Non-Nilpotent Leibniz Algebras with One-Dimensional Derived Subalgebra

Abstract

In this paper we study non-nilpotent non-Lie Leibniz $\mathbb {F}$-algebras with one-dimensional derived subalgebra, where $\mathbb {F}$ is a field with ${\text {char}}(\mathbb {F}) \ne 2$. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by $L_n$, where $n=\dim _\mathbb {F}L_n$. This generalizes the result found in Demir et al. (Algebras and Representation Theory 19:405-417, 2016), which is only valid when $\mathbb {F}=\mathbb {C}$. Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of $L_n$. Eventually, we solve the coquecigrue problem for $L_n$ by integrating it into a Lie rack.

1 Introduction

Leibniz algebras were introduced by Loday in [19] as a non-skew symmetric version of Lie algebras. Earlier such algebraic structures were also considered by A. Blokh, who called them D-algebras [5] for their strict connection with derivations. Leibniz algebras play a significant role in different areas of mathematics and physics.

Many results of Lie algebras are still valid for Leibniz algebras. One of them is the Levi decomposition, which states that any Leibniz algebra over a field $\mathbb {F}$ of characteristic zero is the semidirect sum of its radical and a semisimple Lie algebra. This makes clear the importance of the problem of classification of solvable and nilpotent Lie/Leibniz algebras, which has been dealt with since the early 20th century (see [2,3,4, 9,10,11, 13] and [14], just for giving a few examples).

In [16] and [17] nilpotent Leibniz algebras L with one-dimensional derived subalgebra [L, L] were studied and classified. It was proved that, up to isomorphism, there are three classes of indecomposable Leibniz algebras with these properties, namely the Heisenberg algebras $\mathfrak {l}_{2n+1}^A$, which are parameterized by their dimension $2n+1$ and by a matrix A in canonical form, the Kronecker algebra $\mathfrak {k}_n$ and the Dieudonné algebra $\mathfrak {d}_n$, both parameterized by their dimension only. We want to complete this classification by studying non-nilpotent Leibniz $\mathbb {F}$-algebras with one-dimensional derived subalgebra, where $\mathbb {F}$ is a field with ${\text {char}}(\mathbb {F}) \ne 2$. Using the theory of non-abelian extensions of Leibniz algebras introduced in [18], we prove that a non-nilpotent non-Lie Leibniz algebra L with $\dim _\mathbb {F}L=n$ and $\dim _\mathbb {F}[L,L]=1$ is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra $S_2$, i.e. the algebra with basis $\lbrace e_1,e_2 \rbrace$ and multiplication table given by $[e_2,e_1]=e_1$, and an abelian algebra of dimension $n-2$. We denote it by $L_n$. This generalizes the result found in Theorem 2.6 of [11], where the authors proved that a complex non-split non-nilpotent non-Lie Leibniz algebra with one-dimensional derived subalgebra is isomorphic to $S_2$.

We study in detail the properties of the algebra $L_n$ and we compute the Lie algebra of derivations ${\text {Der}}(L_n)$, its Lie group of automorphism ${\text {Aut}}(L_n$) and the Leibniz algebra of biderivations ${\text {Bider}}(L_n)$.

Finally, we solve the coquecigrue problem for the Leibniz algebra $L_n$. We mean the problem, formulated by J.-L. Loday in [19], of finding a generalization of Lie third theorem to Leibniz algebras. Using M. K. Kinyon’s results for the class of real split Leibniz algebras (see [15]), we show how to explicitly integrate $L_n$ into a Lie rack defined over the vector space $\mathbb {R}^n$.

2 Preliminaries

We assume that $\mathbb {F}$ is a field with ${\text {char}}(\mathbb {F})\ne 2$. For the general theory we refer to [1].

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) $\left[ -,-\right] :L\times L \rightarrow L$ which satisfies the left Leibniz identity

$$\begin{aligned} \left[ x,\left[ y,z\right] \right] =\left[ \left[ x,y\right] ,z\right] +\left[ y,\left[ x,z\right] \right] , \quad \forall x,y,z\in L. \end{aligned}$$

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

$$\begin{aligned} \left[ \left[ x,y\right] ,z\right] =\left[ [x,z],y\right] +\left[ x,\left[ y,z\right] \right] , \quad \forall x,y,z\in L. \end{aligned}$$

Given a left Leibniz algebra L, the multiplication $[x,y]^{{\text {op}}}=[y,x]$ defines a right Leibniz algebra structure on L.

