On non-abelian extensions of 3-Leibniz algebras

Abstract

In this paper, we study non-abelian extensions of 3-Leibniz algebras through Maurer-Cartan elements. We construct a differential graded Lie algebra and prove that there is a one-to-one correspondence between the isomorphism classes of non-abelian extensions in 3-Leibniz algebras and the equivalence classes of Maurer-Cartan elements in this differential graded Lie algebra. And also the Leibniz algebra structure on the space of fundamental elements of 3-Leibniz algebras is analyzed. It is proved that the non-abelian extension of 3-Leibniz algebras induce the non-abelian extensions of Leibniz algebras.

Keywords

3-Leibniz algebras / Leibniz algebra / non-abelian extension / Maurer-Cartan element

1 Introduction

In 1993, Loday and Pirashvili [18] introduced the non-skew-symmetric form of Lie algebra, namely the Leibniz algebra. As Leibniz algebra plays an important role in algebra, geometry, physics and other fields, such as Hochschild homology theory and Nambu mechanics, it has attracted the attention of numerous researchers, hence rapidly developing the theories. [15] studied the properties of restricted Leibniz algebras. In 1985, Filippov [13] first proposed the concept of n-Lie algebras (also known as Filippov algebras), and extensively studied its structure theory. Takhtajan [19, 23] found that n-Lie algebras were the corresponding algebraic structures of Nambu mechanics. 3-Lie algebras were closely related to many areas of mathematical physics (see [2‒3]). As the non-skew-symmetric form of n-Lie algebra, n-Leibniz algebra was proposed by Casas et al. [11]. They also presented the construction between Leibniz algebra and n-Leibniz algebra (n ≥ 3), as well as the cohomology of n-Leibniz algebra. In recent years, domestic and foreign scholars have conducted a lot of researches on the structure and properties of Leibniz algebras and n-Leibniz algebras. [7] introduced the non-abelian tensor products and universal central extensions of n-Leibniz algebras, which generalized the non-abelian tensor products of Leibniz algebras introduced by Kurdiani and Pirashvili. [8] showed the homology of n-Leibniz algebras with trivial coefficients. [1] studied the properties of Cartan subalgebras and normal elements in n-Leibniz algebras. [5, 6, 10] systematically studied the nilpotency and solvability of n-Leibniz algebra. In particular, [4, 16, 17, 21, 22, 24] extensively studied the extension theory of Leibniz algebras and 3-Lie algebras.

We study 3-Leibniz algebras in this paper, which can be considered as the multivariate generalization of Leibniz algebras as well as the non-antisymmetrized generalization of 3-Lie algebra. 3-Leibnitz algebras also plays an important role in mathematical physics. For example, it can represent the duality theory of M2-branes in multiple backgrounds. In this paper, we study non-abelian extensions of 3-Leibniz algebras. First, we characterize the non-abelian extensions of 3-Leibniz algebras and the isomorphism between extensions by selecting the sections. Then, we construct a differential graded Lie algebra and prove that there is a one-to-one correspondence between the gauge equivalence classes of Maurer-Cartan elements in the differential graded Lie algebra and the isomorphism classes of non-abelian extensions in 3-Leibniz algebras. Meanwhile, we also analyze the Leibniz algebra structure on the space of fundamental elements of 3-Leibniz algebras. It is proved that the non-abelian extensions of 3-Leibniz algebras induce the non-abelian extensions of Leibniz algebras. In addition, since 3-Leibniz algebras are non-antisymmetric, the construction of the 3-Leibniz algebra by using sections is more complicated. The challenge of this paper lies in the discussion of the non-abelian extensions of 3-Leibniz algebra, because it is quite different from the previous work.

The paper is organized as follows. Section 2 reviews the definition of 3-Leibniz algebras and their cohomology theories. Section 3 describes the non-abelian extensions of 3-Leibniz algebras and their isomorphisms. Section 4 constructs a differential graded Lie algebra and proves that there is a one-to-one correspondence between the equivalence classes of Maurer-Cartan elements on the differential graded Lie algebra and the isomorphism classes of non-abelian extensions of 3-Leibniz algebras. Section 5 analyzes the structure of Leibniz algebras on the space of fundamental elements of 3-Leibniz algebras, and it is proved that the non-abelian extension of 3-Leibniz algebras induce the non-abelian extensions of Leibniz algebras.

2 Preliminaries

Let \mathbb{K} be an algebraically closed field with characteristic 0. All vector spaces discussed in this section are on the field \mathbb{K}. This section reviews the concepts of representations and cohomology of 3-Leibniz algebras.

Definition 2.1 [11]　Let \mathcal{L} be a vector space. If[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}:\mathcal{L}\otimes \mathcal{L}\otimes \mathcal{L}\to \mathcal{L} is a 3-linear mapping, and for any element x₁, x₂, y₁, y₂, y₃ in \mathcal{L}, it satisfies

\begin{array}{rl}[x_1,x_2,[y_1,y_2,y_3{]}_{\mathcal{L}}{]}_{\mathcal{L}}=& \left[\right[x_1,x_2,y_1{]}_{\mathcal{L}},y_2,y_3{]}_{\mathcal{L}}+[y_1,[x_1,x_2,y_2{]}_{\mathcal{L}},y_3{]}_{\mathcal{L}}\\ & +[y_1,y_2,[x_1,x_2,y_3{]}_{\mathcal{L}}{]}_{\mathcal{L}},\end{array}

then, (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}) will be a 3-Leibniz algebra.

Example 2.1 [11]　From Definition 2.1, it can be seen that any 3-Lie algebra and Lie triple system are 3-Leibniz algebras.

Example 2.2 [10]　Let (\mathcal{L},[\cdot ,\cdot \left]\right) be a Leibniz algebra, and [x_1,x_2,x_3{]}_{\mathcal{L}}=[[x_1,x_2],x_3], where x₁, x₂, x₃\in \mathcal{L}, then (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}) will be a 3-Leibniz algebra.

Definition 2.2 [9]　Let (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}})and ({\mathcal{L}}^{\prime },[\cdot ,\cdot ,\cdot {]}_{{\mathcal{L}}^{\prime }}) be 3-Leibniz algebras. If for any element x₁, x₂, x₃ in \mathcal{L}, linear map f:\mathcal{L}\to {\mathcal{L}}^{\prime } satisfies

f[x_1,x_2,x_3{]}_{\mathcal{L}}=[f\left(x_1\right),f\left(x_2\right),f\left(x_3\right){]}_{\mathcal{L}},

then f is said to be a 3-Leibniz algebra homomorphism.

Remark 2.1 [12]　Let (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}})be a 3-Leibniz algebra. Elements in \mathcal{L}\otimes \mathcal{L} are called fundamental elements. For any element {\mathfrak{X}}_1=x_1\otimes y_1, {\mathfrak{X}}_2=x_2\otimes y_2 in \mathcal{L}\otimes \mathcal{L}, it is defined that

[{\mathfrak{X}}_1,{\mathfrak{X}}_2{]}_{\mathrm{F}}=[x_1,y_1,x_2{]}_{\mathcal{L}}\otimes y_2+x_2\otimes [x_1,y_1,y_2{]}_{\mathcal{L}}.

It is verified that (\mathcal{L}\otimes \mathcal{L},[\cdot ,\cdot {]}_{\mathrm{F}})is a Leibniz algebra.

Definition 2.3 [11]　Let (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}) be a 3-Leibniz algebra, and V be a vector space. If the bilinear maps l:{\otimes }^2\mathcal{L}\to \mathfrak{g}\mathfrak{l}\left(V\right), m:{\otimes }^2\mathcal{L}\to \mathfrak{g}\mathfrak{l}\left(V\right) and r:{\otimes }^2\mathcal{L}\to \mathfrak{g}\mathfrak{l}\left(V\right)) satisfy

\left[r\right(x_1,x_2),l(x_3,x_4\left)\right]=l\left(\right[x_1,x_2,x_3{]}_{\mathcal{L}},x_4)+l(x_3,[x_1,x_2,x_4{]}_{\mathcal{L}}),

\left[r\right(x_1,x_2),m(x_3,x_4\left)\right]=m\left(\right[x_1,x_2,x_3{]}_{\mathcal{L}},x_4)+m(x_3,[x_1,x_2,x_4{]}_{\mathcal{L}}),

\left[r\right(x_1,x_2),r(x_3,x_4\left)\right]=r\left(\right[x_1,x_2,x_3{]}_{\mathcal{L}},x_4)+r(x_3,[x_1,x_2,x_4{]}_{\mathcal{L}}),

l(x_1,[x_2,x_3,x_4{]}_{\mathcal{L}})=l(x_3,x_4)\circ l(x_1,x_2)+m(x_2,x_4)\circ l(x_1,x_3)+r(x_2,x_3)\circ l(x_1,x_4),

m(x_1,[x_2,x_3,x_4{]}_{\mathcal{L}})=l(x_3,x_4)\circ m(x_1,x_2)+m(x_2,x_4)\circ m(x_1,x_3)+r(x_2,x_3)\circ m(x_1,x_4),

and x₁, x₂, x₃, x₄ \in \mathcal{L}, then (V; l, m, r) is called a representation of 3-Leibniz algebra.

Example 2.3　Let (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}) be a 3-Leibniz algebra, \mathrm{\forall }x,y,z\in \mathcal{L}, and define bilinear maps ad₁, ad₂, ad₃ \in \mathrm{H}\mathrm{o}\mathrm{m}({\otimes }^2\mathcal{L},\mathfrak{g}\mathfrak{l}(\mathcal{L}\left)\right) as

{\mathrm{a}\mathrm{d}}_1(x,y)\left(z\right)=[z,x,y{]}_{\mathcal{L}}\text{,}{\mathrm{a}\mathrm{d}}_2(x,y)\left(z\right)=[x,z,y{]}_{\mathcal{L}}\text{,}{\mathrm{a}\mathrm{d}}_3(x,y)\left(z\right)=[x,y,z{]}_{\mathcal{L}}.

It is verified that (\mathcal{L};{\mathrm{a}\mathrm{d}}_1,{\mathrm{a}\mathrm{d}}_2,{\mathrm{a}\mathrm{d}}_3) is a representation of \mathcal{L}. It is also called adjoint representation.

