Cohomology of Modified Rota-Baxter Pre-lie Algebras and Its Applications

Cayley [1] first introduced pre-Lie algebras (also called left-symmetric algebra) 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 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]. See also in [7,8,9,10,11,12,13,14,15] for some interesting related about pre-Lie algebras.

Rota-Baxter operators on associative algebras were first introduced by Baxter [16] in his study of probability fluctuation theory, it was further developed by Rota [17]. Rota-Baxter operator has been widely used in many fields of mathematics and physics, including combinatorics, number theory, operads and quantum field theory [18]. The cohomology and deformation theory of Rota-Baxter operators of weight zero have been studied on various algebraic structures, see [19,20,21,22,23]. Recently, Wang and Zhou [24], Das [25] studied Rota-Baxter associative algebras of any weight by different methods respectively. Inspired by Wang and Zhou’s work, Das [26] considered the cohomology and deformations of weighted Rota-Baxter Lie algebras. The authors [27,28] developed the cohomology, extensions and deformations of Rota-Baxter 3-Lie algebras with any weight. In [29], Chen, Lou and Sun studied the cohomology and extensions of Rota-Baxter Lie triple systems. In [30], Guo and his collaborators explored the cohomology, deformations and extensions of Rota-Baxter pre-Lie algebras of arbitrary weights.

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 [31] as a modification of the operator form of the classical Yang-Baxter equation. Due to the importance of Rota-Baxter algebras and modified Rota-Baxter algebras, Zheng, Guo and Qiu [32] studied properties of extended Rota-Baxter operators. Recently, Jiang and Sheng have been established cohomology and deformation theory of modified r-matrices in [33]. Inspired by [33], modified Rota-Baxter algebraic structures have been widely studied in [34,35,36].

However, there was very few study about the modified Rota-Baxter pre-Lie algebras. The purpose of the paper is to study the cohomology of a modified Rota-Baxter pre-Lie algebra and its applications. In precisely, we introduce the concept of a modified Rota-Baxter pre-Lie algebra, which includes a pre-Lie algebra and a modified Rota-Baxter operator. And then, we propose a representation of a modified Rota-Baxter pre-Lie algebra. We define a cochain map ${\rm Y}$, and then the cohomology of modified Rota-Baxter pre-Lie algebras with coefficients in a representation is constructed. Finally, as applications of our propose 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 prove that any skeletal modified Rota-Baxter pre-Lie 2-algebra can be classified by the third cohomology group.

The paper is organized as follows. In Section 2, we introduce the concept of modified Rota-Baxter pre-Lie algebras, and give its representations. In Section 3, we establish the cohomology theory of modified Rota-Baxter pre-Lie algebras with coefficients in a representation, 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-algebra using the third cohomology group.

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 Rota-Baxter pre-Lie algebras motivated by the modified r-matrices in [33] and give some examples. Next we propose the representation of modified Rota-Baxter pre-Lie algebras. Finally, we establish a new modified Rota-Baxter pre-Lie algebra and give its representation.

First, let’s recall some definitions and results about pre-Lie algebra and its representations from [2].

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 ${\mathcal{P}}^c$.

Recall from [13] that a Nijenhuis operator on a pre-Lie algebra $(\mathcal{P},•)$ is a linear map $N:\mathcal{P}\to \mathcal{P}$ satisfies 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.

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

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

Let us recall the cohomology theory of pre-Lie algebras in [14]. Let $(\mathcal{P},•)$ be a pre-Lie algebra and $(V;{•}_l,{•}_r)$ be a representation of it. Denote the $n-$cochains of $\mathcal{P}$ with coefficients in representation V by

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 Then, it has been proved in [14] that ${\delta }^2=0$. Let us denote by $H_{\mathrm{PLie}}^{*}(\mathcal{P},V)$, 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 representation of it, Recall that Proposition 2.13 and Proposition 2.14 give a new pre-Lie algebra ${\mathcal{P}}_M$ and a new representation $V_M=(V;{•}_l^M,{•}_r^M)$ over ${\mathcal{P}}_M$. Consider the cochain complex of ${\mathcal{P}}_M$ with coefficients in $V_M$:More precisely, $C_{\mathrm{PLie}}^n({\mathcal{P}}_M,V_M):=\mathrm{Hom}({\mathcal{P}}_M^{\otimes n},V_M)$ and its coboundary map ${\delta }_M:C_{\mathrm{PLie}}^n({\mathcal{P}}_M,V_M)\to C_{\mathrm{PLie}}^n({\mathcal{P}}_M,V_M)$, for $a_1,\cdots ,a_{n+1}\in {\mathcal{P}}_R$ and $f\in C_{\mathrm{PLie}}^n({\mathcal{P}}_M,V_M)$, is given as follows:

In particular, when $(\mathcal{P};{•}_l^M={•}_r^M={•}^M,M)$ is the adjoint representation of $({\mathcal{P}}_M,M)$, we denote $({\mathcal{C}}_{\mathrm{MRBO}}^{*}(\mathcal{P},\mathcal{P}),{\delta }_M)$ by $({\mathcal{C}}_{\mathrm{MRBO}}^{*}\left(\mathcal{P}\right),{\delta }_M)$ and call it the cochain complex of modified Rota-Baxter operator M, and denote ${\mathcal{H}}_{\mathrm{MRBO}}^{*}(\mathcal{P},\mathcal{P})$ by ${\mathcal{H}}_{\mathrm{MRBO}}^{*}\left(\mathcal{P}\right)$ 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’s construct the following cochain map. For any $n\ge 1$, we define a linear map ${\rm Y}:C_{\mathrm{PLie}}^n(\mathcal{P},V)\to {\mathcal{C}}_{\mathrm{MRBO}}^n(\mathcal{P},V)$ by among them, when the subscript of $j_{2i-3}$ is negative, f is a zero map. For example, when $n=1$, by Eq. (3.4), the map ${\rm Y}:C_{\mathrm{PLie}}^1(\mathcal{P},V)\to {\mathcal{C}}_{\mathrm{MRBO}}^1(\mathcal{P},V)$ is as follows:

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 infinitesimal deformations 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 representation and a 2-cocycle. It is further proved that they are classified by the second cohomology, as one would expect of a good cohomology theory.

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