# Matching Hom-Setting of Rota-Baxter Algebras, Dendriform Algebras, and Pre-Lie Algebras

Authors: Dan Chen; Xiao-Song Peng; Chia Zargeh; Yi Zhang
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we introduce the Hom-algebra setting of the notions of matching Rota-Baxter algebras, matching (tri)dendriform algebras, and matching pre-Lie algebras. Moreover, we study the properties and relationships between categories of these matching Hom-algebraic structures.


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## Body

## 1. Introduction

### 1.1. Hom-Algebraic Structures

The origin of Hom-structures may be found in the study of Hom-Lie algebras which were first introduced by Hartwig, Larsson, and Silvestrov [1]. Hom-Lie algebras, as a generalization of Lie algebras, are introduced to describe the structures on deformations of the Witt algebra and the Virasoro algebra. More precisely, a Hom-Lie algebra is a triple L,−,−,α consisting of a k-module L, a bilinear skew-symmetric bracket −,−:L⊗L⟶L and an algebra endomorphism α:L⟶L satisfying the following Hom-Jacobi identity:
(1)αx,y,z+αy,z,x+αx,x,y=0for allx,y,z∈L.Recently, there have been several interesting developments of Hom-Lie algebras in mathematics and mathematical physics, including Hom-Lie bialgebras [2, 3], quadratic Hom-Lie algebras [4], involutive Hom-semigroups [5], deformed vector fields and differential calculus [6], representations [7, 8], cohomology and homology theory [9, 10], Yetter-Drinfeld categories [11], Hom-Yang-Baxter equations [12–16], Hom-Lie 2-algebras [17, 18], m,n-Hom-Lie algebras [19], Hom-left-symmetric algebras [20], and enveloping algebras [21]. In particular, the Hom-Lie algebra on semisimple Lie algebras was studied in [22], and the Hom-Lie structure on affine Kac-Moody was constructed in [23].In 2008, Makhlouf and Silvestrov [20] introduced the notation of Hom-associative algebras whose associativity law is twisted by a linear map. Usual functors between the categories of Lie algebras and associative algebras have been extended to the Hom-setting. It is shown that a Hom-associative algebra gives rise to a Hom-Lie algebra using the commutator. Since then, various Hom-analogues of some classical algebraic structures have been introduced and studied intensively, such as Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras [24, 25], Hom-groups [26, 27], Hom-Hopf modules [28], Hom-Lie superalgebras [29, 30], generalize Hom-Lie algebras [31], and Hom-Poisson algebras [32].Dendriform algebras were introduced by Loday [33] with motivation from algebraic K-theory. Latter, tridendriform algebras were proposed by Loday and Ronco [34] in their study of polytopes and Koszul duality. The classical links between Rota-Baxter algebras and (tri)dendriform algebras were given in [35, 36], resembling the structure of Lie algebras on an associative algebra. In 2012, Makhlouf [37] generalized the concepts of dendriform algebras and Rota-Baxter algebras by twisting the identities by mean of a linear map, which were called Hom-dendriform algebras and Rota-Baxter Hom-algebras, respectively. The connections between all these categories of Hom algebras were also investigated in [37]. Due to the fundamental work of Makhlouf [37], we have the following commutative diagram of categories (the arrows will go in the opposite direction for the corresponding operads), see Figure 1.Figure 1