Theorem 2.1　 Let (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}) be a 3-Leibniz algebra, V be a vector space, and l:{\otimes }^2\mathcal{L}\to \mathfrak{g}\mathfrak{l}\left(V\right), m:{\otimes }^2\mathcal{L}\to \mathfrak{g}\mathfrak{l}\left(V\right) and r:{\otimes }^2\mathcal{L}\to \mathfrak{g}\mathfrak{l}\left(V\right)are bilinear maps. Then (V; l, m, r) will be a representation of \mathcal{L} if and only if (\mathcal{L}\oplus V,[\cdot ,\cdot ,\cdot {]}_{(l,m,r)})is a 3-Leibniz algebra, where [\cdot ,\cdot ,\cdot {]}_{(l,m,r)} is defined as

\begin{array}{rl}[x_1+v_1,x_2+v_2,x_3+v_3{]}_{(l,m,r)}=& [x_1,x_2,x_3{]}_{\mathcal{L}}+l(x_2,x_3)\left(v_1\right)\\ & +m(x_1,x_3)\left(v_2\right)+r(x_1,x_2)\left(v_3\right),\\ & \mathrm{\forall }{x}_1,x_2,x_3\in \mathcal{L},v_1,v_2,v_3\in V.\end{array}

Proof　According to Definition 2.1, (\mathcal{L}\oplus V,[\cdot ,\cdot ,\cdot {]}_{(l,m,r)}) is a 3-Leibniz algebra. If and only if for any x_i \in \mathcal{L}, v_i \in V, and i = 1, 2, 3, 4, 5, there is

(2.1)\begin{array}{rl}& {[x_1+v_1,x_2+v_2,{[x_3+v_3,x_4+v_4,x_5+v_5]}_{(l,m,r)}]}_{(l,m,r)}\\ & ={[{[x_1+v_1,x_2+v_2,x_3+v_3]}_{(l,m,r)},x_4+v_4,x_5+v_5]}_{(l,m,r)}\\ & +{[x_3+v_3,{[x_1+v_1,x_2+v_2,x_4+v_4]}_{(l,m,r)},x_5+v_5]}_{(l,m,r)}\\ & +{[x_3+v_3,x_4+v_4,{[x_1+v_1,x_2+v_2,x_5+v_5]}_{(l,m,r)}]}_{(l,m,r)}.\end{array}

Use the definition of [\cdot ,\cdot ,\cdot {]}_{(l,m,r)}to expand the above equation. According to the combination of elements and Definition 2.3, Equation (2.1) holds if and only if (V; l, m, r) is a representation of \mathcal{L}. □

Next, we recall that the cohomology of 3-Leibniz algebra (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}) in the representation (V; l, m, r).

First, a cochain of order p on \mathcal{L} is defined as a linear map

\alpha :({\otimes }^2\mathcal{L}{)}^{\otimes p}\otimes \mathcal{L}\to V,p=0,1,\dots ,

and the space of all p-order cochains is denoted by C^p(\mathcal{L},V), namely

C^p(\mathcal{L},V)=\mathrm{H}\mathrm{o}\mathrm{m}\left(\right({\otimes }^2\mathcal{L}{)}^{\otimes p}\otimes \mathcal{L},V).

Then define the coboundary operator {\delta }^p:C^p(\mathcal{L},V)\to C^{p+1}(\mathcal{L},V) as

where {\mathfrak{X}}_i=x_i\otimes y_i\in {\otimes }^2\mathcal{L}, 1 ≤ i ≤ p + 1, z\in \mathcal{L}, [{\mathfrak{X}}_i,z]=[x_i,y_i,z{]}_{\mathcal{L}}. From [20], it is known that δ satisfies {\delta }^p\circ {\delta }^{p-1}=0.

If δ^p(α) = 0, p-order cochain α will be a p-order closed chain. Let Z^p(\mathcal{L},V) be the set constituting all p-order closed chains. If there exists \beta \in C^{p-1}(\mathcal{L},V) such that \alpha ={\delta }^{p-1}(\beta ), p-order cochain α is said to be a p-order coboundary. Let B^p(\mathcal{L},V) be the set formed by all p-order coboundaries. Then H^p(\mathcal{L},V)=Z^p(\mathcal{L},V)/B^p(\mathcal{L},V). It is said to be the p-order cohomology group of 3-Leibniz algebras (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}})in the representation (V; l, m, r). For cohomology groups of other algebras, refer to [14].

3 Non-abelian extensions of 3-Leibniz algebras

This section gives equivalent characterization conditions for non-abelian extensions and isomorphisms of 3-Leibniz algebras.

Definition 3.1　 \mathrm{L}\mathrm{e}\mathrm{t}(\mathcal{M},[\cdot ,\cdot ,\cdot {]}_{\mathcal{M}}), (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}) and (K,[\cdot ,\cdot ,\cdot {]}_K)be 3-Leibniz algebras. If there exists a short exact sequence of vector space

0⟶\mathcal{M}\stackrel{i}{\to }\mathcal{K}\stackrel{p}{\to }\mathcal{L}⟶0,

where i and p are both 3-Leibniz algebra homomorphisms. Then \mathcal{K} is called a non-abelian extension of \mathcal{L}.

If linear map s: \mathcal{L}→\mathcal{K} satisfies p\circ s={\mathrm{i}\mathrm{d}}_{\mathcal{L}}, s is called a cross section of a non-abelian extension \mathcal{K}.

Definition 3.2　Let {\mathcal{K}}_1 and {\mathcal{K}}_2 be two non-abelian extensions of \mathcal{L}. If there exists a 3-Leibniz algebra homomorphism θ: {\mathcal{K}}_2→ {\mathcal{K}}_1 such that

\begin{array}{ccccc}0& ⟶\mathcal{M}& \stackrel{i_2}{⟶}{\mathcal{K}}_2& \stackrel{p_2}{⟶}\mathcal{L}& ⟶0\\ & \parallel & \downarrow \theta & \parallel & \\ 0& ⟶\mathcal{M}& \stackrel{i_1}{⟶}{\mathcal{K}}_1& \stackrel{p_1}{⟶}\mathcal{L}& ⟶0\end{array}

are commutative, the two non-abelian extensions {\mathcal{K}}_1 and {\mathcal{K}}_2 are said to be isomorphic.

Let \mathcal{K} be a non-abelian extension of \mathcal{L} and s be a section of \mathcal{K}. Then, we will construct the structure of 3-Leibniz algebras on \mathcal{L}\oplus \mathcal{M}.

Define {\rho }_1,{\rho }_2,{\rho }_3:{\otimes }^2\mathcal{L}\to \mathfrak{g}\mathfrak{l}\left(\mathcal{M}\right),{\nu }_1,{\nu }_2,{\nu }_3:\mathcal{L}\to \mathrm{H}\mathrm{o}\mathrm{m}({\otimes }^2\mathcal{M},\mathcal{M}) and \omega :{\otimes }^3\mathcal{L}\to \mathcal{M} as

\begin{array}{rl}{\rho }_1(x,y)\left(m\right)& =[m,s(x),s(y\left)\right]\kappa ,\\ {\rho }_2(x,y)\left(m\right)& =\left[s\right(x),m,s(y\left)\right]\kappa ,\\ {\rho }_3(x,y)\left(m\right)& =\left[s\right(x),s(y),m]\kappa ,\\ {\nu }_1\left(x\right)(m,n)& =\left[s\right(x),m,n{]}_{\mathcal{K}},\\ {\nu }_2\left(x\right)(m,n)& =[m,s(x),n{]}_{\mathcal{K}},\end{array}

\begin{array}{rl}{\nu }_3\left(x\right)(m,n)& =[m,n,s(x){]}_{\mathcal{K}},\\ \omega (x,y,z)& =\left[s\right(x),s(y),s(z){]}_{\mathcal{K}}-s[x,y,z{]}_{\mathcal{L}},\end{array}

\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}, and x,y,z\in \mathcal{L},m,n\in \mathcal{M}.

Define F : \mathcal{K}→\mathcal{L}\oplus \mathcal{M} is F(X) = p(X) + (X ‒ sp(X)). F is a vector space isomorphism; F^{-1}:\mathcal{L}\oplus \mathcal{M}\to \mathcal{K} is F^{-1}(x+m)=s\left(x\right)+m. Then, with the help of isomorphism F, the 3-linear map[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )} on \mathcal{L}\oplus \mathcal{M} is defined as

\begin{array}{rl}& {[x}_1+m_1,x_2+m_2,x_3+m_3{]}_{(\rho ,\nu ,\omega )}\\ & =F\left(\right[F^{-1}(x_1+m_1),F^{-1}(x_2+m_2),F^{-1}(x_3+m_3){]}_{\mathcal{K}})\\ & =F\left[s\right(x_1)+m_1,s(x_2)+m_2,s(x_3)+m_3{]}_{\mathcal{K}}\\ & =F\left[s\right(x_1),s(x_2),s(x_3\left)\right]\chi +F[m_1,m_2,m_3{]}_{\mathcal{M}}\\ & +F[m_1,s(x_2),s(x_3){]}_{\mathcal{K}}+F[s\left(x_1\right),m_2,s\left(x_3\right){]}_{\mathcal{K}}+F\left[s\right(x_1),s(x_2),m_3{]}_{\mathcal{K}}\\ & +F\left[s\right(x_1),m_2,m_3{]}_{\mathcal{K}}+F[m_1,s\left(x_2\right),m_3{]}_{\mathcal{K}}+F[m_1,m_2,s(x_3){]}_{\mathcal{K}}\\ & =[x_1,x_2,x_3{]}_{\mathcal{L}}+[m_1,m_2,m_3{]}_{\mathcal{M}}+\omega (x_1,x_2,x_3)\\ & +{\rho }_1(x_2,x_3)\left(m_1\right)+{\rho }_2(x_1,x_3)\left(m_2\right)+{\rho }_3(x_1,x_2)\left(m_3\right)\\ & +{\nu }_1\left(x_1\right)(m_2,m_3)+{\nu }_2\left(x_2\right)(m_1,m_3)+{\nu }_3\left(x_3\right)(m_1,m_2).\end{array}

Under the above definitions, the following theorem shows equivalent conditions for (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )})being a 3-Leibniz algebra.

Theorem 3.1　 (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )}) is a 3-Leibniz algebra if and only if for any X_i\in \mathcal{L}, m_i \in \mathcal{M}, i = 1, 2, 3, 4, 5, linear maps ρ₁, ρ₂, ρ₃, ν₁, ν₂, ν₃, w satisfy

(3.1)\begin{array}{rl}0=& \omega \left(\right[x_1,x_2,x_3]L,x_4,x_5)+\omega (x_3,[x_1,x_2,x_4]L,x_5)+\omega (x_3,x_4,[x_1,x_2,x_5\left]L\right)\\ & -\omega (x_1,x_2,[x_3,x_4,x_5\left]L\right)+{\rho }_1(x_4,x_5)\omega (x_1,x_2,x_3)+{\rho }_2(x_3,x_5)\omega (x_1,x_2,x_4)\\ & +{\rho }_3(x_3,x_4)\omega (x_1,x_2,x_5)-{\rho }_3(x_1,x_2)\omega (x_3,x_4,x_5),\end{array}

(3.2)\begin{array}{rl}0=& {\rho }_1(x_4,x_5)\left({\rho }_1\right(x_2,x_3\left)\right(m_1\left)\right)+{\rho }_2(x_3,x_5)\left({\rho }_1\right(x_2,x_4\left)\right(m_1\left)\right)\\ & +{\rho }_3(x_3,x_4)\left({\rho }_1\right(x_2,x_5\left)\right(m_1\left)\right)-{\rho }_1(x_2,[x_3,x_4,x_5{]}_{\mathcal{L}}\left)\right(m_1)\\ & -{\nu }_2\left(x_2\right)(m_1,\omega (x_3,x_4,x_5\left)\right),\end{array}

(3.3)\begin{array}{rl}0=& {\rho }_1(x_4,x_5)\left({\rho }_2\right(x_1,x_3\left)\right(m_2\left)\right)+{\rho }_2(x_3,x_5)\left({\rho }_2\right(x_1,x_4\left)\right(m_2\left)\right)\\ & +{\rho }_3(x_3,x_4)\left({\rho }_2\right(x_1,x_5\left)\right(m_2\left)\right)-{\rho }_2(x_1,[x_3,x_4,x_5{]}_{\mathcal{L}}\left)\right(m_2)\\ & -{\nu }_1\left(x_1\right)(m_2,\omega (x_3,x_4,x_5\left)\right),\end{array}

