represented by the set of matrices

$\left(\begin{array}[]{c|c}&0\\\ \textbf{\Large 0}&\vdots\\\ &0\\\ \hline\cr
0\;\nu_{1}\;\mu_{2}\;\nu_{2}\ldots\mu_{n}\;\nu_{n}&0\\\ \end{array}\right)$
(10)

for $\mathfrak{l}_{2n+1}^{J_{1}}$, and by

$\left(\begin{array}[]{c|c}&0\\\ \textbf{\Large 0}&\vdots\\\ &0\\\
\hline\cr\mu_{1}\;\nu_{1}\;\mu_{2}\;\nu_{2}\ldots\mu_{n}\;0\;0&0\\\
\end{array}\right)$ (11)

for the Leibniz algebra $\mathfrak{l}_{2n+1}^{J_{-1}}$. Finally
$\operatorname{AIDer}(\mathfrak{d}_{n})$ consists of the matrices of the type

$\left(\begin{array}[]{c|c}&0\\\ \textbf{\Large 0}&\vdots\\\ &0\\\
\hline\cr\mu_{1}\ldots\mu_{n+1}\;\nu_{1}\ldots\nu_{n}&0\end{array}\right)$
(12)

and an example of almost inner but non-inner derivations is given by the
linear map $d\in\operatorname{gl}(\mathfrak{d}_{n})$ defined by
$d(e_{n+1})=z$.

## References

  * [1] Z. K. Adashev and T. K. Kurbanbaev  “Almost Inner Derivations of Some Nilpotent Leibniz Algebras”  In _Journal of Siberian Federal University. Mathematics and Physics_ , 2020, pp. 733–745  DOI: https://doi.org/10.17516/1997-1397-2020-13-6-733-745
  * [2] S. Ayupov, B. Omirov and I. Rakhimov  “Leibniz Algebras: Structure and Classification”  CRC Press, 2019
  * [3] C. Bartolone, A. Di Bartolo and G. Falcone  “Solvable Extensions of Nilpotent Complex Lie Algebras of Type $\\{$2n, 1, 1$\\}$”  In _Moscow Mathematical Journal_ 18.4, 2018, pp. 607–616  DOI: http://dx.doi.org/10.17323%2F1609-4514-2018-18-4-607-616
  * [4] R. Biggs and G. Falcone  “A class of nilpotent Lie algebras admitting a compact subgroup of automorphisms”  In _Differential Geometry and its Applications_ 54 Elsevier, 2017, pp. 251–263  DOI: https://doi.org/10.1016/j.difgeo.2017.04.009
  * [5] R. Biggs and P. T. Nagy  “On Sub-Riemannian and Riemannian Structures on the Heisenberg Groups”  In _Journal of Dynamical and Control Systems_ 22, 2016, pp. 563–594  DOI: https://doi.org/10.1007/s10883-016-9316-9
  * [6] A. Blokh  “A generalization of the concept of a Lie algebra”  In _Dokl. Akad. Nauk SSSR_ 165.3, 1965, pp. 471–473
  * [7] D. Burde, K. Dekimpe and B. Verbeke  “Almost inner derivations of Lie algebras”  In _Journal of Algebra and Its Applications_ 17.11, 2017  DOI: https://doi.org/10.1142/S0219498818502146
  * [8] D. Burde, K. Dekimpe and B. Verbeke  “Almost inner derivations of 2-step nilpotent Lie algebras of genus 2”  In _Linear Algebra and its Applications_ 608, 2021, pp. 185–202  DOI: https://doi.org/10.1016/j.laa.2020.08.023
  * [9] M. R. Dath and P. Jouan  “Controllability of linear systems on Heisenberg groups”  In _International Journal of Control_ 94.4 Taylor
& Francis, 2021, pp. 1010–1019  DOI:
https://doi.org/10.1080/00207179.2019.1626995

  * [10] K. Erdmann and M. J. Wildon  “Introduction to Lie Algebras”  Springer London, 2006
  * [11] G. Falcone and Á. Figula  “The action of a compact Lie group on nilpotent Lie algebras of type $\\{$n, 2$\\}$”  In _Forum Mathematicum_ 28.4, 2016, pp. 795–806  DOI: http://doi.org/10.1515/forum-2014-0170
  * [12] M. V. Ignatyev, I. Kaygorodov and Y. Popov  “The geometric classification of 2-step nilpotent algebras and applications”  In _Revista Matemática Complutense_ 35.3, 2022, pp. 907–922  DOI: https://doi.org/10.1007/s13163-021-00411-0
  * [13] I. Kaygorodov and Y. Volkov  “Degenerations of noncommutative Heisenberg algebras”  In _Communications in Algebra_ , 2023  DOI: https://doi.org/10.1080/00927872.2023.2202752
  * [14] G. La Rosa and M. Mancini  “Two-step nilpotent Leibniz algebras”  In _Linear Algebra and its Applications_ 637.7, 2022, pp. 119–137  DOI: https://doi.org/10.1016/j.laa.2021.12.013
  * [15] J. Liu, Y. Sheng and Q. Wang  “On non-abelian extensions of Leibniz algebras”  In _Communications in Algebra_ 46.2, 2018, pp. 574–587  DOI: https://doi.org/10.1080/00927872.2017.1324870
  * [16] J.-L. Loday  “Une version non commutative des algebres de Lie: les algebres de Leibniz”  In _L’Enseignement Mathématique_ 39.3-4, 1993, pp. 269–293
  * [17] S. Mac Lane  “Categories for the working mathematician” Graduate Texts in Mathematics  Springer-Verlag, New York, 2013