Cohomology and Crossed Modules of Modified Rota–Baxter Pre-Lie Algebras

Abstract

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with coefficients in a suitable bimodule. Furthermore, we study the infinitesimal deformations and abelian extensions of modified Rota–Baxter pre-Lie algebras and relate them with the second cohomology groups. Finally, we investigate skeletal and strict modified Rota–Baxter pre-Lie 2-algebras. We show that skeletal modified Rota–Baxter pre-Lie 2-algebras can be classified into the third cohomology group, and strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to the crossed modules of modified Rota–Baxter pre-Lie algebras.

1. Introduction

Cayley [1] first introduced pre-Lie algebras (also called left-symmetric algebras) in the context of rooted tree algebras. Independently, Gerstenhaber [2] also introduced pre-Lie algebras in the deformation theory of rings and algebras. Pre-Lie algebras arose from the study of affine manifolds, affine structures on Lie groups and convex homogeneous cones [3], and then appeared in geometry and physics, such as integrable systems, classical and quantum Yang–Baxter equations [4,5], quantum field theory, Poisson brackets, operands, complex and symplectic structures on Lie groups and Lie algebras [6]. Also see [7,8,9,10,11,12,13,14,15,16,17,18] for some interesting related studies about pre-Lie algebras.

Rota–Baxter operators on associative algebras were first introduced by Baxter [19] in his study of probability fluctuation theory, and then it was further developed by Rota [20]. The Rota–Baxter operator has been widely used in many fields of mathematics and physics, including combinatorics, number theory, operands and quantum field theory [21]. The cohomology and deformation theory of Rota–Baxter operators of weight zero have been studied on various algebraic structures; see [22,23,24,25,26]. Recently, Wang and Zhou [27] and Das [28] studied Rota–Baxter associative algebras of any weight using different methods. Inspired by Wang and Zhou’s work, Das [29] considered the cohomology and deformations of weighted Rota–Baxter Lie algebras. The authors in [30,31] developed the cohomology, extensions and deformations of Rota–Baxter 3-Lie algebras with any weight. In [32], Chen, Lou and Sun studied the cohomology and extensions of Rota–Baxter Lie triple systems. See also [33] for weighted Rota–Baxter Lie supertriple systems.

The term modified Rota–Baxter operator stemmed from the notion of the modified classical Yang–Baxter equation, which was also introduced in the work of Semenov-Tian-Shansky [34] as a modification of the operator form of the classical Yang–Baxter equation. Recently, Jiang and Sheng the established cohomology and deformation theory of modified r-matrices in [35]. Inspired by the modified r matrix [34,35], due to the importance of pre-Lie algebras, we naturally study modified Rota–Baxter pre-Lie algebras. More precisely, we introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cochain map, $\mathrm{Y}$, and then the cohomology of modified Rota–Baxter pre-Lie algebras with coefficients in a bimodule is constructed. Finally, as applications of our proposed cohomology theory, we consider the infinitesimal deformations and abelian extensions of a modified Rota–Baxter pre-Lie algebra in terms of second cohomology groups. In addition, we further classify skeletal modified Rota–Baxter pre-Lie 2-algebras using the third cohomology group of a modified Rota–Baxter pre-Lie algebra, and show that strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to crossed modules of modified Rota–Baxter pre-Lie algebras.

This paper is organized as follows. In Section 2, we introduce the concept of modified Rota–Baxter pre-Lie algebras, and give its bimodules. In Section 3, we establish the cohomology theory of modified Rota–Baxter pre-Lie algebras with coefficients in a bimodule, and apply it to the study of infinitesimal deformation. In Section 4, we discuss an abelian extension of the modified Rota–Baxter pre-Lie algebras in terms of our second cohomology groups. Finally, in Section 5, we classify skeletal modified Rota–Baxter pre-Lie 2-algebras using the third cohomology group. Then, we introduce the notion of crossed modules of modified Rota–Baxter pre-Lie algebras, and show that strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to crossed modules of modified Rota–Baxter pre-Lie algebras.

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

2. Bimodules of Modified Rota–Baxter Pre-Lie Algebras

In this section, we introduce the notion of modified Rota–Baxter pre-Lie algebras and give some examples. Next, we propose the bimodule of modified Rota–Baxter pre-Lie algebras. Finally, we establish a new modified Rota–Baxter pre-Lie algebra and give its bimodule.

First, let us recall some definitions and results of pre-Lie algebra and its bimodules from [2,8].

Definition 1 

([2]). A pre-Lie algebra is a pair $(\mathcal{P},•)$ consisting of a vector space, $\mathcal{P}$, and a binary operation, $•,\mathcal{P}\times \mathcal{P}\to \mathcal{P}$, such that for all $a,b,c\in \mathcal{P}$, the associator:

$$\begin{array}{cc}\hfill & (a,b,c)=(a•b)•c-a•(b•c),\hfill \end{array}$$

is symmetric in $a,b$, i.e.,

$$\begin{array}{c}\hfill (a,b,c)=(b,a,c),\mathrm{or}\mathrm{equivalently},(a•b)•c-a•(b•c)=(b•a)•c-b•(a•c).\end{array}$$

(1)

Given a pre-Lie algebra $(\mathcal{P},•)$, the commutator, ${[a,b]}^c=a•b-b•a$, defines a Lie algebra structure on $\mathcal{P}$, which is called the sub-adjacent Lie algebra of $(\mathcal{P},•)$, and we denote it by ${\mathcal{P}}^c$.

Inspired by the modified r-matrix [34,35], we propose the notion of a modified Rota–Baxter operator on pre-Lie algebras.

Definition 2. 

(i) Let $(\mathcal{P},•)$ be a pre-Lie algebra. A modified Rota–Baxter operator on $\mathcal{P}$ is a linear map, $M:\mathcal{P}\to \mathcal{P}$, subject to the following:

$$\begin{array}{cc}\hfill Ma•Mb=& M(Ma•b+a•Mb)-a•b\mathrm{for}\mathrm{all}a,b\in \mathcal{P}.\hfill \end{array}$$

(2)

Furthermore, the triple $(\mathcal{P},•,M)$ is called a modified Rota–Baxter pre-Lie algebra, simply denoted by $(\mathcal{P},M)$.

(ii) A homomorphism between two modified Rota–Baxter pre-Lie algebras $({\mathcal{P}}_1,M_1)$ and $({\mathcal{P}}_2,M_2)$ is a pre-Lie algebra homomorphism, $F:{\mathcal{P}}_1\to {\mathcal{P}}_2$, such that $F\circ M_1=M_2\circ F$. Furthermore, F is called an isomorphism from $({\mathcal{P}}_1,M_1)$ to $({\mathcal{P}}_2,M_2)$ if F is bijective.

Example 1. 

Let $(\mathcal{P},•)$ be a pre-Lie algebra. Then, $(\mathcal{P},•,{\mathrm{id}}_{\mathcal{P}})$ is a modified Rota–Baxter pre-Lie algebra, where ${\mathrm{id}}_{\mathcal{P}}:\mathcal{P}\to \mathcal{P}$ is an identity mapping.

Example 2. 

Let $(\mathcal{P},•)$ be a two-dimensional pre-Lie algebra and $\{{ϵ}_1,{ϵ}_2\}$ be a basis, whose nonzero products are given as follows:

$${ϵ}_1•{ϵ}_2={ϵ}_1,{ϵ}_2•{ϵ}_2={ϵ}_2.$$

Then, the triple $(\mathcal{P},•,M)$ is a two-dimensional modified Rota–Baxter pre-Lie algebra, where $M=\left(\begin{array}{cc}1& k\\ 0& -1\end{array}\right)$, for $k\in \mathbb{K}$.

Example 3. 

Let $(\mathcal{P},•)$ be a pre-Lie algebra. If a linear map, $M:\mathcal{P}\to \mathcal{P}$, is a modified Rota–Baxter operator, then $-M$ is also a modified Rota–Baxter operator.

Definition 3 

([16]). Let $(\mathcal{P},•)$ be a pre-Lie algebra. A Rota–Baxter operator of weight-1 on $\mathcal{P}$ is a linear map, $R:\mathcal{P}\to \mathcal{P}$, subject to the following:

$$\begin{array}{cc}\hfill Ra•Rb=& R(Ra•b+a•Rb-a•b)\mathrm{for}\mathrm{all}a,b\in \mathcal{P}.\hfill \end{array}$$

Then, the triple $(\mathcal{P},•,R)$ is called a Rota–Baxter pre-Lie algebra of weight-1.

Proposition 1. 

Let $(\mathcal{P},•)$ be a pre-Lie algebra. If a linear map, $R:\mathcal{P}\to \mathcal{P}$, is a Rota–Baxter operator of weight-1, then the map, $2R-{\mathrm{id}}_{\mathcal{P}}$, is a modified Rota–Baxter operator on $\mathcal{P}$.

Proof. 

For any $a,b\in \mathcal{P}$, we have the following:

$$\begin{array}{cc}\hfill & (2R-{\mathrm{id}}_{\mathcal{P}})a•(2R-{\mathrm{id}}_{\mathcal{P}})b\hfill \\ \hfill & =(2Ra-a)•(2Rb-b)\hfill \\ \hfill & =4Ra•Rb-2Ra•b-2a•Rb+a•b\hfill \\ \hfill & =4R(Ra•b+a•Rb-a•b)-2Ra•b-2a•Rb+a•b\hfill \\ \hfill & =(2R-{\mathrm{id}}_{\mathcal{P}})\left((2R-{\mathrm{id}}_{\mathcal{P}})a•b+a•(2R-{\mathrm{id}}_{\mathcal{P}})b\right)-a•b.\hfill \end{array}$$

The proposition follows. □

Recall from [16] that a Nijenhuis operator on a pre-Lie algebra $(\mathcal{P},•)$ is a linear map, $N:\mathcal{P}\to \mathcal{P}$, that satisfies the following,

$$\begin{array}{cc}\hfill Na•Nb=& N(Na•b+a•Nb-N(a•b\left)\right),\hfill \end{array}$$

for all $a,b\in \mathcal{P}.$ The relationship between the modified Rota–Baxter operator and Nijenhuis operator is as follows, which proves to be obvious.

Proposition 2. 

Let $(\mathcal{P},•)$ be a pre-Lie algebra and $N:\mathcal{P}\to \mathcal{P}$ be a linear map. If $N^2={\mathrm{id}}_{\mathcal{P}}$, then N is a Nijenhuis operator if, and only if, N is a modified Rota–Baxter operator.

Definition 4 

([8]). Let $(\mathcal{P},•)$ be a pre-Lie algebra and V a vector space. A bimodule of $\mathcal{P}$ on V consists of a pair $({•}_l,{•}_r)$, where ${•}_l:\mathcal{P}\times V\to V$ and ${•}_r:V\times \mathcal{P}\to V$ are two linear maps satisfying the following:

$$\begin{array}{cc}\hfill a{•}_l\left(b{•}_{l}u\right)-(a•b){•}_{l}u& =b{•}_l\left(a{•}_{l}u\right)-(b•a){•}_{l}u,\hfill \\ \hfill a{•}_l\left(u{•}_{r}b\right)-\left(a{•}_{l}u\right){•}_{r}b& =u{•}_r(a•b)-\left(u{•}_{r}a\right){•}_{r}b\mathrm{for}\mathrm{all}a,b\in \mathcal{P},u\in V.\hfill \end{array}$$

Definition 5. 