(3.4)\begin{array}{rl}0=& {\rho }_1\left(\right[x_1,x_2,x_4{]}_{\mathcal{L}},x_5\left)\right(m_3)+{\rho }_1(x_4,[x_1,x_2,x_5{]}_{\mathcal{L}})\left(m_3\right)\\ & +{\rho }_1(x_4,x_5)\left({\rho }_3\right(x_1,x_2\left)\right(m_3\left)\right)-{\rho }_3(x_1,x_2)\left({\rho }_1\right(x_4,x_5\left)\right(m_3\left)\right)\\ & +{\nu }_2\left(x_4\right)(m_3,\omega (x_1,x_2,x_5\left)\right)+{\nu }_3\left(x_5\right)(m_3,\omega (x_1,x_2,x_4\left)\right),\end{array}

(3.5)\begin{array}{rl}0=& {\rho }_2\left(\right[x_1,x_2,x_3{]}_{\mathcal{L}},x_5\left)\right(m_4)+{\rho }_2(x_3,[x_1,x_2,x_5{]}_{\mathcal{L}})\left(m_4\right)\\ & +{\rho }_2(x_3,x_5)\left({\rho }_3\right(x_1,x_2\left)\right(m_4\left)\right)-{\rho }_3(x_1,x_2)\left({\rho }_2\right(x_3,x_5\left)\right(m_4\left)\right)\\ & +{\nu }_1\left(x_3\right)(m_4,\omega (x_1,x_2,x_5\left)\right)+{\nu }_3\left(x_5\right)\left(\omega \right(x_1,x_2,x_3),m_4),\end{array}

(3.6)\begin{array}{rl}0=& {\rho }_3\left(\right[x_1,x_2,x_3{]}_{\mathcal{L}},x_4\left)\right(m_5)+{\rho }_3(x_3,[x_1,x_2,x_4{]}_{\mathcal{L}})\left(m_5\right)\\ & +{\rho }_3(x_3,x_4)\left({\rho }_3\right(x_1,x_2\left)\right(m_5\left)\right)-{\rho }_3(x_1,x_2)\left({\rho }_3\right(x_3,x_4\left)\right(m_5\left)\right)\\ & +{\nu }_1\left(x_3\right)\left(\omega \right(x_1,x_2,x_4),m_5)+{\nu }_2\left(x_4\right)\left(\omega \right(x_1,x_2,x_3),m_5),\end{array}

(3.7)\begin{array}{rl}0=& {\rho }_1(x_4,x_5)\left({\nu }_3\right(x_3\left)\right(m_1,m_2\left)\right)+{\rho }_2(x_3,x_5)\left({\nu }_3\right(x_4\left)\right(m_1,m_2\left)\right)\\ & +{\rho }_3(x_3,x_4)\left({\nu }_3\right(x_5\left)\right(m_1,m_2\left)\right)-{\nu }_3\left(\right[x_3,x_4,x_5{]}_{\mathcal{L}}\left)\right(m_1,m_2)\\ & -[m_1,m_2,\omega (x_3,x_4,x_5){]}_{\mathcal{M}},\end{array}

(3.8)\begin{array}{rl}0=& {\rho }_1(x_4,x_5)\left({\nu }_2\right(x_2\left)\right(m_1,m_3\left)\right)+{\nu }_2\left(x_4\right)(m_3,{\rho }_1(x_2,x_5\left)\right(m_1\left)\right)\\ & +{\nu }_3\left(x_5\right)(m_3,{\rho }_1(x_2,x_4\left)\right(m_1\left)\right)-{\nu }_2\left(x_2\right)(m_1,{\rho }_1(x_4,x_5\left)\right(m_3\left)\right),\end{array}

(3.9)\begin{array}{rl}0=& {\rho }_2(x_3,x_5)\left({\nu }_2\right(x_2\left)\right(m_1,m_4\left)\right)+{\nu }_1\left(x_3\right)(m_4,{\rho }_1(x_2,x_5\left)\right(m_1\left)\right)\\ & +{\nu }_3\left(x_5\right)\left({\rho }_1\right(x_2,x_3\left)\right(m_1),m_4)-{\nu }_2\left(x_2\right)(m_1,{\rho }_2(x_3,x_5\left)\right(m_4\left)\right),\end{array}

(3.10)\begin{array}{rl}0=& {\rho }_3(x_3,x_4)\left({\nu }_2\right(x_2\left)\right(m_1,m_5\left)\right)+{\nu }_1\left(x_3\right)\left({\rho }_1\right(x_2,x_4\left)\right(m_1),m_5)\\ & +{\nu }_2\left(x_4\right)\left({\rho }_1\right(x_2,x_3\left)\right(m_1),m_5)-{\nu }_2\left(x_2\right)(m_1,{\rho }_3(x_3,x_4\left)\right(m_5\left)\right),\end{array}

(3.11)\begin{array}{rl}0=& {\rho }_1(x_4,x_5)\left({\nu }_1\right(x_1\left)\right(m_2,m_3\left)\right)+{\nu }_2\left(x_4\right)(m_3,{\rho }_2(x_1,x_5\left)\right(m_2\left)\right)\\ & +{\nu }_3\left(x_5\right)(m_3,{\rho }_2(x_1,x_4\left)\right(m_2\left)\right)-{\nu }_1\left(x_1\right)(m_2,{\rho }_1(x_4,x_5\left)\right(m_3\left)\right),\end{array}

(3.12)\begin{array}{rl}0=& {\rho }_2(x_3,x_5)\left({\nu }_1\right(x_1\left)\right(m_2,m_4\left)\right)+{\nu }_1\left(x_3\right)(m_4,{\rho }_2(x_1,x_5\left)\right(m_2\left)\right)\\ & +{\nu }_3\left(x_5\right)\left({\rho }_2\right(x_1,x_3\left)\right(m_2),m_4)-{\nu }_1\left(x_1\right)(m_2,{\rho }_2(x_3,x_5\left)\right(m_4\left)\right),\end{array}

(3.13)\begin{array}{rl}0=& {\rho }_3(x_3,x_4)\left({\nu }_1\right(x_1\left)\right(m_2,m_5\left)\right)+{\nu }_1\left(x_3\right)\left({\rho }_2\right(x_1,x_4\left)\right(m_2,m_5\left)\right)\\ & +{\nu }_2\left(x_4\right)\left({\rho }_2\right(x_1,x_3\left)\right(m_2),m_5)-{\nu }_1\left(x_1\right)(m_2,{\rho }_3(x_3,x_4\left)\right(m_5\left)\right),\end{array}

(3.14)\begin{array}{rl}0=& {\nu }_3\left(x_5\right)(m_3,{\rho }_3(x_1,x_2\left)\right(m_4\left)\right)+{\nu }_3\left(x_5\right)\left({\rho }_3\right(x_1,x_2\left)\right(m_3),m_4)\\ & +{\nu }_3\left(\right[x_1,x_2,x_5\left]L\right)(m_3,m_4)+[m_3,m_4,\omega (x_1,x_2,x_5\left)\right]M\\ & -{\rho }_3(x_1,x_2)\left({\nu }_3\right(x_5\left)\right(m_3,m_4\left)\right),\end{array}

(3.15)\begin{array}{rl}0=& {\nu }_2\left(x_4\right)(m_3,{\rho }_3(x_1,x_2\left)\right(m_5\left)\right)+{\nu }_2\left(x_4\right)\left({\rho }_3\right(x_1,x_2\left)\right(m_3),m_5)\\ & +{\nu }_2\left(\right[x_1,x_2,x_4\left]L\right)(m_3,m_5)+[m_3,\omega (x_1,x_2,x_4),m_5]M\\ & -{\rho }_3(x_1,x_2)\left({\nu }_2\right(x_4\left)\right(m_3,m_5\left)\right),\end{array}

(3.16)\begin{array}{rl}0=& {\nu }_1\left(x_3\right)(m_4,{\rho }_3(x_1,x_2\left)\right(m_5\left)\right)+{\nu }_1\left(x_3\right)\left({\rho }_3\right(x_1,x_2\left)\right(m_4),m_5)\\ & +{\nu }_1\left(\right[x_1,x_2,x_3\left]L\right)(m_4,m_5)+\left[\omega \right(x_1,x_2,x_3),m_4,m_5]M\\ & -{\rho }_3(x_1,x_2)\left({\nu }_1\right(x_3\left)\right(m_4,m_5\left)\right),\end{array}

(3.17)\begin{array}{rl}0=& {\rho }_1(x_4,x_5)\left(\right[m_1,m_2,m_3{]}_{\mathcal{M}})+{\nu }_2(x_4\left)\right(m_3,{\nu }_3\left(x_5\right)(m_1,m_2))\\ & +{\nu }_3\left(x_5\right)(m_3,{\nu }_3(x_4\left)\right(m_1,m_2\left)\right)-[m_1,m_2,{\rho }_1(x_4,x_5\left)\right(m_3){]}_{\mathcal{M}},\end{array}

(3.18)\begin{array}{rl}0=& {\rho }_2(x_3,x_5)\left(\right[m_1,m_2,m_4{]}_{\mathcal{M}})+{\nu }_1(x_3\left)\right(m_4,{\nu }_3\left(x_5\right)(m_1,m_2))\\ & +{\nu }_3\left(x_5\right)\left({\nu }_3\right(x_3\left)\right(m_1,m_2),m_4)-[m_1,m_2,{\rho }_2(x_3,x_5\left)\right(m_4){]}_{\mathcal{M}},\end{array}

(3.19)\begin{array}{rl}0=& {\rho }_3(x_3,x_4)\left(\right[m_1,m_2,m_5{]}_{\mathcal{M}})+{\nu }_1(x_3\left)\right({\nu }_3\left(x_4\right)(m_1,m_2),m_5)\\ & +{\nu }_2\left(x_4\right)\left({\nu }_3\right(x_3\left)\right(m_1,m_2),m_5)-[m_1,m_2,{\rho }_3(x_3,x_4\left)\right(m_5){]}_{\mathcal{M}},\end{array}

(3.20)\begin{array}{rl}0=& {\nu }_3\left(x_5\right)(m_3,{\nu }_2(x_2\left)\right(m_1,m_4\left)\right)+{\nu }_3\left(x_5\right)\left({\nu }_2\right(x_2\left)\right(m_1,m_3),m_4)\\ & +[m_3,m_4,{\rho }_1(x_2,x_5\left)\right(m_1){]}_{\mathcal{M}}-{\nu }_2(x_2\left)\right(m_1,{\nu }_3\left(x_5\right)(m_3,m_4)),\end{array}

(3.21)\begin{array}{rl}0=& {\nu }_2\left(x_4\right)(m_3,{\nu }_2(x_2\left)\right(m_1,m_5\left)\right)+{\nu }_2\left(x_4\right)\left({\nu }_2\right(x_2\left)\right(m_1,m_3),m_5)\\ & +[m_3,{\rho }_1(x_2,x_4\left)\right(m_1),m_5{]}_{\mathcal{M}}-{\nu }_2(x_2\left)\right(m_1,{\nu }_2\left(x_4\right)(m_3,m_5)),\end{array}