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

We have a full inclusion functor $i:{\textbf {Lie}} \rightarrow {\textbf {Leib}}$ that embeds Lie algebras over $\mathbb {F}$ into Leibniz algebras over $\mathbb {F}$. Its left adjoint is the functor $\pi :{\textbf {Leib}} \rightarrow {\textbf {Lie}}$, which associates to each Leibniz algebra L the quotient $L/{\text {Leib}}(L)$, where ${\text {Leib}}(L)$ is the smallest bilateral ideal of L such that the quotient $L/{\text {Leib}}(L)$ becomes a Lie algebra. ${\text {Leib}}(L)$ is defined as the subalgebra generated by all elements of the form $\left[ x,x\right]$, for any $x\in L$, and it is called the Leibniz kernel of L.

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

$$\begin{aligned} {\text {Z}}_l(L)=\left\{ x\in L \,|\,\left[ x,L\right] =0\right\} ,\,\,\, {\text {Z}}_r(L)=\left\{ x\in L \,|\,\left[ L,x\right] =0\right\} . \end{aligned}$$

The intersection of the left and right center is called the center of L and it is denoted by ${\text {Z}}(L)$. In general for a left Leibniz algebra L, the left center ${\text {Z}}_l(L)$ is a bilateral ideal, meanwhile the right center is not even a subalgebra. Furthermore, one can check that ${\text {Leib}}(L) \subseteq {\text {Z}}_l(L)$.

The definition of derivation for a Leibniz algebra is the same as in the case of Lie algebras.

Definition 1.2

A linear map $d :L \rightarrow L$ is a derivation of L if

$$\begin{aligned} d([x,y])=[d(x),y]+[x,d(y)], \quad \forall x,y \in L. \end{aligned}$$

An equivalent way to define a left Leibniz algebra L is to saying that the left adjoint maps ${\text {ad}}_x=[x,-]$ are derivations. Meanwhile the right adjoint maps ${\text {Ad}}_x=[-,x]$ are not derivations in general. The set ${\text {Der}}(L)$ of all derivations of L is a Lie algebra with the usual bracket $[d,d']=d \circ d' - d' \circ d$ and the set ${\text {Inn}}(L)$ spanned by the left adjoint maps, which are called inner derivations, is an ideal of ${\text {Der}}(L)$. Moreover ${\text {Aut}}(L)$ is a Lie group and its Lie algebra is precisely ${\text {Der}}(L)$.

In [19] Loday introduced the notion of anti-derivation and biderivation for a Leibniz algebra.

Definition 1.3

A linear map $D :L \rightarrow L$ is an anti-derivation of L if

$$\begin{aligned} D([x,y])=[x,D(y)]-[y,D(x)], \quad \forall x,y \in L. \end{aligned}$$

The space ${\text {ADer}}(L$) of anti-derivations of L has a ${\text {Der}}(L)$-module structure with the extra multiplication $d \cdot D = d \circ D - D \circ d$, for any derivation d and for any anti-derivation D, and one can check that the right adjoint maps ${\text {Ad}}_x$ are anti-derivations.

Definition 1.4

A biderivation of L is a pair $(d,D) \in {\text {Der}}(L) \times {\text {ADer}}(L)$ such that

$$\begin{aligned} {[}d(x)+D(x),y]=0, \quad \forall x,y \in L. \end{aligned}$$

The set ${\text {Bider}}(L)$ of all biderivations of L has a Leibniz algebra structure with the bracket

$$\begin{aligned} {[}(d,D),(d',D')]=([d,d'],d \cdot D') \end{aligned}$$

and it is defined a Leibniz algebra homomorphism

$$\begin{aligned} L \rightarrow {\text {Bider}}(L), \; x \mapsto ({\text {ad}}_x,{\text {Ad}}_x). \end{aligned}$$

The pair $({\text {ad}}_x,{\text {Ad}}_x)$ is called the inner biderivation associated with $x \in L$ and the set of all inner biderivations of L forms a Leibniz subalgebra of Bider(L).

We recall the definitions of solvable and nilpotent Leibniz algebras.

Definition 1.5

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

$$\begin{aligned} L^{0}=L,\,\,L^{k+1}=[L^{k},L^k],\quad \forall k\ge 0 \end{aligned}$$

be the derived series of L. L is $n-$ step solvable if $L^{n-1}\ne 0$ and $L^{n}=0$.

Definition 1.6

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

$$\begin{aligned} L^{(0)}=L,\,\,L^{(k+1)}=[L,L^{(k)}],\quad \forall k\ge 0 \end{aligned}$$

be the lower central series of L. L is $n-$ step nilpotent if $L^{(n-1)}\ne 0$ and $L^{(n)}=0$.