A bimodule of the modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$ is a quadruple $(V;{•}_l,{•}_r,M_V)$ such that the following conditions are satisfied:

(i) $(V;{•}_l,{•}_r)$ is a bimodule of the pre-Lie algebra $(\mathcal{P},•)$;

(ii) $M_V:V\to V$ is a linear map satisfying the following equations,

$$\begin{array}{cc}\hfill Ma{•}_l{M}_{V}u=& M_V(Ma{•}_{l}u+a{•}_l{M}_{V}u)-a{•}_{l}u,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill M_{V}u{•}_{r}Ma=& M_V(M_{V}u{•}_{r}a+u{•}_{r}Ma)-u{•}_{r}a,\hfill \end{array}$$

(4)

for $a\in \mathcal{P}$ and $u\in V.$ In this case, the quadruple $(V;{•}_l,{•}_r,M_V)$ is also called a representation over $(\mathcal{P},•,M)$.

Example 4. 

$(\mathcal{P};{•}_l={•}_r=•,M)$ is an adjoint bimodule of the modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$.

Next, we construct the semidirect product of the modified Rota–Baxter pre-Lie algebra.

Proposition 3. 

The quadruple $(V;{•}_l,{•}_r,M_V)$ is a bimodule of a modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$ if, and only if, $\mathcal{P}\oplus V$ is a modified Rota–Baxter pre-Lie algebra with the following maps,

$$\begin{array}{ccc}& & (a+u){•}_{⋉}(b+v):=a•b+a{•}_{l}v+u{•}_{r}b,\hfill \\ & & M\oplus M_V(a+u)=Ma+M_{V}u,\hfill \end{array}$$

for $a\in \mathcal{P}$ and $u\in V$. In the case, the modified Rota–Baxter pre-Lie algebra $\mathcal{P}\oplus V$ is called a semidirect product of $\mathcal{P}$ and V, denoted by $\mathcal{P}⋉V=(\mathcal{P}\oplus V,{•}_{⋉},M\oplus M_V)$.

Proof. 

Firstly, it is easy to verify that $(\mathcal{P}\oplus V,{•}_{⋉})$ is a pre-Lie algebra. In addition, for any $a,b\in \mathcal{P}$ and $u,v\in V$, via Equations (2)–(4), we have

$$\begin{array}{cc}\hfill & M\oplus M_V(a+u){•}_{⋉}M\oplus M_V(b+v)\hfill \\ \hfill & =(Ma+M_{V}u){•}_{⋉}(Mb+M_{V}v)\hfill \\ \hfill & =Ma•Mb+Ma{•}_l{M}_{V}v+M_{V}u{•}_{r}Mb\hfill \\ \hfill & =M(Ma•b+a•Mb)-a•b+M_V(Ma{•}_{l}u+a{•}_l{M}_{V}u)-a{•}_{l}u\hfill \\ \hfill & +M_V(M_{V}u{•}_{r}b+u{•}_{r}Mb)-u{•}_{r}b\hfill \\ \hfill & =M\oplus M_V\left((a+u){•}_{⋉}M\oplus M_V(b+v)+M\oplus M_V(a+u){•}_{⋉}(b+v)\right)-(a+u){•}_{⋉}(b+v),\hfill \end{array}$$

which means that $(\mathcal{P}\oplus V,{•}_{⋉},M\oplus M_V)$ is a modified Rota–Baxter pre-Lie algebra. □

Proposition 4. 

Let $(\mathcal{P},•,M)$ be a modified Rota–Baxter pre-Lie algebra. Define a new operation as follows:

$$\begin{array}{cc}\hfill a{•}_{M}b=& Ma•b+a•Mb\mathrm{for}\mathrm{all}a,b\in \mathcal{P}.\hfill \end{array}$$

(5)

Then, (i) $(\mathcal{P},{•}_M)$ is a pre-Lie algebra. We denote this pre-Lie algebra as ${\mathcal{P}}_M.$

(ii) $({\mathcal{P}}_M,M)$ is a modified Rota–Baxter pre-Lie algebra.

Proof. 

(i) For any $a,b,c\in \mathcal{P},$ according to Equations (1) and (2), we have the following:

$$\begin{array}{cc}\hfill & \left(a{•}_{M}b\right){•}_{M}c-a{•}_M\left(b{•}_{M}c\right)\hfill \\ \hfill & =M(Ma•b+a•Mb)•c+(Ma•b+a•Mb)•Mc-Ma•(Mb•c+b•Mc)\hfill \\ \hfill & -a•M(Mb•c+b•Mc)\hfill \\ \hfill & =M(Mb•a+b•Ma)•c+(Mb•a+b•Ma)•Mc-Mb•(Ma•c+a•Mc)\hfill \\ \hfill & -b•M(Ma•c+a•Mc)\hfill \\ \hfill & =\left(b{•}_{M}a\right){•}_{M}c-b{•}_M\left(a{•}_{M}c\right).\hfill \end{array}$$

Thus, $(\mathcal{P},{•}_M)$ is a pre-Lie algebra.

(ii) For any $a,b\in \mathcal{P},$ according to Equation (2), we have

$$\begin{array}{cc}\hfill Ma{•}_{M}Mb=& M^{2}a•Mb+Ma•M^{2}b\hfill \\ \hfill =& M(M^{2}a•b+Ma•Mb)-Ma•b+M(Ma•Mb+a•M^{2}b)-a•Mb\hfill \\ \hfill =& M(Ma{•}_{M}b+Ma{•}_{M}b)-a{•}_{M}b.\hfill \end{array}$$

Hence, $({\mathcal{P}}_M,M)$ is a modified Rota–Baxter pre-Lie algebra. □

Proposition 5. 

Let $(V;{•}_l,{•}_r,M_V)$ be a bimodule of the modified Rota–Baxter pre-Lie algebra, $(\mathcal{P},•,M)$. Define two bilinear maps, ${•}_l^M:\mathcal{P}\times V\to V$ and ${•}_r^M:V\times \mathcal{P}\to V$, via the following:

$$\begin{array}{cc}\hfill & a{•}_l^{M}u:=Ma{•}_{l}u-M_V\left(a{•}_{l}u\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & u{•}_r^{M}a:=u{•}_{r}Ma-M_V\left(u{•}_{r}a\right)\mathrm{for}\mathrm{all}a\in \mathcal{P},u\in V.\hfill \end{array}$$

(7)

Then, $(V;{•}_l^M,{•}_r^M)$ is a bimodule of a pre-Lie algebra ${\mathcal{P}}_M$. Moreover, $(V;{•}_l^M,{•}_r^M,M_V)$ is a bimodule of a modified Rota–Baxter pre-Lie algebra $({\mathcal{P}}_M,M)$.

Proof. 

First, by direct verification, we determine that $(V;{•}_l^M,{•}_r^M)$ is a bimodule of the pre-Lie algebra ${\mathcal{P}}_M$. Further, for any $a\in \mathcal{P}$ and $u\in V$, according to Equation (3), we have the following:

$$\begin{array}{cc}\hfill & Ma{•}_l^M{M}_{V}u\hfill \\ \hfill & =M^{2}a{•}_l{M}_{V}u-M_V\left(Ma{•}_l{M}_{V}u\right)\hfill \\ \hfill & =M_V(M^{2}a{•}_{l}u+Ma{•}_l{M}_{V}u)-Ma{•}_{l}u-M_V^2(Ma{•}_{l}u+a{•}_l{M}_{V}u)+M_V\left(a{•}_{l}u\right)\hfill \\ \hfill & =M_V\left(M^{2}a{•}_{l}u+Ma{•}_l{M}_{V}u-M_V(Ma{•}_{l}u+a{•}_l{M}_{V}u)\right)-\left(Ma{•}_{l}u-M_V\left(a{•}_{l}u\right)\right)\hfill \\ \hfill & =M_V(Ma{•}_l^{M}u+a{•}_l^M{M}_{V}u)-a{•}_l^{M}u.\hfill \end{array}$$

Similarly, according to Equation (4), there is also $M_{V}u{•}_r^{M}Ma=M_V(M_{V}u{•}_r^{M}a+u{•}_r^{M}Ma)-u{•}_r^{M}a$. Hence, $(V;{•}_l^M,{•}_r^M,M_V)$ is a bimodule of $({\mathcal{P}}_M,M)$. □

Example 5 

$(\mathcal{P};{•}_l^M,{•}_r^M,M)$ is an adjoint bimodule of the modified Rota–Baxter pre-Lie algebra $({\mathcal{P}}_M,M)$, where

$$\begin{array}{cc}\hfill a{•}_l^{M}b: =& Ma•b-M(a•b),\hfill \\ \hfill a{•}_r^{M}b: =& a•Mb-M(a•b),\hfill \end{array}$$

for any $a,b\in \mathcal{P}$.

3. Cohomology of Modified Rota–Baxter Pre-Lie Algebras

In this section, we develop the cohomology of a modified Rota–Baxter pre-Lie algebra with coefficients in its bimodule.

Let us recall the cohomology theory of pre-Lie algebras in [17]. Let $(\mathcal{P},•)$ be a pre-Lie algebra and $(V;{•}_l,{•}_r)$ be a bimodule of it. Denote the $n–$cochains of $\mathcal{P}$ with coefficients in representation V via the following:

$$\begin{array}{c}\hfill C_{\mathrm{PLie}}^n(\mathcal{P},V):=\mathrm{Hom}({\mathcal{P}}^{\otimes n},V).\end{array}$$

The coboundary operator $\delta :C_{\mathrm{PLie}}^n(\mathcal{P},V)\to C_{\mathrm{PLie}}^{n+1}(\mathcal{P},V)$, for $a_1,\cdots ,a_{n+1}\in \mathcal{P}$ and $g\in C_{\mathrm{PLie}}^n(\mathcal{P},V)$, as follows:

$$\begin{array}{cc}\hfill & \delta g(a_1,\dots ,a_{n+1})\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^n}{(-1)}^{i+1}{a}_i{•}_{l}g(a_1,\dots ,{\widehat{a}}_i,\cdots ,a_{n+1})+{\displaystyle \sum _{i=1}^n}{(-1)}^{i+1}g(a_1,\dots ,{\widehat{a}}_i,\dots ,a_n,a_i){•}_r{a}_{n+1}\hfill \\ \hfill & -{\displaystyle \sum _{i=1}^n}{(-1)}^{i+1}g(a_1,\dots ,{\widehat{a}}_i,\dots ,a_n,a_i•a_{n+1})+{\displaystyle \sum _{1\le i<j\le n}}{(-1)}^{i+j}g({[a_i,a_j]}^c,a_1,\dots ,{\widehat{a}}_i,\dots ,{\widehat{a}}_j,\dots ,a_{n+1}).\hfill \end{array}$$

(8)

Then, it is proven in [17] that ${\delta }^2=0$. Let us denote, via $H_{\mathrm{PLie}}^{*}(\mathcal{P},V)$, the cohomology group associated to the cochain complex $(C_{\mathrm{PLie}}^{*}(\mathcal{P},V),\delta )$.

