Abelian Extensions of Modified λ-Differential Left-Symmetric Algebras and Crossed Modules

Abstract

In this paper, we define a cohomology theory of a modified $\lambda$-differential left-symmetric algebra. Moreover, we introduce the notion of modified $\lambda$-differential left-symmetric 2-algebras, which is the categorization of a modified $\lambda$-differential left-symmetric algebra. As applications of cohomology, we classify linear deformations and abelian extensions of modified $\lambda$-differential left-symmetric algebras using the second cohomology group and classify skeletal modified $\lambda$-differential left-symmetric 2-algebra using the third cohomology group. Finally, we show that strict modified $\lambda$-differential left-symmetric 2-algebras are equivalent to crossed modules of modified $\lambda$-differential left-symmetric algebras.

1. Introduction

Derivation, also known as differential operator, plays an important role in mathematical physics, such as homotopy Lie algebras [1], differential Galois theory [2], control theory and gauge theories of quantumj field theory [3]. In [4,5], the authors studied associative algebras with derivations from the operadic point of view. Recently, in [6], Tang and their collaborators considered Lie algebras with derivations from the cohomological point of view. Inspired by the work of [6], associative algebras with derivations and pre-algebras with derivations have been studied in [7,8], respectively.

The solution of the modified classical Yang-Baxter equation, called modified r-matrix, was introduced by Semenov-Tian-Shansky in [9]. Recently, Jiang and Sheng [10] developed the deformations of modified r-matrices. Inspired from [9,10], the notion of modified $\lambda$-differential Lie algebras was introduced in [11]. Subsequently, the algebraic structures with modified operators were widely studied in [12,13,14,15,16].

However, there have been very few studies about the modified $\lambda$-differential left-symmetric algebras. Left-symmetric algebras (also called pre-Lie algebras) are nonassociative algebras, which were introduced by Cayley [17] as a kind of rooted tree algebras and also introduced by Gerstenhaber [18] when studying the deformation theory of rings and algebras. Left symmetric algebras have been widely used in geometry and physics, such as affine manifolds [19], integrable systems, quantum Yang-Baxter Equations [20,21], Poisson brackets, operands, and complex and symplectic structures on Lie groups [22]. See also [23,24,25,26,27,28,29,30,31,32] for more details. Thus, it is natural and necessary to study the modified $\lambda$-differential left symmetric algebras.

Motivated by the work in [8,11,12,13], our main purpose is to study the representation and cohomology of the modified $\lambda$-differential left symmetric algebras and applied them to the linear deformation, abelian extension, and skeletal modified $\lambda$-differential left symmetric 2-algebras. The paper is organized as follows. Section 2 introduces the representations of modified $\lambda$-differential left-symmetric algebras. In Section 3, we define a cohomology theory of modified $\lambda$-differential left-symmetric algebras with coefficients in a representation, and apply it to the study of linear deformation. In Section 4, we investigate abelian extensions of the modified $\lambda$-differential left-symmetric algebras in terms of second cohomology groups. Finally, in Section 5, we classify skeletal modified $\lambda$-differential left-symmetric 2-algebras by using the third cohomology group. We then prove that strict modified $\lambda$-differential left-symmetric 2-algebras are equivalent to the crossed modules of modified $\lambda$-differential left-symmetric algebras.

All tensor products, vector spaces, and (multi)linear maps are over a field $\mathbb{K}$ of characteristic 0.

2. Modified $\mathbf{\lambda }$-Differential Left-Symmetric Algebras and Their Representations

This section introduces the notion of a modified $\lambda$-differential left-symmetric algebra and gives their representations.

Now let us recall some basic concepts of left-symmetric algebras from [18,24].

Definition 1

([18]). Left-symmetric algebra (LSA in short) is a vector space $\mathfrak{p}$ with a bilinear product $★:\mathfrak{p}\times \mathfrak{p}\to \mathfrak{p}$ such that for $p_1,p_2,p_3\in \mathfrak{p}$, the associator:

$$\begin{array}{cc}\hfill & (p_1,p_2,p_3)=(p_1★p_2)★p_3-p_1★(p_2★p_3),\hfill \end{array}$$

is symmetric in $p_1,p_2$, i.e., $(p_1,p_2,p_3)=(p_2,p_1,p_3),\mathit{or}\mathit{equivalently}$:

$$\begin{array}{c}\hfill (p_1★p_2)★p_3-p_1★(p_2★p_3)=(p_2★p_1)★p_3-p_2★(p_1★p_3).\end{array}$$

(1)

Denote it by $(\mathfrak{p},★)$.

Remark 1.

Let $(\mathfrak{p},★)$ be a LSA. If we define a bilinear bracket ${[,]}^c:\mathfrak{p}\times \mathfrak{p}\to \mathfrak{p}$ as:

$${[p_1,p_2]}^c=p_1★p_2-p_2★p_1,$$

then $(\mathfrak{p},{[,]}^c)$ is a Lie algebra.

Example 1.

Let $(\mathfrak{p},[,\left]\right)$ be a Lie algebra and $R:\mathfrak{p}\to \mathfrak{p}$ be a linear map satisfying the Rota–Baxter equation:

$$[R{p}_1,R{p}_2]=R([R{p}_1,p_2]+[p_1,R{p}_2]),\forall p_1,p_2\in \mathfrak{p},$$

then, $(\mathfrak{p},{★}_R)$ is a LSA, in which the LSA operation is $p_1{★}_R{p}_2=[R{p}_1,p_2]$.

Definition 2.

Let $(\mathfrak{p},★)$ be a LSA and $\lambda \in \mathbb{K}$. If the linear map $\partial :\mathfrak{p}\to \mathfrak{p}$ satisfies

$$\begin{array}{cc}\hfill \partial (p_1★p_2)=& \partial p_1★p_2+p_1★\partial p_2+\lambda (p_1★p_2),\forall p_1,p_2\in \mathfrak{p},\hfill \end{array}$$

(2)

then ∂ is called a modified λ-differential operator (MλD operator in short). Moreover, the triple $(\mathfrak{p},★,\partial )$ is called modified λ-differential left-symmetric algebra (MλDLSA in short), simply denoted by $(\mathfrak{p},\partial )$.

Remark 2.

Let ∂ be an MλD operator on a LSA $(\mathfrak{p},★)$. If $\lambda =0,$ then ∂ is a derivation on $(\mathfrak{p},★)$ and $(\mathfrak{p},★,\partial )$ is a LSA with a derivation. See [8] for LSAs with derivations.

Definition 3.

The homomorphism between MλDLSAs $({\mathfrak{p}}_1,{★}_1,{\partial }_1)$ and $({\mathfrak{p}}_2,{★}_2,{\partial }_2)$ is a linear map $\Phi :{\mathfrak{p}}_1\to {\mathfrak{p}}_2$ that satisfies $\Phi \left(p_1{★}_1{p}_2\right)=\Phi \left(p_1\right){★}_2\Phi \left(p_2\right)$ and $\Phi \circ {\partial }_1={\partial }_2\circ \Phi$. In addition, if Φ is bijective, it is said that Φ is isomorphic from $({\mathfrak{p}}_1,{\partial }_1)$ to $({\mathfrak{p}}_2,{\partial }_2)$.

Example 2.

Let $(\mathfrak{p},★)$ be a LSA. Then $(\mathfrak{p},★,{\mathrm{id}}_{\mathfrak{p}})$ is an MλDLSA, where ${\mathrm{id}}_{\mathfrak{p}}:\mathfrak{p}\to \mathfrak{p}$ is an identity map.

Example 3.

Let $(\mathfrak{p},[,],\partial )$ be an MλD Lie algebra (see [11], Definition 2.5). By Example 1, if $R\circ \partial =\partial \circ R$, then $(\mathfrak{p},{★}_R,\partial )$ is an MλDLSA.

Example 4.

Let $(\mathfrak{p},★)$ be a two-dimensional LSA and $\{e_1,e_2\}$ be a basis, whose nonzero products are given as follows:

$$e_1★e_2=e_1,e_2★e_2=e_2.$$

Then the triple $(\mathfrak{p},★,\partial )$ is a two-dimensional MλDLSA, where $\partial =\left(\begin{array}{cc}{k}_1& k_2\\ 0& -\lambda \end{array}\right)$, for $k_1,k_2\in \mathbb{K}$.

Example 5.

If $(\mathfrak{p},★,\partial )$ is an MλDLSA, and for any $k\in \mathbb{K}$, then $(\mathfrak{p},★,k\partial )$ is also an M$k\lambda$DLSA.

Definition 4

([24]). A representation of a LSA $(\mathfrak{p},★)$ is a triple $(\mathcal{V};{★}_l,{★}_r)$, where $\mathcal{V}$ is a vector space, ${★}_l:\mathfrak{p}\times \mathcal{V}\to \mathcal{V}$ and ${★}_r:\mathcal{V}\times \mathfrak{p}\to \mathcal{V}$ are two linear maps such that for all $p_1,p_2\in \mathfrak{p},u\in \mathcal{V}$:

$$\begin{array}{cc}\hfill p_1{★}_l\left(p_2{★}_{l}u\right)-(p_1★p_2){★}_{l}u& =p_2{★}_l\left(p_1{★}_{l}u\right)-(p_2★p_1){★}_{l}u,\hfill \\ \hfill p_1{★}_l\left(u{★}_r{p}_2\right)-\left(p_1{★}_{l}u\right){★}_r{p}_2& =u{★}_r(p_1★p_2)-\left(u{★}_r{p}_1\right){★}_r{p}_2.\hfill \end{array}$$

Definition 5.