(3.22)\begin{array}{rl}0=& {\nu }_1\left(x_3\right)(m_4,{\nu }_2(x_2\left)\right(m_1,m_5\left)\right)+{\nu }_1\left(x_3\right)\left({\nu }_2\right(x_2\left)\right(m_1,m_4),m_5)\\ & +\left[{\rho }_1\right(x_2,x_3\left)\right(m_1),m_4,m_5{]}_{\mathcal{M}}-{\nu }_2(x_2\left)\right(m_1,{\nu }_1\left(x_3\right)(m_4,m_5)),\end{array}

(3.23)\begin{array}{rl}0=& {\nu }_3\left(x_5\right)(m_3,{\nu }_1(x_1\left)\right(m_2,m_4\left)\right)+{\nu }_3\left(x_5\right)\left({\nu }_1\right(x_1\left)\right(m_2,m_3),m_4)\\ & +[m_3,m_4,{\rho }_2(x_1,x_5\left)\right(m_2){]}_{\mathcal{M}}-{\nu }_1(x_1\left)\right(m_2,{\nu }_3\left(x_5\right)(m_3,m_4)),\end{array}

(3.24)\begin{array}{rl}0=& {\nu }_2\left(x_4\right)(m_3,{\nu }_1(x_1\left)\right(m_2,m_5\left)\right)+{\nu }_2\left(x_4\right)\left({\nu }_1\right(x_1\left)\right(m_2,m_3),m_5)\\ & +[m_3,{\rho }_2(x_1,x_4\left)\right(m_2),m_5{]}_{\mathcal{M}}-{\nu }_1(x_1\left)\right(m_2,{\nu }_2\left(x_4\right)(m_3,m_5)),\end{array}

(3.25)\begin{array}{rl}0=& {\nu }_1\left(x_3\right)(m_4,{\nu }_1(x_1\left)\right(m_2,m_5\left)\right)+{\nu }_1\left(x_3\right)\left({\nu }_1\right(x_1\left)\right(m_2,m_4),m_5)\\ & +\left[{\rho }_2\right(x_1,x_3\left)\right(m_2),m_4,m_5{]}_{\mathcal{M}}-{\nu }_1(x_1\left)\right(m_2,{\nu }_1\left(x_3\right)(m_4,m_5)),\end{array}

(3.26)\begin{array}{rl}0=& \left[{\rho }_3\right(x_1,x_2\left)\right(m_3),m_4,m_5{]}_{\mathcal{M}}+[m_3,{\rho }_3(x_1,x_2)\left(m_4\right),m_5{]}_{\mathcal{M}}\\ & +[m_3,m_4,{\rho }_3(x_1,x_2\left)\right(m_5){]}_{\mathcal{M}}-{\rho }_3(x_1,x_2\left)\right([m_3,m_4,m_5{]}_{\mathcal{M}}),\end{array}

(3.27)\begin{array}{rl}0=& {\nu }_3\left(x_5\right)\left(\right[m_1,m_2,m_3{]}_{\mathcal{M}},m_4)+{\nu }_3(x_5\left)\right(m_3,[m_1,m_2,m_4{]}_{\mathcal{M}})\\ & +[m_3,m_4,{\nu }_3(x_5\left)\right(m_1,m_2){]}_{\mathcal{M}}-[m_1,m_2,{\nu }_3\left(x_5\right)(m_3,m_4){]}_{\mathcal{M}},\end{array}

(3.28)\begin{array}{rl}0=& {\nu }_2\left(x_4\right)\left(\right[m_1,m_2,m_3{]}_{\mathcal{M}},m_5)+{\nu }_2(x_4\left)\right(m_3,[m_1,m_2,m_5{]}_{\mathcal{M}})\\ & +[m_3,{\nu }_3(x_4\left)\right(m_1,m_2),m_5{]}_{\mathcal{M}}-[m_1,m_2,{\nu }_2\left(x_4\right)(m_3,m_5){]}_{\mathcal{M}},\end{array}

(3.29)\begin{array}{rl}0=& {\nu }_1\left(x_3\right)\left(\right[m_1,m_2,m_4{]}_{\mathcal{M}},m_5)+{\nu }_1(x_3\left)\right(m_4,[m_1,m_2,m_5{]}_{\mathcal{M}})\\ & +\left[{\nu }_3\right(x_3\left)\right(m_1,m_2),m_4,m_5{]}_{\mathcal{M}}-[m_1,m_2,{\nu }_1\left(x_3\right)(m_4,m_5){]}_{\mathcal{M}},\end{array}

(3.30)\begin{array}{rl}0=& \left[{\nu }_2\right(x_2\left)\right(m_1,m_3),m_4,m_5{]}_{\mathcal{M}}+[m_3,{\nu }_2\left(x_2\right)(m_1,m_4),m_5{]}_{\mathcal{M}}\\ & +[m_3,m_4,{\nu }_2(x_2\left)\right(m_1,m_5){]}_{\mathcal{M}}-{\nu }_2(x_2\left)\right(m_1,[m_3,m_4,m_5{]}_{\mathcal{M}}),\end{array}

(3.31)\begin{array}{rl}0=& \left[{\nu }_1\right(x_1\left)\right(m_2,m_3),m_4,m_5{]}_{\mathcal{M}}+[m_3,{\nu }_1\left(x_1\right)(m_2,m_4),m_5{]}_{\mathcal{M}}\\ & +[m_3,m_4,{\nu }_1(x_1\left)\right(m_2,m_5){]}_{\mathcal{M}}-{\nu }_1(x_1\left)\right(m_2,[m_3,m_4,m_5{]}_{\mathcal{M}}).\end{array}

Proof　From the definition of a 3-Leibniz algebra, (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )})is a 3-Leibniz algebra if and only if for any x_i \in \mathcal{L}, m_i \in \mathcal{M}, i = 1, 2, 3, 4, 5, there is

(3.32)\begin{array}{rl}& [x_1+m_1,x_2+m_2,[x_3+m_3,x_4+m_4,x_5+m_5{]}_{(\rho ,\nu ,\omega )}{]}_{(\rho ,\nu ,\omega )}\\ & =\left[\right[x_1+m_1,x_2+m_2,x_3+m_3{]}_{(\rho ,\nu ,\omega )},x_4+m_4,x_5+m_5{]}_{(\rho ,\nu ,\omega )}\\ & +[x_3+m_3,[x_1+m_{,}{x}_2+m_2,x_4+m_4{]}_{(\rho ,\nu ,\omega )},x_5+m_5{]}_{(\rho ,\nu ,\omega )}\\ & +[x_3+m_3,x_4+m_4,[x_1+m_1,x_2+m_2,x_5+m_5{]}_{(\rho ,\nu ,\omega )}{]}_{(\rho ,\nu ,\omega )}\end{array}.

After calculation and based on the combination of elements, Equation (3.32) holds if and only if Equations (3.1)‒(3.31) hold. □

Remark 3.1　From the above theorem, it is known that for the given section s of any non-abelian extension \mathcal{K} in 3-Leibniz algebra, (K,[\cdot ,\cdot ,\cdot {]}_K) can be isomorphic to (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )}). Therefore, the form of (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )}) will be considered in the following study of the non-abelian extensions in 3-Leibniz algebras.

Proposition 3.1　 Let (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )})and (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{({\rho }^{\prime },{\nu }^{\prime },{\omega }^{\prime })})be two non-abelian extensions of \mathcal{L}. Then the two non-abelian extensions are isomorphic if and only if there exists a linear map ξ: \mathcal{L}→\mathcal{M} such that for any x, y, z ∈, m, n \in \mathcal{M}, the following equations hold:

(3.33)\begin{array}{rl}{\rho }_1(x,y)\left(m\right)-{\rho }_1^{\prime }(x,y)\left(m\right)=& [m,\xi (x)\text{Д}\xi (y){]}_{\mathcal{M}}\\ & +{\nu }_2^{\prime }(x\left)\right(m,\xi \left(y\right))+{\nu }_3^{\prime }(y\left)\right(m,\xi \left(x\right)),\end{array}

(3.34)\begin{array}{rl}{\rho }_2(x,y)\left(m\right)-{\rho }_2^{\prime }(x,y)\left(m\right)=& \left[\xi \right(x),m,\xi (y){]}_{\mathcal{M}}\\ & +{\nu }_1^{\prime }(x\left)\right(m,\xi \left(y\right))+{\nu }_3^{\prime }(y\left)\right(\xi \left(x\right),m),\end{array}

(3.35)\begin{array}{rl}{\rho }_3(x,y)\left(m\right)-{\rho }_3^{\prime }(x,y)\left(m\right)=& \left[\xi \right(x),\xi (y),m{]}_{\mathcal{M}}\\ & +{\nu }_1^{\prime }(x\left)\right(\xi \left(y\right),m)+{\nu }_2^{\prime }(y\left)\right(\xi \left(x\right),m),\end{array}

(3.36){\nu }_1\left(x\right)(m,n)-{\nu }_1^{\prime }\left(x\right)(m,n)=\left[\xi \right(x),m,n{]}_{\mathcal{M}},

(3.37){\nu }_2\left(x\right)(m,n)-{\nu }_2^{\prime }\left(x\right)(m,n)=[m,\xi (x),n{]}_{\mathcal{M}},

(3.38){\nu }_3\left(x\right)(m,n)-{\nu }_3^{\prime }\left(x\right)(m,n)=[m,n,\xi (x){]}_{\mathcal{M}},

(3.39)\begin{array}{rl}\omega (x,y,z)-{\omega }^{\prime }(x,y,z)=& {\nu }_1^{\prime }\left(x\right)\left(\xi \right(y),\xi (z\left)\right)\\ & +{\nu }_2^{\prime }\left(y\right)\left(\xi \right(x),\xi (z\left)\right)+{\nu }_3^{\prime }\left(z\right)\left(\xi \right(x),\xi (y\left)\right)\\ & +{\rho }_1^{\prime }(y,z)\left(\xi \right(x\left)\right)+{\rho }_2^{\prime }(x,z)\left(\xi \right(y\left)\right)\\ & +{\rho }_3^{\prime }(x,y)\left(\xi \right(z\left)\right)+\left[\xi \right(x),\xi (y),\xi (z){]}_{\mathcal{M}}\\ & -\xi [x,y,z{]}_{\mathcal{L}}.\end{array}

Proof　On one hand, if (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )}) and (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{({\rho }^{\prime },{\nu }^{\prime },{\omega }^{\prime })}) are isomorphic, namely there exists a 3-Leibniz algebra homomorphism\theta :\mathcal{L}\oplus {\mathcal{M}}_{({\rho }^{\prime },{\nu }^{\prime },{\omega }^{\prime })}\to \mathcal{L}\oplus {\mathcal{M}}_{(\rho ,\nu ,\omega )} such that

\begin{array}{ccccc}0⟶\mathcal{M}⟶& \mathcal{L}\oplus {\mathcal{M}}_{({\rho }^{\prime },{\nu }^{\prime },{\omega }^{\prime })}& \stackrel{p}{\to }& \mathcal{L}⟶0& \\ \parallel & {\downarrow }_{\theta }& & \parallel & \\ 0⟶\mathcal{M}⟶& \mathcal{L}\oplus {\mathcal{M}}_{(\rho ,\nu ,\omega )}& \stackrel{p}{\to }& \mathcal{L}⟶0& \end{array}

are commutative. Among them i is embedding, namely for any m \in \mathcal{M}, i(m) = m, p is projection. For any x + m \in \mathcal{L}\oplus \mathcal{M} and p(x + m) = x, there exists a linear map ξ: \mathcal{L}\to \mathcal{M} such that θ(x + m) = x + (m − ξ(x)). After calculation, it is obtained that ξ satisfies Equations (3.33)−(3.39). On the other hand, if there exists a linear map ξ: \mathcal{L}\to \mathcal{M} such that Equations (3.33)−(3.39) hold, then with the help of linear map ξ, θ: \mathcal{L}\oplus \mathcal{M}\to \mathcal{L}\oplus \mathcal{M} can be defined as

\theta (x+m)=x+(m-\xi (x\left)\right)\text{,}\mathrm{\forall }x+m\in \mathcal{L}\oplus \mathcal{M}.