When L is two-step nilpotent, it lies in different varieties of non-associative algebras, such as associative, alternative and Zinbiel algebras. In this case we refer at L as a two-step nilpotent algebra and we have the following.

Proposition 1.7

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

• (ii) If L is a left nilpotent Leibniz algebra with $\dim _\mathbb {F}[L,L]=1$, then L is two-step nilpotent.

In [16] the classification of nilpotent Leibniz algebras with one-dimensional derived subalgebra was established. The classification revealed that, up to isomorphism, there exist only three classes of indecomposable nilpotent Leibniz algebras of this type.

Definition 1.8

[16]. 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 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 brackets are given by

$$\begin{aligned} {[}e_i,f_j]=(\delta _{ij}+a_{ij})z, \quad [f_j,e_i]=(-\delta _{ij}+a_{ij})z, \quad \forall i,j=1,\ldots ,n. \end{aligned}$$

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

Definition 1.9

[16]. Let $n \in \mathbb {N}$. The Kronecker 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 brackets are given by

$$\begin{aligned}&{[}e_{i},f_{i}] = [f_{i,}e_{i}{]}=z, \quad \forall i=1,\ldots ,n \\&{[}e_{i},f_{i-1}] = z, [f_{i-1},e_{i}] = -z, \quad \forall i= 2,\ldots ,n. \end{aligned}$$

Definition 1.10

[16] Let $n \in \mathbb {N}$. The Dieudonn$\acute{e}$ algebra $\mathfrak {d}_n$ is the $(2n+2)$-dimensional Leibniz algebra with basis $\left\{ e_1,\ldots ,e_{2n+1},z\right\}$ and the brackets are given by

$$\begin{aligned}&\left[ e_1,e_{n+2}\right] =z,\\&\left[ e_i,e_{n+i}\right] =\left[ e_i,e_{n+i+1}\right] =z, \quad \forall i=2,\ldots ,n,\\&\left[ e_{n+1},e_{2n+1}\right] =z,\\&\left[ e_i,e_{i-n}\right] =z, \quad \left[ e_i,e_{i-n-1}\right] =-z, \quad \forall i=n+2,\ldots ,2n+1. \end{aligned}$$

We want to extend this classification by studying non-nilpotent Leibniz algebras with one-dimensional derived subalgebra.

3 Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

Let L be a non-nilpotent left Leibniz algebra over $\mathbb {F}$ with $\dim _\mathbb {F}L=n$ and $\dim _\mathbb {F}\left[ L,L\right] =1$. We observe that such an algebra is two-step solvable since the derived subalgebra $\left[ L,L\right]$ is abelian.

It is well known that a non-nilpotent Lie algebra with one-dimensional derived subalgebra is isomorphic to the direct sum of the two-dimensional non-abelian Lie algebra and an abelian algebra (see [12, Sect. 3]). Thus we are interested in the classification of non-Lie Leibniz algebras with these properties.

In [11, Theorem 2.6] the authors prove that a complex non-split non-nilpotent non-Lie Leibniz algebra with one-dimensional derived subalgebra is isomorphic to the two-dimensional algebra with basis $\lbrace e_1,e_2 \rbrace$ and multiplication table $[e_2,e_1]=[e_2,e_2]=e_1$. Here we generalize this result when $\mathbb {F}$ is a general field with ${\text {char}}(\mathbb {F}) \ne 2$.

Proposition 2.1

Let L be a non-nilpotent left Leibniz algebra over $\mathbb {F}$ with $\dim _\mathbb {F}\left[ L,L\right] =1$. Then L has a two-dimensional bilateral ideal S which is isomorphic to one of the following Leibniz algebras:

• (i) $S_1=\langle e_1,e_2 \rangle$ with $\left[ e_2,e_1\right] =-\left[ e_1,e_2\right] =e_1$;

• (ii) $S_2=\langle e_1,e_2 \rangle$ with $\left[ e_2, e_1\right] =\left[ e_2,e_2\right] =e_1$.

