Pythagorean triples and quadratic residues modulo an odd prime

1.   Introduction

Rota-Baxter operators, which were initially introduced by Baxter in probability theory ^[1] and were later developed by Rota and Cartier in combinatorics ^[2,3]. The significance of Rota-Baxter operators has been established in various research areas, such as Connes-Kreimer's algebraic method for renormalizing perturbative quantum field theory and dendriform algebras ^[4,5]. Furthermore, the study of inverse scattering theory, integrable systems and quantum groups reveals a close association between Rota-Baxter operators on Lie algebras and the classical Yang-Baxter equation; see the book by Guo for more details ^[6].

In order to gain a deeper understanding of the classical Yang-Baxter equation, Kupershmidt introduced a broader concept of an $\mathcal{O}$-operator (also known as a relative Rota-Baxter operator) on a Lie algebra ^[7]. Recently, there has been the establishment of cohomologies and deformations of relative Rota-Baxter operators on different algebraic structures, including Lie algebras, Leibniz algebras, 3-Lie algebras, $n$-Lie algebras, Lie conformal algebras and others ^{[8,9,10,11,12]}.

In addition, the close relationship with other operators also reflects the importance of (relative) Rota-Baxter operators. One of them is the Reynolds operator, known as a time-average operator in fluid dynamics, which was initially presented by Reynolds in his renowned work on fluctuation theory in 1895 ^[13], and subsequently named by Kampé de Fériet to provide a comprehensive analysis of Reynolds operators in general ^[14]. Reynolds operators were also widely used in functional analysis, invariant theory and have a close relation with algebra endomorphisms, derivations, rational G-modules, geometry and operads ^{[15,16,17,18,19]}.

Motivated by the twisted Poisson structures introduced and studied in ^[20,21], Uchino introduced a twisted version of Rota-Baxter operators on associative algebras, known as twisted Rota-Baxter operators (or generalized Reynolds operators), and examined its correlation with NS-algebras ^[22]. Based on Uchino's work, Das conducted an additional investigation into the cohomology and deformations of twisted Rota-Baxter operators on associative algebras and Lie algebras ^[23,24]. Note that twisted Rota-Baxter operators can be seen as extensions of Reynolds operators [22, Example 3.5]. Hou and Sheng employed the terminology of a generalized Reynolds operator instead of a twisted Rota-Baxter operator on 3-Lie algebras ^[25]. In ^[26], Gharbi et al. delved into the investigation of generalized Reynolds operators on Lie triple systems, while also introducing NS-Lie triple systems as the fundamental framework of generalized Reynolds operators.

In this paper, we consider twisted Rota-Baxter operators on Hom-Lie algebra s. Hartwig et al. were the first to introduce Hom-Lie algebra s in 2006 for the purpose of studying the deformation of the Witt and the Virasoro algebras ^[27], which can be traced back to $q$-deformations of algebras of vector fields in the field of physics. Since then, other algebras of the Hom type (e.g., Hom-associative algebras, Hom-Leibniz algebras), as well as their $n$-ary generalizations, have been widely studied both in mathematics and mathematical physics ^{[28,29,30,31,32]}. Additionally, it is noteworthy noting that Wang and his collaborators investigated twisted Rota-Baxter operators on 3-Hom-Lie algebras and Reynolds operators on Hom-Leibniz algebras by using cohomology and deformation theory ^[33,34].

This paper is organized as follows. Section 2 provides an overview of the concepts and properties related to Hom-Lie algebra s, including representation and cohomology. In Section 3, we delve into the topic of twisted Rota-Baxter operator s on Hom-Lie algebra s, exploring their connection to Reynolds operators and derivations. Furthermore, we present a novel approach to the construction of twisted Rota-Baxter operator s on Hom-Lie algebra s by using $R$-admissible 1-cocycles. Moving on to Section 4, we construct a new $L_\infty$-algebra, whose Maurer-Cartan elements correspond precisely to twisted Rota-Baxter operator s on Hom-Lie algebra s. With this foundation, we define the cohomology of twisted Rota-Baxter operator s by using the technique of constructing twisting $L_\infty$-algebras pioneered by Getzler. Additionally, we establish an intriguing relationship between the cohomology of a twisted Rota-Baxter operator and the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation. Section 5 is devoted to the realm of linear and formal deformations of twisted Rota-Baxter operator s, demonstrating that the linear term in such deformations of a twisted Rota-Baxter operator $R$ manifests as a 1-cocycle in the cohomology of $R$. Finally, we introduce Nijenhuis elements as a means of characterizing the rigidity of twisted Rota-Baxter operator s.

In this paper, all vector spaces, linear maps and tensor products are assumed to be over a field $\mathbb{K}$ of characteristic $0$.

2.   Preliminaries

In this section, we recall several fundamental concepts, including the representation and cohomology of Hom-Lie algebra s. The material can be found in the literature ^[31,35,36].

Definition 2.1. A Hom algebra is a triple $(\mathfrak g, [\cdot, \cdot], \alpha)$ consisting of a vector space $\mathfrak g$, a bilinear map (bracket) $[\cdot, \cdot]:\wedge^2 \mathfrak g\rightarrow \mathfrak g$ and a linear map $\alpha: \mathfrak g\rightarrow \mathfrak g$ such that $\alpha([x, y]) = [\alpha(x), \alpha(y)]$. Moreover, if a Hom algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ also satisfies the following Hom-Jacobi identity:

then $(\mathfrak g, [\cdot, \cdot], \alpha)$ will be called a Hom-Lie algebra.

Definition 2.2. A morphism of Hom-Lie algebra s $\phi:(\mathfrak g_1, [\cdot, \cdot], \alpha)\rightarrow (\mathfrak g_2, [[\cdot, \cdot]], \beta)$ is a linear map $\phi: \mathfrak g_1\rightarrow \mathfrak g_2$ such that

In particular, if $\phi$ is invertible, we say that $(\mathfrak g_1, [\cdot, \cdot], \alpha)$ and $(\mathfrak g_2, [[\cdot, \cdot]], \beta)$ are isomorphic.

Definition 2.3. A linear map $D: \mathfrak g\rightarrow \mathfrak g$ on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ is called an $\alpha^k$-derivation if it satisfies that

for all $x, y\in \mathfrak g$, where $k$ is a nonnegative integer.