We first study the cohomology of the modified Rota–Baxter operator.

Let $(\mathcal{P},•,M)$ be a modified Rota–Baxter pre-Lie algebra and $(V;{•}_l,{•}_r,M_V)$ be a bimodule of it. Recall that Proposition 4 and Proposition 5 give a new pre-Lie algebra, ${\mathcal{P}}_M$, and a new bimodule, $V_M=(V;{•}_l^M,{•}_r^M)$, over ${\mathcal{P}}_M$. Consider the cochain complex of ${\mathcal{P}}_M$ with coefficients in $V_M$:

$$(C_{\mathrm{PLie}}^{*}({\mathcal{P}}_M,V_M),{\delta }_M)=({\oplus }_{n=1}^{\infty }{C}_{\mathrm{PLie}}^n({\mathcal{P}}_M,V_M),{\delta }_M).$$

More precisely, $C_{\mathrm{PLie}}^n({\mathcal{P}}_M,V_M):=\mathrm{Hom}({\mathcal{P}}_M^{\otimes n},V_M)$ and its coboundary map, ${\delta }_M:C_{\mathrm{PLie}}^n({\mathcal{P}}_M,V_M)\to C_{\mathrm{PLie}}^{n+1}({\mathcal{P}}_M,V_M)$, for $a_1,\cdots ,a_{n+1}\in {\mathcal{P}}_R$ and $f\in C_{\mathrm{PLie}}^n({\mathcal{P}}_M,V_M)$, are given as follows:

$$\begin{array}{cc}\hfill & {\delta }_{M}f(a_1,\dots ,a_{n+1})\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^n}{(-1)}^{i+1}\left(M{a}_i{•}_{l}f(a_1,\dots ,{\widehat{a}}_i,\cdots ,a_{n+1})-M_V\left(a_i{•}_{l}f(a_1,\dots ,{\widehat{a}}_i,\cdots ,a_{n+1})\right)\right)\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}{(-1)}^{i+1}\left(f(a_1,\dots ,{\widehat{a}}_i,\dots ,a_n,a_i){•}_{r}M{a}_{n+1}-M_V\left(f(a_1,\dots ,{\widehat{a}}_i,\dots ,a_n,a_i){•}_r{a}_{n+1}\right)\right)\hfill \\ \hfill & -{\displaystyle \sum _{i=1}^n}{(-1)}^{i+1}f(a_1,\dots ,{\widehat{a}}_i,\dots ,a_n,M{a}_i•a_{n+1}+a_i•M{a}_{n+1})\hfill \\ \hfill & +{\displaystyle \sum _{1\le i<j\le n}}{(-1)}^{i+j}f(M{a}_i•a_j+a_i•M{a}_j-M{a}_j•a_i-a_j•M{a}_i,a_1,\dots ,{\widehat{a}}_i,\dots ,{\widehat{a}}_j,\dots ,a_{n+1}).\hfill \end{array}$$

(9)

In particular, for $n=1$,

$$\begin{array}{cc}\hfill & {\delta }_{M}f(a_1,a_2)\hfill \\ \hfill & =M{a}_1{•}_{l}f\left(a_2\right)-M_V\left(a_1{•}_{l}f\left(a_2\right)\right)+f\left(a_1\right){•}_{r}M{a}_2-M_V\left(f\left(a_1\right){•}_r{a}_2\right)-f(M{a}_1•a_2+a_1•M{a}_2).\hfill \end{array}$$

Definition 6. 

Let $(\mathcal{P},•,M)$ be a modified Rota–Baxter pre-Lie algebra and $(V;{•}_l,{•}_r,M_V)$ be a bimodule of it. Then, the cochain complex $(C_{\mathrm{PLie}}^{*}({\mathcal{P}}_M,V_M),{\delta }_M)$ is called the cochain complex of the modified Rota–Baxter operator, M, with coefficients in $V_M$, denoted by $({\mathcal{C}}_{\mathrm{MRBO}}^{*}(\mathcal{P},V),{\delta }_M)$. The cohomology of $({\mathcal{C}}_{\mathrm{MRBO}}^{*}(\mathcal{P},V),{\delta }_M)$, denoted by ${\mathcal{H}}_{\mathrm{MRBO}}^{*}(\mathcal{P},V)$, is called the cohomology of modified Rota–Baxter operator M with coefficients in $V_M$.

In particular, when $(\mathcal{P};{•}_l^M={•}_r^M={•}^M,M)$ is the adjoint bimodule of $({\mathcal{P}}_M,M)$, we denote $({\mathcal{C}}_{\mathrm{MRBO}}^{*}(\mathcal{P},\mathcal{P}),{\delta }_M)$ as $({\mathcal{C}}_{\mathrm{MRBO}}^{*}\left(\mathcal{P}\right),{\delta }_M)$ and call it the cochain complex of modified Rota–Baxter operator M, denote ${\mathcal{H}}_{\mathrm{MRBO}}^{*}(\mathcal{P},\mathcal{P})$ as ${\mathcal{H}}_{\mathrm{MRBO}}^{*}\left(\mathcal{P}\right)$ and call it the cohomology of modified Rota–Baxter operator M.

Next, we will combine the cohomology of pre-Lie algebras and the cohomology of modified Rota–Baxter operators to construct a cohomology theory for modified Rota–Baxter pre-Lie algebras.

Let us construct the following cochain map. For any $n\ge 1$, we define a linear map, $\mathrm{Y}:C_{\mathrm{PLie}}^n(\mathcal{P},V)\to {\mathcal{C}}_{\mathrm{MRBO}}^n(\mathcal{P},V)$, via the following:

$$\begin{array}{cc}\hfill & \left(\mathrm{Y}f\right)(a_1,\dots ,a_n)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^{\lfloor {\displaystyle \frac{n}{2}}\rfloor +1}}({\displaystyle \sum _{1⩽j_1<\cdots <j_{2i-2}⩽n}}f(a_1,\dots ,M{a}_{j_1},\dots ,M{a}_{j_{2i-2}},\dots ,a_n)\hfill \\ \hfill & -{\displaystyle \sum _{1⩽j_1<\cdots <j_{2i-3}⩽n}}{M}_{V}f(a_1,\dots ,M{a}_{j_1},\dots ,M{a}_{j_{2i-3}},\dots ,a_n)),\mathrm{if}n\mathrm{is}\mathrm{an}\mathrm{even},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \left(\mathrm{Y}f\right)(a_1,\dots ,a_n)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^{\lfloor {\displaystyle \frac{n}{2}}\rfloor +1}}({\displaystyle \sum _{1⩽j_1<\cdots <j_{2i-1}⩽n}}f(a_1,\dots ,M{a}_{j_1},\dots ,M{a}_{j_{2i-1}},\dots ,a_n)\hfill \\ \hfill & -{\displaystyle \sum _{1⩽j_1<\cdots <j_{2i-2}⩽n}}{M}_{V}f(a_1,\dots ,M{a}_{j_1},\dots ,M{a}_{j_{2i-2}},\dots ,a_n)),\mathrm{if}n\mathrm{is}\mathrm{an}\mathrm{odd},\hfill \end{array}$$

(11)

Among them, when the subscript of $j_{2i-3}$ is negative, f is a zero map. For example, when $n=1$, according to Equation (11), the map $\mathrm{Y}:C_{\mathrm{PLie}}^1(\mathcal{P},V)\to {\mathcal{C}}_{\mathrm{MRBO}}^1(\mathcal{P},V)$ is as follows:

$$\begin{array}{c}\hfill \left(\mathrm{Y}f\right)\left(a_1\right)=f\left(M{a}_1\right)-M_{V}f\left(a_1\right).\end{array}$$

(12)

Lemma 1. 

Map Y is a cochain map, i.e., $\mathrm{Y}\circ \delta ={\delta }_M\circ \mathrm{Y}.$ In other words, the following diagram is commutative:

$$\begin{array}{cccccccc}{C}_{\mathrm{PLie}}^1(\mathcal{P},V)& \stackrel{\delta }{\to }& C_{\mathrm{PLie}}^2(\mathcal{P},V)& \dots \dots \dots \dots \dots & C_{\mathrm{PLie}}^n(\mathcal{P},V)& \stackrel{\delta }{\to }& C_{\mathrm{PLie}}^{n+1}(\mathcal{P},V)& \dots \dots \dots \dots \dots \\ \downarrow \mathrm{Y}& & \downarrow \mathrm{Y}& & \downarrow \mathrm{Y}& & \downarrow \mathrm{Y}& \\ {\mathcal{C}}_{\mathrm{MRBO}}^1(\mathcal{P},V)& \stackrel{{\delta }_M}{\to }& {\mathcal{C}}_{\mathrm{MRBO}}^2(\mathcal{P},V)& \dots \dots \dots \dots \dots & {\mathcal{C}}_{\mathrm{MRBO}}^n(\mathcal{P},V)& \stackrel{{\delta }_M}{\to }& {\mathcal{C}}_{\mathrm{MRBO}}^{n+1}(\mathcal{P},V)& \dots \dots \dots \dots \dots .\end{array}$$

Proof. 

It can be proven by using similar arguments to those in Appendix A in [31]. Here, we only prove the case of $n=1$. For any $f\in C_{\mathrm{PLie}}^1(\mathcal{P},V)$ and $a,b\in \mathcal{P},$ according to Equations (2)–(10) and (12), we have the following:

$$\begin{array}{cc}\hfill & \mathrm{Y}\left(\delta f\right)(a,b)\hfill \\ \hfill & =\left(\delta f\right)(Ma,Mb)-M_V\left(\left(\delta f\right)(Ma,b)+\left(\delta f\right)(a,Mb)\right)+\left(\delta f\right)(a,b)\hfill \\ \hfill & =Ma{•}_{l}f\left(Mb\right)+f\left(Ma\right){•}_{r}Mb-f(Ma•Mb)-M_V(Ma{•}_{l}f\left(b\right)+f\left(Ma\right){•}_{r}b-f(Ma•b)\hfill \\ \hfill & +a{•}_{l}f\left(Mb\right)+f\left(a\right){•}_{r}Mb-f(a•Mb))+a{•}_{l}f\left(b\right)+f\left(a\right){•}_{r}b-f(a•b)\hfill \end{array}$$

(13)

and

$$\begin{array}{cc}\hfill & {\delta }_M\left(\mathrm{Y}f\right)(a,b)\hfill \\ \hfill & =Ma{•}_l(f\left(Mb\right)-M_{V}f\left(b\right))-M_V\left(a{•}_l(f\left(Mb\right)-M_{V}f\left(b\right))\right)+(f\left(Ma\right)-M_{V}f\left(a\right)){•}_{r}Mb\hfill \\ \hfill & -M_V\left((f\left(Ma\right)-M_{V}f\left(a\right)){•}_{r}b\right)-f(Ma•Mb+a•b)+M_{V}f(Ma•b+a•Mb)\hfill \end{array}$$

(14)

Further comparing Equations (13) and (14), we have (13) = (14). Therefore, $\mathrm{Y}\circ \delta ={\delta }_M\circ \mathrm{Y}.$ □

Definition 7. 