Proof

Let $\left[ L,L\right] =\mathbb {F}z$. Since L is not nilpotent, then

$$\begin{aligned} {[}L,\left[ L,L\right] ]\ne 0, \end{aligned}$$

i.e. $z \notin \textrm{Z}_r(L)$. Since $\left[ L,L\right]$ is an abelian algebra, there exists a vector $x\in L$, which is linearly independent than z, such that $\left[ x, z\right] \ne 0$. Thus

$$\begin{aligned} \left[ x, z\right] =\gamma z, \end{aligned}$$

for some $\gamma \in \mathbb {F}^*$. The subspace $S=\langle x, z\rangle$ is an ideal of L and it is not nilpotent: in fact

$$\begin{aligned} 0\ne \gamma z=\left[ x,z\right] \in \left[ S,\left[ S,S\right] \right] . \end{aligned}$$

Thus S is a non-nilpotent Leibniz algebra. Using the classification of two-dimensional Leibniz algebras given by C. Cuvier in [8], S is isomorphic either to $S_1$ or to $S_2$.$\square$

Remark 2.1

The algebras $S_1$ and $S_2$ are respectively the Leibniz algebras $L_2$ and $L_4$ of Sect. 3.1 in [1]. We observe that $S_1$ is a Lie algebra, meanwhile $S_2$ is a non-right left Leibniz algebra.

One can see L as an extension of the abelian algebra $L_0 = L/S \cong \mathbb {F}^{n-2}$ by S [18]

(1)

It turns out that there exists an equivalence of Leibniz algebra extensions

where $L_0 \ltimes _{\omega } S$ is the Leibniz algebra defined on the direct sum of vector spaces $L_0 \oplus S$ with the bilinear operation given by

$$\begin{aligned} {[}(x,a),(y,b)]_{(l,r,\omega )}=(0,[a,b]+l_x(b)+r_y(a)+\omega (x,y)), \end{aligned}$$

where

$$\begin{aligned} \omega (x,y)=[\sigma (x),\sigma (y)]_L-\sigma ([x,y]_{L_0})=[\sigma (x),\sigma (y)]_L \end{aligned}$$

is the Leibniz algebra 2-cocycle associated with (1) and

$$\begin{aligned} l_x(b)=[\sigma (x),i(b)]_L, \; \; r_y(a)=[i(a),\sigma (y)]_L \end{aligned}$$

define the action of $L_0$ on S; $i_1,i_2,\pi _1$ are the canonical injections and projection. The Leibniz algebra isomorphism $\theta$ is defined by $\theta (x,a)= \sigma (x)+i(a)$, for every $(x,a) \in L_0 \oplus S$.

By [18, Proposition 4.2], the 2-cocycle $\omega :L_0 \times L_0 \rightarrow S$ and the linear maps $l,r :L_0 \rightarrow {\text {gl}}(S)$ must satisfy the following set of equations

• (L1) $l_x([a,b])=[l_x(a),b]+[x,l_x(b)]$;

• (L2) $r_x([a,b])=[a,r_x(b)]-[b,r_x(a)]$;

• (L3) $[l_x(a)+r_x(a),b]=0$;

• (L4) $[l_x,l_y]_{{\text {gl}}(S)}-l_{[x,y]_{L_0}}={\text {ad}}_{\omega (x,y)}$;

• (L5) $[l_x,r_y]_{{\text {gl}}(S)}-r_{[x,y]_{L_0}}={\text {Ad}}_{\omega (x,y)}$;

• (L6) $r_y(r_x(a)+l_x(a))=0$;

• (L7) $l_x(\omega (y,z))-l_y(\omega (x,z))-r_z(\omega (x,y))$

  $=\omega ([x,y]_{L_0},z)-\omega (x,[y,z]_{L_0})+\omega (y,[x,z]_{L_0})$

for any $x,y \in L_0$ and for any $a,b \in S$. Notice that these equations where also studied in [6] in the case of Leibniz algebra split extensions.

Remark 2.2

The first three equations state that the pair $(l_x,r_x)$ is a biderivation of the Leibniz algebra S, for any $x \in L_0$. Biderivations of low-dimensional Leibniz algebras were classified in [20] and it turns out that

• ${\text {Bider}}(S_1)=\left\{ (d,-d) \; | \; d \in {\text {Der}}(S_1) \right\}$ and

  ${\text {Der}}(S_1)=\left\{ \begin{pmatrix} \alpha &{}\quad \beta \\ 0 &{}\quad 0 \end{pmatrix} \Bigg \vert \; \alpha ,\beta \in \mathbb {F} \right\} ;$

• ${\text {Bider}}(S_2)=\left\{ \left( \begin{pmatrix} \alpha &{}\quad \alpha \\ 0 &{}\quad 0 \end{pmatrix},\begin{pmatrix} 0 &{}\quad \beta \\ 0 &{}\quad 0 \end{pmatrix} \right) \Bigg \vert \; \alpha ,\beta \in \mathbb {F} \right\}$.

