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

: 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.

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, Υ, 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, K denotes a field of characteristic zero.All the vector spaces and (multi)linear maps are taken over K.

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 (P, •) consisting of a vector space, P, and a binary operation, •, P × P → P, such that for all a, b, c ∈ P, the associator: defines a Lie algebra structure on P, which is called the sub-adjacent Lie algebra of (P, •), and we denote it by 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 (P, •) be a pre-Lie algebra.A modified Rota-Baxter operator on P is a linear map, M : P → P, subject to the following: Furthermore, the triple (P, •, M) is called a modified Rota-Baxter pre-Lie algebra, simply denoted by (P, M).
(ii) A homomorphism between two modified Rota-Baxter pre-Lie algebras (P 1 , M 1 ) and (P 2 , M 2 ) is a pre-Lie algebra homomorphism, F : Example 1.Let (P, •) be a pre-Lie algebra.Then, (P, •, id P ) is a modified Rota-Baxter pre-Lie algebra, where id P : P → P is an identity mapping.
Example 2. Let (P, •) be a two-dimensional pre-Lie algebra and {ϵ 1 , ϵ 2 } be a basis, whose nonzero products are given as follows: Then, the triple (P, •, M) is a two-dimensional modified Rota-Baxter pre-Lie algebra, where Example 3. Let (P, •) be a pre-Lie algebra.If a linear map, M : P → P, is a modified Rota-Baxter operator, then −M is also a modified Rota-Baxter operator.
Definition 3 ([16]).Let (P, •) be a pre-Lie algebra.A Rota-Baxter operator of weight-1 on P is a linear map, R : P → P, subject to the following: Then, the triple (P, •, R) is called a Rota-Baxter pre-Lie algebra of weight-1.
Proposition 1.Let (P, •) be a pre-Lie algebra.If a linear map, R : P → P, is a Rota-Baxter operator of weight -1, then the map, 2R − id P , is a modified Rota-Baxter operator on P.
Proof.For any a, b ∈ P, we have the following: The proposition follows.
Recall from [16] that a Nijenhuis operator on a pre-Lie algebra (P, •) is a linear map, N : P → P, that satisfies the following, for all a, b ∈ P. The relationship between the modified Rota-Baxter operator and Nijenhuis operator is as follows, which proves to be obvious.Proposition 2. Let (P, •) be a pre-Lie algebra and N : P → P be a linear map.If N 2 = id P , then N is a Nijenhuis operator if, and only if, N is a modified Rota-Baxter operator.

Definition 4 ([8]
).Let (P, •) be a pre-Lie algebra and V a vector space.A bimodule of P on V consists of a pair (• l , • r ), where • l : P × V → V and • r : V × P → V are two linear maps satisfying the following: Definition 5. A bimodule of the modified Rota-Baxter pre-Lie algebra (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 (P, •) ; (ii) M V : V → V is a linear map satisfying the following equations, for a ∈ P and u ∈ V.In this case, the quadruple (V; • l , • r , M V ) is also called a representation over (P, •, M).
Example 4. (P; for a ∈ P and u ∈ V.In the case, the modified Rota-Baxter pre-Lie algebra P ⊕ V is called a semidirect product of P and V, denoted by P ⋉ Proof.Firstly, it is easy to verify that (P ⊕ V, • ⋉ ) is a pre-Lie algebra.In addition, for any a, b ∈ P and u, v ∈ V, via Equations ( 2)-( 4), we have Then, (i) (P, • M ) is a pre-Lie algebra.We denote this pre-Lie algebra as P M .
(ii) (P M , M) is a modified Rota-Baxter pre-Lie algebra.
Proof.(i) For any a, b, c ∈ P, according to Equations ( 1) and ( 2), we have the following: Thus, (P, • M ) is a pre-Lie algebra.
(ii) For any a, b ∈ P, according to Equation ( 2), we have Hence, (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, (P, •, M).Define two bilinear maps, Then, (V; Proof.First, by direct verification, we determine that(V; • M l , • M r ) is a bimodule of the pre-Lie algebra P M .Further, for any a ∈ P and u ∈ V, according to Equation (3), we have the following: Similarly, according to Equation ( 4), there is also ) is an adjoint bimodule of the modified Rota-Baxter pre-Lie algebra (P M , M), where for any a, b ∈ P.