Let $(\mathcal{V};{★}_l,{★}_r)$ be a representation of a LSA $(\mathfrak{p},★)$. Then $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$ is called a representation of an MλDLSA $(\mathfrak{p},★,\partial )$ if $(\mathcal{V};{★}_l,{★}_r)$ is endowed with a linear map ${\partial }_{\mathcal{V}}:\mathcal{V}\to \mathcal{V}$ satisfying the following equations:

$$\begin{array}{cc}\hfill {\partial }_{\mathcal{V}}\left(p_1{★}_{l}u\right)=& \partial p_1{★}_{l}u+p_1{★}_l{\partial }_{\mathcal{V}}u+\lambda \left(p_1{★}_{l}u\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\partial }_{\mathcal{V}}\left(u{★}_r{p}_1\right)=& {\partial }_{\mathcal{V}}u{★}_r{p}_1+u{★}_r\partial p_1+\lambda \left(u{★}_r{p}_1\right).\hfill \end{array}$$

(4)

For example, given an M$\lambda$DLSA $(\mathfrak{p},★,\partial )$, there is a natural adjoint representation on itself. The corresponding representation maps ${★}_l,{★}_r$ and ${\partial }_{\mathcal{V}}$ are given by ${★}_l={★}_r=★$ and ${\partial }_{\mathcal{V}}=\partial$.

Proposition 1.

The quadruple $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$ is a representation of an MλDLSA $(\mathfrak{p},★,\partial )$ if and only if $\mathfrak{p}\oplus \mathcal{V}$ is an MλDLSA with the following maps:

$$\begin{array}{cc}\hfill (p_1+u_1){★}_{⋉}(p_2+u_2):=& p_1★p_2+p_1{★}_l{u}_2+u_1{★}_r{p}_2,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \partial \oplus {\partial }_{\mathcal{V}}(p_1+u_1)=& \partial p_1+{\partial }_{\mathcal{V}}{u}_1,\hfill \end{array}$$

(6)

for all $p_1,p_2\in \mathfrak{p}$ and $u_1,u_2\in \mathcal{V}$.

Proof. 

Firstly, it is easy to verify that $(\mathfrak{p}\oplus \mathcal{V},{★}_{⋉})$ is a LSA. Furthermore, for any $p_1,p_2\in \mathfrak{p}$ and $u_1,u_2\in \mathcal{V}$, by Equations (2)–(6) we have:

$$\begin{array}{cc}\hfill & \partial \oplus {\partial }_{\mathcal{V}}\left((p_1+u_1){★}_{⋉}(p_2+u_2)\right)\hfill \\ \hfill & =\partial \oplus {\partial }_{\mathcal{V}}\left(p_1★p_2+p_1{★}_l{u}_2+u_1{★}_r{p}_2\right)\hfill \\ \hfill & =\partial (p_1★p_2)+{\partial }_{\mathcal{V}}(p_1{★}_l{u}_2+u_1{★}_r{p}_2)\hfill \\ \hfill & =\partial p_1★p_2+p_1★\partial p_2+\lambda (p_1★p_2)+\partial p_1{★}_l{u}_2+p_1{★}_l{\partial }_{\mathcal{V}}{u}_2+\lambda \left(p_1{★}_l{u}_2\right)\hfill \\ \hfill & +{\partial }_{\mathcal{V}}{u}_1{★}_r{p}_2+u_1{★}_r\partial p_2+\lambda \left(u_1{★}_r{p}_2\right)\hfill \\ \hfill & =\partial \oplus {\partial }_{\mathcal{V}}(p_1+u_1){★}_{⋉}(p_2+u_2)+(p_1+u_1){★}_{⋉}\partial \oplus {\partial }_{\mathcal{V}}(p_2+u_2)+\lambda \left((p_1+u_1){★}_{⋉}(p_2+u_2)\right).\hfill \end{array}$$

Hence, $(\mathfrak{p}\oplus \mathcal{V},{★}_{⋉},\partial \oplus {\partial }_{\mathcal{V}})$ is an M$\lambda$DLSA.

Conversely, suppose $(\mathfrak{p}\oplus \mathcal{V},{★}_{⋉},\partial \oplus {\partial }_{\mathcal{V}})$ is an M$\lambda$DLSA, then for any $p\in \mathfrak{p}$ and $u\in \mathcal{V}$, we have:

$$\begin{array}{cc}\hfill & \partial \oplus {\partial }_{\mathcal{V}}\left((p+0){★}_{⋉}(0+u)\right)\hfill \\ \hfill & =\partial \oplus {\partial }_{\mathcal{V}}(p+0){★}_{⋉}(0+u)+(p+0){★}_{⋉}\partial \oplus {\partial }_{\mathcal{V}}(0+u)+\lambda \left((p+0){★}_{⋉}(0+u)\right),\hfill \\ \hfill & \partial \oplus {\partial }_{\mathcal{V}}\left((0+u){★}_{⋉}(p+0)\right)\hfill \\ \hfill & =\partial \oplus {\partial }_{\mathcal{V}}(0+u){★}_{⋉}(p+0)+(0+u){★}_{⋉}\partial \oplus {\partial }_{\mathcal{V}}(p+0)+\lambda \left((0+u){★}_{⋉}(p+0)\right),\hfill \end{array}$$

which implies that ${\partial }_{\mathcal{V}}\left(p{★}_{l}u\right)=\partial p{★}_{l}u+p{★}_l{\partial }_{\mathcal{V}}u+\lambda \left(p{★}_{l}u\right)$ and ${\partial }_{\mathcal{V}}\left(u{★}_{r}p\right)={\partial }_{\mathcal{V}}u{★}_{r}p+u{★}_r\partial p+\lambda \left(u{★}_{r}p\right)$. Therefore, $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$ is a representation of $(\mathfrak{p},★,\partial )$. □

3. Cohomology and Linear Deformations of Modified $\mathbf{\lambda }$-Differential Left-Symmetric Algebras

This section defines the cohomology theory of an M$\lambda$DLSA with coefficients in a representation. Then the linear deformation of M$\lambda$DLSAs is studied by using low-order cohomology groups.

Let us first recall the cohomology theory of LSAs in [30]. Let $(\mathfrak{p},★)$ be a LSA and $(\mathcal{V};{★}_l,{★}_r)$ be a representation of it. Denote the $n-$cochains of $\mathfrak{p}$ with coefficients in $\mathcal{V}$ by ${\mathfrak{C}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V}):=\mathrm{Hom}({\mathfrak{p}}^{\otimes n},\mathcal{V}).$

The coboundary map $\delta :{\mathfrak{C}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V})\to {\mathfrak{C}}_{\mathrm{LSA}}^{n+1}(\mathfrak{p},\mathcal{V})$, for $p_1,\cdots ,p_{n+1}\in \mathfrak{p}$ and $\theta \in {\mathfrak{C}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V})$, as:

$$\begin{array}{cc}\hfill & \delta \theta (p_1,\dots ,p_{n+1})\hfill \\ \hfill =& \sum _{i=1}^n{(-1)}^{i+1}{p}_i{★}_l\theta (p_1,\dots ,{\widehat{p}}_i,\dots ,p_{n+1})+\sum _{i=1}^n{(-1)}^{i+1}\theta (p_1,\dots ,{\widehat{p}}_i,\dots ,p_n,p_i){★}_r{p}_{n+1}\hfill \\ \hfill & -\sum _{i=1}^n{(-1)}^{i+1}\theta (p_1,\dots ,{\widehat{p}}_i,\dots ,p_n,p_i★p_{n+1})\hfill \\ \hfill & +\sum _{1\le i<j\le n}{(-1)}^{i+j}\theta (p_i★p_j-p_j★p_i,p_1,\dots ,{\widehat{p}}_i,\dots ,{\widehat{p}}_j,\dots ,p_{n+1}).\hfill \end{array}$$

(7)

It has been proved in [30] that $\delta \circ \delta =0$. The cohomology group of cochain complex $({\mathfrak{C}}_{\mathrm{LSA}}^{•}(\mathfrak{p},\mathcal{V}),\delta )$ is denoted as ${\mathcal{H}}_{\mathrm{LSA}}^{•}(\mathfrak{p},\mathcal{V})$.

Let $(\mathfrak{p},★,\partial )$ be an M$\lambda$DLSA and $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$ be a representation of it. Inspired by [12,13], a linear map $\Gamma :{\mathfrak{C}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V})\to {\mathfrak{C}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V})$ is defined as:

$$\begin{array}{cc}\hfill & (\Gamma \theta )(p_1,\dots ,p_n)\hfill \\ \hfill & =\sum _{i=1}^n\theta (p_1,\dots ,\partial p_i,\dots ,p_n)+(n-1)\lambda \theta (p_1,\dots ,p_n)-{\partial }_{\mathcal{V}}\theta (p_1,\dots ,p_n)\mathrm{for}n\ge 1.\hfill \end{array}$$

(8)

Lemma 1.

The map Γ defined above is a cochain map, that is, the diagram:

is commutative.

Proof. 

For any $\theta \in {\mathfrak{C}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V})$ and $p_1,\dots ,p_{n+1}\in \mathfrak{p},$ we have:

$$\begin{array}{cc}\hfill & \Gamma \left(\delta \theta \right)(p_1,\dots ,p_{n+1})\hfill \\ \hfill =& \sum _{i=1}^{n+1}\left(\delta \theta \right)(p_1,\dots ,\partial p_i,\dots ,p_{n+1})+n\lambda \left(\delta \theta \right)(p_1,\dots ,p_{n+1})-{\partial }_{\mathcal{V}}\left(\delta \theta \right)(p_1,\dots ,p_{n+1})\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill & \delta (\Gamma \theta )(p_1,\dots ,p_{n+1})\hfill \\ \hfill =& \sum _{i=1}^n{(-1)}^{i+1}{p}_i{★}_l(\Gamma \theta )(p_1,\dots ,{\widehat{p}}_i,\dots ,p_{n+1})+\sum _{i=1}^n{(-1)}^{i+1}(\Gamma \theta )(p_1,\dots ,{\widehat{p}}_i,\dots ,p_n,p_i){★}_r{p}_{n+1}\hfill \\ \hfill & -\sum _{i=1}^n{(-1)}^{i+1}(\Gamma \theta )(p_1,\dots ,{\widehat{p}}_i,\dots ,p_n,p_i★p_{n+1})\hfill \\ \hfill & +\sum _{1\le i<j\le n}{(-1)}^{i+j}(\Gamma \theta )(p_i★p_j-p_j★p_i,p_1,\dots ,{\widehat{p}}_i,\dots ,{\widehat{p}}_j,\dots ,p_{n+1}).\hfill \end{array}$$

(10)

From Equations (1)–(4) and expanding Equations (9) and (10), we can deduce that $\Gamma \circ \delta =\delta \circ \Gamma .$ □

Definition 6.