In the sequel, an $\alpha^0$-derivation on a Hom-Lie algebra will be called a derivation for simplicity.

Definition 2.4. A representation of a Hom algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ on a vector space $V$ with respect to $A\in \mathfrak {gl}(V)$ is a linear map $\rho: \mathfrak{g} \rightarrow$ $\mathfrak{g l}(V)$ such that, for any $x, y\in \mathfrak g$, it holds that

We denote a representation of a Hom-Lie algebra $\mathfrak g$ with respect to $A$ by $(V, \rho, A)$.

Example 2.5. Let $(\mathfrak g, [\cdot, \cdot], \alpha)$ be a Hom-Lie algebra. Define ${\rm ad}: \mathfrak g\rightarrow \mathfrak {gl}(\mathfrak g)$ by ${\rm ad}(x)(y) = [x, y]$ for all $x, y\in \mathfrak g$; sometimes, we may write ${\rm ad}(x)$ as ${\rm ad}_x$. Then, $(\mathfrak g, {\rm ad}, \alpha)$ is a representation of $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to $\alpha$, which is called the adjoint representation of $(\mathfrak g, [\cdot, \cdot], \alpha)$.

Next, we recall the cohomology of Hom-Lie algebras. Let $(V, \rho, A)$ be a representation of the Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$. Denote the space of $p$-cochains by

where $A\circ f = f\circ\alpha^{\otimes^p}$ means that

For any $x_1, \cdots, x_{p+1}\in \mathfrak g$, define the coboundary operator $d_\rho: C_{HLie}^p(\mathfrak{g}, V) \rightarrow C_{HLie}^{p+1}(\mathfrak{g}, V), p\geq 1$ by

and when $p = 0$, define $d_\rho: C_{HLie}^0(\mathfrak{g}, V) \rightarrow \operatorname{Hom}(\mathfrak g, V)$ by

Since

we deduce that $d_\rho$ is a map from $C_{HLie}^0(\mathfrak{g}, V)$ to $C_{HLie}^{1}(\mathfrak{g}, V)$, indeed. Thus, we have that $d_\rho\circ d_\rho = 0$ and, hence, $\left(\oplus_{p = 0}^{+\infty} C_{HLie}^p(\mathfrak{g}, V), d_\rho\right)$ is a cochain complex. Denote the set of $p$-cocycles by $Z_{HLie}^{p}(\mathfrak g, V)$ and the set of $p$-coboundaries by $B_{HLie}^{p}(\mathfrak g, V)$. Then, the corresponding $p$-th cohomology group is

In view of (2.6), a 1-cochain $f\in C_{HLie}^1(\mathfrak{g}, V)$ is a 1-cocycle on $\mathfrak g$ with coefficients in $(V, \rho, A)$ if $f$ satisfies

and a 2-cochain $\Phi\in C_{HLie}^2(\mathfrak{g}, V)$ is a 2-cocycle if $\Phi$ satisfies

3.   Twisted Rota-Baxter operators on Hom-Lie algebras

In this section, we introduce the notion of twisted Rota-Baxter operator s on Hom-Lie algebra s. We establish the relation between Reynolds operators and derivations. We also show that a linear map is a twisted Rota-Baxter operator if and only if its graph is a subalgebra of the $\Phi$-twisted semi-direct Hom-Lie algebra. Moreover, we provide a method for constructing twisted Rota-Baxter operator s by using $R$-admissible 1-cocycles.

First, by a direct check, we have the following result.

Proposition 3.1. Let $(\mathfrak g, [\cdot, \cdot], \alpha)$ be a Hom-Lie algebra and $(V, \rho, A)$ a representation of $\mathfrak g$. Given a 2-cocycle $\Phi\in C_{HLie}^2(\mathfrak{g}, V)$, there exists a Hom-Lie algebra structure on the direct sum $\mathfrak g\oplus V$ that is defined by

This Hom-Lie algebra is called the $\Phi$-twisted semi-direct Hom-Lie algebra and will be denoted by $(\mathfrak g\ltimes_\Phi\! V, \alpha\oplus A)$.

A twisted Rota-Baxter operator on a Hom-Lie algebra is defined as follows, the weight of which is a 2-cocycle instead of a scalar, as in the classical case.

Definition 3.2. Let $(\mathfrak g, [\cdot, \cdot], \alpha)$ be a Hom-Lie algebra and $(V, \rho, A)$ a representation of $\mathfrak g$. A linear map $R:V\rightarrow \mathfrak g$ is called a twisted Rota-Baxter operator on $\mathfrak g$ associated with a 2-cocycle $\Phi$ with respect to $(V, \rho, A)$ (of weight $\Phi$) if

Remark 3.3. A twisted Rota-Baxter operator is also called a generalized Reynolds operator; see ^[25,26] for more details, where the authors considered it on Lie triple systems and 3-Lie algebras, respectively. Furthermore, it was also named a relative cocycle weighted Reynolds operator by Guo and Zhang in the setting of pre-Lie algebras ^[37].

Example 3.4. Any Rota-Baxter operator or relative Rota-Baxter operator of weight 0 is a twisted Rota-Baxter operator with $\Phi = 0$.

Example 3.5. Let $(\mathfrak g, [\cdot, \cdot], \alpha)$ be a Hom-Lie algebra and $(V, \rho, A)$ a representation of $\mathfrak g$. Assume that a linear map $f\in C_{HLie}^{1}(\mathfrak{g}, V)$ is invertible. Set $\Phi = -d_\rho f$ and $R = f^{-1}$. Then, $\Phi$ is a 2-cocycle. Since

we obtain that $R$ is a twisted Rota-Baxter operator of weight $\Phi = -d_\rho f$.

Example 3.6. Let $(\mathfrak g, [\cdot, \cdot], \alpha)$ be a Hom-Lie algebra and $(\mathfrak g, {\rm ad}, \alpha)$ the adjoint representation. Set $\Phi = -[\cdot, \cdot]$; then, a linear transformation $R: \mathfrak g\rightarrow \mathfrak g$ defined by (3.3) and (3.4) is called a Reynolds operator; more specifically, $R$ satisfies that

Note that the authors of ^[34] defined Reynolds operators on Hom-Leibniz algebras. Furthermore, twisted Rota-Baxter operators on Hom-Lie algebras are extensions of both twisted Rota-Baxter operators and 0-weighted Rota-Baxter operators on Lie algebras.