After calculation, it is obtained that

\begin{array}{rl}& \theta [x_1+m_1,x_2+m_2,x_3+m_3{]}_{({\rho }^{\prime },{\nu }^{\prime },{\omega }^{\prime })}\\ & =[x_1,x_2,x_3{]}_{\mathcal{L}}+[m_1,m_2,m_3{]}_{\mathcal{M}}+{\omega }^{\prime }(x_1,x_2,x_3)\\ & +{\rho }_1^{\prime }\left(x_2,x_3\right)\left(m_1\right)+{\rho }_2^{\prime }\left(x_1,x_3\right)\left(m_2\right)+{\rho }_3^{\prime }\left(x_1,x_2\right)\left(m_3\right)\\ & +{\nu }_1^{\prime }\left(x_1\right)(m_2,m_3)+{\nu }_2^{\prime }\left(x_2\right)(m_1,m_3)+{\nu }_3^{\prime }\left(x_3\right)(m_1,m_2)-\xi [x_1,x_2,x_3{]}_{\mathcal{L}},\end{array}

\begin{array}{rl}& \left[\theta \right(x_1+m_1),\theta (x_2+m_2),\theta (x_3+m_3){]}_{(\rho ,\nu ,\omega )}\\ & =[x_1+(m_1-\xi \left(x_1\right)),x_2+(m_2-\xi \left(x_2\right)),x_3+(m_3-\xi \left(x_3\right)){]}_{(\rho ,\nu ,\omega )}\\ & =[x_1,x_2,x_3{]}_{\mathcal{L}}+\omega (x_1,x_2,x_3)+[m_1-\xi \left(x_1\right),m_2-\xi \left(x_2\right),m_3-\xi \left(x_3\right){]}_{\mathcal{M}}\\ & +{\rho }_1(x_2,x_3)(m_1-\xi (x_1\left)\right)+{\rho }_2(x_1,x_3)(m_2-\xi (x_2\left)\right)\\ & +{\rho }_3(x_1,x_2)(m_3-\xi (x_3\left)\right)+{\nu }_1\left(x_1\right)(m_2-\xi (x_2),m_3-\xi (x_3\left)\right)\\ & +{\nu }_2\left(x_2\right)(m_1-\xi (x_1),m_3-\xi (x_3\left)\right)+{\nu }_3\left(x_3\right)(m_1-\xi (x_1),m_2-\xi (x_2\left)\right).\end{array}

From Equations (3.33)‒(3.39), it follows that

\theta [x_1+m_1,x_2+m_2,x_3+m_3{]}_{({\rho }^{\prime },{\nu }^{\prime },{\omega }^{\prime })}=[\theta (x_1+m_1),\theta (x_2+m_2),\theta (x_3+m_3){]}_{(\rho ,\nu ;\omega )},

so θ is a 3-Leibniz algebra homomorphism. \Theta satisfies the condition that

\begin{array}{ccccc}0⟶\mathcal{M}⟶& \mathcal{L}\oplus {\mathcal{M}}_{({\rho }^{\prime },{\nu }^{\prime },{\omega }^{\prime })}& \stackrel{p}{\to }& \mathcal{L}⟶0& \\ \parallel & {\downarrow }_{\theta }& \parallel & & \\ 0⟶\mathcal{M}⟶& \mathcal{L}\oplus {\mathcal{M}}_{(\rho ,\nu ,\omega )}& \stackrel{p}{\to }& \mathcal{L}⟶0& \end{array}

are commutative. Therefore, non-abelian extension (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )}) and (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{({\rho }^{\prime },{\nu }^{\prime },{\omega }^{\prime })}) are isomorphic. □

4 Non-abelian extensions of 3-Leibniz algebras and Maurer-Cartan elements

Reference [20] constructed a differential graded Lie algebra such that its Maurer-Cartan element is an n-Leibniz algebra. In this section, we apply this method to the non-abelian extensions of 3-Leibniz algebras and construct a differential graded Lie algebra. It is proved that there is a one-to-one correspondence between the equivalence classes of Maurer-Cartan elements on this differential graded Lie algebras and the non-abelian extension isomorphism classes of 3-Leibniz algebras.

Define C^{\ast }(\mathcal{L},\mathcal{L})={\oplus }_P{C}^p(\mathcal{L},\mathcal{L}) and C^p(\mathcal{L},\mathcal{L})=\mathrm{H}\mathrm{o}\mathrm{m}\left(\right({\otimes }^2\mathcal{L}{)}^{\otimes p}\otimes \mathcal{L},\mathcal{L}). If the permutation \sigma \in S_n satisfies σ(1) σ(i) and σ(i + 1) σ(n), then σ is called a (i, n−i)-unshuffle. Let the set formed by all (i, n−i)-unshuffle be unsh (i, n−i). With definitions mentioned above, [20] shows the following.

Theorem 4.1 [20]　 Define on the graded vector space C^{\ast }(\mathcal{L},\mathcal{L})

[\alpha ,\beta {]}^{3\mathcal{L}}=\alpha \circ \beta -(-1{)}^{pq}\beta \circ \alpha ,

where α \in C^p(\mathcal{L},\mathcal{L}), β \in C^q(\mathcal{L},\mathcal{L}), and p, q ≥ 0. α ◦ β \in C^{p+q}(\mathcal{L},\mathcal{L}) is defined as

\begin{array}{rl}\alpha \circ \beta ({\mathfrak{X}}_1,\dots ,{\mathfrak{X}}_{p+q},x)=& \sum _{k=0}^{p-1}(-1{)}^{kq}(\sum _{\sigma \in \mathrm{u}\mathrm{n}\mathrm{s}\mathrm{h}\left(k,q\right)}(-1{)}^{\sigma }\\ & \times (\alpha ({\mathfrak{X}}_{\sigma \left(1\right)},\dots ,{\mathfrak{X}}_{\sigma \left(k\right)},\beta ({\mathfrak{X}}_{\sigma (k+1)},\dots ,{\mathfrak{X}}_{\sigma (k+q)},x_{k+q+1})\\ & \otimes y_{k+q+1},{\mathfrak{X}}_{k+q+2},\dots ,{\mathfrak{X}}_{p+q},x)\\ & +\alpha ({\mathfrak{X}}_{\sigma \left(1\right)},\dots ,{\mathfrak{X}}_{\sigma \left(k\right)},x_{k+q+1}\otimes \beta ({\mathfrak{X}}_{\sigma \left(k+1\right)},\dots ,{\mathfrak{X}}_{\sigma \left(k+q\right)},\\ & y_{k+q+1}),{\mathfrak{X}}_{k+q+2},\dots ,{\mathfrak{X}}_{p+q},x))\\ & +\sum _{\sigma \in \mathrm{u}\mathrm{n}\mathrm{s}\mathrm{h}(p,q)}(-1{)}^{pq}(-1{)}^{\sigma }\alpha ({\mathfrak{X}}_{\sigma \left(1\right)},\dots ,{\mathfrak{X}}_{\sigma \left(p\right)},\\ & \beta ({\mathfrak{X}}_{\sigma (p+1)},\dots ,{\mathfrak{X}}_{\sigma (p+q)},x)),\end{array}

where {\mathcal{X}}_i=x_i\otimes y_i\in {\otimes }^2\mathcal{L}, x_i, y_i \in \mathcal{L}, i=1,2,\dots ,p+q. Then \left(C^{\ast }\right(\mathcal{L},\mathcal{L}),[\cdot ,\cdot {]}^{3\mathcal{L}}) is a graded Lie algebra. In addition, {\mu }_{\mathcal{L}}\in \mathrm{H}\mathrm{o}\mathrm{m}({\otimes }^3\mathcal{L},\mathcal{L}) defines a 3-Leibniz algebra if and only if [{\mu }_{\mathcal{L}},{\mu }_{\mathcal{L}}{]}^{3\mathcal{L}}=0.

Remark 4.1　For any α \in C^p(\mathcal{L},\mathcal{L}), let \overline{\delta }\alpha =[{\mu }_{\mathcal{L}},\alpha {]}^{3\mathcal{L}}, where {\mu }_{\mathcal{L}} is a structure in 3-Leibniz algebra on \mathcal{L}. Then \left(C^{\ast }\right(\mathcal{L},\mathcal{L}),[\cdot ,\cdot {]}^{3\mathcal{L}},\overline{\delta }) will be a differential graded Lie algebra. In addition, there is \overline{\delta }\alpha =(-1{)}^p\delta \alpha, where δ is the coboundary operator of \mathcal{L} on adjoint representation .

Next, we describe the non-abelain extensions of 3-Leibniz algebras by using Maurer-Cartan elements. First, the set of all Maurer-Cartan elements on a differential graded Lie algebra(L,[\cdot ,\cdot ],d)is written as

\mathrm{M}\mathrm{C}\left(L\right)=\left\{P\in L_1|\mathrm{d}P+\frac{1}{2}[P,P]=0\right\}.

For P, P^{\prime }\in \mathrm{MC}(L), if there exists \xi \in L_0 such that

P^{\prime }={\mathrm{e}}^{{\mathrm{a}\mathrm{d}}_{\xi }}P-\frac{{\mathrm{e}}^{{\mathrm{a}\mathrm{d}}_{\xi }}-1}{{\mathrm{a}\mathrm{d}}_{\xi }}\mathrm{d}\xi ,

where {\mathrm{e}}^{{\mathrm{a}\mathrm{d}}_{\xi }}={\mathrm{e}}^{[\xi ,\cdot ]}, then P^{\prime } in P is said to be gauge equivalent

Let (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}) and (\mathcal{M},[\cdot ,\cdot ,\cdot {]}_{\mathcal{M}})be two 3-Leibniz algebras. Define the 3-Leibniz bracket {\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}} on the direct sum \mathcal{L}\oplus \mathcal{M} of \mathcal{L} and\mathcal{M}, that is

[x_1+m_1,x_2+m_2,x_3+m_3{]}_{\mathcal{L}\oplus \mathcal{M}}=[x_1,x_2,x_3{]}_{\mathcal{L}}+[m_1,m_2,m_3{]}_{\mathcal{M}}.