Let $(\mathfrak{p},★,\partial )$ be an MλDLSA and $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$ be a representation of it. We define the cochain complex $({\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^{•}(\mathfrak{p},\mathcal{V}),\mathfrak{D})$ of $(\mathfrak{p},★,\partial )$ with coefficients in $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$ to the negative shift of the mapping cone of Γ, that is, let:

$${\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^1(\mathfrak{p},\mathcal{V})={\mathfrak{C}}_{\mathrm{LSA}}^1(\mathfrak{p},\mathcal{V})\mathrm{and}{\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^n(\mathfrak{p},\mathcal{V}):={\mathfrak{C}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V})\oplus {\mathfrak{C}}_{\mathrm{LSA}}^{n-1}(\mathfrak{p},\mathcal{V}),\forall n\ge 2,$$

and the coboundary operator $\mathfrak{D}:{\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^1(\mathfrak{p},\mathcal{V})\to {\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^2(\mathfrak{p},\mathcal{V})$ is given by:

$$\begin{array}{c}\hfill \mathfrak{D}\left(\theta \right)=(\delta \theta ,-\Gamma \theta ),\forall \theta \in {\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^1(\mathfrak{p},\mathcal{V});\end{array}$$

for $n\ge 2$, the coboundary operator $\mathfrak{D}:{\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^n(\mathfrak{p},\mathcal{V})\to {\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^{n+1}(\mathfrak{p},\mathcal{V})$ is given by:

$$\begin{array}{c}\hfill \mathfrak{D}({\theta }_1,{\theta }_2)=(\delta {\theta }_1,\delta {\theta }_2+{(-1)}^n\Gamma {\theta }_1),\forall ({\theta }_1,{\theta }_2)\in {\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^n(\mathfrak{p},\mathcal{V}).\end{array}$$

The cohomology of $({\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^{•}(\mathfrak{p},\mathcal{V}),\mathfrak{D})$, denoted by ${\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^{•}(\mathfrak{p},\mathcal{V})$, is called the cohomology of the MλDLSA $(\mathfrak{p},★,\partial )$ with coefficients in $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$. In particular, when $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})=(\mathfrak{p};{★}_l={★}_r=★,\partial )$, we just denote $({\mathcal{C}}_{\mathrm{M}\cup \mathrm{D}\mathrm{LSA}}^{•}(\mathfrak{p},\mathfrak{p}),\mathfrak{D})$, ${\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^{•}(\mathfrak{p},\mathfrak{p})$ by $({\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^{•}\left(\mathfrak{p}\right),\mathfrak{D})$, ${\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^{*}\left(\mathfrak{p}\right)$, respectively, and call them the cochain complex, the cohomology of an MλDLSA $(\mathfrak{p},★,\partial )$, respectively.

Corollary 1.

Let $(\mathfrak{p},★,\partial )$ be an MλDLSA. Then, there is a short exact sequence of cochain complexes:

$$\begin{array}{c}\hfill 0\to {\mathfrak{C}}_{\mathrm{LSA}}^{•-1}(\mathfrak{p},\mathcal{V})⟶{\mathcal{C}}_{\mathrm{M}\cup \mathrm{D}\mathrm{LSA}}^{•}(\mathfrak{p},\mathcal{V})⟶{\mathfrak{C}}_{\mathrm{LSA}}^{•}(\mathfrak{p},\mathcal{V})\to 0.\end{array}$$

Consequently, it induces a long exact sequence of cohomology groups:

$$\begin{array}{c}\hfill \cdots \to {\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^n(\mathfrak{p},\mathcal{V})\to {\mathcal{H}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V})\to {\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^{n+1}(\mathfrak{p},\mathcal{V})\to {\mathcal{H}}_{\mathrm{LSA}}^{n+1}(\mathfrak{p},\mathcal{V})\to \cdots .\end{array}$$

Next, we use the established cohomology theory to characterize linear deformations of M$\lambda$DLSAs.

Definition 7.

Let $(\mathfrak{p},★,\partial )$ be an MλDLSA. If for all $t\in \mathbb{K}$, $(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t=★+t{★}_1,{\partial }_t=\partial +t{\partial }_1)$ is still an MλDLSA over $\mathbb{K}\left[\left[t\right]\right]/\left(t^2\right)$, where $({★}_1,{\partial }_1)\in {\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^2\left(\mathfrak{p}\right).$ We say that $({★}_1,{\partial }_1)$ generates a linear deformation of an MλDLSA $(\mathfrak{p},★,\partial )$.

Proposition 2.

If $({★}_1,{\partial }_1)$ generates a linear deformation of an MλDLSA $(\mathfrak{p},★,\partial )$, then $({★}_1,{\partial }_1)$ is a 2-cocycle of the MλDLSA $(\mathfrak{p},★,\partial )$.

Proof. 

If $({★}_1,{\partial }_1)$ generates a linear deformation of an M$\lambda$DLSA $(\mathfrak{p},★,\partial )$, then for any $p_1,p_2,p_3\in \mathfrak{p}$, we have:

$$\begin{array}{cc}\hfill \left(p_1{★}_t{p}_2\right){★}_t{p}_3-p_1{★}_t\left(p_2{★}_t{p}_3\right)=& \left(p_2{★}_t{p}_1\right){★}_t{p}_3-p_2{★}_t\left(p_1{★}_t{p}_3\right),\hfill \\ \hfill {\partial }_t\left(p_1{★}_t{p}_2\right)=& {\partial }_t{p}_1{★}_t{p}_2+p_1{★}_t{\partial }_t{p}_2+\lambda \left(p_1{★}_t{p}_2\right).\hfill \end{array}$$

Comparing coefficients of $t^1$ on both sides of the above equations, we have:

$$\begin{array}{cc}\hfill & \left(p_1{★}_1{p}_2\right)★p_3+(p_1★p_2){★}_1{p}_3-p_1★\left(p_2{★}_1{p}_3\right)-p_1{★}_1(p_2★p_3)\hfill \\ \hfill & =\left(p_2{★}_1{p}_1\right)★p_3+(p_2★p_1){★}_1{p}_3-p_2{★}_1(p_1★p_3)-p_2★\left(p_1{★}_1{p}_3\right)\hfill \end{array}$$

(11)

and

$$\begin{array}{cc}\hfill & {\partial }_1(p_1★p_2)+\partial \left(p_1{★}_1{p}_2\right)=\partial p_1{★}_1{p}_2+{\partial }_1{p}_1★p_2+p_1★{\partial }_1{p}_2+p_1{★}_1\partial p_2+\lambda p_1{★}_1{p}_2.\hfill \end{array}$$

(12)

Note that Equation (11) is equivalent to $\delta {★}_1=0$ and that Equation (12) is equivalent to $\delta {\partial }_1+\Gamma {★}_1=0.$ Therefore, $\mathfrak{D}({★}_1,{\partial }_1)=(\delta {★}_1,\delta {\partial }_1+\Gamma {★}_1)=0$, that is, $({★}_1,{\partial }_1)$ is a 2-cocycle. □

Definition 8.

Let $(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t=★+t{★}_1,{\partial }_t=\partial +t{\partial }_1)$ and $(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t^{\prime }=★+t{★}_1^{\prime },{\partial }_t^{\prime }=\partial +t{\partial }_1^{\prime })$ be two linear deformations of MλDLSA $(\mathfrak{p},★,\partial )$. We call them equivalent if there exists ${\Phi }_1:\mathfrak{p}\to \mathfrak{p}$ such that ${\Phi }_t={\mathrm{id}}_{\mathfrak{p}}+t{\Phi }_1$ is a homomorphism from $(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t^{\prime },{\partial }_t^{\prime })$ to $(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t,{\partial }_t)$, i.e., for all $p_1,p_1\in \mathfrak{p}$, the following equations hold:

$$\begin{array}{cc}\hfill {\Phi }_t\left(p_1{★}_t^{\prime }{p}_2\right)=& {\Phi }_t\left(p_1\right){★}_t{\Phi }_t\left(p_2\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\Phi }_t\left({\partial }_t^{\prime }{p}_1\right)=& {\partial }_t{\Phi }_t\left(p_1\right).\hfill \end{array}$$

(14)

Proposition 3.

If two linear deformations $(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t=★+t{★}_1,{\partial }_t=\partial +t{\partial }_1)$ and $\left(\mathfrak{p}\right[\left[t\right]]/$$\left(t^2\right),{★}_t^{\prime }=★+t{★}_1^{\prime },{\partial }_t^{\prime }=\partial +t{\partial }_1^{\prime })$ are equivalent, then $({★}_1,{\partial }_1)$ and $({★}_1^{\prime },{\partial }_1^{\prime })$ are in the same cohomology class of ${\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^2\left(\mathfrak{p}\right)$.

Proof. 