Next, we establish the connection between derivations and Reynolds operators on Hom-Lie algebra s. First, a derivation can induce a Reynolds operator on a Hom-Leibniz algebra ^[34]. Specifically, we have the following:

Proposition 3.7. Assume that $D: \mathfrak g\rightarrow \mathfrak g$ is a derivation on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$. If $(D+{\rm Id}): \mathfrak g\rightarrow \mathfrak g$ has an inverse, then $(D+{\rm Id})^{-1}$ is a Reynolds operator.

Conversely, a derivation on a Hom-Lie algebra can be derived from a Reynolds operator.

Proposition 3.8. Assume that $R: \mathfrak g\rightarrow \mathfrak g$ is a Reynolds operator on the Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$. If $R$ has an inverse, then $(R^{-1}-{\rm Id}): \mathfrak g\rightarrow \mathfrak g$ is a derivation on $(\mathfrak g, [\cdot, \cdot], \alpha)$.

Proof. Suppose that $R: \mathfrak g\rightarrow \mathfrak g$ is an invertible Reynolds operator. Then, $\alpha\circ R = R\circ \alpha$. Moreover, thanks to (3.6), we obtain

for any $x_1, x_2\in \mathfrak g$, which is equivalent to

This completes the proof. □

In the sequel, $\Phi$ denotes a 2-cocycle, and a twisted Rota-Baxter operator is always endowed with the weight $\Phi$ unless otherwise specified elsewhere.

Let $(\mathfrak g, [\cdot, \cdot], \alpha)$ be a Hom-Lie algebra and $(V, \rho, A)$ a representation of $\mathfrak g$. Suppose that $R:V\rightarrow \mathfrak g$ is a linear map which satisfies that $\alpha\circ R = R\circ A$. Then, we call the set $Gr(R) = \{Rv+v|v\in V\}$ the graph of $R$.

Theorem 3.9. Let $(\mathfrak g, [\cdot, \cdot], \alpha)$ be a Hom-Lie algebra and $(V, \rho, A)$ a representation of $\mathfrak g$. A linear map $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator if and only if the graph $Gr(R) = \{Rv+v|v\in V\}$ is a subalgebra of the $\Phi$-twisted semi-direct Hom-Lie algebra $(\mathfrak g\ltimes_\Phi\! V, \alpha\oplus A)$.

Proof. Set $v_1, v_2\in V$. Then, we have

Hence, the graph $Gr(R) = \{Rv+v|v\in V\}$ is a subalgebra of $\mathfrak g\ltimes_\Phi\! V$ if and only if

which is precisely (3.4). The proof is finished. □

The following corollary is straightforward given $Gr(R)\cong V$ as vector spaces.

Corollary 3.10. Suppose that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Then, $(V, [\cdot, \cdot]_R, A)$ is a Hom-Lie algebra, called the sub-adjacent Hom-Lie algebra of $R$, where the operation $[\cdot, \cdot]_R$ is given by

Moreover, $R$ is a homomorphism of Hom-Lie algebras from $(V, [\cdot, \cdot]_R, A)$ to $(\mathfrak g, [\cdot, \cdot], \alpha)$.

We will now present a method for constructing twisted Rota-Baxter operators on Hom-Lie algebra s by introducing the concept of an $R$-admissible 1-cocycle. Consider a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$, a representation $(V, \rho, A)$ of $\mathfrak g$ and a linear map $\theta\in C_{HLie}^1(\mathfrak{g}, V)$. Define $\Omega_{\theta}: \mathfrak g\oplus V\rightarrow \mathfrak g\oplus V$ by $\Omega_{\theta} = \left(\begin{array}{cc}{\rm Id} & 0 \\ {\theta} & {\rm Id}\end{array}\right)$. Then, $\Omega_{\theta}$ is invertible. Note that $\Phi-d_\rho {\theta}$ is a 2-cocycle.

Lemma 3.11. Consider the above notations. Then, $\Omega_{\theta}$ serves as an isomorphism between the $\Phi$-twisted semi-direct Hom-Lie algebra $(\mathfrak g\ltimes_\Phi\! V, \alpha\oplus A)$ and the $(\Phi-d_\rho {\theta})$-twisted semi-direct Hom-Lie algebra $(\mathfrak g\ltimes_{\Phi-d_\rho {\theta}}\! V, \alpha\oplus A)$.

Proof. Set $x_1, x_2\in \mathfrak g$ and $v_1, v_2\in V$. Then,

as required. □

Proposition 3.12. Let $R:V\rightarrow \mathfrak g$ be a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to a representation $(V, \rho, A)$. Suppose that ${\theta}\in C_{HLie}^1(\mathfrak{g}, V)$. If the linear map $({\rm Id}_V+{\theta}\circ R):V\rightarrow V$ has an inverse, then the map $R\circ ({\rm Id}_V+{\theta}\circ R)^{-1}$ is a twisted Rota-Baxter operator of weight $(\Phi-d_\rho {\theta})$.

Proof. Thanks to Theorem 3.9, $Gr(R)$ is a subalgebra of the $\Phi$-twisted semi-direct Hom-Lie algebra $(\mathfrak g\ltimes_\Phi\! V, \alpha\oplus A)$. In view of Lemma 3.11, $\Omega_{\theta}(Gr(R))\subseteq (\mathfrak g\ltimes_{\Phi-d_\rho {\theta}}\! V, \alpha\oplus A)$ is also a subalgebra. Given that the linear map $({\rm Id}_V+{\theta}\circ R):V\rightarrow V$ has an inverse, by a direct check, we see that

and, hence, $\Omega_{\theta}(Gr(R))$ is the graph of $R\circ ({\rm Id}_V+{\theta}\circ R)^{-1}:V\rightarrow \mathfrak g$. Then, by Theorem 3.9, again, we deduce that the map $R\circ ({\rm Id}_V+{\theta}\circ R)^{-1}$ is a twisted Rota-Baxter operator of weight $(\Phi-d_\rho {\theta})$, we have the conclusion. □

Definition 3.13. Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. An $R$-admissible 1-cocycle is a 1-cocycle ${\theta}\in Z_{HLie}^1(\mathfrak{g}, V)$ such that the map $({\rm Id}_V+{\theta}\circ R):V\rightarrow V$ is invertible.

The following corollary is straightforward due to Proposition 3.12 and Definition 3.13.