Thus, we can obtain a differential graded Lie algebra \left(C^{\ast }\right(\mathcal{L}\oplus \mathcal{M},\mathcal{L}\oplus \mathcal{M}),[\cdot ,\cdot {]}^{3\mathcal{L}},\overline{\delta }), where C^{\ast }(\mathcal{L}\oplus \mathcal{M},\mathcal{L}\oplus \mathcal{M})=\underset{p\ge 0}{\oplus }{C}^p(\mathcal{L}\oplus \mathcal{M},\mathcal{L}\oplus \mathcal{M}). According to

C^k(\mathcal{L}\oplus \mathcal{M},\mathcal{M})=C_{>}^k(\mathcal{L}\oplus \mathcal{M},\mathcal{M})\oplus C^k(\mathcal{M},\mathcal{M}),

define C_{>}^k(\mathcal{L}\oplus \mathcal{M},\mathcal{M})\subseteq C^k(\mathcal{L}\oplus \mathcal{M},\mathcal{M}), and

C_{>}(\mathcal{L}\oplus \mathcal{M},\mathcal{M})=\underset{k\ge 0}{\oplus }{C}_{>}^k(\mathcal{L}\oplus \mathcal{M},\mathcal{M}).

This is a graded vector space.

Under the descriptions mentioned above, we obtain the following.

Theorem 4.2　 Let \mathcal{L} and \mathcal{M} be two 3-Leibniz algebras, and \overline{\delta }(C_{>}^k(\mathcal{L}\oplus \mathcal{M},\mathcal{M}))\subseteq C_{>}^{k+1}(\mathcal{L}\oplus \mathcal{M},\mathcal{M}).(C_{>}(\mathcal{L}\oplus \mathcal{M},\mathcal{M}),[\cdot ,\cdot {]}^{3\mathcal{L}},\overline{\delta }) is a differential graded Lie sub-algebra of \left(C^{\ast }\right(\mathcal{L}\oplus \mathcal{M},\mathcal{L}\oplus \mathcal{M}),[\cdot ,\cdot {]}^{3\mathcal{L}},\overline{\delta }). Meanwhile, C_{>}^0(\mathcal{L}\oplus \mathcal{M},\mathcal{M})is commutative.

Proof　For any \alpha \in C_{>}^p(\mathcal{L}\oplus \mathcal{M},\mathcal{M}), α can be regarded as \alpha \in C^p(\mathcal{L}\oplus \mathcal{M},\mathcal{M}), which satisfies \alpha {|}_{({\otimes }^2\mathcal{M}{)}^{\otimes p}\otimes \mathcal{M}}=0. Therefore, for any \alpha \in C_{>}^p(\mathcal{L}\oplus \mathcal{M},\mathcal{M}), there is

\begin{array}{rl}\overline{\delta }\left(\alpha \right){|}_{({\otimes }^2\mathcal{M}{)}^{\otimes (p+1)}\otimes \mathcal{M}}& =[{\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}},\alpha {]}^{3\mathcal{L}}{|}_{({\otimes }^2\mathcal{M}{)}^{\otimes (p+1)}\otimes \mathcal{M}}\\ & =\left(\right({\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}})\circ \alpha -(-1{)}^p\alpha \circ ({\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}})){|}_{({\otimes }^2\mathcal{M}{)}^{\otimes (p+1)}\otimes \mathcal{M}}=0,\end{array}

so \overline{\delta }\left(\alpha \right)\in C_{>}^{k+1}(\mathcal{L}\oplus \mathcal{M},\mathcal{M}), namely \overline{\delta }\left(C_{>}^k\right(\mathcal{L}\oplus \mathcal{M},\mathcal{M}\left)\right)\subseteq C_{>}^{k+1}(\mathcal{L}\oplus \mathcal{M},\mathcal{M}). For any \alpha \in C_{>}^p(\mathcal{L}\oplus \mathcal{M},\mathcal{M}), \beta \in C_{>}^q(\mathcal{L}\oplus \mathcal{M},\mathcal{M}), as

[\alpha ,\beta {]}^{3\mathcal{L}}{|}_{({\otimes }^2\mathcal{M}{)}^{\otimes (p+q)}\otimes \mathcal{M}}=(\alpha \circ \beta -(-1{)}^{pq}\beta \circ \alpha ){|}_{({\otimes }^2\mathcal{M}{)}^{\otimes (p+q)}\otimes \mathcal{M}}=0,

[\alpha ,\beta {]}^{3\mathcal{L}}\in C_{>}^{p+q}(\mathcal{L}\oplus \mathcal{M},\mathcal{M}). Therefore \left(C_{>}\right(\mathcal{L}\oplus \mathcal{M},\mathcal{M}),[\cdot ,\cdot {]}^{3\mathcal{L}},\overline{\delta })is a differential graded Lie sub-algebra.

For any \alpha ,\beta \in C_{>}^0(\mathcal{L}\oplus \mathcal{M},\mathcal{M})=\mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{L},\mathcal{M}), there are \alpha {|}_{\mathcal{M}}=0, \beta {|}_{\mathcal{M}}=0. Therefore, for any x\in \mathcal{L}[\alpha ,\beta {]}^{3\mathcal{L}}(x)=(\alpha \circ \beta -\beta \circ \alpha \left)\right(x)=\alpha (\beta \left(x\right))-\beta (\alpha \left(x\right))=0. From the arbitrariness of x, it is known that [\alpha ,\beta {]}^{3\mathcal{L}}=0. From the arbitrariness of α and β, it is known that C_{>}^0(\mathcal{L}\oplus \mathcal{M},\mathcal{M})is commutative. □

Proposition 4.1　 Let \mathcal{L} and \mathcal{M} be two 3-Leibniz algebra. Then (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )})be a 3-Leibniz algebra if and only if ρ₁ + ρ₂ + ρ₃ + ν₁ + v₂ + v₃ + w is a Maurer-Cartan element of the differential graded Lie algebra \left(C_{>}\right(\mathcal{L}\oplus \mathcal{M},\mathcal{M}),[\cdot ,\cdot {]}^{3\mathcal{L}},\overline{\delta }).

Proof　Let ρ₁ + ρ₂ + ρ₃ + ν₁ + v₂ + v₃ + w. From the definition of Maurer-Cartan element, c is a Maurer-Cartan element of a differential graded Lie algebra \left(C_{>}\right(\mathcal{L}\oplus \mathcal{M},\mathcal{M}),[\cdot ,\cdot {]}^{3\mathcal{L}},\overline{\delta }) if and only if

\overline{\delta }c+\frac{1}{2}[c,c{]}^{3\mathcal{L}}=0.

As \overline{\delta }=[{\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}},\cdot {]}^{3\mathcal{L}},

[{\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}},c{]}^{3\mathcal{L}}+\frac{1}{2}[c,c{]}^{3\mathcal{L}}=0.

Since {\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}} is the 3-Leibniz algebra structure on \mathcal{L}\oplus \mathcal{M}, [{\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}},{\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}}{]}^{3\mathcal{L}}=0. From the bilinearity and graded antisymmetryof [\cdot ,\cdot {]}^{3\mathcal{L}}, it is known that the equation [{\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}},c{]}^{3\mathcal{L}}+\frac{1}{2}[c,c{]}^{3\mathcal{L}}=0 is equivalent to

[{\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}}+c,{\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}}+c{]}^{3\mathcal{L}}=0.

Therefore, by Theorem 4.1 it can be seen that (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )}) will be a 3-Leibniz algebra if and only if ρ₁ + ρ₂ + ρ₃ + ν₁ + v₂ + v₃ + w is a Maurer-Cartan element of the differential graded Lie algebra \left(C_{>}\right(\mathcal{L}\oplus \mathcal{M},\mathcal{M}),[\cdot ,\cdot {]}^{3\mathcal{L}},\overline{\delta }). □

Theorem 4.3　 Let \mathcal{L} and \mathcal{M} be two 3-Leibniz algebras. The isomorphism classes of non-abelian extensions in 3-Leibniz algebras of \mathcal{L} are in one-to-one correspondence with the gauge equivalence classes of Maurer-Cartan elements in differential graded Lie algebras \left(C_{>}\right(\mathcal{L}\oplus \mathcal{M},\mathcal{M}),[\cdot ,\cdot {]}^{3\mathcal{L}},\overline{\delta }).

Proof　Let c = ρ₁ + ρ₂ + ρ₃ + ν₁ + v₂ + v₃ + w and c^{\prime }={\rho }_1^{\prime }+{\rho }_2^{\prime }+{\rho }_3^{\prime }+{\nu }_1^{\prime }+{\nu }_2^{\prime }+{\nu }_3^{\prime }+{\omega }^{\prime }\in C_{>}^1(\mathcal{L}\oplus \mathcal{M},\mathcal{M}) be gauge equivalences. There exists \xi \in C_{>}^0(\mathcal{L}\oplus \mathcal{M},\mathcal{M})=\mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{L},\mathcal{M}) such that c^{\prime }={\mathrm{e}}^{{\mathrm{a}\mathrm{d}}_{\xi }}c-\frac{{\mathrm{e}}^{{\mathrm{a}\mathrm{d}}_{\xi }}-1}{{\mathrm{a}\mathrm{d}}_{\xi }}\bar{\delta }\xi, namely for any e_i=x_i+m_i\in \mathcal{L}\oplus \mathcal{M}, i = 1, 2, 3, there is

\begin{array}{rl}{c}^{\prime }(e_1\otimes e_2,e_3)=& \left(\left(\mathrm{i}\mathrm{d}+{\mathrm{a}\mathrm{d}}_{\xi }+\frac{{\mathrm{a}\mathrm{d}}_{\xi }^2}{2!}+\cdots +\frac{{\mathrm{a}\mathrm{d}}_{\xi }^n}{n!}+\cdots \right)c\right)(e_1\otimes e_2,e_3)\\ & -\left(\left(\mathrm{i}\mathrm{d}+\frac{{\mathrm{a}\mathrm{d}}_{\xi }}{2!}+\frac{{\mathrm{a}\mathrm{d}}_{\xi }^2}{3!}+\cdots +\frac{{\mathrm{a}\mathrm{d}}_{\xi }^{n-1}}{n!}+\cdots \right)\overline{\delta }\xi \right)(e_1\otimes e_2,e_3).\end{array}