Let $(\mathcal{P},•,M)$ be a modified Rota–Baxter pre-Lie algebra and $(V;{•}_l,{•}_r,M_V)$ be a bimodule of it. We attribute the cochain complex $({\mathcal{C}}_{\mathrm{MRBPLie}}^{*}(\mathcal{P},V),\partial )$ of a modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$ with coefficients in $(V;{•}_l,{•}_r,M_V)$ to the negative shift in the mapping cone of Υ, that is, let

$${\mathcal{C}}_{\mathrm{MRBPLie}}^1(\mathcal{P},V)=C_{\mathrm{PLie}}^1(\mathcal{P},V)\mathrm{and}{\mathcal{C}}_{\mathrm{MRBPLie}}^n(\mathcal{P},V):=C_{\mathrm{PLie}}^n(\mathcal{P},V)\oplus {\mathcal{C}}_{\mathrm{MRBO}}^{n-1}(\mathcal{P},V)\mathrm{for}n\ge 2,$$

The coboundary map $\partial :{\mathcal{C}}_{\mathrm{MRBPLie}}^1(\mathcal{P},V)\to {\mathcal{C}}_{\mathrm{MRBPLie}}^2(\mathcal{P},V)$ is given by the following:

$$\begin{array}{c}\hfill \partial \left(f\right)=(\delta f,-\mathrm{Y}f)\mathrm{for}\mathrm{all}f\in {\mathcal{C}}_{\mathrm{MRBPLie}}^1(\mathcal{P},V);\end{array}$$

For $n\ge 2$, the coboundary map $\partial :{\mathcal{C}}_{\mathrm{MRBPLie}}^n(\mathcal{P},V)\to {\mathcal{C}}_{\mathrm{MRBPLie}}^{n+1}(\mathcal{P},V)$ is given by the following:

$$\begin{array}{c}\hfill \partial (f,g)=(\delta f,-{\delta }_{M}g-\mathrm{Y}f)\mathrm{for}\mathrm{all}(f,g)\in {\mathcal{C}}_{\mathrm{MRBPLie}}^n(\mathcal{P},V).\end{array}$$

The cohomology of $({\mathcal{C}}_{\mathrm{MRBPLie}}^{*}(\mathcal{P},V),\partial )$, denoted by ${\mathcal{H}}_{\mathrm{MRBPLie}}^{*}(\mathcal{P},V)$, is called the cohomology of the modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$ with coefficients in $(V;{•}_l,{•}_r,M_V)$. In particular, when $(V;{•}_l,{•}_r,M_V)=(\mathcal{P};{•}_l={•}_r=•,M)$, we just denote $({\mathcal{C}}_{\mathrm{MRBPLie}}^{*}(\mathcal{P},\mathcal{P}),\partial )$ and ${\mathcal{H}}_{\mathrm{MRBPLie}}^{*}(\mathcal{P},\mathcal{P})$ by $({\mathcal{C}}_{\mathrm{MRBPLie}}^{*}\left(\mathcal{P}\right),\partial )$, ${\mathcal{H}}_{\mathrm{MRBPLie}}^{*}\left(\mathcal{P}\right)$, and call them the cochain complex and the cohomology of the modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$, respectively.

It is obvious that there is a short exact sequence of cochain complexes:

$$\begin{array}{c}\hfill 0\to {\mathcal{C}}_{\mathrm{MRBO}}^{*-1}(\mathcal{P},V)⟶{\mathcal{C}}_{\mathrm{MRBPLie}}^{*}(\mathcal{P},V)⟶C_{\mathrm{PLie}}^{*}(\mathcal{P},V)\to 0.\end{array}$$

This induces a long exact sequence of cohomology groups:

$$\begin{array}{c}\hfill \cdots \to {\mathcal{H}}_{\mathrm{MRBPLie}}^n(\mathcal{P},V)\to H_{\mathrm{PLie}}^n(\mathcal{P},V)\to {\mathcal{H}}_{\mathrm{MRBO}}^n(\mathcal{P},V)\to {\mathcal{H}}_{\mathrm{MRBPLie}}^{n+1}(\mathcal{P},V)\to H_{\mathrm{PLie}}^{n+1}(\mathcal{P},V)\to \cdots .\end{array}$$

At the end of this section, we use the established cohomology theory to characterize infinitesimal deformations of modified Rota–Baxter pre-Lie algebras.

Definition 8. 

An infinitesimal deformation of the modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$ is a pair $({•}_t,M_t)$ of the following forms,

$${•}_t=•+{•}_{1}t,M_t=M+M_{1}t,$$

such that the following conditions are satisfied:

(i) 

$({•}_1,M_1)\in {\mathcal{C}}_{\mathrm{MRBPLie}}^2\left(\mathcal{P}\right)$;

(ii) 

$(\mathcal{P}\left[\left[t\right]\right],{•}_t,M_t)$ is a modified Rota–Baxter pre-Lie algebra over $\mathbb{K}\left[\right[t\left]\right]$.

Proposition 6. 

Let $(\mathcal{P}\left[\left[t\right]\right],{•}_t,M_t)$ be an infinitesimal deformation of modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$. Then, $({•}_1,M_1)$ is a 2-cocycle in the cochain complex $({\mathcal{C}}_{\mathrm{MRBPLie}}^{*}\left(\mathcal{P}\right),\partial )$.

Proof. 

Suppose $(\mathcal{P}\left[\left[t\right]\right],{•}_t,M_t)$ is a modified Rota–Baxter pre-Lie algebra. Then, for any $a,b,c\in \mathcal{P}$, we have

$$\begin{array}{cc}\hfill \left(a{•}_{t}b\right){•}_{t}c-a{•}_t\left(b{•}_{t}c\right)=& \left(b{•}_{t}a\right){•}_{t}c-b{•}_t\left(a{•}_{t}c\right),\hfill \\ \hfill M_{t}a{•}_t{M}_{t}b=& M_t(M_{t}a{•}_{t}b+a{•}_t{M}_{t}b)-a{•}_{t}b.\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill & \left(a{•}_{1}b\right)•c+(a•b){•}_{1}c-a•\left(b{•}_{1}c\right)-a{•}_1(b•c)\hfill \\ \hfill & =\left(b{•}_{1}a\right)•c+(b•a){•}_{1}c-b{•}_1(a•c)-b•\left(a{•}_{1}c\right),\hfill \\ \hfill & M_{1}a•Mb+Ma•M_{1}b+Ma{•}_{1}Mb\hfill \\ \hfill & =M(M_{1}a•b+Ma{•}_{1}b+a•M_{1}b+a{•}_{1}Mb)+M_1(Ma•b+a•Mb)-a{•}_{1}b.\hfill \end{array}$$

Therefore, $\partial ({•}_1,M_1)=(\delta {•}_1,-{\delta }_M{M}_1-\mathrm{Y}{•}_1)=0$, that is, $({•}_1,M_1)$ is a 2-cocycle. □

Definition 9. 

The 2-cocycle $({•}_1,M_1)$ is called the infinitesimal of the infinitesimal deformation $(\mathcal{P}\left[\left[t\right]\right],{•}_t,M_t)$ of the modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$.

Definition 10. 

Let $(\mathcal{P}\left[\left[t\right]\right],{•}_t,M_t)$ and $(\mathcal{P}\left[\left[t\right]\right],{•}_t^{\prime },M_t^{\prime })$ be two infinitesimal deformations of a modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$. An isomorphism from $(\mathcal{P}\left[\left[t\right]\right],{•}_t^{\prime },M_t^{\prime })$ to $(\mathcal{P}\left[\left[t\right]\right],{•}_t,M_t)$ is a linear map, ${\phi }_t=\mathrm{id}+t{\phi }_1$, where ${\phi }_1:\mathcal{P}\to \mathcal{P}$ is a linear map, such that:

$$\begin{array}{cc}\hfill {\phi }_t\circ {•}_t^{\prime }=& {•}_t\circ ({\phi }_t\otimes {\phi }_t),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\phi }_t\circ M_t^{\prime }=& M_t\circ {\phi }_t.\hfill \end{array}$$

(16)

In this case, we say that the two infinitesimal deformations $(\mathcal{P}\left[\left[t\right]\right],{•}_t,M_t)$ and $(\mathcal{P}\left[\left[t\right]\right],{•}_t^{\prime },M_t^{\prime })$ are equivalent.

Proposition 7. 

The infinitesimals of two equivalent infinitesimal deformations of $(\mathcal{P},•,M)$ are in the same cohomology class in ${\mathcal{H}}_{\mathrm{MRBPLie}}^2\left(\mathcal{P}\right)$.

Proof. 

Let ${\phi }_t:(\mathcal{P}\left[\left[t\right]\right],{•}_t^{\prime },M_t^{\prime })\to (\mathcal{P}\left[\left[t\right]\right],{•}_t,M_t)$ be an isomorphism. By expanding Equations (15) and (16) and comparing the coefficients of $t^1$ on both sides, we have

$$\begin{array}{cc}\hfill {•}_1^{\prime }-{•}_1=•\circ ({\phi }_1\otimes \mathrm{id})+•\circ (\mathrm{id}\otimes {\phi }_1)-{\phi }_1\circ •=& \delta {\phi }_1,\hfill \\ \hfill M_1^{\prime }-M_1=M\circ {\phi }_1-{\phi }_1\circ M=& -\mathrm{Y}{\phi }_1,\hfill \end{array}$$

that is, we have the following:

$$({•}_1^{\prime },M_1^{\prime })-({•}_1,M_1)=(\delta {\phi }_1,-\mathrm{Y}{\phi }_1)=\partial \left({\phi }_1\right)\in {\mathcal{B}}_{\mathrm{MRBPLie}}^2\left(\mathcal{P}\right).$$

Therefore, $({•}_1^{\prime },M_1^{\prime })$ and $({•}_1,M_1)$ are cohomologous and belong to the same cohomology class in ${\mathcal{H}}_{\mathrm{MRBPLie}}^2\left(\mathcal{P}\right)$. □

4. Abelian Extensions of Modified Rota–Baxter Pre-Lie Algebras

In this section, we prove that any abelian extension of a modified Rota–Baxter pre-Lie algebra has a bimodule and a 2-cocycle. It is further proven that they are classified by the second cohomology, as one would expect of a good cohomology theory.

Definition 11. 

Let $(\mathcal{P},•,M)$ be a modified Rota–Baxter pre-Lie algebra and $(V,{•}_V,M_V)$ be an abelian modified Rota–Baxter pre-Lie algebra with the trivial product ${•}_V$. An abelian extension $(\widehat{\mathcal{P}},\widehat{•},\widehat{M})$ of $(\mathcal{P},•,M)$ by $(V,{•}_V,M_V)$ is a short exact sequence of morphisms of modified Rota–Baxter pre-Lie algebras,

$$\begin{array}{ccccccccc}0& \to & (V,{•}_V,M_V)& \stackrel{\mathbf{i}}{\to }& (\widehat{\mathcal{P}},\widehat{•},\widehat{M})& \stackrel{\mathbf{p}}{\to }& (\mathcal{P},•,M)& \to & 0\end{array}$$

such that $\widehat{M}u=M_{V}u$ and $u\widehat{•}v=0$, for $u,v\in V,$ i.e., V is an abelian ideal of $\widehat{\mathcal{P}}.$

Definition 12. 