Corollary 3.14. Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$, and let ${\theta}: \mathfrak g\rightarrow V$ denote an $R$-admissible 1-cocycle. The composition of the map $R$ with the inverse of $({\rm Id}_V+{\theta}\circ R)$ forms a twisted Rota-Baxter operator. Denote this twisted Rota-Baxter operator by $R_{\theta}$.

With the help of Corollary 3.2, $(V, [\cdot, \cdot]_R)$ and $(V, [\cdot, \cdot]_{R_{\theta}})$ are Hom-Lie algebra s. We conclude this section by pointing out that $(V, [\cdot, \cdot]_R)$ and $(V, [\cdot, \cdot]_{R_{\theta}})$ are isomorphic as Hom-Lie algebra s.

Proposition 3.15. Let $R:V\rightarrow \mathfrak g$ be a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$ and ${\theta}$ an $R$-admissible 1-cocycle. Then, $(V, [\cdot, \cdot]_R)\cong(V, [\cdot, \cdot]_{R_{\theta}})$ denotes Hom-Lie algebras.

Proof. It suffices to show that the invertible map $({\rm Id}_V+{\theta}\circ R)$ is an isomorphism between $(V, [\cdot, \cdot]_R)$ and $(V, [\cdot, \cdot]_{R_{\theta}})$. For any $v_1, v_2\in V$, it holds that

which finishes the proof. □

4.   Cohomology of twisted Rota-Baxter operators on Hom-Lie algebras

In this section, we first recall the concept of an $L_\infty$-algebra and the Nijenhuis-Richardson bracket for Hom-Lie algebra s. Subsequently, we construct an $L_\infty$-algebra whose Maurer-Cartan elements are given by twisted Rota-Baxter operator s on Hom-Lie algebra s. Following this, we delve into the study of twisting theory for Hom-Lie algebra s and establish the cohomology of twisted Rota-Baxter operator s. Furthermore, we demonstrate that this cohomology can be perceived as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation.

4.1.   Maurer-Cartan characterization and the controlling $L_\infty$-algebra of twisted Rota-Baxter operators

In this subsection, we construct an explicit $L_\infty$-algebra whose Maurer-Cartan elements are twisted Rota-Baxter operators on Hom-Lie algebra s by using Voronov's higher derived bracket ^[38]. Using Getzler's method in ^[39], we also establish a twisted $L_\infty$-algebra which governs the deformations of twisted Rota-Baxter operators on Hom-Lie algebra s.

An $(i, n-i)$-shuffle is a permutation $\sigma\in\mathbb{S}_n$ such that $\sigma (1) < \cdots < \sigma (i)$ and $\sigma (i+1) < \cdots < \sigma (n)$. In the case in which $i = 0$ or $i = n$, we make the assumption that $\sigma = {\rm Id}$. $\mathbb{S}_{ (i, n-i)}$ will represent the collection of all $(i, n-i)$-shuffles.

Definition 4.1. (^[40]) A $\mathbb{Z}$-graded vector space $\mathfrak g = \oplus_{k\in\mathbb{Z}} \mathfrak g^k$ with a collection $(k\geq1)$ of linear maps $l_k:\otimes^k \mathfrak g\rightarrow \mathfrak g$ of degree 1 is called an $L_\infty$-algebra if, for all homogeneous elements $x_1, \cdots, x_n\in \mathfrak g$, it holds that

(ⅰ) (graded symmetry) for any $\sigma\in\mathbb{S}_n$,

(ⅱ) (generalized Jacobi identity) for any $n\geq1$,

where $\varepsilon(\sigma): = \varepsilon(\sigma; x_1, \cdots, x_n)\in\{-1, 1\}$ is the Koszul sign.

Definition 4.2. A Maurer-Cartan element of an $L_\infty$-algebra $(\mathfrak g = \oplus_{k\in\mathbb{Z}} \mathfrak g^k, \{l_i\}_{i = 1}^{+\infty})$ is an element $x\in \mathfrak g^0$ such that $\sum_{n = 1}^{+\infty} \frac{1}{n!} l_n(x, \cdots, x)$ converges to 0, that is, $x$ obeys the Maurer-Cartan equation

Before proceeding further, let us recall the higher derived brackets due to Voronov, which can be utilized for the construction of explicit $L_\infty$-algebras.

Definition 4.3. (^[38]) A V-data consists of a quadruple $(L, \mathfrak h, \mathcal{P}, \Delta)$, where the following holds:

● $(L, [\cdot, \cdot])$ is a graded Lie algebra.

● $\mathfrak h$ is an abelian graded Lie subalgebra of $(L, [\cdot, \cdot])$.

● $\mathcal{P}:L\rightarrow L$ is a projection, i.e., $\mathcal{P}\circ \mathcal{P} = \mathcal{P}$, where $\mathfrak h$ is the image and the kernel is a graded Lie subalgebra of $(L, [\cdot, \cdot])$.

● $\Delta$ is an element of $\ker\mathcal{P}$ with degree 1 satisfying that $[\Delta, \Delta] = 0$.

Theorem 4.4. (^[38]) Assume $(L, \mathfrak h, \mathcal{P}, \Delta)$ to be a V-data. Then, $(\mathfrak h, \{l_i\}_{i = 1}^{+\infty})$ forms an $L_\infty$-algebra, where

We call $\{l_i\}_{i = 1}^{+\infty}$ the higher derived brackets of the V-data $(L, \mathfrak h, \mathcal{P}, \Delta)$.

Let $\mathfrak g$ be a vector space and $\alpha: \mathfrak g\rightarrow \mathfrak g$ a linear map. Denote by $V_ \alpha^p(\mathfrak g) = C_ \alpha^{p+1}(\mathfrak g, \mathfrak g), \ p\geq0$ the space of all linear maps $P: \mathfrak g^{\otimes(p+1)}\rightarrow \mathfrak g$ satisfying that $\alpha\circ P = P\circ\alpha^{\otimes^{p+1}}$, that is,

Set $C_ \alpha^{0}(\mathfrak g, \mathfrak g) = \mathfrak g$. Recall from ^[28,30] that the graded space $V_ \alpha^*(\mathfrak g) = \oplus _{p\geq -1}C_ \alpha^{p+1}(\mathfrak g, \mathfrak g)$ carries a graded Lie algebra structure $[\cdot, \cdot]_ \alpha:V_ \alpha^p(\mathfrak g)\ \times\ V_ \alpha^q(\mathfrak g)\rightarrow V_ \alpha^{p+q}(\mathfrak g)$ (called Nijenhuis-Richardson bracket), defined by

where $\{P, Q\}_ \alpha\in V_ \alpha^{p+q}(\mathfrak g)$ is given by

and the above notation $|\sigma|$ denotes the signature of the permutation $\sigma$.