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 (P, •) be a pre-Lie algebra and (V; • l , • r ) be a bimodule of it.Denote the n−cochains of P with coefficients in representation V via the following:
The coboundary operator δ : PLie (P, V), as follows: Then, it is proven in [17] that δ 2 = 0. Let us denote, via H * PLie (P, V), the cohomology group associated to the cochain complex (C * PLie (P, V), δ).We first study the cohomology of the modified Rota-Baxter operator.Let (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, P M , and a new bimodule, V M = (V; • M l , • M r ), over P M .Consider the cochain complex of P M with coefficients in V M : More precisely, C n PLie (P M , V M ) := Hom(P ⊗n M , V M ) and its coboundary map, ), are given as follows: In particular, for n = 1, Definition 6.Let (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 * PLie (P M , V M ), δ M ) is called the cochain complex of the modified Rota-Baxter operator, M, with coefficients in V M , denoted by (C * MRBO (P, V), δ M ).The cohomology of (C * MRBO (P, V), δ M ), denoted by H * MRBO (P, V), is called the cohomology of modified Rota-Baxter operator M with coefficients in V M .
In particular, when (P; is the adjoint bimodule of (P M , M), we denote (C * MRBO (P, P ), δ M ) as (C * MRBO (P ), δ M ) and call it the cochain complex of modified Rota-Baxter operator M, denote H * MRBO (P, P ) as H * MRBO (P ) 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 ≥ 1, we define a linear map, Υ : C n PLie (P, V) → C n MRBO (P, V), via the following: 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 Υ : Lemma 1. Map Υ is a cochain map, i.e., Υ • δ = δ M • Υ.In other words, the following diagram is commutative: 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 ∈ C 1 PLie (P, V) and a, b ∈ P, according to Equations ( 2)-( 10) and ( 12), we have the following: and Further comparing Equations ( 13) and ( 14), we have (13) = (14).Therefore, Definition 7. Let (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 (C * MRBPLie (P, V), ∂) of a modified Rota-Baxter pre-Lie algebra (P, •, M) with coefficients in (V; • l , • r , M V ) to the negative shift in the mapping cone of Υ, that is, let MRBPLie (P, V) is given by the following: For n ≥ 2, the coboundary map ∂ : C n MRBPLie (P, V) → C n+1 MRBPLie (P, V) is given by the following: The cohomology of (C * MRBPLie (P, V), ∂), denoted by H * MRBPLie (P, V), is called the cohomology of the modified Rota-Baxter pre-Lie algebra (P, •, M) with coefficients in (V; • l , • r , M V ).In particular, when (V; It is obvious that there is a short exact sequence of cochain complexes: This induces a long exact sequence of cohomology groups: 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 (P, •, M) is a pair (• t , M t ) of the following forms, such that the following conditions are satisfied: • t , M t ) be an infinitesimal deformation of modified Rota-Baxter pre-Lie algebra (P, •, M).Then, (• 1 , M 1 ) is a 2-cocycle in the cochain complex (C * MRBPLie (P ), ∂).

Proof. Suppose (P [[t]],
• t , M t ) is a modified Rota-Baxter pre-Lie algebra.Then, for any a, b, c ∈ P, we have Comparing coefficients of t 1 on both sides of the above equations, we have • t , M t ) is a linear map, φ t = id + tφ 1 , where φ 1 : P → P is a linear map, such that: In this case, we say that the two infinitesimal deformations (P Proposition 7. The infinitesimals of two equivalent infinitesimal deformations of (P, •, M) are in the same cohomology class in H 2 MRBPLie (P ).
• t , M t ) be an isomorphism.By expanding Equations ( 15) and ( 16) and comparing the coefficients of t 1 on both sides, we have that is, we have the following: Therefore, (• ′ 1 , M ′ 1 ) and (• 1 , M 1 ) are cohomologous and belong to the same cohomology class in H 2 MRBPLie (P ).