Let ${\Phi }_t:(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t^{\prime },{\partial }_t^{\prime })\to (\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t,{\partial }_t)$ be an isomorphism. Expanding the equations and collecting coefficients of t, we get from Equations (13) and (14):

$$\begin{array}{cc}\hfill p_1{★}_1^{\prime }{p}_2-p_1{★}_1{p}_2={\Phi }_1\left(p_1\right)★p_2+p_1★{\Phi }_1\left(p_2\right)-{\Phi }_1(p_1★p_2)=& \delta {\Phi }_1(p_1,p_2),\hfill \\ \hfill {\partial }_1^{\prime }{p}_1-{\partial }_1{p}_1=\partial {\Phi }_1\left(p_1\right)-{\Phi }_1(\partial p_1)=& -\Gamma {\Phi }_1\left(p_1\right),\hfill \end{array}$$

that is, $({★}_1^{\prime },{\partial }_1^{\prime })-({★}_1,{\partial }_1)=(\delta {\Phi }_1,-\Gamma {\Phi }_1)=\mathfrak{D}\left({\Phi }_1\right)\in {\mathcal{B}}_{\mathrm{M}\cup \mathrm{DLSA}}^2\left(\mathfrak{p}\right).$ So, $({★}_1,{\partial }_1)$ and $({★}_1^{\prime },{\partial }_1^{\prime })$ are in the same cohomology class of ${\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^2\left(\mathfrak{p}\right)$. □

Remark 3.

If $(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t,{\partial }_t)$ is equivalent to the undeformed deformation $(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),★,\partial )$, we call the linear deformation $(\mathfrak{p}\left[\left[t\right]\right]/\left(t^2\right),{★}_t=★+t{★}_1,{\partial }_t=\partial +t{\partial }_1)$ of an MλDLSA $(\mathfrak{p},★,\partial )$ trivial.

4. Abelian Extensions of Modified $\lambda$-Differential Left-Symmetric Algebras

This section mainly studies the abelian extensions of an M$\lambda$DLSA.

Definition 9.

Let $(\mathfrak{p},★,\partial )$ be an MλDLSA and $(\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})$ an abelian MλDLSA with the trivial product ${★}_{\mathcal{V}}$. An abelian extension $(\widehat{\mathfrak{p}},\widehat{★},\widehat{\partial })$ of $(\mathfrak{p},★,\partial )$ by $(\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})$ is a short exact sequence of morphisms of MλDLSAs:

$$\begin{array}{c}\hfill \begin{array}{ccccccccc}0& \to & (\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})& \stackrel{\mathrm{i}}{\to }& (\widehat{\mathfrak{p}},\widehat{★},\widehat{\partial })& \stackrel{\mathrm{p}}{\to }& (\mathfrak{p},★,\partial )& \to & 0,\end{array}\end{array}$$

(15)

that is, there exists a commutative diagram:

$$\begin{array}{c}\hfill \begin{array}{ccccccccc}0& \to & \mathcal{V}& \stackrel{\mathrm{i}}{\to }& \widehat{\mathfrak{p}}& \stackrel{\mathrm{p}}{\to }& \mathfrak{p}& \to & 0\\ & & {\partial }_{\mathcal{V}}\downarrow \phantom{{\partial }_{\mathcal{V}}}& & \widehat{\partial }\downarrow \phantom{\widehat{\partial }}& & \partial \downarrow \phantom{\partial }& & & & \\ 0& \to & \mathcal{V}& \stackrel{\mathrm{i}}{\to }& \widehat{\mathfrak{p}}& \stackrel{\mathrm{p}}{\to }& \mathfrak{p}& \to & 0,\end{array}\end{array}$$

such that $\widehat{\partial }u={\partial }_{V}u$ and $u\widehat{★}v=0$, for $u,v\in \mathcal{V},$ i.e., $\mathcal{V}$ is an abelian ideal of $\widehat{\mathfrak{p}}.$

A section of an abelian extension $(\widehat{\mathfrak{p}},\widehat{★},\widehat{\partial })$ of $(\mathfrak{p},★,\partial )$ by $(\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})$ is a linear map $\mathrm{s}:\mathfrak{p}\to \widehat{\mathfrak{p}}$ such that $\mathrm{p}\circ \mathrm{s}={\mathrm{id}}_{\mathfrak{p}}$.

Definition 10.

Let $({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)$ and $({\widehat{\mathfrak{p}}}_2,{\widehat{★}}_2,{\widehat{\partial }}_2)$ be two abelian extensions of $(\mathfrak{p},★,\partial )$ by $(\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})$. They are said to be isomorphic if there exists an MλDLSA isomorphism $\Phi :({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)\to ({\widehat{\mathfrak{p}}}_2,{\widehat{★}}_2,{\widehat{\partial }}_2)$, such that the following diagram is commutative:

$$\begin{array}{c}\hfill \begin{array}{ccccccccc}0& \to & (\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})& \stackrel{{\mathrm{i}}_1}{\to }& ({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)& \stackrel{{\mathrm{p}}_1}{\to }& (\mathfrak{p},★,\partial )& \to & 0\\ & & {\mathrm{id}}_{\mathcal{V}}\downarrow \phantom{{\mathrm{id}}_{\mathcal{V}}}& & \Phi \downarrow \phantom{\Phi }& & {\mathrm{id}}_{\mathfrak{p}}\downarrow \phantom{{\mathrm{id}}_{\mathfrak{p}}}& & & & \\ 0& \to & (\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})& \stackrel{{\mathrm{i}}_2}{\to }& ({\widehat{\mathfrak{p}}}_2,{\widehat{★}}_2,{\widehat{\partial }}_2)& \stackrel{{\mathrm{p}}_2}{\to }& (\mathfrak{p},★,\partial )& \to & 0.\end{array}\end{array}$$

(16)

Let $(\widehat{\mathfrak{p}},\widehat{★},\widehat{\partial })$ be an abelian extension of an M$\lambda$DLSA $(\mathfrak{p},★,\partial )$ by $(\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})$ and $\mathrm{s}:\mathfrak{p}\to \widehat{\mathfrak{p}}$ be a section of it. For any $p\in \mathfrak{p},u\in \mathcal{V}$, define ${★}_l:\mathfrak{p}\times \mathcal{V}\to \mathcal{V}$ and ${★}_r:\mathcal{V}\times \mathfrak{p}\to \mathcal{V}$, respectively, by:

$$p{★}_{l}u=\mathrm{s}\left(p\right)\widehat{★}u,u{★}_{r}p=u\widehat{★}\mathrm{s}\left(p\right).$$

We further define linear maps $\omega :\mathfrak{p}\times \mathfrak{p}\to \mathcal{V}$ and $\chi :\mathfrak{p}\to \mathcal{V}$, respectively, by:

$$\begin{array}{cc}\hfill \omega (p_1,p_2)& =\mathrm{s}\left(p_1\right)\widehat{★}\mathrm{s}\left(p_2\right)-\mathrm{s}(p_1★p_2),\hfill \\ \hfill \chi \left(p_1\right)& =\widehat{\partial }\mathrm{s}\left(p_1\right)-\mathrm{s}(\partial p_1),\forall p_1,p_2\in \mathfrak{p}.\hfill \end{array}$$

Obviously, $\widehat{\mathfrak{p}}$ is isomorphic to $\mathfrak{p}\oplus \mathcal{V}$ as vector spaces. Transfer the M$\lambda$DLSA structure on $\widehat{\mathfrak{p}}$ to that on $\mathfrak{p}\oplus \mathcal{V}$, we obtain an M$\lambda$DLSA $(\mathfrak{p}\oplus \mathcal{V},{★}_{\omega },{\partial }_{\chi })$, where ${★}_{\omega }$ and ${\partial }_{\chi }$ are given by:

$$\begin{array}{cc}\hfill (p_1+u_1){★}_{\omega }(p_2+u_2)& =p_1★p_2+p_1{★}_l{u}_2+u_1{★}_r{p}_2+\omega (p_1,p_2),\hfill \\ \hfill {\partial }_{\chi }(p_1+u_1)& =\partial p_1+\chi \left(p_1\right)+{\partial }_{\mathcal{V}}{u}_1,\forall p_1,p_2\in \mathfrak{p},u_1,u_2\in \mathcal{V}.\hfill \end{array}$$

In addition, we have an abelian extension

$$\begin{array}{c}\hfill \begin{array}{ccccccccc}0& \to & (\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})& \stackrel{\mathrm{i}}{\to }& (\mathfrak{p}\oplus \mathcal{V},{★}_{\omega },{\partial }_{\chi })& \stackrel{\mathrm{p}}{\to }& (\mathfrak{p},★,\partial )& \to & 0,\end{array}\end{array}$$

(17)

which is isomorphic to the original abelian extension (15).

Proposition 4.

With the above notations, $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$ is a representation of the MλDLSA $(\mathfrak{p},★,\partial )$.

Proof. 

For any $p_1,p_2\in \mathfrak{p}$ and $u\in \mathcal{V}$, $\mathcal{V}$ is an abelian ideal of $\widehat{\mathfrak{p}}$ and $\mathrm{s}(p_1★p_2)-\mathrm{s}\left(p_1\right)\widehat{★}\mathrm{s}\left(p_2\right)\in \mathcal{V}$, and we have:

$$\begin{array}{cc}\hfill p_1{★}_l\left(p_2{★}_{l}u\right)-(p_1★p_2){★}_{l}u& =\mathrm{s}\left(p_1\right)\widehat{★}\left(\mathrm{s}\left(p_2\right)\widehat{★}u\right)-\mathrm{s}(p_1★p_2)\widehat{★}u\hfill \\ \hfill & =\mathrm{s}\left(p_1\right)\widehat{★}\left(\mathrm{s}\left(p_2\right)\widehat{★}u\right)-\left(\mathrm{s}\left(p_1\right)\widehat{★}\mathrm{s}\left(p_2\right)\right)\widehat{★}u\hfill \\ \hfill & =\mathrm{s}\left(p_2\right)\widehat{★}\left(\mathrm{s}\left(p_1\right)\widehat{★}u\right)-\left(\mathrm{s}\left(p_2\right)\widehat{★}\mathrm{s}\left(p_1\right)\right)\widehat{★}u\hfill \\ \hfill & =p_2{★}_l\left(p_1{★}_{l}u\right)-(p_2★p_1){★}_{l}u.\hfill \end{array}$$

It is similar to see $p_1{★}_l\left(u{★}_r{p}_2\right)-\left(p_1{★}_{l}u\right){★}_r{p}_2=u{★}_r(p_1★p_2)-\left(u{★}_r{p}_1\right){★}_r{p}_2.$ Hence, this shows that $(\mathcal{V};{★}_l,{★}_r)$ is a representation of the LSA $(\mathfrak{p},★)$.