Now, let $(\mathfrak g, [\cdot, \cdot], \alpha)$ be a Hom algebra. For simplicity, we shall use $\mu:\wedge^2 \mathfrak g\rightarrow \mathfrak g$ to denote the bilinear bracket $[\cdot, \cdot]$, and a Hom algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ can be rewritten as $(\mathfrak g, \mu, \alpha)$. Note that $\mu\in V_ \alpha^1(\mathfrak g)$. Thus, a Hom algebra $(\mathfrak g, \mu, \alpha)$ becomes a Hom-Lie algebra if and only if $[\mu, \mu]_ \alpha = 0$, that is, $\mu$ is a Maurer-Cartan element of the graded Lie algebra $(V_ \alpha^*(\mathfrak g), [\cdot, \cdot]_ \alpha)$.

Let $\mathfrak g$ and $V$ be vector spaces with linear maps $\alpha: \mathfrak g\rightarrow \mathfrak g$ and $A:V\rightarrow V$. Suppose that $\mu:\wedge^2 \mathfrak g\rightarrow \mathfrak g$, $\rho: \mathfrak g\rightarrow {\rm End}(V)$ and $\Phi:\wedge^2 \mathfrak g\rightarrow V$ are linear maps. Define $\mu+\rho+\Phi\in{\rm Hom}(\wedge^2(\mathfrak g\oplus V), \mathfrak g\oplus V)$ by

Proposition 4.5. The map $\mu$ defines a Hom-Lie algebra structure on the pair $(\mathfrak g, \alpha)$, the map $\rho$ defines a representation of the Hom-Lie algebra $(\mathfrak g, \mu, \alpha)$ on the pair $(V, A)$ and the map $\Phi$ defines a 2-cocycle with respect to the representation $(V, \rho, A)$ if and only if $\mu+\rho+\Phi$ is a Maurer-Cartan element of the graded Lie algebra $(V_{ \alpha\oplus A}^*(\mathfrak g\oplus V), [\cdot, \cdot]_{ \alpha\oplus A})$.

Proof. Due to (4.3), $\mu+\rho+\Phi\in V_{ \alpha\oplus A}^1(\mathfrak g\oplus V)$ if and only if

for all $x_1, x_2\in \mathfrak g$ and $v_1, v_2\in V$, that is,

In addition, the map $\mu+\rho+\Phi$ is a Maurer-Cartan element if and only if

for $x_1, x_2, x_3\in \mathfrak g$ and $v_1, v_2, v_3\in V$. Equivalently,

Owing to Definition 2.1, Definition 2.4 and (2.9), the conclusion follows. □

Proposition 4.6. Let $(V, \rho, A)$ be a representation of a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ and $\Phi\in C_{HLie}^2(\mathfrak{g}, V)$ a 2-cocycle with respect to $(V, \rho, A)$. Thus we have a V-data $(L, \mathfrak h, \mathcal{P}, \Delta)$, as follows:

● the graded Lie algebra $(L, [\cdot, \cdot])$ is given by $(V_{ \alpha\oplus A}^*(\mathfrak g\oplus V), [\cdot, \cdot]_{ \alpha\oplus A})$;

● the abelian graded Lie subalgebra $\mathfrak h$ is defined by

● $\mathcal{P}:L\rightarrow L$ is the projection onto the space $\mathfrak h$;

● $\Delta = \mu+\rho+\Phi$.

Therefore, we get an $L_\infty$-algebra $(C_{HLie}^*(V, \mathfrak g), l_2, l_3)$, where

for $P\in C_{HLie}^p(V, \mathfrak g)$, $Q\in C_{HLie}^q(V, \mathfrak g)$ and $S\in C_{HLie}^s(V, \mathfrak g)$.

Proof. First note that $\Delta = \mu+\rho+\Phi\in\ker\mathcal{P}$ with degree 1 and $[\Delta, \Delta]_{ \alpha\oplus A} = 0$ due to Proposition 4.5. Thus, we have that $(L, \mathfrak h, \mathcal{P}, \Delta)$ is a V-data. Define the higher derived brackets $\{l_i\}_{i = 1}^{+\infty}$ as (4.2). Then, for any $P\in C_{HLie}^p(V, \mathfrak g)$, $Q\in C_{HLie}^q(V, \mathfrak g)$ and $S\in C_{HLie}^s(V, \mathfrak g)$ we have

and, hence, $l_1 = 0$. Similarly, we obtain that $l_k = 0$ for $k\geq4$. Therefore, the graded vector space $C_{HLie}^*(V, \mathfrak g)$ is an $L_\infty$-algebra with nontrivial $l_2, l_3$, and the other higher derived brackets are trivial. □

With the aid of Proposition 4.6, we have the main theorem in this subsection.

Theorem 4.7. Let $(V, \rho, A)$ be a representation of a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ and $\Phi\in C_{HLie}^2(\mathfrak{g}, V)$ a 2-cocycle with respect to $(V, \rho, A)$. Then, a linear map $R: V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator if and only if $R$ is a Maurer-Cartan element of the $L_\infty$-algebra $(C_{HLie}^*(V, \mathfrak g), l_2, l_3)$.

Proof. Set $v_1, v_2\in V$. By direct computation, we have

Therefore, according to Definition 4.2 and (4.3), $R$ is a Maurer-Cartan element of $(C_{HLie}^*(V, \mathfrak g), l_2, l_3)$ if and only if $\alpha\circ R = R\circ \alpha$ and

which is equivalent to $R:V\rightarrow \mathfrak g$ being a twisted Rota-Baxter operator on the Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. This completes the proof. □

4.2.   Cohomology of twisted Rota-Baxter operators

In this subsection, we establish the cohomology of twisted Rota-Baxter operators on Hom-Lie algebra s by utilizing twisted $L_\infty$-algebra structures from a given $L_\infty$-algebra and a Maurer-Cartan element, as introduced by Getzler ^[39].