We study now in detail the non-abelian extension (1) in both cases that S is isomorphic either to $S_1$ or to $S_2$.

3.1 S is a Lie algebra

When $S \cong S_1$, we have that $r_y=-l_y$, for any $y \in L_0$ and the bilinear operation of $L_0 \ltimes _{\omega } S_1$ becomes

$$\begin{aligned} {[}(x,a),(y,b)]_{(l,\omega )}=(0,[a,b]+l_x(b)-l_y(a)+\omega (x,y)). \end{aligned}$$

The linear map $l_x$ is represented by a $2 \times 2$ matrix

$$\begin{aligned} \begin{pmatrix} \alpha _x &{}\quad \beta _x \\ 0 &{}\quad 0 \end{pmatrix} \end{aligned}$$

with $\alpha _x$,$\beta _x \in \mathbb {F}$. From equations (L4)-(L5) it turns out that

$$\begin{aligned} \omega (x,y)=(\alpha _x \beta _y - \alpha _y \beta _x)e_1, \; \; \forall x,y \in L_0 \end{aligned}$$

and the 2-cocycle $\omega$ is skew-symmetric. Moreover, equations (L6)-(L7) are automatically satisfied and the resulting algebra $L_0 \ltimes _{\omega } S_1 \cong L$ is a Lie algebra. We conclude that L is isomorphic to the direct sum of $S_1$ and $L_0 \cong \mathbb {F}^{n-2}$.

3.2 S is not a Lie algebra

With the change of basis $e_2 \mapsto e_2 - e_1$, $S_2$ becomes the Leibniz algebra with basis $\lbrace e_1,e_2 \rbrace$ and the only non-trivial bracket given by $[e_2,e_1]=e_1$. Now a biderivation of $S_1$ is represented by a pair of matrices

$$\begin{aligned} \left( \begin{pmatrix} \alpha &{}\quad 0\\ 0 &{}\quad 0 \end{pmatrix},\begin{pmatrix} 0 &{} \beta \\ 0 &{} 0 \end{pmatrix} \right) \end{aligned}$$

with $\alpha , \beta \in \mathbb {F}$ and the pair $(l_x,r_x) \in {\text {Bider}}(S_2)$ is defined by $l_x(e_1)=\alpha _x e_1$ and $r_x(e_2)=\beta _x e_1$, for any $x \in L_0$.

Equation (L4) states that $\left[ l_x,l_y\right] _{{\text {gl}}(S_2)}=\left[ \omega (x,y), -\right]$, with

$$\begin{aligned} \left[ l_x, l_y\right] _{{\text {gl}}(S_2)}=l_x\circ l_y-l_y\circ l_x&= \begin{pmatrix} \alpha _x &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix} \begin{pmatrix} \alpha _y &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}- \begin{pmatrix} \alpha _y &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix} \begin{pmatrix} \alpha _x &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}\\&=\begin{pmatrix} \alpha _x\alpha _y &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}- \begin{pmatrix} \alpha _x\alpha _y &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}= \begin{pmatrix} 0 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}, \end{aligned}$$

for any $x,y\in L_0$. Thus $\omega (x,y)\in \textrm{Z}_l(S_2)=\mathbb {F}e_1$.

From equation (L5) we have $\left[ l_x,r_y\right] _{{\text {gl}}(S_2)}=\left[ -,\omega (x,y)\right] _{S_2}$, with

$$\begin{aligned} \left[ l_x, r_y\right] _{{\text {gl}}(S_2)}=l_x\circ r_y-r_y\circ l_x= \begin{pmatrix} 0 &{}\quad \alpha _x\beta _y\\ 0 &{}\quad 0 \end{pmatrix}- \begin{pmatrix} 0 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}= \begin{pmatrix} 0 &{}\quad \alpha _x\beta _y\\ 0 &{}\quad 0 \end{pmatrix}. \end{aligned}$$

Thus, for every $a=a_1e_1 + a_2 e_2 \in S_2$ and for every $x,y\in L_0$, we have

$$\begin{aligned} {[}a,\omega (x,y)]=\left[ l_x, r_y\right] (a)=\alpha _x\beta _ya_2 e_1, \end{aligned}$$

i.e. $\omega (x,y)=\alpha _x \beta _y e_1$. Finally, equations (L6) and (L7) are identically satisfied.