A section of an abelian extension $(\widehat{\mathcal{P}},\widehat{•},\widehat{M})$ of $(\mathcal{P},•,M)$ by $(V,{•}_V,M_V)$ is a linear map, $\mathbf{s}:\mathcal{P}\to \widehat{\mathcal{P}}$, such that $\mathbf{p}\circ \mathbf{s}={\mathrm{id}}_{\mathcal{P}}$ and $\mathbf{s}\circ M=\widehat{M}\circ \mathbf{s}$.

Definition 13. 

Let $({\widehat{\mathcal{P}}}_1,{\widehat{•}}_1,{\widehat{M}}_1)$ and $({\widehat{\mathcal{P}}}_2,{\widehat{•}}_2,{\widehat{M}}_2)$ be two abelian extensions of $(\mathcal{P},•,M)$ by $(V,{•}_V,M_V)$. They are said to be equivalent if there is an isomorphism of modified Rota–Baxter pre-Lie algebras, $F:({\widehat{\mathcal{P}}}_1,{\widehat{•}}_1,{\widehat{M}}_1)\to ({\widehat{\mathcal{P}}}_2,{\widehat{•}}_2,{\widehat{M}}_2)$ such that the following diagram is commutative:

$$\begin{array}{c}\hfill \begin{array}{ccccccccc}0& \to & (V,{•}_V,M_V)& \stackrel{{\mathbf{i}}_1}{\to }& ({\widehat{\mathcal{P}}}_1,{\widehat{•}}_1,{\widehat{M}}_1)& \stackrel{{\mathbf{p}}_1}{\to }& (\mathcal{P},•,M)& \to & 0\\ & & \parallel & & F\downarrow \phantom{F}& & \parallel & & & & \\ 0& \to & (V,{•}_V,M_V)& \stackrel{{\mathbf{i}}_2}{\to }& ({\widehat{\mathcal{P}}}_2,{\widehat{•}}_2,{\widehat{M}}_2)& \stackrel{{\mathbf{p}}_2}{\to }& (\mathcal{P},•,M)& \to & 0.\end{array}\end{array}$$

(17)

Now for an abelian extension $(\widehat{\mathcal{P}},\widehat{•},\widehat{M})$ of $(\mathcal{P},•,M)$ by $(V,{•}_V,M_V)$ with a section, $\mathbf{s}:\mathcal{P}\to \widehat{\mathcal{P}}$, we define two bilinear maps, ${•}_l:\mathcal{P}\times V\to V$, ${•}_r:V\times \mathcal{P}\to V$, by

$$a{•}_{l}u=\mathbf{s}\left(a\right)\widehat{•}u,u{•}_{r}a=u\widehat{•}\mathbf{s}\left(a\right)\mathrm{for}\mathrm{all}a\in \mathcal{P},u\in V.$$

Proposition 8. 

With the above notations, $(V;{•}_l,{•}_r,M_V)$ is a bimodule of the modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$ and does not depend on the choice of $\mathbf{s}$.

Proof. 

First, for any other section, ${\mathbf{s}}^{\prime }:\mathcal{P}\to \widehat{\mathcal{P}}$, for any $a\in \mathcal{P}$, we have the following:

$$\mathbf{p}(\mathbf{s}\left(a\right)-{\mathbf{s}}^{\prime }\left(a\right))=\mathbf{p}\left(\mathbf{s}\left(a\right)\right)-\mathbf{p}\left({\mathbf{s}}^{\prime }\left(a\right)\right)=a-a=0.$$

Thus, there exists an element, $u\in V$, such that ${\mathbf{s}}^{\prime }\left(a\right)=\mathbf{s}\left(a\right)+u$. Note that V is an abelian ideal of $\widehat{\mathcal{P}}$; this yields the following:

$${\mathbf{s}}^{\prime }\left(x\right)\widehat{•}u=(\mathbf{s}\left(x\right)+v)\widehat{•}u=\mathbf{s}\left(x\right)\widehat{•}u,u\widehat{•}{\mathbf{s}}^{\prime }\left(x\right)=u\widehat{•}(\mathbf{s}\left(x\right)+v)=u\widehat{•}\mathbf{s}\left(x\right).$$

This means that ${•}_l,{•}_r$ does not depend on the choice of $\mathbf{s}$.

Next, for any $a,b\in \mathcal{P}$ and $u\in V$, V is an abelian ideal of $\widehat{\mathcal{P}}$ and $\mathbf{s}\left(a\right)\widehat{•}\mathbf{s}\left(b\right)-\mathbf{s}(a•b)\in V$; we have the following:

$$\begin{array}{cc}\hfill a{•}_l\left(b{•}_{l}u\right)-(a•b){•}_{l}u& =\mathbf{s}\left(a\right)\widehat{•}\left(\mathbf{s}\left(b\right)\widehat{•}u\right)-\mathbf{s}(a•b)\widehat{•}u\hfill \\ \hfill & =\mathbf{s}\left(a\right)\widehat{•}\left(\mathbf{s}\left(b\right)\widehat{•}u\right)-\left(\mathbf{s}\left(a\right)\widehat{•}\mathbf{s}\left(b\right)\right)\widehat{•}u\hfill \\ \hfill & =\mathbf{s}\left(b\right)\widehat{•}\left(\mathbf{s}\left(a\right)\widehat{•}u\right)-\left(\mathbf{s}\left(b\right)\widehat{•}\mathbf{s}\left(a\right)\right)\widehat{•}u\hfill \\ \hfill & =b{•}_l\left(a{•}_{l}u\right)-(b•a){•}_{l}u.\hfill \end{array}$$

By the same token, there is also $a{•}_l\left(u{•}_{r}b\right)-\left(a{•}_{l}u\right){•}_{r}b=u{•}_r(a•b)-\left(u{•}_{r}a\right){•}_{r}b.$ This shows that $(V;{•}_l,{•}_r)$ is a bimodule of the pre-Lie algebra $(\mathcal{P},•)$

On the other hand, according to $\widehat{M}\mathbf{s}\left(a\right)-\mathbf{s}\left(Ma\right)\in V,$ we have the following:

$$\begin{array}{cc}\hfill Ma{•}_l{M}_{V}u=& \mathbf{s}\left(Ma\right)\widehat{•}{M}_{V}u=\widehat{M}\mathbf{s}\left(a\right)\widehat{•}{M}_{V}u=\widehat{M}\mathbf{s}\left(a\right)\widehat{•}\widehat{M}u\hfill \\ \hfill =& \widehat{M}(\widehat{M}\mathbf{s}\left(a\right)\widehat{•}u+\mathbf{s}\left(a\right)\widehat{•}\widehat{M}u)-\mathbf{s}\left(a\right)\widehat{•}u\hfill \\ \hfill =& M_V(\mathbf{s}\left(Ma\right)\widehat{•}u+\mathbf{s}\left(a\right)\widehat{•}{M}_{V}u)-\mathbf{s}\left(a\right)\widehat{•}u\hfill \\ \hfill =& M_V(Ma{•}_{l}u+a{•}_l{M}_{V}u)-a{•}_{l}u.\hfill \end{array}$$

In the same way, there is also $M_{V}u{•}_{r}Ma=M_V(M_{V}u{•}_{r}a+u{•}_{r}Ma)-u{•}_{r}a$. Hence, $(V;{•}_l,{•}_r,M_V)$ is a bimodule of $(\mathcal{P},•,M)$. □

Let $(\widehat{\mathcal{P}},\widehat{•},\widehat{M})$ be an abelian extension of $(\mathcal{P},•,M)$ by $(V,{•}_V,M_V)$ and $\mathbf{s}:\mathcal{P}\to \widehat{\mathcal{P}}$ be a section of it. Define the maps $\omega :\mathcal{P}\times \mathcal{P}\to V$ and $\chi :\mathcal{P}\to V$ by the following, respectively:

$$\begin{array}{cc}\hfill \omega (a,b)& =\mathbf{s}\left(a\right)\widehat{•}\mathbf{s}\left(b\right)-\mathbf{s}(a•b),\hfill \\ \hfill \chi \left(a\right)& =\widehat{M}\mathbf{s}\left(a\right)-\mathbf{s}\left(Ma\right)\mathrm{for}\mathrm{all}a,b\in \mathcal{P}.\hfill \end{array}$$

We transfer the modified Rota–Baxter pre-Lie algebra structure on $\widehat{\mathcal{P}}$ to $\mathcal{P}\oplus V$ by endowing $\mathcal{P}\oplus V$ with a multiplication, ${•}_{\omega }$, and a modified Rota–Baxter operator, $M_{\chi }$, defined by the following:

$$\begin{array}{cc}\hfill (a+u){•}_{\omega }(b+v)& =a•b+a{•}_{l}v+u{•}_{r}b+\omega (a,b),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill M_{\chi }(a+u)& =Ma+\chi \left(a\right)+M_{V}u\mathrm{for}\mathrm{all}a,b\in \mathcal{P},u,v\in V.\hfill \end{array}$$

(19)

Proposition 9. 

The triple $(\mathcal{P}\oplus V,{•}_{\omega },M_{\chi })$ is a modified Rota–Baxter pre-Lie algebra if, and only if, $(\omega ,\chi )$ is a 2-cocycle of the modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$ with the coefficient in $(V,{•}_V,M_V)$. In this case,

$$\begin{array}{ccccccccc}0& \to & (V,{•}_V,M_V)& \stackrel{\mathbf{i}}{\to }& (\mathcal{P}\oplus V,{•}_{\omega },M_{\chi })& \stackrel{\mathbf{p}}{\to }& (\mathcal{P},•,M)& \to & 0\end{array}$$

is an abelian extension.

Proof. 

The triple $(\mathcal{P}\oplus V,{•}_{\omega },M_{\chi })$ is a modified Rota–Baxter pre-Lie algebra if, and only if, for any $a,b,c\in \mathcal{P}$ and $u,v,w\in V$, the following equations hold true:

$$\begin{array}{cc}\hfill & \left((a+u){•}_{\omega }(b+v)\right){•}_{\omega }(c+w)-(a+u){•}_{\omega }\left((b+v){•}_{\omega }(c+w)\right)\hfill \\ \hfill & =\left((b+v){•}_{\omega }(a+u)\right){•}_{\omega }(c+w)-(b+v){•}_{\omega }\left((a+u){•}_{\omega }(c+w)\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & M_{\chi }(a+u){•}_{\omega }{M}_{\chi }(b+v)\hfill \\ \hfill & =M_{\chi }(M_{\chi }(a+u){•}_{\omega }(b+v)+(a+u){•}_{\omega }{M}_{\chi }(b+v))-(a+u){•}_{\omega }(b+v).\hfill \end{array}$$

(21)

Further, Equations (20) and (21) are equivalent to the following equations:

$$\begin{array}{cc}\hfill & \omega (a,b){•}_{r}c+\omega (a•b,c)-a{•}_l\omega (b,c)-\omega (a,b•c)\hfill \\ \hfill & =\omega (b,a){•}_{r}c+\omega (b•a,c)-b{•}_l\omega (a,c)-\omega (b,a•c),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & Ma{•}_l\chi \left(b\right)+\chi \left(a\right){•}_{r}Mb+\omega (Ma,Mb)\hfill \\ \hfill & =\chi (Ma•b+a•Mb)+M_V\left(\chi \left(a\right){•}_{r}b+a{•}_l\chi \left(b\right)+\omega (Ma,b)+\omega (a,Mb)\right)-\omega (a,b).\hfill \end{array}$$

(23)

Using Equations (22) and (23), we have $\delta \omega =0$ and $-{\delta }_M\chi -\mathrm{Y}\omega =0$, respectively. Therefore, $\partial (\omega ,\chi )=(\delta \omega ,-{\delta }_M\chi -\mathrm{Y}\omega )=0,$ that is, $(\omega ,\chi )$ is a 2-cocycle.