Let $\omega$ be a Maurer-Cartan element of an $L_\infty$-algebra $(\mathfrak g = \oplus_{k\in\mathbb{Z}} \mathfrak g^k, \{l_i\}_{i = 1}^{+\infty})$. Define a series of twisted linear maps $l_k^{\!\ \omega}:\otimes^k \mathfrak g\rightarrow \mathfrak g$ of degree 1, $k\geq1$ by

Theorem 4.8. (^[39]) Keeping the notations as above, $(\mathfrak g, \{l_k^{\!\ \omega}\}_{k = 1}^{+\infty})$ is an $L_\infty$-algebra, obtained from $\mathfrak g$ by twisting with the Maurer-Cartan element $\omega$. Furthermore, $\omega+\omega'$ is a Maurer-Cartan element of $(\mathfrak g, \{l_i\}_{i = 1}^{+\infty})$ if and only if $\omega'$ is a Maurer-Cartan element of the twisted $L_\infty$-algebra $(\mathfrak g, \{l_k^{\!\ \omega}\}_{k = 1}^{+\infty})$.

Applying Theorem 4.8 to the $L_\infty$-algebra $(C_{HLie}^*(V, \mathfrak g), l_2, l_3)$, we get the following proposition.

Proposition 4.9. Let $R:V\rightarrow \mathfrak g$ be a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Then, $C_{HLie}^*(V, \mathfrak g)$ carries a twisted $L_{\infty}$-algebra structure, as follows:

for all $k\geq4$, $P\in C_{HLie}^p(V, \mathfrak g)$, $Q\in C_{HLie}^q(V, \mathfrak g)$ and $S\in C_{HLie}^s(V, \mathfrak g)$.

Proof. In view of Theorem 4.7, $R$ is a Maurer-Cartan element of the $L_\infty$-algebra $(C_{HLie}^*(V, \mathfrak g), l_2, l_3)$. Then, the result follows due to Theorem 4.8. □

Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Denote the twisted $L_\infty$-algebra in the above proposition by $(C_{HLie}^*(V, \mathfrak g), l_1^R, l_2^R, l_3^R)$. Therefore, we obtain that the twisted Rota-Baxter operator $R$ generates a differential $l_1^R: C_{HLie}^p(V, \mathfrak g)\rightarrow C_{HLie}^{p+1}(V, \mathfrak g), p\geq 1$. Define the set of $p$-cochains by

Define $d_R = l_1^R$ for all $p\geq1$. Moreover, if $p = 0$, define $d_R: C_{R}^0(V, \mathfrak{g}) \rightarrow C_{R}^1(V, \mathfrak{g})$ by

which is well defined since $\alpha(x) = x$. Thus, we have that $d_R\circ d_R = 0$; hence, $\left(\oplus_{p = 0}^{+\infty} C_{R}^p(V, \mathfrak{g}), d_R\right)$ is a cochain complex. Then, the corresponding cohomology groups are

which represents the cohomology of the twisted Rota-Baxter operator $R$.

For the last part of this subsection, we give a description showing that the above twisted $L_\infty$-algebra governs the deformations of twisted Rota-Baxter operators on Hom-Lie algebra s.

Theorem 4.10. Let $R:V\rightarrow \mathfrak g$ be a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Then, for a linear map $R':V\rightarrow \mathfrak g$, $R+R'$ becomes a twisted Rota-Baxter operator if and only if $R'$ is a Maurer-Cartan element of the twisted $L_\infty$-algebra $(C_{HLie}^*(V, \mathfrak g), l_1^R, l_2^R, l_3^R)$.

Proof. According to Definition 4.2, $R'$ is a Maurer-Cartan element of $(C_{HLie}^*(V, \mathfrak g), l_1^R, l_2^R, l_3^R)$ if and only if

By Proposition 4.9, the above formula is equivalent to

Since $R$ is a twisted Rota-Baxter operator, by Theorem 4.7, we have

Collecting the two equalities (4.7) and (4.8) gives that (4.7) is equivalent to

that is, $R+R'$ is a Maurer-Cartan element of $(C_{HLie}^*(V, \mathfrak g), l_2, l_3)$. By Theorem 4.7, again, it is equivalent to $R+R'$ being a twisted Rota-Baxter operator; thus, we have the conclusion. □

4.3.   Cohomology of twisted Rota-Baxter operators as Chevalley-Eilenberg cohomology

In this subsection, we offer an alternative understanding of the cohomology of twisted Rota-Baxter operators. It turns out that this cohomology can be perceived as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation.

Recall that $(V, [\cdot, \cdot]_R)$ is the sub-adjacent Hom-Lie algebra of $R$ (see Corollary 3.10). First, we construct the representation of $(V, [\cdot, \cdot]_R)$ as follows.

Lemma 4.11. Let $R:V\rightarrow \mathfrak g$ be a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Define ${\rho_R}:V\rightarrow \mathfrak {gl}(\mathfrak g)$ by

Then, $(\mathfrak g, \rho_R, \alpha)$ is a representation of the Hom-Lie algebra $(V, [\cdot, \cdot]_R, A)$.

Proof. Set $v_1, v_2\in V$ and $x\in \mathfrak g$. By a direct check, we get that $\rho_R(Av_1)\circ \alpha = \alpha\circ\rho_R(v_1)$. Moreover, since $\Phi$ is a 2-cocycle, by (2.2) and (2.3), (3.3) and (3.4), as well as (2.9), we have

Therefore, $(\mathfrak g, \rho_R, \alpha)$ is a representation of the Hom-Lie algebra $(V, [\cdot, \cdot]_R, A)$. □

The above lemma allows us to consider the Chevalley-Eilenberg cohomology of the Hom-Lie algebra $(V, [\cdot, \cdot]_R, A)$ with coefficients in the representation $(\mathfrak g, \rho_R, \alpha)$. Let $\delta_{CE}:C_{HLie}^p(V, \mathfrak{g}) \rightarrow C_{HLie}^{p+1}(V, \mathfrak{g}), (p\geq1)$ be the corresponding coboundary operator of the Hom-Lie algebra $(V, [\cdot, \cdot]_R, A)$ with coefficients in the representation $(\mathfrak g, \rho_R, \alpha)$, where $C_{HLie}^p(V, \mathfrak{g})$ is given in Proposition 4.6. More precisely, $\delta_{CE}:C_{HLie}^p(V, \mathfrak{g}) \rightarrow C_{HLie}^{p+1}(V, \mathfrak{g})$ is given by

for any $f\in C_{HLie}^p(V, \mathfrak{g})$ and $v_1, \cdots, v_{p+1}\in V$. If $p = 0$, according to (2.7), the coboundary map $\delta_{CE}: C_{HLie}^0(V, \mathfrak{g}) \rightarrow C_{HLie}^1(V, \mathfrak{g})$ is given by

in view of (4.6), we obtain that

Denote the Chevalley-Eilenberg cohomology group correspondng to the cochain complex $\left(\oplus_{p = 0}^{+\infty} C_{HLie}^p(V, \mathfrak{g}), \delta_{CE}\right)$ by $H_{CE}^*(V, \mathfrak g)$.