Summarizing we have

$$\begin{aligned} {\left\{ \begin{array}{ll} l_x\equiv \begin{pmatrix} \alpha _x &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}\\ r_y\equiv \begin{pmatrix} 0 &{}\quad \beta _y \\ 0 &{}\quad 0 \end{pmatrix}\\ \omega (x,y)=\alpha _x\beta _ye_1 \end{array}\right. } \end{aligned}$$

for every $x,y\in L_0$ and the bilinear operation $[-,-]_{(l,r,\omega )}$ becomes

$$\begin{aligned} {[}(x,a),(y,b)]_{(l,r,\omega )}=(0,(a_2 b_1+\alpha _x b_1 + \beta _y a_2 + \alpha _x \beta _y)e_1), \end{aligned}$$

for any x, $y \in L_0$ and for any $a=a_1 e_1 + a_2 e_2$, $b=b_1 e_1 + b_2 e_2 \in S_2$.

If we fix a basis $\lbrace f_3,\ldots ,f_n\rbrace$ of $L_0$ and we denote by

$$\begin{aligned} \alpha _i=\alpha _{f_i}, \; \; \beta _i=\beta _{f_i}, \; \; \forall i=3,\ldots ,n \end{aligned}$$

then L is isomorphic to the Leibniz algebra with basis $\left\{ e_1, e_2, f_3, \ldots , f_n\right\}$ and non-zero brackets

$$\begin{aligned}&\left[ e_2,e_1\right] =e_1 \\&\left[ e_2, f_i\right] =\beta _ie_1, \quad \forall i=3,\ldots , n\\&\left[ f_i, e_1\right] =\alpha _ie_1, \quad \forall i=3,\ldots , n\\&\left[ f_i, f_j\right] =\alpha _i\beta _je_1, \quad \forall i,j=3,\ldots , n. \end{aligned}$$

With the change of basis $f_i \mapsto f_i' = \dfrac{f_i}{\beta _i}-e_1$, if $\beta _i \ne 0$, we obtain that

$$\begin{aligned}&[e_2,f_i']=e_1-[e_2,e_1]=0,\\&[f_i',e_1]=\gamma _i e_1, \; \; \text {where } \gamma _i = \frac{\alpha _i}{\beta _i},\\&[f_i,f_j']=\alpha _i e_1 - [f_i,e_1]=0,\\&[f_i',f_j']=\gamma _i e_1 - \frac{1}{\beta _i} [f_i,e_1]=0. \end{aligned}$$

If we denote again $f_i \equiv f_i'$ and $\alpha _i \equiv \gamma _i$ when $\beta _i \ne 0$, then L has basis $\lbrace e_1,e_2,f_3,\ldots ,f_n \rbrace$ and non-trivial brackets

$$\begin{aligned} {[}e_2,e_1]=e_1, \; \; [f_i,e_1]=\alpha _i e_1, \; \; \forall i=3,\ldots ,n. \end{aligned}$$

Finally, when $\alpha _i \ne 0$, we can operate the change of basis

$$\begin{aligned} f_i \mapsto \dfrac{f_i}{\alpha _i} - e_2. \end{aligned}$$

One can check that the only non-trivial bracket now is $[e_2,e_1]=e_1$ and L is isomorphic to the direct sum of $S_2$ and the abelian algebra $L_0 \cong \mathbb {F}^{n-2}$. This allows us to conclude with the following.

Theorem 2.2

Let $\mathbb {F}$ be a field with ${\text {char}}(\mathbb {F}) \ne 2$. Let L be a non-nilpotent non-Lie left Leibniz algebra over $\mathbb {F}$ with $\dim _\mathbb {F}L= n$ and $\dim _\mathbb {F}[L,L]=1$. Then L is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra $S_2$ and an abelian algebra of dimension $n-2$. We denote this algebra by $L_n$. $\square$

If we suppose that L is a non-split algebra, i.e. L cannot be written as the direct sum of two proper ideals, then we obtain the following result, that is a generalization of [11, Theorem 2.6] and which is valid over a general field $\mathbb {F}$ with ${\text {char}}(\mathbb {F}) \ne 2$.

Corollary 2.3

Let L be a non-split non-nilpotent non-Lie left Leibniz algebra over $\mathbb {F}$ with $\dim _\mathbb {F}L= n$ and $\dim _\mathbb {F}[L,L]=1$. Then $n=2$ and $L\cong S_2$. $\square$

Now we study in detail the algebra $L_n=S_2 \oplus \mathbb {F}^{n-2}$ by describing the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations. Moreover, when $\mathbb {F}=\mathbb {R}$, we solve the coquegigrue problem (see [7] and [15]) for $L_n$ by integrating it into a Lie rack.