Conversely, if $(\omega ,\chi )$ is a 2-cocycle of $(\mathcal{P},•,M)$ with the coefficient in $(V,{•}_V,M_V)$, then we have $\partial (\omega ,\chi )=(\delta \omega ,-{\delta }_M\chi -\mathrm{Y}\omega )=0,$ in which case Equations (20) and (21) hold true. Hence, $(\mathcal{P}\oplus V,{•}_{\omega },M_{\chi })$ is a modified Rota–Baxter pre-Lie algebra. □

Proposition 10. 

Let $(\widehat{\mathcal{P}},\widehat{•},\widehat{M})$ be an abelian extension of $(\mathcal{P},•,M)$ by $(V,{•}_V,M_V)$ and $\mathbf{s}$ be a section of it. If the pair $(\omega ,\chi )$ is a 2-cocycle of $(\mathcal{P},•,M)$ with the coefficient in $(V,{•}_V,M_V)$ constructed using the section $\mathbf{s}$, then its cohomology class does not depend on the choice of $\mathbf{s}$.

Proof. 

Let ${\mathbf{s}}_1,{\mathbf{s}}_2:\mathcal{P}\to \widehat{\mathcal{P}}$ be two distinct sections; according to Proposition 9, we have two corresponding 2-cocycles, $({\omega }_1,{\chi }_1)$ and $({\omega }_2,{\chi }_2)$, respectively. Define a linear map, $\gamma :\mathcal{P}\to V$, by $\gamma \left(a\right)={\mathbf{s}}_1\left(a\right)-{\mathbf{s}}_2\left(a\right)$. Then,

$$\begin{array}{cc}\hfill {\omega }_1(a,b)& ={\mathbf{s}}_1\left(a\right){\widehat{•}}_1{\mathbf{s}}_1\left(b\right)-{\mathbf{s}}_1(a•b)\hfill \\ \hfill & =({\mathbf{s}}_2\left(a\right)+\gamma \left(a\right)){\widehat{•}}_1({\mathbf{s}}_2\left(b\right)+\gamma \left(b\right))-({\mathbf{s}}_2(a•b)+\gamma (a•b))\hfill \\ \hfill & ={\mathbf{s}}_2\left(a\right){\widehat{•}}_2{\mathbf{s}}_2\left(b\right)-{\mathbf{s}}_2(a•b)+{\mathbf{s}}_2\left(a\right){\widehat{•}}_2\gamma \left(b\right)+\gamma \left(a\right){\widehat{•}}_2{\mathbf{s}}_2\left(b\right)+\gamma \left(a\right){\widehat{•}}_2\gamma \left(b\right)-\gamma (a•b)\hfill \\ \hfill & ={\mathbf{s}}_2\left(a\right){\widehat{•}}_2{\mathbf{s}}_2\left(b\right)-{\mathbf{s}}_2(a•b)+a{•}_l\gamma \left(b\right)+\gamma \left(a\right){•}_{r}b-\gamma (a•b)\hfill \\ \hfill & ={\omega }_2(a,b)+\delta \gamma (a,b)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\chi }_1\left(a\right)& =\widehat{M}{\mathbf{s}}_1\left(a\right)-{\mathbf{s}}_1\left(Ma\right)\hfill \\ \hfill & =\widehat{M}({\mathbf{s}}_2\left(a\right)+\gamma \left(a\right))-({\mathbf{s}}_2\left(Ma\right)+\gamma \left(Ma\right))\hfill \\ \hfill & =\widehat{M}{\mathbf{s}}_2\left(a\right)-{\mathbf{s}}_2\left(Ma\right)+\widehat{M}\gamma \left(a\right)-\gamma \left(Ma\right)\hfill \\ \hfill & ={\chi }_2\left(a\right)+M_V\gamma \left(a\right)-\gamma \left(Ma\right)\hfill \\ \hfill & ={\chi }_2\left(a\right)-\mathrm{Y}\gamma \left(a\right).\hfill \end{array}$$

Hence, $({\omega }_1,{\chi }_1)-({\omega }_2,{\chi }_2)=(\delta \gamma ,-\mathrm{Y}\gamma )=\partial \left(\gamma \right)\in {\mathcal{B}}_{\mathrm{MRBPLie}}^2(\mathcal{P},V)$, that is $({\omega }_1,{\chi }_1)$ and $({\omega }_2,{\chi }_2)$ form the same cohomological class in ${\mathcal{H}}_{\mathrm{MRBPLie}}^2(\mathcal{P},V)$. □

Next, we are ready to classify abelian extensions of a modified Rota–Baxter pre-Lie algebra.

Theorem 1. 

Abelian extensions of a modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$ by $(V,{•}_V,M_V)$ are classified by the second cohomology group, ${\mathcal{H}}_{\mathrm{MRBPLie}}^2(\mathcal{P},V)$.

Proof. 

Assume that $({\widehat{\mathcal{P}}}_1,{\widehat{•}}_1,{\widehat{M}}_1)$ and $({\widehat{\mathcal{P}}}_2,{\widehat{•}}_2,{\widehat{M}}_2)$ are equivalent abelian extensions of $(\mathcal{P},•,M)$ by $(V,{•}_V,M_V)$ with the associated isomorphism $F:({\widehat{\mathcal{P}}}_1,{\widehat{•}}_1,{\widehat{M}}_1)\to ({\widehat{\mathcal{P}}}_2,{\widehat{•}}_2,{\widehat{M}}_2)$ such that the diagram in (17) is commutative. Let ${\mathbf{s}}_1$ be a section of $({\widehat{\mathcal{P}}}_1,{\widehat{•}}_1,{\widehat{M}}_1)$. As ${\mathbf{p}}_2\circ F={\mathbf{p}}_1$, we have the following:

$${\mathbf{p}}_2\circ (F\circ {\mathbf{s}}_1)={\mathbf{p}}_1\circ {\mathbf{s}}_1={\mathrm{id}}_{\mathcal{P}}.$$

That is, $F\circ {\mathbf{s}}_1$ is a section of $({\widehat{\mathcal{P}}}_2,{\widehat{•}}_2,{\widehat{M}}_2)$. Denote ${\mathbf{s}}_2:=F\circ {\mathbf{s}}_1$. Since F is an isomorphism of modified Rota–Baxter pre-Lie algebras such that ${F|}_V={\mathrm{id}}_V$, we have the following:

$$\begin{array}{cc}\hfill {\omega }_2(a,b)& ={\mathbf{s}}_2\left(a\right){\widehat{•}}_2{\mathbf{s}}_2\left(b\right)-{\mathbf{s}}_2(a•b)\hfill \\ \hfill & =F\circ {\mathbf{s}}_1\left(a\right){\widehat{•}}_{2}F\circ {\mathbf{s}}_1\left(b\right)-F\circ {\mathbf{s}}_1(a•b)\hfill \\ \hfill & =F\left({\mathbf{s}}_1\left(a\right){\widehat{•}}_1{\mathbf{s}}_1\left(b\right)-{\mathbf{s}}_1(a•b)\right)\hfill \\ \hfill & =F\left({\omega }_1(a,b)\right)\hfill \\ \hfill & ={\omega }_1(a,b)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\chi }_2\left(a\right)& =\widehat{M}{\mathbf{s}}_2\left(a\right)-{\mathbf{s}}_2\left(Ma\right)\hfill \\ \hfill & =\widehat{M}(F\circ {\mathbf{s}}_1\left(a\right))-F\circ {\mathbf{s}}_1\left(Ma\right)\hfill \\ \hfill & =\widehat{M}\left({\mathbf{s}}_1\left(a\right)\right)-{\mathbf{s}}_1\left(M\left(a\right)\right)\hfill \\ \hfill & ={\chi }_1\left(a\right).\hfill \end{array}$$

Thus, two isomorphic abelian extensions give rise to the same element in ${\mathcal{H}}_{\mathrm{MRBPLie}}^2(\mathcal{P},V)$.

Conversely, given two 2-cocycles $({\omega }_1,{\chi }_1)$ and $({\omega }_2,{\chi }_2)$, we can construct two abelian extensions, $(\mathcal{P}\oplus V,{•}_{{\omega }_1},M_{{\chi }_1})$ and $(\mathcal{P}\oplus V,{•}_{{\omega }_2},M_{{\chi }_2})$, via Proposition 9. If they represent the same cohomology class in ${\mathcal{H}}_{\mathrm{MRBPLie}}^2(\mathcal{P},V)$, then there is a linear map, $\iota :\mathcal{P}\to V$, such that

$$({\omega }_1,{\chi }_1)-({\omega }_2,{\chi }_2)=\partial \left(\iota \right).$$

Define a linear map, $F_{\iota }:\mathcal{P}\oplus V\to \mathcal{P}\oplus V$, by $F_{\iota }(a+u):=a+\iota \left(a\right)+u,a\in \mathcal{P},u\in V.$ Then, it is easy to verify that $F_{\iota }$ is an isomorphism of the two abelian extensions $(\mathcal{P}\oplus V,{•}_{{\omega }_1},M_{{\chi }_1})$ and $(\mathcal{P}\oplus V,{•}_{{\omega }_2},M_{{\chi }_2})$. □

5. Modified Rota–Baxter Pre-Lie 2-Algebras and Crossed Modules

In this section, we introduce the notion of modified Rota–Baxter pre-Lie 2-algebras and show that skeletal modified Rota–Baxter pre-Lie 2-algebras are classified by 3-cocycles of modified Rota–Baxter pre-Lie algebras.

We first recall the notion of pre-Lie 2-algebras from [18], which is a categorization of a pre-Lie algebra.