Furthermore, comparing the coboundary operator $\delta_{CE}$ given above with the twisted linear map $l_1^R$ introduced in Proposition 4.9, we obtain the following result.

Theorem 4.12. Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Then, we have

Proof. Set $\mu+\rho+\Phi = \Delta$. For any $f\in C_{HLie}^p(V, \mathfrak{g})$ and $v_1, \cdots, v_{p+1}\in V$, by (4.4), we have

Similarly, by direct computation, we obtain

Therefore, we deduce that

This completes the proof. □

Combining the above theorem and (4.11), we arrive at the subsequent corollary.

Corollary 4.13. Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Then,

5.   Deformations of twisted Rota-Baxter operators on Hom-Lie algebras

In this section, we employ the established cohomology to investigate the linear and formal deformations of twisted Rota-Baxter operator s on Hom-Lie algebra s. We establish that equivalent linear deformations define the same cohomology class, as well as formal deformations. Furthermore, we characterize the rigidity of formal deformations based on Nijenhuis elements.

5.1.   Linear deformations of twisted Rota-Baxter operators

Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Recall from (4.5) that $C^0_R(V, \mathfrak g) = \{x\in \mathfrak g\ |\ \alpha(x) = x\}$ and $C^1_R(V, \mathfrak g) = C_{HLie}^1(V, \mathfrak g) = \{f\in\operatorname{Hom}(V, \mathfrak g)\ |\ \alpha\circ f = f\circ A\}.$

Definition 5.1. A linear deformation of a twisted Rota-Baxter operator $R$ consists of $R_t = R+tR_1$ with $R_0 = R$ such that, for all $t\in \mathbb{K}$, $R_t$ is still a twisted Rota-Baxter operator.

Suppose that $R_t = R+tR_1$ is a linear deformation of $R$. Thus, we have that $\alpha\circ R_t = R_t\circ A$ and

By direct computation, we have that $\alpha\circ R_1 = R_1\circ A$ and

Equivalently, $R_1\in C^1_R(V, \mathfrak g)$ and $d_R R_1 = 0$. Then, we get that $R_1$ is a 1-cocycle within the cohomology of $R$.

Definition 5.2. Assume that $R$ and $R'$ are two twisted Rota-Baxter operators on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. A morphism of twisted Rota-Baxter operators from $R$ to $R'$ consists of a pair $(\phi, \psi)$ of the Hom-Lie algebra morphism $\phi: \mathfrak g\rightarrow \mathfrak g$ and a linear map $\psi:V\rightarrow V$ such that, for any $x\in \mathfrak g$, it holds that

Furthermore, $(\phi, \psi)$ will be called an isomorphism from $R$ to $R'$ if $\phi$ and $\psi$ are invertible.

Definition 5.3. Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Two linear deformations, $R_t = R+tR_1$ and $R^{'}_t = R+tR^{'}_{\!1}$, are said to be equivalent if there exists $x\in \mathfrak g$ such that $\alpha(x) = x$ and

is a morphism from $R_t$ to $R^{'}_{\!t}$.

Suppose that $(\phi_t = {\rm Id}_ \mathfrak g+t{ \mathrm{ad}_{x}}, \psi_t = {\rm Id}_V+t\rho(x)+t\Phi(x, R-))$ is a morphism from $R_t$ to $R^{'}_{\!t}$. Then, $\phi_t$ being a Hom-Lie algebra morphism means that

The condition $\psi_t\circ\rho(x) = \rho(\phi_t(x))\circ\psi_t$ is equivalent to

The condition $\psi_t\circ\Phi_t = \Phi\circ({\phi_t\otimes_ \mathbb{K}\phi_t})$ means that

Moreover, the formula $\psi_t\circ A = A\circ\psi_t$ is established automatically because $\alpha(x) = x$. Finally, the condition $\phi\circ R = R'\circ \psi$ is equivalent to

Note that the first condition of (5.4) implies that $R_1-R^{'}_{\!1} = d_R(x)$. Consequently, we have the following result.

Theorem 5.4. Let $R_t = R+tR_1$ and $R^{'}_t = R+tR^{'}_{\!1}$ be two equivalent linear deformations of a twisted Rota-Baxter operator $R$. Then, $R_1$ and $R^{'}_{\!1}$ are in the same cohomology class in $H_R^{1}(V, \mathfrak g)$.

Remark 5.5. A linear deformation $R_t = R+tR_1$ of a twisted Rota-Baxter operator R is called trivial if $R_t$ is equivalent to the unaltered deformation $R^{'}_{\!t} = R$.

5.2.   Formal deformations of twisted Rota-Baxter operators

In this subsection, we investigate formal deformations of twisted Rota-Baxter operators on Hom-Lie algebras.

Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Denote by $V[[t]]$ the formal power series in $t$ with coefficients in $V$. There exists a Hom-Lie algebra structure over the ring $\mathbb{K}[[t]]$ on $\mathfrak g[[t]]$ that is given by

Moreover, there is a representation (denoted also by $\rho$) of the Hom-Lie algebra $\mathfrak g[[t]]$ that is given by

The 2-cocycle $\Phi$ can be extended to a 2-cocycle on the Hom-Lie algebra $\mathfrak g[[t]]$ with coefficients in $V[[t]]$; denote it by using the same notation $\Phi$.

Consider the following formal power series:

Since $R_t\in{\rm Hom}(V, \mathfrak g)[[t]] = {\rm Hom}(V, \mathfrak g[[t]])$, we may extend it to a $\mathbb{K}[[t]]$-module map from $V[[t]]$ to $\mathfrak g[[t]]$, and we still denote it by $R_t$.