### 1.2. Motivations for Matching Hom-Algebraic Structures

The recent concept of a matching or multiple Rota-Baxter [38] came from the study of multiple pre-Lie algebras [39] originated from the pioneering work of Bruned, Hairer, and Zambotti [40] on algebraic renormalization of regularity structures. It is shown that the matching Rota-Baxter algebra was motivated by the studies of associative Yang-Baxter equations, Volterra integral equations, and linear structure of Rota-Baxter operators [38]. More precisely, for exploring the relationship between associative Yang-Baxter equations and classical Yang-Baxter equations, Aguiar [41] proposed a polarized form of the expression on the left-hand side of the associative Yang-Baxter equation:
(2)r,s≔r13s12−r12s23+r23s13,where r,s∈A⊗A and A is a unitary associative algebra. The corresponding equation
(3)r13s12−r12s23+r23s13=0was called polarized associative Yang-Baxter equation (PAYBE) by Guo and etc. [38]. Paralleled to the fact that solutions of the associative Yang-Baxter equation naturally give Rota-Baxter operators, the matching Rota-Baxter operators are determined by solutions of a PAYBE [38].The basic theory of matching Rota-Baxter algebras was originally established in [38, 42], has proven useful not only in (compatible) multiple operations [43–48] but also in other areas of mathematics as well, such as polarized associative Yang-Baxter equation [38], algebraic combinatorics [38, 49], matching shuffle product [42], algebraic integral equation [50], and Gröbner-Shirshov bases and Hopf algebras [49]. Based on the close relationships between matching Rota-Baxter algebras, matching dendriform algebras, and matching pre-Lie algebras, Guo et al. [38] previously showed the following commutative diagram of categories, see Figure 2.Figure 2The main purpose of this paper is to extend these matching algebraic structures to the Hom-algebra setting and study the connections between these categories of Hom-algebras. These results give rise to the following commutative diagram of categories, see Figure3.Figure 3We would like to emphasize that the notation of matching Hom-Lie Rota-Baxter algebras will play a curial role in mathematical physics. The Rota-Baxter equation on a Lie algebra is the operator form of the classical Yang-Baxter equation [51]. Similarly, there should be a close relationship between the matching Hom Rota-Baxter equation in (82) with weight zero and the polarized classical Yang-Baxter equation, as a Hom-Lie algebra variation of the Hom version of the polarized associative Yang-Baxter equation.


### 1.3. Outline of the Paper and Summary of Results

In section2, we provide definitions concerning the generalization of matching associative algebras, matching pre-Lie algebras to Hom-algebras setting and describe some specific cases of matching Hom-algebraic structures. Also, the close relationship between matching Hom-Lie algebras and Hom-Lie algebras will be shown.In section3, we extend the notion of matching Rota-Baxter algebras to the Hom-associative algebra setting. It is also shown that matching Hom-associative Rota-Baxter algebras can be reduced from a matching Rota-Baxter algebra. At the end of this section, the construction of Hom-algebras using elements of the centroid is generalized to the matching Rota-Baxter algebras.Section4 is devoted to the definition of matching Hom-(tri)dendriform algebras and the approach of construction of a matching Hom-(tri)dendiform algebra from a matching (tri)dendiform algebra. Some results related to the connections between matching Hom-(tri)dendiform algebras and compatible Hom-associative algebras as well as between matching Hom-dendriform algebras and matching Hom-preLie algebras will be established.In section5, the concepts of matching Hom-Lie Rota-Baxter algebras and matching Rota-Baxter algebras involving elements of the centroid of matching Lie Rota-Baxter algebras will be established. Also, some results related to the connection between matching Hom-Lie Rota-Baxter algebra of weight zero and matching Hom-preLie algebra will be obtained.Notation. Throughout this paper, letk be a unitary commutative ring unless the contrary is specified, which will be the base ring of all modules, algebras, tensor products, operations as well as linear maps. We always suppose that Ω is a nonempty set. We denote by PΩ≔Pωω∈Ω the collection of operations Pω, ω∈Ω, where Ω is a set indexing the linear operators.