A pre-Lie 2-algebra is a quintuple, $({\mathcal{P}}_0,{\mathcal{P}}_1,h,l_2,l_3)$, where $h:{\mathcal{P}}_1\to {\mathcal{P}}_0$ is a linear map, $l_2:{\mathcal{P}}_i\times {\mathcal{P}}_j\to {\mathcal{P}}_{i+j}$ are bilinear maps and $l_3:{\mathcal{P}}_0\times {\mathcal{P}}_0\times {\mathcal{P}}_0\to {\mathcal{P}}_1$ is a trilinear map, such that for any $a,b,c,x\in {\mathcal{P}}_0$ and $u,v\in {\mathcal{P}}_1$, the following equations are satisfied:

$$\begin{array}{cc}\hfill & h{l}_2(a,u)=l_2(a,h\left(u\right)),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & h{l}_2(u,a)=l_2(h\left(u\right),a),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & l_2(h\left(u\right),v)=l_2(u,h\left(v\right)),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & h{l}_3(a,b,c)=l_2(a,l_2(b,c))-l_2(l_2(a,b),c)-l_2(b,l_2(a,c))+l_2(l_2(b,a),c),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & l_3(a,b,h\left(u\right))=l_2(a,l_2(b,u))-l_2(l_2(a,b),u)-l_2(b,l_2(a,u))+l_2(l_2(b,a),u),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & l_3(h\left(u\right),b,c)=l_2(u,l_2(b,c))-l_2(l_2(u,b),c)-l_2(b,l_2(u,c))+l_2(l_2(b,u),c),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & l_2(x,l_3(a,b,c))-l_2(a,l_3(x,b,c))+l_2(b,l_3(x,a,c))+l_2(l_3(a,b,x),c)-l_2(l_3(x,b,a),c)\hfill \\ \hfill & +l_2(l_3(x,a,b),c)-l_3(a,b,l_2(x,c))+l_3(x,b,l_2(a,c))-l_3(x,a,l_2(b,c))-l_3(l_2(x,a)-l_2(a,x),b,c)\hfill \\ \hfill & +l_3(l_2(x,b)-l_2(b,x),a,c)-l_3(l_2(a,b)-l_2(b,a),x,c)=0.\hfill \end{array}$$

(30)

Motivated by [18] and [26], we propose the notion of a modified Rota–Baxter pre-Lie 2-algebra.

Definition 14. 

A modified Rota–Baxter pre-Lie 2-algebra consists of a pre-Lie 2-algebra, $\mathfrak{P}=({\mathcal{P}}_0,{\mathcal{P}}_1,h,l_2,l_3)$ and a modified Rota–Baxter 2-operator $\mathfrak{M}=(M_0,M_1,M_2)$ on $\mathfrak{P}$, where $M_0:{\mathcal{P}}_0\to {\mathcal{P}}_0$, $M_1:{\mathcal{P}}_1\to {\mathcal{P}}_1$ and $M_2:{\mathcal{P}}_0\times {\mathcal{P}}_0\to {\mathcal{P}}_1$, for any $a,b,c\in {\mathcal{P}}_0,u\in {\mathcal{P}}_1$, satisfying the following equations:

$$\begin{array}{cc}\hfill & M_0\circ h=h\circ M_1,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & h{M}_2(a,b)+l_2(M_{0}a,M_{0}b)=M_0\left(l_2(M_0\left(a\right),b)+l_2(a,M_0\left(b\right))\right)-l_2(a,b),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & M_2(h\left(u\right),b)+l_2(M_{1}u,M_{0}b)=M_1\left(l_2(M_1\left(u\right),b)+l_2(u,M_0\left(b\right))\right)-l_2(u,b),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & M_2(a,h\left(u\right))+l_2(M_{0}a,M_{1}u)=M_1\left(l_2(M_0\left(a\right),u)+l_2(a,M_1\left(u\right))\right)-l_2(a,u),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & M_1{l}_2(a,M_2(b,c))-l_2(M_{0}a,M_2(b,c))+l_2(M_{0}b,M_2(a,c))-M_1{l}_2(b,M_2(a,c))\hfill \\ \hfill & -l_2(M_2(b,a),M_{0}c)+M_1{l}_2(M_2(b,a),c)+l_2(M_2(a,b),M_{0}c)-M_1{l}_2(M_2(a,b),c)\hfill \\ \hfill & +M_2(b,l_2(M_{0}a,c)+l_2(a,M_{0}c))-M_2(a,l_2(M_{0}b,c)+l_2(b,M_{0}c))\hfill \\ \hfill & +M_2(l_2(M_{0}a,b)+l_2(a,M_{0}b)-l_2(M_{0}b,a)-l_2(b,M_{0}a),c)-l_3(M_{0}a,M_{0}b,M_{0}c)\hfill \\ \hfill & +M_1(l_3(a,M_{0}b,M_{0}c)+l_3(M_{0}a,b,M_{0}c)+l_3(M_{0}a,M_{0}b,c))\hfill \\ \hfill & -l_3(M_{0}a,b,c)-l_3(a,M_{0}b,c)-l_3(a,b,M_{0}c)+M_1{l}_3(a,b,c)=0.\hfill \end{array}$$

(35)

We denote a modified Rota–Baxter pre-Lie 2-algebra by $(\mathfrak{P},\mathfrak{M})$.

A modified Rota–Baxter pre-Lie 2-algebra is said to be skeletal (resp. strict) if $h=0$ (resp. $l_3=0,M_2=0$).

Example 6. 

For any modified Rota–Baxter pre-Lie algebra, $(\mathcal{P},•,M)$, $({\mathcal{P}}_0={\mathcal{P}}_1=\mathcal{P},h=0,l_2=•,M_0=M_1=M)$ is a strict modified Rota–Baxter pre-Lie 2-algebra.

Proposition 11. 

Let $(\mathfrak{P},\mathfrak{M})$ be a modified Rota–Baxter pre-Lie 2-algebra.

(i) If $(\mathfrak{P},\mathfrak{M})$ is skeletal or strict, then $({\mathcal{P}}_0,{•}_0,M_0)$ is a modified Rota–Baxter pre-Lie algebra, where $a{•}_{0}b=l_2(a,b)$ for any $a,b\in {\mathcal{P}}_0$.

(ii) If $(\mathfrak{P},\mathfrak{M})$ is strict, then $({\mathcal{P}}_1,{•}_1,M_1)$ is a modified Rota–Baxter pre-Lie algebra, where $u{•}_{1}v=l_2(h\left(u\right),v)=l_2(u,h\left(v\right))$ for any $u,v\in {\mathcal{P}}_1$.

(iii) If $(\mathfrak{P},\mathfrak{M})$ is skeletal or strict, then $({\mathcal{P}}_1;{•}_l,{•}_r,M_1)$ is a bimodule of $({\mathcal{P}}_0,{•}_0,M_0)$ where $a{•}_{l}u=l_2(a,u)$ and $u{•}_{r}a=l_2(u,a)$ for $a\in {\mathcal{P}}_0,u\in {\mathcal{P}}_1$.

Proof. 

Then, (i), (ii) and (iii) can be directly verified by Equations (24)–(29) and (31)–(34). □

Theorem 2. 

There is a one-to-one correspondence between skeletal modified Rota–Baxter pre-Lie 2-algebras and 3-cocycles of modified Rota–Baxter pre-Lie algebras.

Proof. 

Let $(\mathfrak{P},\mathfrak{M})$ be a skeletal modified Rota–Baxter pre-Lie 2-algebra. According to Proposition 11, we can consider the cohomology of modified Rota–Baxter pre-Lie algebra to be $({\mathcal{P}}_0,{•}_0,M_0)$ with coefficients in the bimodule $({\mathcal{P}}_1;{•}_l,{•}_r,M_1)$. For any $a,b,c,x\in {\mathcal{P}}_0$, combining Equations (8) and (30), we have the following:

$$\begin{array}{cc}\hfill & \delta l_3(x,a,b,c)\hfill \\ \hfill =& x{•}_l{l}_3(a,b,c)-a{•}_l{l}_3(x,b,c)+b{•}_l{l}_3(x,a,c)+l_3(a,b,x){•}_{r}c-l_3(x,b,a){•}_{r}c+l_3(x,a,b){•}_{r}c\hfill \\ \hfill & -l_3(a,b,x{•}_{0}c)+l_3(x,b,a{•}_{0}c)-l_3(x,a,b{•}_{0}c)-l_3(x{•}_{0}a-a{•}_{0}x,b,c)+l_3(x{•}_{0}b-b{•}_{0}x,a,c)\hfill \\ \hfill & -l_3(a{•}_{0}b-b{•}_{0}a,x,c)\hfill \\ \hfill =& l_2(x,l_3(a,b,c))-l_2(a,l_3(x,b,c))+l_2(b,l_3(x,a,c))+l_2(l_3(a,b,x),c)-l_2(l_3(x,b,a),c)+l_2(l_3(x,a,b),c)\hfill \\ \hfill & -l_3(a,b,l_2(x,c))+l_3(x,b,l_2(a,c))-l_3(x,a,l_2(b,c))-l_3(l_2(x,a)-l_2(a,x),b,c)\hfill \\ \hfill & +l_3(l_2(x,b)-l_2(b,x),a,c)-l_3(l_2(a,b)-l_2(b,a),x,c)\hfill \\ \hfill =& 0.\hfill \end{array}$$

According to Equations (9) and (35), the following hold true:

$$\begin{array}{cc}\hfill & (-{\delta }_M{M}_2-\mathrm{Y}{l}_3)(a,b,c)=-{\delta }_M{M}_2(a,b,c)-\mathrm{Y}{l}_3(a,b,c)\hfill \\ \hfill =& -M_{0}a{•}_l{M}_2(b,c)+M_1\left(a{•}_l{M}_2(b,c)\right)+M_{0}b{•}_l{M}_2(a,c)-M_1\left(b{•}_l{M}_2(a,c)\right)\hfill \\ \hfill & -M_2(b,a){•}_r{M}_{0}c+M_1\left(M_2(b,a){•}_{r}c\right)+M_2(a,b){•}_r{M}_{0}c-M_1\left(M_2(a,b){•}_{r}c\right)\hfill \\ \hfill & +M_2(b,M_{0}a{•}_{0}c+a{•}_0{M}_{0}c)-M_2(a,M_{0}b{•}_{0}c+b{•}_0{M}_{0}c)\hfill \\ \hfill & +M_2(M_{0}a{•}_{0}b+a{•}_0{M}_{0}b-M_{0}b{•}_{0}a-b{•}_0{M}_{0}a,c)-l_3(M_{0}a,M_{0}b,M_{0}c)\hfill \\ \hfill & +M_1(l_3(a,M_{0}b,M_{0}c)+l_3(M_{0}a,b,M_{0}c)+l_3(M_{0}a,M_{0}b,c))\hfill \\ \hfill & -l_3(M_{0}a,b,c)-l_3(a,M_{0}b,c)-l_3(a,b,M_{0}c)+M_1{l}_3(a,b,c)\hfill \\ \hfill =& -l_2(M_{0}a,M_2(b,c))+M_1{l}_2(a,M_2(b,c))+l_2(M_{0}b,M_2(a,c))-M_1{l}_2(b,M_2(a,c))\hfill \\ \hfill & -l_2(M_2(b,a),M_{0}c)+M_1{l}_2(M_2(b,a),c)+l_2(M_2(a,b),M_{0}c)-M_1{l}_2(M_2(a,b),c)\hfill \\ \hfill & +M_2(b,l_2(M_{0}a,c)+l_2(a,M_{0}c))-M_2(a,l_2(M_{0}b,c)+l_2(b,M_{0}c))\hfill \\ \hfill & +M_2(l_2(M_{0}a,b)+l_2(a,M_{0}b)-l_2(M_{0}b,a)-l_2(b,M_{0}a),c)-l_3(M_{0}a,M_{0}b,M_{0}c)\hfill \\ \hfill & +M_1(l_3(a,M_{0}b,M_{0}c)+l_3(M_{0}a,b,M_{0}c)+l_3(M_{0}a,M_{0}b,c))\hfill \\ \hfill & -l_3(M_{0}a,b,c)-l_3(a,M_{0}b,c)-l_3(a,b,M_{0}c)+M_1{l}_3(a,b,c)\hfill \\ \hfill =& 0.\hfill \end{array}$$

Thus, $\partial (l_3,M_2)=(\delta l_3,-{\delta }_M{M}_2-\mathrm{Y}{l}_3)=0,$ that is $(l_3,M_2)\in {\mathcal{C}}_{\mathrm{MRBPLie}}^3({\mathcal{P}}_0,{\mathcal{P}}_1)$ is a 3-cocycle of a modified Rota–Baxter pre-Lie algebra $({\mathcal{P}}_0,{•}_0,M_0)$ with coefficients in the bimodule $({\mathcal{P}}_1;{•}_l,{•}_r,M_1)$.