Definition 5.6. A formal deformation of a twisted Rota-Baxter operator $R$ consists of a formal power series $R_t = \sum_{i = 0}^{+\infty}R_it^i$, with $R_0 = R$ such that, for all $t\in \mathbb{K}$, $R_t$ remains as a twisted Rota-Baxter operator.

Lemma 5.7. $R_t = \sum\limits_{i = 0}^{+\infty}R_it^i$ is a formal deformation of $R$ if and only if

Proof. Straightforward. □

For $n = 0$, (5.8) gives that $R = R_0$ is a twisted Rota-Baxter operator. For $s = 1$, it follows that

which implies that $d_R R_1 = 0$; hence, $R_1$ is exactly a 1-cocycle of the cohomology of the twisted Rota-Baxter operator $R$. Moreover, by direct computation, we have the following.

Proposition 5.8. Let $R_t = \sum\limits_{i = 0}^{+\infty}R_it^i$ be a formal deformation of a twisted Rota-Baxter operator $R$. If $R_i = 0, 1\leq i < n$, then $R_n$ is a 1-cocycle with respect to the cohomology of $R$, that is, $R_n\in Z_{R}^{1}(V, \mathfrak g)$.

A 1-cochain $R_n$ is called the $n$-infinitesimal of $R_t$ if $R_i = 0$ for all $1\leq i < n$. In particular, the 1-cocycle $R_1$ is called the infinitesimal (or 1-infinitesimal) of $R_t$. Due to Proposition 5.8, the $n$-infinitesimal $R_n$ is a 1-cocycle.

Definition 5.9. Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. Two formal deformations $R_t = R+\sum\limits_{i\geq r}^{+\infty}R_it^i$ and $R^{'}_t = R+\sum\limits_{i\geq r}^{+\infty}R^{'}_{\!i}t^i$ $(r\geq 1)$ are called equivalent if there exist $x\in \mathfrak g$, $\phi_i\in \mathfrak {gl}(\mathfrak g)$ and $\psi_i\in \mathfrak {gl}(V), i\geq r+1$ such that

is a morphism from $R_t$ to $R^{'}_{\!t}$.

Particularly, if $r = 1$, owing to Definition 5.9, two formal deformations $R_t = \sum\limits_{i = 0}^{+\infty}R_it^i$ and $R^{'}_t = \sum\limits_{i = 0}^{+\infty}R^{'}_{\!i}t^i$ are equivalent if there exist $x\in \mathfrak g$, $\phi_i\in \mathfrak {gl}(\mathfrak g)$ and $\psi_i\in \mathfrak {gl}(V), i\geq2$ such that

is a morphism from $R_t$ to $R^{'}_{\!t}$. Then, by extracting the coefficients of $t$ from both sides of $\phi_t\circ R_t = R^{'}_{\!t}\circ\psi_t$, we get

thus, we have the following result:

Proposition 5.10. Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. If two formal deformations $R_t = \sum\limits_{i = 0}^{+\infty}R_it^i$ and $R^{'}_{\!t} = \sum\limits_{i = 0}^{+\infty}R^{'}_{\!i}t^i$ of $R$ are equivalent, then their infinitesimals $R_1$ and $R^{'}_{\!1}$ are in the same cohomology class.

At the end of this subsection, we investigate the rigidity of a twisted Rota-Baxter operator based on Nijenhuis elements.

Definition 5.11. A twisted Rota-Baxter operator $R$ is called rigid if any formal deformation of $R$ is equivalent to the unaltered deformation $R^{'}_{\!t} = R$.

Definition 5.12. Assume that $R:V\rightarrow \mathfrak g$ is a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. An element $x\in \mathfrak g$ with $\alpha(x) = x$ is said to be a Nijenhuis element if $x$ satisfies that

and (5.1)–(5.3) hold. Denote the set of Nijenhuis elements associated with $R$ by $Nij(R)$.

By the above definition and Remark 5.5, any trivial linear deformation induces a Nijenhuis element. Moreover, we give a sufficient condition to characterize the rigidity of a twisted Rota-Baxter operator, as follows.

Theorem 5.13. Let $R:V\rightarrow \mathfrak g$ be a twisted Rota-Baxter operator on a Hom-Lie algebra $(\mathfrak g, [\cdot, \cdot], \alpha)$ with respect to the representation $(V, \rho, A)$. If $Z_R^{1}(V, \mathfrak g) = d_R(Nij(R))$, then $R$ is rigid.

Proof. Suppose that $R_t = \sum\limits_{i = 0}^{+\infty}R_it^i$ is a formal deformation of $R$. Thanks to Proposition 5.8, $R_1$ is a 1-cocycle of $R$. Hence, there exists a Nijenhuis element $x\in \mathfrak g$ such that $R_1 = d_R(x)$. Set

Define $R^{'}_{\!t} = \phi_t\circ R_t\circ \psi_t^{-1}$. Since $x$ is a Nijenhuis element, we deduce that $(\phi_t, \psi_t)$ is a morphism from $R_t$ to $R^{'}_{\!t}$; hence, $R^{'}_t$ is a formal deformation of $R$, which is equivalent to $R_{\!t}$. Furthermore, for $v\in V$, by direct computation, we obtain

which implies that the coefficient of $t$ in the expression of $R^{'}_{\!t}$ is trivial. Continuing by induction, we finally have that $R_t$ is equivalent to $R$. This completes the proof. □

6.   Conclusions

In this article, we introduced the concept of a twisted Rota-Baxter operator on a Hom-Lie algebra and defined its cohomology by constructing a twisting $L_\infty$-algebra associated with the twisted Rota-Baxter operator. We constructed a Chevalley-Eilenberg cohomology for a certain Hom-Lie algebra with coefficients in an appropriate representation. Surprisingly, this Chevalley-Eilenberg cohomology coincides with the cohomology of twisted Rota-Baxter operator s. We also showed that the linear component in a linear or formal deformation of a twisted Rota-Baxter operator $R$ is a 1-cocycle in the cohomology of $R$. At the end, we gave a sufficient condition to characterize the rigidity of formal deformations based on Nijenhuis elements.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence tools in the creation of this article.

Acknowledgments

The authors would like to thank the anonymous referees for valuable comments that have helped to improved the article significantly.

Xu was partially supported by NSF of China (12201253) and NSF of Jiangsu Province (BK20220510).

Conflict of interest

The authors declare that they have no conflicts of interest.