Moreover, by $\widehat{\partial }\mathrm{s}\left(p_1\right)-\mathrm{s}(\partial p_1)\in \mathcal{V},$ we have:

$$\begin{array}{cc}\hfill {\partial }_{\mathcal{V}}\left(p_1{★}_{l}u\right)=& {\partial }_{\mathcal{V}}\left(\mathrm{s}\left(p_1\right)\widehat{★}u\right)=\widehat{\partial }\left(\mathrm{s}\left(p_1\right)\widehat{★}u\right)\hfill \\ \hfill =& \widehat{\partial }\mathrm{s}\left(p_1\right)\widehat{★}u+\mathrm{s}\left(p_1\right)\widehat{★}\widehat{\partial }u+\lambda \left(\mathrm{s}\left(p_1\right)\widehat{★}u\right)\hfill \\ \hfill =& \mathrm{s}(\partial p_1)\widehat{★}u+\mathrm{s}\left(p_1\right)\widehat{★}{\partial }_{\mathcal{V}}u+\lambda \left(\mathrm{s}\left(p_1\right)\widehat{★}u\right)\hfill \\ \hfill =& \partial p_1{★}_{l}u+p_1{★}_l{\partial }_{\mathcal{V}}u+\lambda \left(p_1{★}_{l}u\right).\hfill \end{array}$$

By the same token, ${\partial }_{\mathcal{V}}\left(u{★}_r{p}_1\right)={\partial }_{\mathcal{V}}u{★}_r{p}_1+u{★}_r\partial p_1+\lambda \left(u{★}_r{p}_1\right)$. Hence, we deduce that $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$ is a representation of $(\mathfrak{p},★,\partial )$. □

Proposition 5.

With the above notation, the pair $(\omega ,\chi )$ is a 2-cocycle of the MλDLSA $(\mathfrak{p},★,\partial )$ with coefficients in $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$.

Proof. 

By $(\mathfrak{p}\oplus \mathcal{V},{★}_{\omega },{\partial }_{\chi })$ is an M$\lambda$DLSA, for any $p_1,p_2,p_3\in \mathfrak{p}$ and $u_1,u_2,u_3\in \mathcal{V}$, we have:

$$\begin{array}{cc}\hfill & \left((p_1+u_1){★}_{\omega }(p_2+u_2)\right){★}_{\omega }(p_3+u_3)-(p_1+u_1){★}_{\omega }\left((p_2+u_2){★}_{\omega }(p_3+u_3)\right)\hfill \\ \hfill & =\left((p_2+u_2){★}_{\omega }(p_1+u_1)\right){★}_{\omega }(p_3+u_3)-(p_2+u_2){★}_{\omega }\left((p_1+u_1){★}_{\omega }(p_3+u_3)\right),\hfill \\ \hfill & {\partial }_{\chi }\left((p_1+u_1){★}_{\omega }(p_2+u_2)\right)\hfill \\ \hfill & ={\partial }_{\chi }(p_1+u_1){★}_{\omega }(p_2+u_2)+(p_1+u_1){★}_{\omega }{\partial }_{\chi }(p_2+u_2)+\lambda (p_1+u_1){★}_{\omega }(p_2+u_2).\hfill \end{array}$$

Furthermore, the above two equations are equivalent to the following equations:

$$\begin{array}{c}\omega (p_1,p_2){★}_r{p}_3+\omega (p_1★p_2,p_3)-p_1{★}_l\omega (p_2,p_3)-\omega (p_1,p_2★p_3)\hfill \\ =\omega (p_2,p_1){★}_r{p}_3+\omega (p_2★p_1,p_3)-p_2{★}_l\omega (p_1,p_3)-\omega (p_2,p_1★p_3),\hfill \end{array}$$

(18)

$$\begin{array}{c}\chi (p_1★p_2)+{\partial }_{\mathcal{V}}\omega (p_1,p_2)\hfill \\ =\chi \left(p_1\right){★}_r{p}_2+\omega (\partial p_1,p_2)+p_1{★}_l\chi \left(p_2\right)+\omega (p_1,\partial p_2)+\lambda \omega (p_1,p_2).\hfill \end{array}$$

(19)

Using Equations (18) and (19), we have $\delta \omega =0$ and ${\delta }_M\chi +\Gamma \omega =0$, respectively. Therefore, $\mathfrak{D}(\omega ,\chi )=(\delta \omega ,\delta \chi +\Gamma \omega )=0,$ that is, $(\omega ,\chi )$ is a 2-cocycle. □

Let us now study the influence of different choices of sections.

Proposition 6.

Let $(\widehat{\mathfrak{p}},\widehat{★},\widehat{\partial })$ be an abelian extension of an MλDLSA $(\mathfrak{p},★,\partial )$ by $(\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})$ and $\mathrm{s}:\mathfrak{p}\to \widehat{\mathfrak{p}}$ be a section of it.

(i) Different choices of the section $\mathrm{s}$ give the same representation on $(\mathcal{V},{\partial }_{\mathcal{V}})$. Moreover, isomorphic abelian extensions give rise to the same representation of $(\mathfrak{p},★,\partial )$.

(ii) The cohomology class $(\omega ,\chi )$ of does not depend on the choice of $\mathrm{s}$.

Proof. 

(i) Let ${\mathrm{s}}^{\prime }:\mathfrak{p}\to \widehat{\mathfrak{p}}$ be another section of $(\widehat{\mathfrak{p}},\widehat{★},\widehat{\partial })$ and $(\mathcal{V};{★}_l^{\prime },{★}_r^{\prime },{\partial }_{\mathcal{V}})$ be another representation of $(\mathfrak{p},★,\partial )$ constructed using the section ${\mathrm{s}}^{\prime }$. By $\mathrm{s}\left(p_1\right)-{\mathrm{s}}^{\prime }\left(p_1\right)\in \mathcal{V}$ for $p_1\in \mathfrak{p}$, then we have:

$$p_1{★}_{l}u-p_1{★}_l^{\prime }u=\mathrm{s}\left(p_1\right)\widehat{★}u-{\mathrm{s}}^{\prime }\left(p_1\right)\widehat{★}u=(\mathrm{s}\left(p_1\right)-{\mathrm{s}}^{\prime }\left(p_1\right))\widehat{★}u=0,$$

which implies that ${★}_l={★}_l^{\prime }$. Similarly, there is also ${★}_r={★}_r^{\prime }$. Thus, different choices of the section $\mathrm{s}$ give the same representation on $(\mathcal{V},{\partial }_{\mathcal{V}})$.

Moreover, let $({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)$ and $({\widehat{\mathfrak{p}}}_2,{\widehat{★}}_2,{\widehat{\partial }}_2)$ be two isomorphic abelian extensions of $(\mathfrak{p},★,\partial )$ by $(\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})$ with the associated isomorphism $\Phi :({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)\to ({\widehat{\mathfrak{p}}}_2,{\widehat{★}}_2,{\widehat{\partial }}_2)$ such that the diagram in (16) is commutative. Let ${\mathrm{s}}_1:\mathfrak{p}\to {\widehat{\mathfrak{p}}}_1$ and ${\mathrm{s}}_2:\mathfrak{p}\to {\widehat{\mathfrak{p}}}_2$ be two sections of $({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)$ and $({\widehat{\mathfrak{p}}}_2,{\widehat{★}}_2,{\widehat{\partial }}_2)$, respectively. By Proposition 4, we have $(\mathcal{V};{★}_l^1,{★}_r^1,{\partial }_{\mathcal{V}})$ and $(\mathcal{V};{★}_l^2,{★}_r^2,{\partial }_{\mathcal{V}})$, which are their representations, respectively. Define ${\mathrm{s}}_1^{\prime }:\mathfrak{p}\to {\widehat{\mathfrak{p}}}_1$ by ${\mathrm{s}}_1^{\prime }={\Phi }^{-1}\circ {\mathrm{s}}_2$. As ${\mathrm{p}}_2\circ \Phi ={\mathrm{p}}_1$, we have:

$${\mathrm{p}}_1\circ {\mathrm{s}}_1^{\prime }=({\mathrm{p}}_2\circ \Phi )\circ ({\Phi }^{-1}\circ {\mathrm{s}}_2)={\mathrm{id}}_{\mathfrak{p}}.$$

Thus, we obtain that ${\mathrm{s}}_1^{\prime }$ is a section of $({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)$. By $\Phi$ is an isomorphism of M$\lambda$DLSAs such that ${\Phi |}_{\mathcal{V}}={\mathrm{id}}_{\mathcal{V}}$, for any $p\in \mathfrak{p}$ and $u\in \mathcal{V}$, we have:

$$p{★}_l^{1}u={\mathrm{s}}_1^{\prime }\left(p\right){\widehat{★}}_{1}u={\Phi }^{-1}\circ {\mathrm{s}}_2\left(p\right){\widehat{★}}_{1}u={\Phi }^{-1}\left({\mathrm{s}}_2\left(p\right){\widehat{★}}_{2}u\right)=p{★}_l^{2}u,$$

which implies that ${★}_l^1={★}_l^2$. Similarly, there is also ${★}_r^1={★}_r^2$. Thus, isomorphic abelian extensions give rise to the same representation of $(\mathfrak{p},★,\partial )$.