Conversely, suppose that $(l_3,M_2)\in {\mathcal{C}}_{\mathrm{MRBPLie}}^3(\mathcal{P},V)$ is a 3-cocycle of a modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$ with coefficients in the bimodule $(V;{•}_l,{•}_r,M_V)$. Then, $(\mathfrak{P},\mathfrak{M})$ is a skeletal modified Rota–Baxter pre-Lie 2-algebra, where $\mathfrak{P}=({\mathcal{P}}_0=\mathcal{P},{\mathcal{P}}_1=V,h=0,l_2,l_3)$ and $\mathfrak{M}=(M_0=M,M_1=M_V,M_2)$ with $l_2(a,b)=a•b,l_2(a,u)=a{•}_{l}u,l_2(u,a)=u{•}_{r}a$ for any $a,b\in {\mathcal{P}}_0,u\in {\mathcal{P}}_1$. □

Whitehead [36] introduced the notion of crossed modules in the context of homotopy theory. At the end of this section, we introduce the notion of crossed modules of modified Rota–Baxter pre-Lie algebras and show that they are equivalent to strict modified Rota–Baxter pre-Lie 2-algebras.

Definition 15. 

A crossed module of modified Rota–Baxter pre-Lie algebras is a quadruple $\left(({\mathcal{P}}_0,{•}_0,M_0),({\mathcal{P}}_1,{•}_1,M_1),h,({•}_l,{•}_r)\right)$, where $({\mathcal{P}}_0,{•}_0,M_0)$ and $({\mathcal{P}}_1,{•}_1,M_1)$ are modified Rota–Baxter pre-Lie algebras, $h:{\mathcal{P}}_1\to {\mathcal{P}}_0$ is a homomorphism of modified Rota–Baxter pre-Lie algebras and $({\mathcal{P}}_1;{•}_l,{•}_r,M_1)$ is a bimodule of $({\mathcal{P}}_0,{•}_0,M_0)$, for any $a\in {\mathcal{P}}_0,u,v\in {\mathcal{P}}_1$, satisfying the following equations:

$$\begin{array}{cc}\hfill & h\left(a{•}_{l}u\right)=a{•}_{0}h\left(u\right),h\left(u{•}_{r}a\right)=h\left(u\right){•}_{0}a,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & h\left(u\right){•}_{l}v=u{•}_{r}h\left(v\right)=u{•}_{1}v.\hfill \end{array}$$

(37)

Example 7. 

Let $(\mathcal{P},•,M)$ be a modified Rota–Baxter pre-Lie algebra, $\mathcal{F}$ be its two-sided ideal—that is, $\mathcal{F}$ satisfies $\mathcal{F}•\mathcal{P}\subseteq \mathcal{F},\mathcal{P}•\mathcal{F}\subseteq \mathcal{F}$ and $M\left(\mathcal{F}\right)\subseteq \mathcal{F}$—and $in:\mathcal{F}\to \mathcal{P}$ be the inclusion. Then, $\left((\mathcal{P},•,M),(\mathcal{F},•,M{|}_{\mathcal{F}}),in,({•}_l={•}_r=•)\right)$ is a crossed module of modified Rota–Baxter pre-Lie algebras. In particular, $((\mathcal{P},•,M),(\mathcal{P},•,M),{\mathrm{id}}_{\mathcal{P}},$$({•}_l={•}_r=•))$ is a crossed module of modified Rota–Baxter pre-Lie algebras.

Example 8. 

Let $F:({\mathcal{P}}_1,{•}_1,M_1)\to ({\mathcal{P}}_0,{•}_0,M_0)$ be a homomorphism of modified Rota–Baxter pre-Lie algebras. Then, $\ker \left(F\right)$ is a two-sided ideal of $({\mathcal{P}}_1,{•}_1,M_1)$. Thus, according to Example 7, $\left(({\mathcal{P}}_1,{•}_1,M_1),(\ker \left(F\right),{•}_1,M_1{|}_{\ker \left(F\right)}),in,({•}_l={•}_r={•}_1)\right)$ is a crossed module of modified Rota–Baxter pre-Lie algebras.

Example 9. 

Let $(V;{•}_l,{•}_r,M_V)$ be a bimodule over a modified Rota–Baxter pre-Lie algebra $(\mathcal{P},•,M)$. Endow V with the trivial pre-Lie algebra structure, ${•}_V=0$; in this case, $\left(\right(\mathcal{P},•,M),$$(V,{•}_V,M_V),0,$$({•}_l,{•}_r))$ is a crossed module of modified Rota–Baxter pre-Lie algebras.

Theorem 3. 

There is a one-to-one correspondence between strict modified Rota–Baxter pre-Lie 2-algebras and crossed modules of modified Rota–Baxter pre-Lie algebras.

Proof. 

Let $(({\mathcal{P}}_0,{\mathcal{P}}_1,h,l_2,l_3=0),(M_0,M_1,M_2=0))$ be a strict modified Rota–Baxter pre-Lie 2-algebra. Define the following two operations on ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$:

$$\begin{array}{cc}\hfill & a{•}_{0}b=l_2(a,b),\hfill \\ \hfill & u{•}_{1}v=l_2(h\left(u\right),v)=l_2(u,h\left(v\right))\mathrm{for}\mathrm{all}a,b\in {\mathcal{P}}_0,u,v\in {\mathcal{P}}_1.\hfill \end{array}$$

It is straightforward to see that both $({\mathcal{P}}_0,{•}_0,M_0)$ and $({\mathcal{P}}_1,{•}_1,M_1)$ are modified Rota–Baxter pre-Lie algebras. $l_2$ also gives rise to two maps: ${•}_l:{\mathcal{P}}_0\times {\mathcal{P}}_1\to {\mathcal{P}}_1,{•}_r:{\mathcal{P}}_1\times {\mathcal{P}}_0\to {\mathcal{P}}_1$ by

$$\begin{array}{cc}\hfill & a{•}_{r}u=l_2(a,u),u{•}_{r}a=l_2(u,a)\mathrm{for}\mathrm{all}a\in {\mathcal{P}}_0,u\in {\mathcal{P}}_1.\hfill \end{array}$$

According to (33) and (34), we deduce that $({\mathcal{P}}_1;{•}_l,{•}_r,M_1)$ is a bimodule of $({\mathcal{P}}_0,{•}_0,M_0)$. According to Equation (26), we have

$$h\left(u{•}_{1}v\right)=h{l}_2(h\left(u\right),v)=l_2(h\left(u\right),h\left(v\right))=h\left(u\right){•}_{0}h\left(v\right),$$

which implies that h is a homomorphism of modified Rota–Baxter pre-Lie algebras. Furthermore, we have

$$\begin{array}{cc}\hfill & h\left(a{•}_{l}u\right)=h{l}_2(a,u)=l_2(a,h\left(u\right))=a{•}_{0}h\left(u\right),\hfill \\ \hfill & h\left(u{•}_{r}a\right)=h{l}_2(u,a)=l_2(h\left(u\right),a)=h\left(u\right){•}_{0}a,\hfill \\ \hfill & h\left(u\right){•}_{l}v=l_2(h\left(u\right),v)=l_2(u,h\left(v\right))=u{•}_{r}h\left(v\right)=u{•}_{1}v.\hfill \end{array}$$

Thus, we obtain a crossed module of modified Rota–Baxter pre-Lie algebras.

Conversely, a crossed module of modified Rota–Baxter pre-Lie algebras $\left(\right({\mathcal{P}}_0,{•}_0,M_0),$$({\mathcal{P}}_1,{•}_1,M_1),h,({•}_l,{•}_r))$ gives rise to a strict modified Rota–Baxter pre-Lie 2-algebra $\left(\right({\mathcal{P}}_0,{\mathcal{P}}_1,h,$$l_2,l_3=0),(M_0,M_1,M_2=0))$, in which $l_2:{\mathfrak{g}}_i\times {\mathfrak{g}}_j\to {\mathfrak{g}}_{i+j}$ are given by

$$\begin{array}{cc}\hfill l_2(a,b)& =a{•}_{0}b,\hfill \\ \hfill l_2(u,v)& =u{•}_{1}v,\hfill \\ \hfill l_2(a,u)& =a{•}_{l}u,\hfill \\ \hfill l_2(u,a)& =u{•}_{r}a,\hfill \end{array}$$

for all $a\in {\mathcal{P}}_0,u,v\in {\mathcal{P}}_1$. The crossed module equations give various equations for strict modified Rota–Baxter pre-Lie 2-algebras. The proof is completed. □

6. Conclusions

In the current research, we mainly study a modified Rota–Baxter pre-Lie algebra, which includes a modified Rota–Baxter operator and a pre-Lie algebra. More precisely, we introduce the bimodule of a modified Rota–Baxter pre-Lie algebra. We show that a modified Rota–Baxter pre-Lie algebra induces a pre-Lie algebra, and the bimodule of a modified Rota–Baxter pre-Lie algebra induces the bimodule of a pre-Lie algebra. Considering this fact, we define the cohomology of a modified Rota–Baxter operator on a pre-Lie algebra. Using the cohomology of pre-Lie algebras, we construct a cochain map, and the cohomology of modified Rota–Baxter pre-Lie algebras is defined. We study infinitesimal deformations of modified Rota–Baxter pre-Lie algebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We investigate abelian extensions of modified Rota–Baxter pre-Lie algebras by using the second cohomology group. Additionally, the notion of modified Rota–Baxter pre-Lie 2-algebra is introduced, which is the categorization of a modified Rota–Baxter pre-Lie algebra. We study the skeletal modified Rota–Baxter pre-Lie 2-algebras using the third cohomology group. Finally, we introduce the notion of crossed modules of modified Rota–Baxter pre-Lie algebras, give some examples, and prove that strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to crossed modules of modified Rota–Baxter pre-Lie algebras.