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 quantum field theory [3]. In [4,5], the authors studied associative algebras with derivations from the operadic point of view. Recently, in [6], Tang and his collaborators investigated the deformation and extension of 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 term modified r-matrix stemmed from the concept of modified classical Yang-Baxter equation, which was introduced by Semenov-Tian-Shansky [9]. Recently, Jiang and Sheng [10] developed the deformations of modified r-matrices and cohomologies of related algebraic structures. Motivated by the modified r-matrices, in [11], Peng and his collaborators introduced the concept of modified $\lambda$-differential Lie algebras. Subsequently, the algebraic structures with modified operators have been widely studied in [12,13,14,15,16,17,18].

However, there was very few study about the modified $\lambda$-differential left-symmetric algebras. Left-symmetric algebras (also called pre-Lie algebras) are nonassociative algebras, which were introduced by Cayley [19] as a kind of rooted tree algebras and also introduced by Gerstenhaber [20] when studying the deformation theory of rings and algebras. Left symmetric algebras have been widely used in geometry and physics, such as affine manifolds, affine structures on Lie groups and convex homogeneous cones [21], integrable systems, classical and quantum Yang-Baxter equations [22,23], quantum field theory, Poisson brackets, operands, complex and symplectic structures on Lie groups and Lie algebras [24]. See also [25,26,27,28,29,30,31,32,33,34] for more details. So it should be quite interesting and necessary to study the modified $\lambda$-differential left-symmetric algebras.

In this paper, we commence to study the modified $\lambda$-differential left-symmetric algebraic version, which includes a left-symmetric algebra and a modified $\lambda$-differential operator. We introduce the cohomology of modified $\lambda$-differential left-symmetric algebras with coefficients in a representation. As applications of cohomology theory, we study the linear deformations and abelian extensions of a modified $\lambda$-differential left-symmetric algebra by using the second cohomology groups. Additionally, we investigate the skeletal modified $\lambda$-differential left-symmetric 2-algebras by using the third cohomology group. Finally, we prove that strict modified $\lambda$-differential left-symmetric 2-algebras are equivalent to crossed modules of modified $\lambda$-differential left-symmetric algebras.

The paper is organized as follows. Section 2 introduces the representations of modified $\lambda$-differential left-symmetric algebras. In Section 3, we define the 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 Rota-Baxter pre-Lie 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 skeletal modified $\lambda$-differential left-symmetric 2-algebras are equivalent to the crossing modules of modified $\lambda$-differential left-symmetric algebras.

Throughout this paper, $\mathbb{K}$ denotes a field of characteristic zero. All the vector spaces and (multi)linear maps are taken over $\mathbb{K}$.

In this section, we introduce the concept of modified $\lambda$-differential left-symmetric algebra and give some examples. Next we propose the representation of modified $\lambda$-differential left-symmetric algebras.

First, let’s recall some definitions and results about left-symmetric algebras and its representations from [20,26].

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

In this section, we define the cohomology of a modified $\lambda$-differential left-symmetric algebra with coefficients in its representation.

Let us recall the cohomology theory of left-symmetric algebras in [32]. Let $(\mathfrak{p},★)$ be a left-symmetric algebra and $(\mathcal{V};{★}_l,{★}_r)$ be a representation of it. Denote the $n—$cochains of $\mathfrak{p}$ with coefficients in representation $\mathcal{V}$ 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 Then, it was proved that $\delta \circ \delta =0$. Let us denote by ${\mathcal{H}}_{\mathrm{LSA}}^{•}(\mathfrak{p},\mathcal{V})$, the cohomology group associated to the cochain complex $({\mathfrak{C}}_{\mathrm{LSA}}^{•}(\mathfrak{p},\mathcal{V}),\delta )$.

For any $n\ge 1$, we define a linear map $\Gamma :{\mathfrak{C}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V})\to {\mathfrak{C}}_{\mathrm{LSA}}^n(\mathfrak{p},\mathcal{V})$ by

It is obvious that there is a short exact sequence of cochain complexes:It induces a long exact sequence of cohomology groups:

At the end of this section, we use the established cohomology theory to characterize linear deformations of modified $\lambda$-differential left-symmetric algebras.

In this section, we study abelian extensions of a modified $\lambda$-differential left-symmetric algebra. It is proved that they are classified by the second cohomology group.

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 $\mathrm{p}\circ \mathrm{s}={\mathrm{id}}_{\mathfrak{p}}$.

Let $(\widehat{\mathfrak{p}},\widehat{★},\widehat{\partial })$ be an abelian extension of a modified $\lambda$-differential left-symmetric algebra $(\mathfrak{p},★,\partial )$ 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

We further define linear maps $\omega :\mathfrak{p}\times \mathfrak{p}\to \mathcal{V}$ and $\chi :\mathfrak{p}\to \mathcal{V}$ respectively by Obviously, $\widehat{\mathfrak{p}}$ is isomorphic to $\mathfrak{p}\oplus \mathcal{V}$ as vector spaces. Transfer the modified $\lambda$-differential left-symmetric algebra structure on $\widehat{\mathfrak{p}}$ to that on $\mathfrak{p}\oplus \mathcal{V}$, we obtain a modified $\lambda$-differential left-symmetric algebra $(\mathfrak{p}\oplus \mathcal{V},{★}_{\omega },{\partial }_{\chi })$, where ${★}_{\omega }$ and ${\partial }_{\chi }$ are given by Moreover, we get an abelian extension which is easily seen to be isomorphic to the original one (4.1).