(ii) Let ${\mathrm{s}}^{\prime }:\mathfrak{p}\to \widehat{\mathfrak{p}}$ be another section of $(\widehat{\mathfrak{p}},\widehat{★},\widehat{\partial })$, by Proposition 5, we get another corresponding 2-cocycle $({\omega }^{\prime },{\chi }^{\prime })$. Define $\tau :\mathfrak{p}\to \mathcal{V}$ by $\tau \left(p_1\right)=\mathrm{s}\left(p_1\right)-{\mathrm{s}}^{\prime }\left(p_1\right)$, for any $p_1,p_2\in \mathfrak{p}$, we have:

$$\begin{array}{cc}\hfill \omega (p_1,p_2)& =\mathrm{s}\left(p_1\right)\widehat{★}\mathrm{s}\left(p_2\right)-\mathrm{s}(p_1★p_2)\hfill \\ \hfill & =({\mathrm{s}}^{\prime }\left(p_1\right)+\tau \left(p_1\right))\widehat{★}({\mathrm{s}}^{\prime }\left(p_2\right)+\tau \left(p_2\right))-({\mathrm{s}}^{\prime }(p_1★p_2)+\tau (p_1★p_2))\hfill \\ \hfill & ={\mathrm{s}}^{\prime }\left(p_1\right)\widehat{★}{\mathrm{s}}^{\prime }\left(p_2\right)+{\mathrm{s}}^{\prime }\left(p_1\right)\widehat{★}\tau \left(p_2\right)+\tau \left(p_1\right)\widehat{★}{\mathrm{s}}^{\prime }\left(p_2\right)+\tau \left(p_1\right)\widehat{★}\tau \left(p_2\right)-{\mathrm{s}}^{\prime }(p_1★p_2)-\tau (p_1★p_2)\hfill \\ \hfill & ={\mathrm{s}}^{\prime }\left(p_1\right)\widehat{★}{\mathrm{s}}^{\prime }\left(p_2\right)-{\mathrm{s}}^{\prime }(p_1★p_2)+p_1{★}_l\tau \left(p_2\right)+\tau \left(p_1\right){★}_r{p}_2-\tau (p_1★p_2)\hfill \\ \hfill & ={\omega }^{\prime }(p_1,p_2)+\delta \tau (p_1,p_2),\hfill \\ \hfill \chi \left(p_1\right)& =\widehat{\partial }\mathrm{s}\left(p_1\right)-\mathrm{s}(\partial p_1)\hfill \\ \hfill & =\widehat{\partial }({\mathrm{s}}^{\prime }\left(p_1\right)+\tau \left(p_1\right))-({\mathrm{s}}^{\prime }(\partial p_1)+\tau (\partial p_1))\hfill \\ \hfill & =\widehat{\partial }{\mathrm{s}}^{\prime }\left(p_1\right)+\widehat{\partial }\tau \left(p_1\right)-{\mathrm{s}}^{\prime }(\partial p_1)-\tau (\partial p_1)\hfill \\ \hfill & =\widehat{\partial }{\mathrm{s}}^{\prime }\left(p_1\right)-{\mathrm{s}}^{\prime }(\partial p_1)+{\partial }_{\mathcal{V}}\tau \left(p_1\right)-\tau (\partial p_1)\hfill \\ \hfill & ={\chi }^{\prime }\left(p_1\right)-\Gamma \tau \left(p_1\right).\hfill \end{array}$$

Hence, $(\omega ,\chi )-({\omega }^{\prime },{\chi }^{\prime })=(\delta \tau ,-\Gamma \tau )=\mathfrak{D}\left(\tau \right)\in {\mathcal{B}}_{\mathrm{M}\cup \mathrm{DLSA}}^2(\mathfrak{p},\mathcal{V})$, that is $(\omega ,\chi )$ and $({\omega }^{\prime },{\chi }^{\prime })$ are in the same cohomological class in ${\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^2(\mathfrak{p},\mathcal{V})$. □

Next, we are ready to classify abelian extensions of an M$\lambda$DLSA.

Theorem 1.

Abelian extensions of an MλDLSA $(\mathfrak{p},★,\partial )$ by $(\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})$ are classified by the second cohomology group ${\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^2(\mathfrak{p},\mathcal{V})$.

Proof. 

Assume that $({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)$ and $({\widehat{\mathfrak{p}}}_2,{\widehat{★}}_2,{\widehat{\partial }}_2)$ are two isomorphic abelian extensions of $(\mathfrak{p},★,\partial )$ by $(\mathcal{V},{★}_{\mathcal{V}},{\partial }_{\mathcal{V}})$ with the associated isomorphism $\Phi :({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)\to ({\widehat{\mathfrak{p}}}_2,{\widehat{★}}_2,{\widehat{\partial }}_2)$ such that the diagram in (16) is commutative. Let ${\mathrm{s}}_1$ be a section of $({\widehat{\mathfrak{p}}}_1,{\widehat{★}}_1,{\widehat{\partial }}_1)$. As ${\mathrm{p}}_2\circ \Phi ={\mathrm{p}}_1$, we have:

$${\mathrm{p}}_2\circ (\Phi \circ {\mathrm{s}}_1)={\mathrm{p}}_1\circ {\mathrm{s}}_1={\mathrm{id}}_{\mathfrak{p}}.$$

Thus, we obtain that $\Phi \circ {\mathrm{s}}_1$ is a section of $({\widehat{\mathfrak{p}}}_2,{\widehat{★}}_2,{\widehat{\partial }}_2)$. Denote ${\mathrm{s}}_2:=\Phi \circ {\mathrm{s}}_1$. Since $\Phi$ is an isomorphism of M$\lambda$DLSAs such that ${\Phi |}_{\mathcal{V}}={\mathrm{id}}_{\mathcal{V}}$, we have:

$$\begin{array}{cc}\hfill {\omega }_2(p_1,p_2)& ={\mathrm{s}}_2\left(p_1\right){\widehat{★}}_2{\mathrm{s}}_2\left(p_2\right)-{\mathrm{s}}_2(p_1★p_2)\hfill \\ \hfill & =\Phi \circ {\mathrm{s}}_1\left(p_1\right){\widehat{★}}_2\Phi \circ {\mathrm{s}}_1\left(p_2\right)-\Phi \circ {\mathrm{s}}_1(p_1★p_2)\hfill \\ \hfill & =\Phi \left({\mathrm{s}}_1\left(p_1\right){\widehat{★}}_1{\mathrm{s}}_1\left(p_2\right)-{\mathrm{s}}_1(p_1★p_2)\right)\hfill \\ \hfill & =\Phi \left({\omega }_1(p_1,p_2)\right)\hfill \\ \hfill & ={\omega }_1(p_1,p_2)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\chi }_2\left(p_1\right)& ={\widehat{\partial }}_2{\mathrm{s}}_2\left(p_1\right)-{\mathrm{s}}_2(\partial p_1)\hfill \\ \hfill & ={\widehat{\partial }}_2(\Phi \circ {\mathrm{s}}_1\left(p_1\right))-\Phi \circ {\mathrm{s}}_1(\partial p_1)\hfill \\ \hfill & =\Phi ({\widehat{\partial }}_1{\mathrm{s}}_1\left(p_1\right)-{\mathrm{s}}_1(\partial p_1))\hfill \\ \hfill & ={\chi }_1\left(p_1\right).\hfill \end{array}$$

Thus, isomorphic abelian extensions gives rise to the same element in ${\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^2(\mathfrak{p},\mathcal{V})$.

Conversely, given two 2-cocycles $({\omega }_1,{\chi }_1)$ and $({\omega }_2,{\chi }_2)$, we can construct two abelian extensions $(\mathfrak{p}\oplus \mathcal{V},{★}_{{\omega }_1},{\partial }_{{\chi }_1})$ and $(\mathfrak{p}\oplus \mathcal{V},{★}_{{\omega }_2},{\partial }_{{\chi }_2})$ via (17). If they represent the same cohomology class in ${\mathcal{H}}_{\mathrm{M}\cup \mathrm{DLSA}}^2(\mathfrak{p},\mathcal{V})$, then there exists $\tau :\mathfrak{p}\to \mathcal{V}$ such that:

$$({\omega }_1,{\chi }_1)-({\omega }_2,{\chi }_2)=\mathfrak{D}\left(\tau \right).$$

We define ${\Phi }_{\tau }:\mathfrak{p}\oplus \mathcal{V}\to \mathfrak{p}\oplus \mathcal{V}$ by ${\Phi }_{\tau }(p_1+u):=p_1+\tau \left(p_1\right)+u$ for all $p_1\in \mathfrak{p},u\in \mathcal{V}.$ Then, it is easy to verify that ${\Phi }_{\tau }$ is an isomorphism of these two abelian extensions $(\mathfrak{p}\oplus \mathcal{V},{★}_{{\omega }_1},{\partial }_{{\chi }_1})$ and $(\mathfrak{p}\oplus \mathcal{V},{★}_{{\omega }_2},{\partial }_{{\chi }_2})$ such that the diagram in (16) is commutative. □

5. Skeletal Modified $\lambda$-Differential Left-Symmetric Algebras and Crossed Modules

In this section, first we classify skeletal M$\lambda$D left-symmetric 2-algebras via the third cohomology group. Then, we introduce the notion of a crossed module of M$\lambda$DLSAs, and show that they are equivalent to strict M$\lambda$D left-symmetric 2-algebras.

We first recall the definition of left-symmetric 2-algebras from [31], which is the categorization of a LSA.