3.3 Derivations, Automorphisms and Biderivations of $L_n$

Let $n \ge 2$ and let $L_n=S_2 \oplus \mathbb {F}^{n-2}$. We fix the basis $\mathcal {B}_n=\lbrace e_1,e_2,f_3,\ldots ,f_n \rbrace$ of $L_n$ and we recall that the only non-trivial commutator is $[e_2,e_1]=e_1$. A straightforward application of the algorithm proposed in [20] for finding derivations and anti-derivations of a Leibniz algebra as pair of matrices with respect to a fixed basis produces the following.

Theorem 2.4

• (i) A derivation of $L_n$ is represented, with respect to the basis $\mathcal {B}_n$, by a matrix

  $$\begin{aligned} \left( \begin{array}{c|c} \begin{array}{cc} \alpha &{}\quad 0 \\ 0 &{}\quad 0 \\ \end{array} &{} \begin{array}{c c c c} 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \end{array} \\ \hline \begin{array}{c c} 0 &{}\quad a_3 \\ 0 &{}\quad a_4 \\ \vdots &{}\quad \vdots \\ 0 &{}\quad a_n \\ \end{array}&A \end{array} \right) \end{aligned}$$

  where $A \in {\text {M}}_{n-2}(\mathbb {F})$.

• (ii) The group of automorphisms ${\text {Aut}}(L_n)$ is the Lie subgroup of ${\text {GL}}_n(\mathbb {F})$ of matrices of the form

  $$\begin{aligned} \left( \begin{array}{c | c} \begin{array}{c c} \beta &{}\quad 0 \\ 0 &{}\quad 1 \\ \end{array} &{} \begin{array}{c c c c} 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \end{array} \\ \hline \begin{array}{c c} 0 &{}\quad b_3 \\ 0 &{}\quad b_4 \\ \vdots &{}\quad \vdots \\ 0 &{}\quad b_n \\ \end{array}&B \end{array} \right) \end{aligned}$$

  where $\beta \ne 0$ and $B \in {\text {GL}}_{n-2}(\mathbb {F})$.

• (iii) The Leibniz algebra of biderivations of $L_n$ consists of the pairs (d, D) of linear endomorphisms of $L_n$ which are represented by the pair of matrices

  $$\begin{aligned} \left( \left( \begin{array}{c | c} \begin{array}{c c} \alpha &{}\quad 0 \\ 0 &{}\quad 0 \\ \end{array} &{} \begin{array}{c c c c} 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \end{array} \\ \hline \begin{array}{c c} 0 &{}\quad a_3 \\ 0 &{}\quad a_4 \\ \vdots &{}\quad \vdots \\ 0 &{}\quad a_n \\ \end{array}&A \end{array} \right) , \left( \begin{array}{c | c} \begin{array}{c c} 0 &{}\quad \alpha ' \\ 0 &{}\quad 0 \\ \end{array} &{} \begin{array}{c c c c} 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \end{array} \\ \hline \begin{array}{c c} 0 &{}\quad a_3' \\ 0 &{}\quad a_4' \\ \vdots &{}\quad \vdots \\ 0 &{}\quad a_n' \\ \end{array}&A' \end{array} \right) \right) \end{aligned}$$

  where A,$A' \in {\text {M}}_{n-2}(\mathbb {F})$.$\square$

4 The Integration of the Leibniz Algebra $L_n$

The coquecigrue problem is the problem formulated by Loday in [19] of finding a generalization of Lie third theorem to Leibniz algebras. Given a real Leibniz algebra L, one wants to find a manifold endowed with a smooth map, which plays the role of the adjoint map for Lie groups, such that the tangent space at a distinguished element, endowed with the differential of this map, gives a Leibniz algebra isomorphic to L. Moreover, when L is a Lie algebra, we want to obatin the simply connected Lie group associated with L. From now on, we assume that the underlying field of any algebra is $\mathbb {F}=\mathbb {R}$.

In [15] M. K. Kinyon shows that it is possible to define an algebraic structure, called rack, whose operation, differentiated twice, defines on its tangent space at the unit element a Leibniz algebra structure.

Definition 3.1

A rack is a set X with a binary operation $\rhd :X \times X \rightarrow X$ which is left autodistributive

$$\begin{aligned} x \rhd (y \rhd z) = (x \rhd y) \rhd (x \rhd z), \quad \forall x,y,z \in X \end{aligned}$$

and such that the left multiplications $x \rhd -$ are bijections.

A rack is pointed if there exists an element $1 \in X$ such that $1 \rhd x = x$ and $x \rhd 1 = 1$, for any $x \in X$.