## 1.1. Hom-Algebraic Structures

The origin of Hom-structures may be found in the study of Hom-Lie algebras which were first introduced by Hartwig, Larsson, and Silvestrov [1]. Hom-Lie algebras, as a generalization of Lie algebras, are introduced to describe the structures on deformations of the Witt algebra and the Virasoro algebra. More precisely, a Hom-Lie algebra is a triple L,−,−,α consisting of a k-module L, a bilinear skew-symmetric bracket −,−:L⊗L⟶L and an algebra endomorphism α:L⟶L satisfying the following Hom-Jacobi identity:
(1)αx,y,z+αy,z,x+αx,x,y=0for allx,y,z∈L.Recently, there have been several interesting developments of Hom-Lie algebras in mathematics and mathematical physics, including Hom-Lie bialgebras [2, 3], quadratic Hom-Lie algebras [4], involutive Hom-semigroups [5], deformed vector fields and differential calculus [6], representations [7, 8], cohomology and homology theory [9, 10], Yetter-Drinfeld categories [11], Hom-Yang-Baxter equations [12–16], Hom-Lie 2-algebras [17, 18], m,n-Hom-Lie algebras [19], Hom-left-symmetric algebras [20], and enveloping algebras [21]. In particular, the Hom-Lie algebra on semisimple Lie algebras was studied in [22], and the Hom-Lie structure on affine Kac-Moody was constructed in [23].In 2008, Makhlouf and Silvestrov [20] introduced the notation of Hom-associative algebras whose associativity law is twisted by a linear map. Usual functors between the categories of Lie algebras and associative algebras have been extended to the Hom-setting. It is shown that a Hom-associative algebra gives rise to a Hom-Lie algebra using the commutator. Since then, various Hom-analogues of some classical algebraic structures have been introduced and studied intensively, such as Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras [24, 25], Hom-groups [26, 27], Hom-Hopf modules [28], Hom-Lie superalgebras [29, 30], generalize Hom-Lie algebras [31], and Hom-Poisson algebras [32].Dendriform algebras were introduced by Loday [33] with motivation from algebraic K-theory. Latter, tridendriform algebras were proposed by Loday and Ronco [34] in their study of polytopes and Koszul duality. The classical links between Rota-Baxter algebras and (tri)dendriform algebras were given in [35, 36], resembling the structure of Lie algebras on an associative algebra. In 2012, Makhlouf [37] generalized the concepts of dendriform algebras and Rota-Baxter algebras by twisting the identities by mean of a linear map, which were called Hom-dendriform algebras and Rota-Baxter Hom-algebras, respectively. The connections between all these categories of Hom algebras were also investigated in [37]. Due to the fundamental work of Makhlouf [37], we have the following commutative diagram of categories (the arrows will go in the opposite direction for the corresponding operads), see Figure 1.Figure 1


## 1.2. Motivations for Matching Hom-Algebraic Structures

The recent concept of a matching or multiple Rota-Baxter [38] came from the study of multiple pre-Lie algebras [39] originated from the pioneering work of Bruned, Hairer, and Zambotti [40] on algebraic renormalization of regularity structures. It is shown that the matching Rota-Baxter algebra was motivated by the studies of associative Yang-Baxter equations, Volterra integral equations, and linear structure of Rota-Baxter operators [38]. More precisely, for exploring the relationship between associative Yang-Baxter equations and classical Yang-Baxter equations, Aguiar [41] proposed a polarized form of the expression on the left-hand side of the associative Yang-Baxter equation:
(2)r,s≔r13s12−r12s23+r23s13,where r,s∈A⊗A and A is a unitary associative algebra. The corresponding equation
(3)r13s12−r12s23+r23s13=0was called polarized associative Yang-Baxter equation (PAYBE) by Guo and etc. [38]. Paralleled to the fact that solutions of the associative Yang-Baxter equation naturally give Rota-Baxter operators, the matching Rota-Baxter operators are determined by solutions of a PAYBE [38].The basic theory of matching Rota-Baxter algebras was originally established in [38, 42], has proven useful not only in (compatible) multiple operations [43–48] but also in other areas of mathematics as well, such as polarized associative Yang-Baxter equation [38], algebraic combinatorics [38, 49], matching shuffle product [42], algebraic integral equation [50], and Gröbner-Shirshov bases and Hopf algebras [49]. Based on the close relationships between matching Rota-Baxter algebras, matching dendriform algebras, and matching pre-Lie algebras, Guo et al. [38] previously showed the following commutative diagram of categories, see Figure 2.Figure 2The main purpose of this paper is to extend these matching algebraic structures to the Hom-algebra setting and study the connections between these categories of Hom-algebras. These results give rise to the following commutative diagram of categories, see Figure3.Figure 3We would like to emphasize that the notation of matching Hom-Lie Rota-Baxter algebras will play a curial role in mathematical physics. The Rota-Baxter equation on a Lie algebra is the operator form of the classical Yang-Baxter equation [51]. Similarly, there should be a close relationship between the matching Hom Rota-Baxter equation in (82) with weight zero and the polarized classical Yang-Baxter equation, as a Hom-Lie algebra variation of the Hom version of the polarized associative Yang-Baxter equation.