A left-symmetric 2-algebra is a quintuple $({\mathfrak{p}}_0,{\mathfrak{p}}_1,d,l_2,l_3)$, where $d:{\mathfrak{p}}_1\to {\mathfrak{p}}_0$ is a linear map, $l_2:{\mathfrak{p}}_i\times {\mathfrak{p}}_j\to {\mathfrak{p}}_{i+j}$ are bilinear maps, and $l_3:{\mathfrak{p}}_0\times {\mathfrak{p}}_0\times {\mathfrak{p}}_0\to {\mathfrak{p}}_1$ is a trilinear map, such that for any $p_1,p_2,p_3,p_4\in {\mathfrak{p}}_0$ and $u,v\in {\mathfrak{p}}_1$, the following equations are satisfied:

$$\begin{array}{}\mathrm{(20)}& d{l}_2(p_1,u)=l_2(p_1,d\left(u\right)),\hfill \mathrm{(21)}& d{l}_2(u,p_1)=l_2(d\left(u\right),p_1),\hfill \mathrm{(22)}& l_2(d\left(u\right),v)=l_2(u,d\left(v\right)),\hfill \mathrm{(23)}& d{l}_3(p_1,p_2,p_3)=l_2(p_1,l_2(p_2,p_3))-l_2(l_2(p_1,p_2),p_3)-l_2(p_2,l_2(p_1,p_3))+l_2(l_2(p_2,p_1),p_3),\hfill \mathrm{(24)}& l_3(p_1,p_2,d\left(u\right))=l_2(p_1,l_2(p_2,u))-l_2(l_2(p_1,p_2),u)-l_2(p_2,l_2(p_1,u))+l_2(l_2(p_2,p_1),u),\hfill \mathrm{(25)}& l_3(d\left(u\right),p_2,p_3)=l_2(u,l_2(p_2,p_3))-l_2(l_2(u,p_2),p_3)-l_2(p_2,l_2(u,p_3))+l_2(l_2(p_2,u),p_3),\hfill \mathrm{(26)}& \begin{array}{c}{l}_2(p_1,l_3(p_2,p_3,p_4))-l_2(p_2,l_3(p_1,p_3,p_4))+l_2(p_3,l_3(p_1,p_2,p_4))+l_2(l_3(p_2,p_3,p_1),p_4)\hfill \\ -l_2(l_3(p_1,p_3,p_2),p_4)+l_2(l_3(p_1,p_2,p_3),p_4)-l_3(p_2,p_3,l_2(p_1,p_4))+l_3(p_1,p_3,l_2(p_2,p_4))\hfill \\ -l_3(p_1,p_2,l_2(p_3,p_4))-l_3(l_2(p_1,p_2)-l_2(p_2,p_1),p_3,p_4)+l_3(l_2(p_1,p_3)-l_2(p_3,p_1),p_2,p_4)\hfill \\ -l_3(l_2(p_2,p_3)-l_2(p_3,p_2),p_1,p_4)=0.\hfill \end{array}\hfill \end{array}$$

Motivated by [31,32], we propose the concept of an M$\lambda$D left-symmetric 2-algebra.

Definition 11.

An MλD left-symmetric 2-algebra consists of a left-symmetric 2-algebra $\mathcal{P}=({\mathfrak{p}}_0,{\mathfrak{p}}_1,d,l_2,l_3)$ and an MλD 2-operator $\tilde{\partial }=({\partial }_0,{\partial }_1,{\partial }_2)$ on $\mathcal{P}$, where ${\partial }_0:{\mathfrak{p}}_0\to {\mathfrak{p}}_0$, ${\partial }_1:{\mathfrak{p}}_1\to {\mathfrak{p}}_1$ and ${\partial }_2:{\mathfrak{p}}_0\times {\mathfrak{p}}_0\to {\mathfrak{p}}_1$, for any $p_1,p_2,p_3\in {\mathfrak{p}}_0,u\in {\mathfrak{p}}_1$, satisfying the following equations:

$$\begin{array}{}\mathrm{(27)}& {\partial }_0\circ d=d\circ {\partial }_1,\hfill \mathrm{(28)}& d{\partial }_2(p_1,p_2)+{\partial }_0{l}_2(p_1,p_2)=l_2({\partial }_0{p}_1,p_2)+l_2(p_1,{\partial }_0{p}_2)+\lambda l_2(p_1,p_2),\hfill \mathrm{(29)}& {\partial }_2(p_1,d\left(u\right))+{\partial }_1{l}_2(p_1,u)=l_2({\partial }_0{p}_1,u)+l_2(p_1,{\partial }_{1}u)+\lambda l_2(p_1,u),\hfill \mathrm{(30)}& {\partial }_2(d\left(u\right),p_2)+{\partial }_1{l}_2(u,p_2)=l_2({\partial }_{1}u,p_2)+l_2(u,{\partial }_0{p}_2)+\lambda l_2(u,p_2),\hfill \mathrm{(31)}& \begin{array}{c}{l}_2(p_1,{\partial }_2(p_2,p_3))-l_2(p_2,{\partial }_2(p_1,p_3))+l_2({\partial }_2(p_2,p_1),p_3)-l_2({\partial }_2(p_1,p_2),p_3)\hfill \\ -{\partial }_2(p_2,l_2(p_1,p_3))+{\partial }_2(p_1,l_2(p_2,p_3))-{\partial }_2(l_2(p_1,p_2)-l_2(p_2,p_1),p_3)-l_3({\partial }_0{p}_1,p_2,p_3)\hfill \\ -l_3(p_1,{\partial }_0{p}_2,p_3)-l_3(p_1,p_2,{\partial }_0{p}_3)-2\lambda l_3(p_1,p_2,p_3)+{\partial }_1{l}_3(p_1,p_2,p_3)=0.\hfill \end{array}\hfill \end{array}$$

We denote an MλD left-symmetric 2-algebra by $(\mathcal{P},\tilde{\partial })$.

An M$\lambda$D left-symmetric 2-algebra is said to be skeletal (resp. strict) if $d=0$ (resp. $l_3=0,{\partial }_2=0$).

Example 6.

For any MλDLSA $(\mathfrak{p},★,\partial )$, $({\mathfrak{p}}_0={\mathfrak{p}}_1=\mathfrak{p},d=0,l_2=★,{\partial }_0={\partial }_1=\partial )$ is a strict MλD left-symmetric 2-algebra.

Proposition 7.

Let $(\mathcal{P},\tilde{\partial })$ be an MλD left-symmetric 2-algebra.

(i) If $(\mathcal{P},\tilde{\partial })$ is skeletal or strict, then $({\mathfrak{p}}_0,{★}_0,{\partial }_0)$ is an MλDLSA, where $p_1{★}_0{p}_2=l_2(p_1,p_2)$ for $p_1,p_2\in {\mathfrak{p}}_0$.

(ii) If $(\mathcal{P},\tilde{\partial })$ is strict, then $({\mathfrak{p}}_1,{★}_1,{\partial }_1)$ is an MλDLSA, where $u{★}_{1}v=l_2(d\left(u\right),v)=l_2(u,d\left(v\right))$ for $u,v\in {\mathfrak{p}}_1$.

(iii) If $(\mathcal{P},\tilde{\partial })$ is skeletal or strict, then $({\mathfrak{p}}_1;{★}_l,{★}_r,{\partial }_1)$ is a representation of $({\mathfrak{p}}_0,{★}_0,{\partial }_0)$, where $p_1{★}_{l}u=l_2(p_1,u)$ and $u{★}_r{p}_1=l_2(u,p_1)$ for $p_1\in {\mathfrak{p}}_0,u\in {\mathfrak{p}}_1$.

Proof. 

From Equations (20)–(25) and (27)–(30), (i), (ii), and (iii) can be obtained by direct verification. □

Theorem 2.

There is a one-to-one correspondence between skeletal MλD left-symmetric 2-algebras and 3-cocycles of MλDLSAs.

Proof. 

Let $(\mathcal{P},\tilde{\partial })$ be an M$\lambda$D left-symmetric 2-algebra. By Proposition 7, we can consider the cohomology of M$\lambda$DLSA $({\mathfrak{p}}_0,{★}_0,{\partial }_0)$ with coefficients in the representation $({\mathfrak{p}}_1;{★}_l,{★}_r,{\partial }_1)$. For any $p_1,p_2,p_3,p_4\in {\mathfrak{p}}_0$, by Equation (26), we have:

$$\begin{array}{cc}\hfill & \delta l_3(p_1,p_2,p_3,p_4)\hfill \\ \hfill =& l_2(p_1,l_3(p_2,p_3,p_4))-l_2(p_2,l_3(p_1,p_3,p_4))+l_2(p_3,l_3(p_1,p_2,p_4))+l_2(l_3(p_2,p_3,p_1),p_4)\hfill \\ \hfill & -l_2(l_3(p_1,p_3,p_2),p_4)+l_2(l_3(p_1,p_2,p_3),p_4)-l_3(p_2,p_3,l_2(p_1,p_4))+l_3(p_1,p_3,l_2(p_2,p_4))\hfill \\ \hfill & -l_3(p_1,p_2,l_2(p_3,p_4))-l_3(l_2(p_1,p_2)-l_2(p_2,p_1),p_3,p_4)+l_3(l_2(p_1,p_3)-l_2(p_3,p_1),p_2,p_4)\hfill \\ \hfill & -l_3(l_2(p_2,p_3)-l_2(p_3,p_2),p_1,p_4)\hfill \\ \hfill =& 0.\hfill \end{array}$$