A rack is a quandle if the binary operation $\rhd$ is idempotent.

The first example of a rack is any group G endowed with its conjugation

$$\begin{aligned} x \rhd y = x y x^{-1}, \quad \forall x,y \in G. \end{aligned}$$

We denote this rack by ${\text {Conj}}(G)$ and we observe that it is a quandle.

Definition 3.2

A pointed rack $(X, \rhd , 1)$ is said to be a Lie rack if X is a smooth manifold, $\rhd$ is a smooth map and the left multiplications are diffeomorphisms.

M. K. Kinyon proved that the tangent space ${\text {T}}_1 X$ at the unit element 1 of a Lie rack X, endowed with the bilinear operation

$$\begin{aligned} {[}x,y] = \frac{\partial ^2}{\partial s\partial t}\bigg |_{s,t=0} \gamma _1(s) \rhd \gamma _2(t) \end{aligned}$$

where $\gamma _1,\gamma _2 :[0,1] \rightarrow X$ are smooth paths such that $\gamma _1(0)=\gamma _2(0)=1$, $\gamma _1'(0)=x$ and $\gamma _2'(0)=y$, is a Leibniz algebra.

He also solved the coquecigrue problem for the class of split Leibniz algebras. Here a Leibniz algebra is said to be split if there exists an ideal

$$\begin{aligned} {\text {Leib}}(L) \subseteq I \subseteq {\text {Z}}_l (L) \end{aligned}$$

and a Lie subalgebra M of L such that $L \cong (M \oplus I, \lbrace -,- \rbrace )$, where the bilinear operation $\lbrace -,- \rbrace$ is defined by

$$\begin{aligned} \lbrace (x,a),(y,b) \rbrace = ([x,y],\rho _x(b)) \end{aligned}$$

and $\rho :M \times I \rightarrow I$ is the action on the M-module I. L is said to be the demisemidirect product of M and I. More precisely, we have the following.

Theorem 3.3

[15]. Let L be a split Leibniz algebra. Then a Lie rack integrating L is $X=(H \oplus I, \rhd )$, where H is the simply connected Lie group integrating M and the binary operation is defined by

$$\begin{aligned} (g,a) \rhd (h,b)=(ghg^{-1},\phi _g(b)), \end{aligned}$$

where $\phi$ is the exponentiation of the Lie algebra action $\rho$.

Some years later S. Covez generalized M. K. Kinyon’s results proving that every real Leibniz algebra admits an integration into a Lie local rack (see [7]). More recently it was showed in [16] that the integration proposed by S. Covez is global for any nilpotent Leibniz algebra. Moreover, when a Leibniz algebra L is integrated into a Lie quandle X, it turns out that L is a Lie algebra and $X={\text {Conj}}(G)$, where G is the simply connected Lie group integrating L.

Our aim here is to solve the coquecigrue problem for the non-nilpotent Leibniz algebra $L_n=S_2 \oplus \mathbb {F}^{n-2}$. One can check that $S_2$ is a split Leibniz algebra, in the sense of M. K. Kinyon, with $I={\text {Z}}_l(S_2)\cong \mathbb {R}$ and $M \cong \mathbb {R}$. Thus $L \cong (\mathbb {R}^2, \lbrace -,- \rbrace )$ with the bilinear operation defined by

$$\begin{aligned} \lbrace (x_1,x_2),(y_1,y_2) \rbrace = (0,\rho _{x_1}(y_2)) \end{aligned}$$

and $\rho _{x_1}(y_2)=x_1 y_2$, for any $x_1,y_2 \in \mathbb {R}$. It turns out that a Lie rack integrating $S_2$ is $(\mathbb {R}^2, \rhd )$, where

$$\begin{aligned} (x_1,x_2) \rhd (y_1,y_2) = (y_1,y_2 + e^{x_1}y_2). \end{aligned}$$

and the unit element is (0, 0). Finally, one can check that the binary operation

$$\begin{aligned} (x_1,x_2,x_3,\ldots ,x_n) \rhd (y_1,y_2,y_3,\ldots ,y_n) = (y_1,y_2 + e^{x_1}y_2,y_3,\ldots ,y_n) \end{aligned}$$

defines on $\mathbb {R}^n$ a Lie rack structure with unit element $1=(0,\ldots ,0)$, such that $({\text {T}}_1 \mathbb {R}^n, \rhd )$ is a Leibniz algebra isomorphic to $L_n$. This result, combined with the ones of [16, Section 4], completes the classification of Lie racks whose tangent space at the unit element gives a Leibniz algebra with one-dimensional derived subalgebra.