## 1.3. Outline of the Paper and Summary of Results

In section2, we provide definitions concerning the generalization of matching associative algebras, matching pre-Lie algebras to Hom-algebras setting and describe some specific cases of matching Hom-algebraic structures. Also, the close relationship between matching Hom-Lie algebras and Hom-Lie algebras will be shown.In section3, we extend the notion of matching Rota-Baxter algebras to the Hom-associative algebra setting. It is also shown that matching Hom-associative Rota-Baxter algebras can be reduced from a matching Rota-Baxter algebra. At the end of this section, the construction of Hom-algebras using elements of the centroid is generalized to the matching Rota-Baxter algebras.Section4 is devoted to the definition of matching Hom-(tri)dendriform algebras and the approach of construction of a matching Hom-(tri)dendiform algebra from a matching (tri)dendiform algebra. Some results related to the connections between matching Hom-(tri)dendiform algebras and compatible Hom-associative algebras as well as between matching Hom-dendriform algebras and matching Hom-preLie algebras will be established.In section5, the concepts of matching Hom-Lie Rota-Baxter algebras and matching Rota-Baxter algebras involving elements of the centroid of matching Lie Rota-Baxter algebras will be established. Also, some results related to the connection between matching Hom-Lie Rota-Baxter algebra of weight zero and matching Hom-preLie algebra will be obtained.Notation. Throughout this paper, letk be a unitary commutative ring unless the contrary is specified, which will be the base ring of all modules, algebras, tensor products, operations as well as linear maps. We always suppose that Ω is a nonempty set. We denote by PΩ≔Pωω∈Ω the collection of operations Pω, ω∈Ω, where Ω is a set indexing the linear operators.