By Equations (8) and (31), it holds that

$$\begin{array}{cc}\hfill & (\delta {\partial }_2-\Gamma l_3)(p_1,p_2,p_3)=\delta {\partial }_2(p_1,p_2,p_3)-\Gamma l_3(p_1,p_2,p_3)\hfill \\ \hfill =& p_1{★}_l{\partial }_2(p_2,p_3)-p_2{★}_l{\partial }_2(p_1,p_3)+{\partial }_2(p_2,p_1){★}_r{p}_3-{\partial }_2(p_1,p_2){★}_r{p}_3\hfill \\ \hfill & -{\partial }_2(p_2,p_1★p_3)+{\partial }_2(p_1,p_2★p_3)-{\partial }_2(p_1★p_2-p_2★p_1,p_3)+l_3({\partial }_0{p}_1,p_2,p_3)\hfill \\ \hfill & +l_3(p_1,{\partial }_0{p}_2,p_3)+l_3(p_1,p_2,{\partial }_0{p}_3)+2\lambda l_3(p_1,p_2,p_3)-{\partial }_1{l}_3(p_1,p_2,p_3)\hfill \\ \hfill =& l_2(p_1,{\partial }_2(p_2,p_3))-l_2(p_2,{\partial }_2(p_1,p_3))+l_2({\partial }_2(p_2,p_1),p_3)-l_2({\partial }_2(p_1,p_2),p_3)\hfill \\ \hfill & -{\partial }_2(p_2,l_2(p_1,p_3))+{\partial }_2(p_1,l_2(p_2,p_3))-{\partial }_2(l_2(p_1,p_2)-l_2(p_2,p_1),p_3)-l_3({\partial }_0{p}_1,p_2,p_3)\hfill \\ \hfill & -l_3(p_1,{\partial }_0{p}_2,p_3)-l_3(p_1,p_2,{\partial }_0{p}_3)-2\lambda l_3(p_1,p_2,p_3)+{\partial }_1{l}_3(p_1,p_2,p_3)\hfill \\ \hfill =& 0.\hfill \end{array}$$

Thus, $\mathfrak{D}(l_3,{\partial }_2)=(\delta l_3,\delta {\partial }_2-\Gamma l_3)=0,$ which implies that $(l_3,{\partial }_2)\in {\mathcal{C}}_{\mathrm{M}\lambda \mathrm{DLSA}}^3({\mathfrak{p}}_0,{\mathfrak{p}}_1)$ is a 3-cocycle of M$\lambda$DLSA $({\mathfrak{p}}_0,{★}_0,{\partial }_0)$ with coefficients in the representation $({\mathfrak{p}}_1;{★}_l,{★}_r,{\partial }_1)$.

Conversely, assume that $(l_3,{\partial }_2)\in {\mathcal{C}}_{\mathrm{M}\cup \mathrm{DLSA}}^3(\mathfrak{p},\mathcal{V})$ is a 3-cocycle of M$\lambda$DLSA $(\mathfrak{p},★,\partial )$ with coefficients in the representation $(\mathcal{V};{★}_l,{★}_r,{\partial }_{\mathcal{V}})$. Then, $(\mathcal{P},\tilde{\partial })$ is a skeletal M$\lambda$D left-symmetric 2-algebra, where $\mathcal{P}=({\mathfrak{p}}_0=\mathfrak{p},{\mathfrak{p}}_1=\mathcal{V},d=0,l_2,l_3)$ and $\tilde{\partial }=({\partial }_0=\partial ,{\partial }_1={\partial }_{\mathcal{V}},{\partial }_2)$ with $l_2(p_1,p_2)=p_1★p_2,l_2(p_1,u)=p_1{★}_{l}u,l_2(u,p_1)=u{★}_r{p}_1$ for any $p_1,p_2\in {\mathfrak{p}}_0,$$u\in {\mathfrak{p}}_1$. □

Next we introduce the concept of crossed modules of M$\lambda$DLSAs, which are equivalent to strict M$\lambda$D left-symmetric 2-algebras.

Definition 12.

A crossed module of MλDLSAs is a quadruple $\left(({\mathfrak{p}}_0,{★}_0,{\partial }_0),({\mathfrak{p}}_1,{★}_1,{\partial }_1),d,({★}_l,{★}_r)\right)$, where $({\mathfrak{p}}_0,{★}_0,{\partial }_0)$ and $({\mathfrak{p}}_1,{★}_1,{\partial }_1)$ are MλDLSAs, $d:{\mathfrak{p}}_1\to {\mathfrak{p}}_0$ is a homomorphism of MλDLSAs and $({\mathfrak{p}}_1,{★}_l,{★}_r,{\partial }_1)$ is a representation of $({\mathfrak{p}}_0,{★}_0,{\partial }_0)$, for any $p\in {\mathfrak{p}}_0,u,v\in {\mathfrak{p}}_1$, satisfying the following equations:

$$\begin{array}{}\mathrm{(32)}& d\left(p{★}_{l}u\right)=p{★}_{0}d\left(u\right),d\left(u{★}_{r}p\right)=d\left(u\right){★}_{0}p,\hfill \mathrm{(33)}& d\left(u\right){★}_{l}v=u{★}_{r}d\left(v\right)=u{★}_{1}v.\hfill \end{array}$$

Theorem 3.

There is a one-to-one correspondence between strict MλD left-symmetric 2-algebras and crossed modules of MλDLSAs.

Proof. 

Let $(\mathcal{P},\tilde{\partial })=\left(({\mathfrak{p}}_0,{\mathfrak{p}}_1,d,l_2,l_3=0),({\partial }_0,{\partial }_1,{\partial }_2=0)\right)$ be a strict M$\lambda$D left-symmetric 2-algebra. By Proposition 7, we may construct a crossed module of M$\lambda$DLSA $\left(\right({\mathfrak{p}}_0,{★}_0,{\partial }_0),$$({\mathfrak{p}}_1,{★}_1,{\partial }_1),d,({★}_l,{★}_r))$, where, $p_1{★}_0{p}_2=l_2(p_1,p_2),u_1{★}_1{u}_2=l_2(d\left(u_1\right),u_2)=l_2(u_1,d\left(u_2\right)),$ $p_1{★}_l{u}_1=l_2(p_1,u_1)$ and $u_1{★}_r{p}_1=l_2(u_1,p_1)$, for $p_1,p_2\in {\mathfrak{p}}_0,u_1,u_2\in {\mathfrak{p}}_1$. Based on Proposition 7, we only need to check that Equations (32) and (33) hold and d is a homomorphism of M$\lambda$DLSAs. In fact, by Equation (20), we have:

$$d\left(u_1{★}_1{u}_2\right)=d{l}_2(d\left(u_1\right),u_2)=l_2(d\left(u_1\right),d\left(u_2\right))=d\left(u_1\right){★}_{0}d\left(u_2\right),$$

combining with Equation (27), which implies that d is a homomorphism of M$\lambda$DLSAs. Furthermore, we have:

$$\begin{array}{cc}\hfill & d\left(p_1{★}_l{u}_1\right)=d{l}_2(p_1,u_1)=l_2(p_1,d\left(u_1\right))=p_1{★}_{0}d\left(u_1\right),\hfill \\ \hfill & d\left(u_1{★}_r{p}_1\right)=d{l}_2(u_1,p_1)=l_2(d\left(u_1\right),p_1)=d\left(u_1\right){★}_0{p}_1,\hfill \\ \hfill & d\left(u_1\right){★}_l{u}_2=l_2(d\left(u_1\right),u_2)=l_2(u_1,d\left(u_2\right))=u_1{★}_{r}d\left(u_2\right)=u_1{★}_1{u}_2.\hfill \end{array}$$

Thus, we obtain a crossed module of M$\lambda$DLSAs.

Conversely, a crossed module of M$\lambda$DLSA $\left(\right({\mathfrak{p}}_0,{★}_0,{\partial }_0),$$({\mathfrak{p}}_1,{★}_1,{\partial }_1),d,({★}_l,{★}_r))$ gives rise to a strict M$\lambda$D left-symmetric 2-algebra $(\mathcal{P},\tilde{\partial })$ = $\left(({\mathfrak{p}}_0,{\mathfrak{p}}_1,d,l_2,l_3=0),({\partial }_0,{\partial }_1,{\partial }_2=0)\right)$, where $l_2:{\mathfrak{p}}_i\times {\mathfrak{p}}_j\to {\mathfrak{p}}_{i+j}$ are given by:

$$l_2(p_1,p_2)=p_1{★}_0{p}_2,l_2(u_1,u_2)=u_1{★}_1{u}_2,l_2(p_1,u_1)=p_1{★}_l{u}_1,l_2(u_1,p_1)=u_1{★}_r{p}_1,$$

for all $p_1,p_2\in {\mathfrak{p}}_0,u_1,u_2\in {\mathfrak{p}}_1$. Direct verification shows that $\left(\right({\mathfrak{p}}_0,{\mathfrak{p}}_1,d,l_2,l_3=0),$$({\partial }_0,{\partial }_1,{\partial }_2=0))$ is a strict M$\lambda$D left-symmetric 2-algebra. □

6. Conclusions

In the current study, we mainly study the modified $\lambda$-differential left-symmetric algebra, which includes a left-symmetric algebra and a modified $\lambda$-differential operator. More precisely, the cohomology theory of modified $\lambda$-differential left-symmetric algebras with coefficients in a representation is proposed. The linear deformations and abelian extensions of modified $\lambda$-differential left-symmetric algebras are studied by using the second cohomology groups. Additionally, the notion of modified $\lambda$-differential left-symmetric 2-algebras is introduced, which is the categorization of a modified $\lambda$-differential left-symmetric algebra. We investigate the skeletal modified $\lambda$-differential left-symmetric 2-algebras by using the third cohomology group. Finally, we introduce the notion of a crossed module of modified $\lambda$-differential left-symmetric algebras, and prove that they are equivalent to strict modified $\lambda$-differential left-symmetric 2-algebras. It is worth noting that modified $\lambda$-differential left-symmetric algebras are a generalization of left-symmetric algebras with derivations (see Remark 2), so the conclusion in this paper can be adopted by left-symmetric algebras with derivations.