From the definition of [\cdot ,\cdot {]}^{3\mathcal{L}}, it is known that

\begin{array}{rl}{\mathrm{a}\mathrm{d}}_{\xi }\left(c\right)(e_1\otimes e_2,e_3)=& [\xi ,c{]}^{3\mathcal{L}}(e_1\otimes e_2,e_3)\\ =& -{\rho }_1(x_2,x_3)\left(\xi \right(x_1\left)\right)-{\rho }_2(x_1,x_3)\left(\xi \right(x_2\left)\right)-{\rho }_3(x_1,x_2)\left(\xi \right(x_3\left)\right)\\ & -{\nu }_1\left(x_1\right)(m_2,\xi (x_3\left)\right)-{\nu }_1\left(x_1\right)\left(\xi \right(x_2),m_3)\\ & -{\nu }_2\left(x_2\right)(m_1,\xi (x_3\left)\right)-{\nu }_2\left(x_2\right)\left(\xi \right(x_1),m_3)\\ & -{\nu }_3\left(x_3\right)(m_1,\xi (x_2\left)\right)-{\nu }_3\left(x_3\right)\left(\xi \right(x_1),m_2),\end{array}

\begin{array}{rl}{\mathrm{a}\mathrm{d}}_{\xi }^2\left(c\right)(e_1\otimes e_2,e_3)=& [\xi ,[\xi ,c{]}^{3\mathcal{L}}{]}^{3\mathcal{L}}(e_1\otimes e_2,e_3)\\ =& -[\xi ,c{]}^{3\mathcal{L}}(\xi \left(x_1\right)\otimes e_2,e_3)-[\xi ,c{]}^{3\mathcal{L}}(e_1\otimes \xi (x_2),e_3)\\ & -[\xi ,c{]}^{3\mathcal{L}}(e_1\otimes e_2,\xi \left(x_3\right))\\ =& 2{\nu }_1\left(x_1\right)\left(\xi \right(x_2),\xi (x_3\left)\right)+2{\nu }_2\left(x_2\right)\left(\xi \right(x_1),\xi (x_3\left)\right)\\ & +2{\nu }_3\left(x_3\right)\left(\xi \right(x_1),\xi (x_2\left)\right).\end{array}

When n ≥ 3 , {\mathrm{a}\mathrm{d}}_{\xi }^n\left(c\right)(e_1\otimes e_2,e_3)=0.

\begin{array}{rl}\overline{\delta }\xi (e_1\otimes e_2,e_3)=& [{\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}},\xi {]}^{3\mathcal{L}}(e_1\otimes e_2,e_3)\\ =& ({\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}})\left(\xi \right(x_1)\otimes e_2,e_3)+({\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}})(e_1\otimes \xi (x_2),e_3)\\ & +({\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}})(e_1\otimes e_2,\xi (x_3\left)\right)-\xi \left(\right({\mu }_{\mathcal{L}}+{\mu }_{\mathcal{M}}\left)\right(e_1\otimes e_2,e_3\left)\right)\\ =& \left[\xi \right(x_1),m_2,m_3{]}_{\mathcal{M}}+[m_1,\xi \left(x_2\right),m_3{]}_{\mathcal{M}}\\ & +[m_1,m_2,\xi (x_3){]}_{\mathcal{M}}-\xi [x_1,x_2,x_3{]}_{\mathcal{L}},\end{array}

\begin{array}{rl}{\mathrm{a}\mathrm{d}}_{\xi }\left(\overline{\delta }\xi \right)(e_1\otimes e_2,e_3)=& [\xi ,\overline{\delta }\xi {]}^{3\mathcal{L}}(e_1\otimes e_2,e_3)\\ =& -\overline{\delta }\xi \left(\xi \right(x_1)\otimes e_2,e_3)-\overline{\delta }\xi (e_1\otimes \xi (x_2),e_3)\\ & -\overline{\delta }\xi (e_1\otimes e_2,\xi (x_3\left)\right)\\ =& -2\left[\xi \right(x_1),\xi (x_2),m_3{]}_{\mathcal{M}}\\ & -2[\xi \left(x_1\right),m_2,\xi \left(x_3\right){]}_{\mathcal{M}}-2[m_1,\xi (x_2),\xi (x_1){]}_{\mathcal{M}},\end{array}

\begin{array}{rl}{\mathrm{a}\mathrm{d}}_{\xi }^2\left(\overline{\delta }\xi \right)(e_1\otimes e_2,e_3)=& [\xi ,[\xi ,\overline{\delta }\xi {]}^{3\mathcal{L}}{]}^{3\mathcal{L}}(e_1\otimes e_2,e_3)\\ =& -[\xi ,\overline{\delta }\xi {]}^{3\mathcal{L}}(\xi \left(x_1\right)\otimes e_2,e_3)\\ & -[\xi ,\overline{\delta }\xi {]}^{3\mathcal{L}}(e_1\otimes \xi (x_2),e_3)-[\xi ,\overline{\delta }\xi {]}^{3\mathcal{L}}(e_1\otimes e_2,\xi \left(x_3\right))\\ =& 6\left[\xi \right(x_1),\xi (x_2),\xi (x_3){]}_{\mathcal{M}}.\end{array}

When n ≥ 3, {\mathrm{a}\mathrm{d}}_{\xi }^n(\overline{\delta }\xi )(e_1\otimes e_2,e_3)=0.

From the equations mentioned above, it follows that c and c′are gauge equivalence if and only if

c^{\prime }=\left(c+[\xi ,c{]}^{3\mathcal{M}}+\frac{1}{2!}[\xi ,[\xi ,c{]}^{3\mathcal{L}}{]}^{3\mathcal{L}}\right)-\left(\overline{\delta }\xi +\frac{1}{2!}[\xi ,\overline{\delta }\xi {]}^{3\mathcal{L}}+\frac{1}{3!}[\xi ,[\xi ,\overline{\delta }\xi {]}^{3\mathcal{L}}{]}^{3\mathcal{L}}\right),

if and only if Equations (3.33)‒(3.39) hold. With the help of Proposition 3.1, the theorem is proved. □

5 Relations to non-abelian extensions of leibniz algebra

Let (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )}) be a non-abelian extension of a 3-Leibniz algebra \mathcal{L}. In this section, we will analyze the structure of Leibniz algebra on the space{\otimes }^2\left(\mathcal{L}\oplus \mathcal{M}\right) formed by all the fundamental elements in the 3-Leibniz algebra(\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )}), and then prove that the non-abelian extension of a Leibnitz algebra can be given by the non-abelian extension of the 3-Leibnitz algebra. First, we review some results about non-abelian extensions of Leibniz algebras in [17].

Definition 5.1　Let \mathfrak{l} be a vector space. If the bilinear map[\cdot ,\cdot {]}_{\mathfrak{t}}:\mathfrak{l}\otimes \mathfrak{l}\to \mathfrak{l} satisfies

[x,[y,z{]}_{\mathfrak{l}}{]}_{\mathfrak{l}}=\left[\right[x,y{]}_{\mathfrak{l}},z{]}_{\mathfrak{l}}+[y,[x,z{]}_{\mathfrak{l}}{]}_{\mathfrak{l}}\text{,}\mathrm{\forall }x,y,z\in \mathfrak{l},

then (\mathfrak{l},[\cdot ,\cdot {]}_{\mathfrak{l}}) is called a Leibniz algebra.

Definition 5.2　Let (\mathfrak{l},[\cdot ,\cdot {]}_{\mathfrak{l}}) be a Leibniz algebra. Define the set formed by all left derivations as Der^L (\mathfrak{l}), and the set formed by all right derivations as Der^R(\mathfrak{l}), namely

{\mathrm{D}\mathrm{e}\mathrm{r}}^L\left(\mathfrak{l}\right)=\{D\in \mathfrak{g}\mathfrak{l}(\mathfrak{l})\mid D[x,y{]}_{\mathfrak{l}}=\left[D\right(x),y{]}_{\mathfrak{l}}+[x,D\left(y\right){]}_{\mathfrak{l}},\mathrm{\forall }x,y\in \mathfrak{l}\},

{\mathrm{D}\mathrm{e}\mathrm{r}}^R\left(\mathfrak{l}\right)=\{D\in \mathfrak{g}\mathfrak{l}(\mathfrak{l})\mid D[x,y{]}_{\mathfrak{l}}=[x,D(y){]}_{\mathfrak{l}}-[y,D\left(x\right){]}_{\mathfrak{l}},\mathrm{\forall }x,y\in \mathfrak{l}\}.

Example 5.1　Let (\mathfrak{l},[\cdot ,\cdot {]}_{\mathfrak{l}})be a Leibniz algebra. For any element x in \mathfrak{l}, it is defined as {\mathrm{a}\mathrm{d}}_x^L,{\mathrm{a}\mathrm{d}}_x^R\in \mathfrak{g}\mathfrak{l}\left(\mathfrak{l}\right), and

{\mathrm{a}\mathrm{d}}_x^L\left(y\right)=[x,y{]}_{\mathfrak{l}}\text{,}{\mathrm{a}\mathrm{d}}_x^R\left(y\right)=[y,x{]}_{\mathfrak{l}}\text{,}\mathrm{\forall }y\in \mathfrak{l}.

Then {\mathrm{a}\mathrm{d}}_x^L\in {\mathrm{D}\mathrm{e}\mathrm{r}}^L\left(\mathfrak{l}\right), {\mathrm{a}\mathrm{d}}_x^R\in {\mathrm{D}\mathrm{e}\mathrm{r}}^R\left(\mathfrak{l}\right).

Definition 5.3　Let (\mathfrak{s},[\cdot ,\cdot {]}_{\mathfrak{s}}), (\mathfrak{l},[\cdot ,\cdot {]}_{\mathfrak{l}}) and (\hat{\mathfrak{l}},[\cdot ,\cdot {]}_{\hat{\mathfrak{l}}}) be Leibniz algebras. If there is a short exact sequence of Leibniz algebra homomorphisms

0\to \mathfrak{s}\stackrel{i}{\to }\hat{\mathfrak{l}}\stackrel{p}{\to }\mathfrak{l}\to 0,

\hat{\mathfrak{l}} is said to be a non-abelain extension of \mathfrak{l}.

If linear map s:\mathfrak{l}\to \hat{\mathfrak{l}} satisfies p\circ s={\mathrm{i}\mathrm{d}}_{\mathfrak{l}}, s is called a section of non-abelian extension \hat{\mathfrak{l}}.

Let \hat{\mathfrak{l}} be a non-abelian extension of \mathfrak{l}, and s:\mathfrak{l}\to \hat{\mathfrak{l}} is a section of non-abelian extension \hat{\mathfrak{l}}. Define l:\mathfrak{l}\to \mathfrak{g}\mathfrak{l}\left(\mathfrak{s}\right), r:\mathfrak{l}\to \mathfrak{g}\mathfrak{l}\left(\mathfrak{s}\right) and \omega :\mathfrak{l}\otimes \mathfrak{l}\to \mathfrak{s} as follows:

l_x\left(\beta \right)=\left[s\right(x),\beta {]}_{\hat{\mathfrak{l}}}\text{,}\mathrm{\forall }x\in \mathfrak{l},\beta \in \mathfrak{s},

r_y\left(\alpha \right)=[\alpha ,s(y){]}_{\hat{l}}\text{,}\mathrm{\forall }y\in \mathfrak{l},\alpha \in \mathfrak{s},

\omega (x,y)=\left[s\right(x),s(y){]}_{\uparrow }-s[x,y{]}_{}\text{,}\mathrm{\forall }x,y\in \mathfrak{l}.

As vector space \hat{\mathfrak{l}}\cong \mathfrak{l}\oplus \mathfrak{s}, we can transfer the Leibniz algebra structure on \hat{\mathfrak{l}} to\mathfrak{l}\oplus \mathfrak{s}, namely

[x+\alpha ,y+\beta {]}_{(l,r,\omega )}=[x,y{]}_l+\omega (x,y)+l_x\left(\beta \right)+r_y\left(\alpha \right)+[\alpha ,\beta {]}_s.