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 (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 ( P, •, M) of (P, •, M) by (V, • V , M V ) is a short exact sequence of morphisms of modified Rota-Baxter pre-Lie algebras, is an abelian ideal of P. Definition 12.A section of an abelian extension ( P, •, M) of (P, •, M) by (V, • V , M V ) is a linear map, s : P → P, such that p • s = id P and s • M = M • s.Definition 13.Let ( P1 , •1 , M1 ) and ( P2 , •2 , M2 ) be two abelian extensions of (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 : ( P1 , •1 , M1 ) → ( P2 , •2 , M2 ) such that the following diagram is commutative: Now for an abelian extension ( P, •, M) of (P, •, M) by (V, • V , M V ) with a section, s : P → P, we define two bilinear maps, Proposition 8.With the above notations, (V; • l , • r , M V ) is a bimodule of the modified Rota-Baxter pre-Lie algebra (P, •, M) and does not depend on the choice of s.
Proof.First, for any other section, s ′ : P → P, for any a ∈ P, we have the following: Thus, there exists an element, u ∈ V, such that s ′ (a) = s(a) + u.Note that V is an abelian ideal of P; this yields the following: This means that • l , • r does not depend on the choice of s.Next, for any a, b ∈ P and u ∈ V, V is an abelian ideal of P and s(a) •s(b) − s(a • b) ∈ V; we have the following: By the same token, there is also a On the other hand, according to Ms(a) − s(Ma) ∈ V, we have the following: In the same way, there is also Let ( P, •, M) be an abelian extension of (P, •, M) by (V, • V , M V ) and s : P → P be a section of it.Define the maps ω : P × P → V and χ : P → V by the following, respectively: We transfer the modified Rota-Baxter pre-Lie algebra structure on P to P ⊕ V by endowing P ⊕ V with a multiplication, • ω , and a modified Rota-Baxter operator, M χ , defined by the following: Proposition 9.The triple (P ⊕ V, • ω , M χ ) is a modified Rota-Baxter pre-Lie algebra if, and only if, (ω, χ) is a 2-cocycle of the modified Rota-Baxter pre-Lie algebra (P, •, M) with the coefficient in (V, • V , M V ).In this case, Proof.The triple (P ⊕ V, • ω , M χ ) is a modified Rota-Baxter pre-Lie algebra if, and only if, for any a, b, c ∈ P and u, v, w ∈ V, the following equations hold true: Further, Equations ( 20) and ( 21) are equivalent to the following equations: Using Equations ( 22) and ( 23), we have δω = 0 and −δ M χ − Υω = 0, respectively.Therefore, ∂(ω, χ) = (δω, −δ M χ − Υω) = 0, that is, (ω, χ) is a 2-cocycle.Conversely, if (ω, χ) is a 2-cocycle of (P, •, M) with the coefficient in (V, • V , M V ), then we have ∂(ω, χ) = (δω, −δ M χ − Υω) = 0, in which case Equations ( 20) and ( 21) hold true.Hence, (P ⊕ V, • ω , M χ ) is a modified Rota-Baxter pre-Lie algebra.Proposition 10.Let ( P, •, M) be an abelian extension of (P, •, M) by (V, • V , M V ) and s be a section of it.If the pair (ω, χ) is a 2-cocycle of (P, •, M) with the coefficient in (V, • V , M V ) constructed using the section s, then its cohomology class does not depend on the choice of s.Proof.Let s 1 , s 2 : P → P be two distinct sections; according to Proposition 9, we have two corresponding 2-cocycles, (ω 1 , χ 1 ) and (ω 2 , χ 2 ), respectively.Define a linear map, γ : P → V, by γ(a) = s 1 (a) − s 2 (a).Then, and Hence, (ω 1 , χ 1 ) − (ω 2 , χ 2 ) = (δγ, −Υγ) = ∂(γ) ∈ B 2 MRBPLie (P, V), that is (ω 1 , χ 1 ) and (ω 2 , χ 2 ) form the same cohomological class in H 2 MRBPLie (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 (P, •, M) by (V, • V , M V ) are classified by the second cohomology group, H 2 MRBPLie (P, V).
Example 9. Let (V; • l , • r , M V ) be a bimodule over a modified Rota-Baxter pre-Lie algebra (P, •, M).Endow V with the trivial pre-Lie algebra structure, • V = 0; in this case, (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.

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.

Proposition 4 .
Let (P, •, M) be a modified Rota-Baxter pre-Lie algebra.Define a new operation as follows: a • M b =Ma • b + a • Mb for all a, b ∈ P.
an adjoint bimodule of the modified Rota-Baxter pre-Lie algebra (P, •, M).The quadruple (V; • l , • r , M V ) is a bimodule of a modified Rota-Baxter pre-Lie algebra (P, •, M) if, and only if, P ⊕ V is a modified Rota-Baxter pre-Lie algebra with the following maps, we just denote (C MRBPLie (P ), and call them the cochain complex and the cohomology of the modified Rota-Baxter pre-Lie algebra (P, •, M), respectively.