Proposition 5.1　 Under the definitions mentioned above, (\mathfrak{l}\oplus \mathfrak{s},[\cdot ,\cdot {]}_{(l,r,\omega )})is a Leibniz algebra if and only if l, γ, w satisfy

(5.1)0=l_x\left(\right[\alpha ,\beta {]}_{\mathfrak{s}})-[l_x\left(\alpha \right),\beta {]}_{\mathfrak{s}}-[\alpha ,l_x(\beta ){]}_{\mathfrak{s}},

(5.2)0=r_x\left(\right[\alpha ,\beta {]}_{\mathfrak{s}})-[\alpha ,r_x\left(\beta \right){]}_{\mathfrak{s}}+[\beta ,r_x(\alpha ){]}_{\mathfrak{s}},

(5.3)0=\left[l_x\right(\alpha )+r_x(\alpha ),\beta {]}_{\mathfrak{s}},

(5.4)0=[l_x,l_y]-l_{[x,y{]}_{\mathfrak{l}}}-{\mathrm{a}\mathrm{d}}_{\omega (x,y)}^L,

(5.5)0=[l_x,r_y]-r_{[x,y{]}_t}-{\mathrm{a}\mathrm{d}}_{\omega (x,y)}^R,

(5.6)0=r_y\left(r_x\right(\alpha )+l_x(\alpha \left)\right),

(5.7)\begin{array}{rl}0=& l_x\left(\omega \right(y,z\left)\right)-l_y\left(\omega \right(x,z\left)\right)-r_z\left(\omega \right(x,y\left)\right)\\ & +\omega (x,[y,z{]}_{\mathfrak{l}})-\omega ([x,y{]}_{\mathfrak{l}},z)-\omega (y,[x,z{]}_{\mathfrak{l}}).\end{array}

{\left[\cdot ,\cdot \right]}_{\hat{\mathrm{F}}} refers to the Leibniz bracket on {\otimes }^2\left(\mathcal{L}\oplus \mathcal{M}\right), and there is

{\otimes }^2\left(\mathcal{L}\oplus \mathcal{M}\right)\cong \left(\left({\otimes }^2\mathcal{M}\right)\oplus (\mathcal{L}\otimes \mathcal{M})\oplus (\mathcal{M}\otimes \mathcal{L})\right)\oplus \left({\otimes }^2\mathcal{L}\right).

Let \mathfrak{L}:=\left({\otimes }^2\mathcal{M}\right)\oplus (\mathcal{L}\otimes \mathcal{M})\oplus (\mathcal{M}\otimes \mathcal{L}). Then we give a Leibniz algebra structure on \mathcal{L}. Define the bilinear map \{\cdot ,\cdot \}:\mathfrak{L}\otimes \mathfrak{L}\to \mathfrak{L} as

\begin{array}{rl}& \{m_1\otimes m_2+x_1\otimes m_3+m_4\otimes x_2,n_1\otimes n_2+y_1\otimes n_3+n_4\otimes y_2\}\\ & =[m_1,m_2,n_1{]}_{\mathcal{M}}\otimes n_2+n_1\otimes [m_1,m_2,n_2{]}_{\mathcal{M}}\\ & +{\nu }_3\left(y\right)(m_1,m_2)\otimes n_3+y_1\otimes [m_1,m_2,n_3{]}_{\mathcal{M}}\\ & +[m_1,m_2,n_4{]}_{\mathcal{M}}\otimes y_2+n_4\otimes {\nu }_3(y_2\left)\right(m_1,m_2)\\ & +{\nu }_1\left(x\right)(m_3,n_1)\otimes n_2+n_1\otimes {\nu }_1\left(x\right)(m_3,n_2)\\ & +{\rho }_2(x_1,y_1)\left(m_3\right)\otimes n_3+y_1\otimes {\nu }_1\left(x_1\right)(m_3,n_3)\\ & +{\nu }_1\left(x_1\right)(m_3,n_4)\otimes y_2+n_4\otimes {\rho }_2(x_1,y_2)\left(m_3\right)\\ & +{\nu }_2\left(x_2\right)(m_4,n_1)\otimes n_2+n_1\otimes {\nu }_2\left(x_2\right)(m_4,n_2)\\ & +{\rho }_1(x_2,y_1)\left(m_4\right)\otimes n_3+y_1\otimes {\nu }_2\left(x_2\right)(m_4,n_3)\\ & +{\nu }_2\left(x_2\right)(m_4,n_4)\otimes y_2+n_4\otimes {\rho }_1(x_2,y_2)\left(m_4\right),\end{array}

where x₁, x₂, y₁, y₂ \in \mathcal{L}, m_i, n_i \in \mathcal{M}, i = 1, 2, 3, 4.

Proposition 5.2　 Under the definitions mentioned above, (\mathfrak{L},\{\cdot ,\cdot \left\}\right) is a Leibniz algebra.

Proof　After calculation, it is known that

\begin{array}{rl}& \{m_1\otimes m_2+x_1\otimes m_3+m_4\otimes x_2,n_1\otimes n_2+y_1\otimes n_3+n_4\otimes y_2\}\\ & =[m_1\otimes m_2+x_1\otimes m_3+m_4\otimes x_2,n_1\otimes n_2+y_1\otimes n_3+n_4\otimes y_2{]}_{\hat{\mathrm{F}}}.\end{array}

Therefore, (\mathfrak{L},\{\cdot ,\cdot \left\}\right) a sub-algebra of the Leibniz algebra\left({\otimes }^2\right(\mathcal{L}\oplus \mathcal{M}),[\cdot ,\cdot {]}_{\hat{\mathrm{F}}}). □

Then define \tilde{l}:{\otimes }^2\mathcal{L}\to \mathfrak{g}\mathfrak{l}\left(\mathfrak{L}\right), \tilde{r}:{\otimes }^2\mathcal{L}\to \mathfrak{g}\mathfrak{l}\left(\mathfrak{L}\right)] and \tilde{\omega }:\left({\otimes }^2\mathcal{L}\right)\otimes \left({\otimes }^2\mathcal{L}\right)\to \mathfrak{L} as

\begin{array}{rl}& \tilde{l}(x_1\otimes x_2)(m_1\otimes m_2+x_3\otimes m_3+m_4\otimes x_4)\\ & ={\rho }_3(x_1,x_2)\left(m_1\right)\otimes m_2+m_1\otimes {\rho }_3(x_1,x_2)\left(m_2\right)\\ & +x_3\otimes {\rho }_3(x_1,x_2)\left(m_3\right)+[x_1,x_2,x_3{]}_{\mathcal{L}}\otimes m_3\\ & +\omega (x_1,x_2,x_3)\otimes m_3+{\rho }_3(x_1,x_2)\left(m_4\right)\otimes x_4\\ & +m_4\otimes [x_1,x_2,x_4]L+m_4\otimes \omega (x_1,x_2,x_4),\end{array}

\begin{array}{rl}& \tilde{r}(x_1\otimes x_2)(m_1\otimes m_2+x_3\otimes m_3+m_4\otimes x_4)\\ & ={\nu }_3\left(x_1\right)(m_1,m_2)\otimes x_2+x_1\otimes {\nu }_3\left(x_2\right)(m_1,m_2)\\ & +{\rho }_2(x_3,x_1)\left(m_3\right)\otimes x_2+x_1\otimes {\rho }_2(x_3,x_2)\left(m_3\right)\\ & +{\rho }_1(x_4,x_1)\left(m_4\right)\otimes x_2+x_1\otimes {\rho }_1(x_4,x_2)\left(m_4\right),\end{array}

\tilde{\omega }(x_1\otimes x_2,x_3\otimes x_4)=\omega (x_1,x_2,x_3)\otimes x_4+x_3\otimes \omega (x_1,x_2,x_4),

where x₁, x₂, x₃, x₄ \in \mathcal{L}, m₁, m₂, m₃, m₄ \in \mathcal{M}. After calculation, it follows that \tilde{l}, \tilde{r}, \tilde{w} satisfy Equations (5.1)‒(5.7). According to Proposition 5.1, (({\otimes }^2\mathcal{L})\oplus \mathcal{L},[\cdot ,\cdot {]}_{(\tilde{l},\tilde{r},\tilde{\omega })})is a Leibniz algebra.

Theorem 5.1　 Let (\mathcal{L},[\cdot ,\cdot ,\cdot {]}_{\mathcal{L}}) and (\mathcal{M},[\cdot ,\cdot ,\cdot {]}_{\mathcal{M}}) be two 3-Leibniz algebras and (\mathcal{L}\oplus \mathcal{M},[\cdot ,\cdot ,\cdot {]}_{(\rho ,\nu ,\omega )}) is a non-abelian extension of \mathcal{L}. Then \left({\otimes }^2\right(\mathcal{L}\oplus \mathcal{M}),[\cdot ,\cdot {]}_{\hat{\mathrm{F}}}) is a non-abelian extension of the Leibniz algebra ({\otimes }^2\mathcal{L},[\cdot ,\cdot {]}_{\mathrm{F}}).

Proof　As {\otimes }^2\left(\mathcal{L}\oplus \mathcal{M}\right)\cong \left(\right({\otimes }^2\mathcal{M})\oplus (\mathcal{L}\otimes \mathcal{M})\oplus (\mathcal{M}\otimes \mathcal{L}\left)\right)\oplus \left({\otimes }^2\mathcal{L}\right), the Leibniz algebra structure on{\otimes }^2\left(\mathcal{L}\oplus \mathcal{M}\right) is

\begin{array}{rl}& [m_1\otimes m_2+x_1\otimes m_3+m_4\otimes x_2+x_3\otimes x_4,n_1\otimes n_2+y_1\otimes n_3+n_4\otimes y_2+y_3\otimes y_4{]}_{\hat{\text{F}}}\\ & =[m_1\otimes m_2+x_1\otimes m_3+m_4\otimes x_2,n_1\otimes n_2+y_1\otimes n_3+n_4\otimes y_2{]}_{\hat{\text{F}}}\\ & +[x_3\otimes x_4,n_1\otimes n_2+y_1\otimes n_3+n_4\otimes y_2{]}_{\hat{\text{F}}}\\ & +[m_1\otimes m_2+x_1\otimes m_3+m_4\otimes x_2,y_3\otimes y_4{]}_{\hat{\mathrm{F}}}+[x_3\otimes x_4,y_3\otimes y_4{]}_{\hat{\mathrm{F}}}\\ & =\{m_1\otimes m_2+x_1\otimes m_3+m_4\otimes x_2,n_1\otimes n_2+y_1\otimes n_3+n_4\otimes y_2\}\\ & +\tilde{r}(x_3\otimes x_4)(n_1\otimes n_2+y_1\otimes n_3+n_4\otimes y_2)\\ & +\tilde{l}(y_3\otimes y_4)(m_1\otimes m_2+x_1\otimes m_3+m_4\otimes x_2)\\ & +\tilde{\omega }(x_3\otimes x_4,y_3\otimes y_4)+[x_3\otimes x_4,y_3\otimes y_4{]}_{\text{F}}.\end{array}

Therefore, \left({\otimes }^2\right(\mathcal{L}\oplus \mathcal{M}),[\cdot ,\cdot {]}_{\hat{\mathrm{F}}}\left)\right)is a non-abelian extension of Leibniz algebra ({\otimes }^2\mathcal{L},[\cdot ,\cdot {]}_{\mathrm{F}}). □

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