## 2. Matching Hom-Associative, Matching Hom-preLie and Matching Hom-Lie Algebras

In this section, we give the definitions of matching Hom-associative algebras, compatible Hom-associative algebras, compatible Hom-preLie algebras, and compatible Hom-Lie algebras, which generalize the corresponding matching algebraic structures introduced in [38]. Then, we explore the relationships between these categories from the point of view of Hom-algebras.Definition 1.
A matching Hom-associative algebra is a k-moduleA together with a collection of binary operations ·ω:A⊗A⟶A,ω∈Ω and a linear map p:A⟶A such that
(4)x·αy·βpz=px·αy·βzforallx,y,z∈Aandα,β∈Ω.
A matching Hom-associative algebra is called totally compatible if it satisfies(5)x·αy·βpz=px·βy·αzforallx,y,z∈Aandα,β∈Ω.More generally,Definition 2.
A compatible Hom-associative algebra is a k-moduleA together with a collection of binary operations ·ω:A⊗A⟶A,ω∈Ωand a linear map p:A⟶A such that
(6)x·αy·βpz+x·βy·αpz=px·αy·βz+px·βy·αz
for allx,y,z∈A and α,β∈Ω. For simplicity, we denote it by A,·Ω,p.Remark 3.
(a)
Any matching Hom-associative algebra or totally compatible Hom-associative algebra is a compatible Hom-associative algebra(b)
By takingp=id, we recover to the definition of matching associative algebras, totally compatible associative algebra and compatible associative algebra given in [38](c)
IfΩ is a singleton and the characteristic of k is not 2, then the notation of matching Hom-associative algebras and the notation of compatible Hom-associative algebras are equivalent and recover to the Hom-associative algebras introduced in [20]Definition 4.
A matching Hom-Lie algebra is a k-moduleg equipped with a collection of binary operations ,ω:g⊗g⟶g, ω∈Ω and a linear map p:g⟶g such that
(7)x,xω=0(8)px,y,zβα+py,z,xαβ+pz,x,yαβ=0for all x,y,z∈g and α,β,ω∈Ω.Remark 5.
A totally compatible Hom-associative algebraA,·Ω,p has a natural matching Hom-Lie algebra structure with the Lie bracket defined by
(9)x,yω≔x·ωy−y·ωx,forx,y∈Aandω∈Ω.The matching Hom-Lie algebra has a close relationship with Hom-Lie algebras. We first record a lemma for a preparation.Lemma 6.
Letg,,Ω,p be a matching Hom-Lie algebra. Consider linear combinations
(10),A≔∑α∈Ωaα,αand,B≔∑β∈Ωbβ,β,where aα,bβ∈k for α,β∈Ω with finite supports. Then
(11)px,y,zBA+py,z,xAB+pz,x,yAB=0forx,y,z∈g.Proof.
By Eq. (10), for x,y,z∈g, we have
(12)px,y,zBA=px,∑β∈Ωbβy,zβA=∑α∈Ωaαpx,∑β∈Ωbβy,zβα=∑α∈Ω∑β∈Ωaαbβpx,y,zβα.
Similarly, we also have(13)py,z,xAB=∑α∈Ω∑β∈Ωbβaαpy,z,xαβandpz,x,yAB=∑α∈Ω∑β∈Ωbβaαpz,x,yαβ.
Sinceg,,Ω,p is a matching Hom-Lie algebra, then
(14)px,y,zβα+py,z,xαβ+pz,x,yαβ=0forallx,y,z∈gandα,β∈Ω.
Thus(15)px,y,zBA+py,z,xAB+pz,x,yAB=0,as desired.Proposition 7.
Letg,,Ω,p be a matching Hom-Lie algebra. Consider linear combinations
(16),A≔∑ω∈Ωaω,ω,aω∈k,with a finite support. Then, g,,A is a Hom-Lie algebra.Proof.
It follows from Lemma6 by taking aωω∈Ω=bωω∈Ω.More generally, we proposeDefinition 8.
A compatible Hom-Lie algebra is a k-moduleg together with a set of binary operations ,ω:g⊗g⟶g,ω∈Ω and a linear map p:g⟶g such that
(17)x,xω=0(18)px,y,zαβ+py,z,xαβ+pz,x,yαβ+px,y,zβα+py,z,xβα+pz,x,yβα=0
for allx,y,z∈g and ω,α,β∈Ω.Remark 9.
(a)
Every matching Hom-Lie algebra is a compatible Hom-Lie algebra.(b)
Given two Hom-Lie algebrasg,,α,p and g,,β,p. Define a new bracket ,:g⊗g⟶g as follows:(19)x,y≔aαx,yα+bβx,yβforsomeaα,bβ∈k.
Clearly, this new bracket is both skew symmetric and bilinear. Then,g,,,p is further a Hom-Lie algebra if , satisfies the Hom-Jacobi identity
(20)px,y,z+py,z,x+pz,x,y=0.
By a direct calculation, we get that this condition is equivalent to Eq. (18).Proposition 10.
Letg,,Ω,p be a matching Hom-Lie algebra. Then for x,y,z∈g and α,β∈Ω, we have
(21)px,y,zαβ=px,y,zβα,px,y,zαβ+py,z,xαβ+pz,x,yαβ=0.Proof.
Since Eq. (8) holds for any x,y,z∈A and α,β∈Ω, we get
(22)py,z,xαβ+pz,x,yβα+px,y,zβα=0.
Eqs. (8) and (22) result in
(23)pz,x,yαβ−pz,x,yβα=0.
By the arbitrariness ofx,y,z, we have
(24)px,y,zαβ=px,y,zβαand so
(25)px,y,zαβ+py,z,xαβ+pz,x,yαβ=0.Generalizing the well-known result that an associative algebra has a Lie algebra structure via the commutator bracket, we show that a compatible Hom-associative algebra has a compatible Hom-Lie algebra structure.Proposition 11.
LetA,·Ω,p be a compatible Hom-associative algebra. Then A,,Ω,p is a compatible Hom-Lie algebra, where
(26),ω:A⊗A⟶A,x,yω≔x·ωy−y·ωxforx,y∈Aandω∈Ω.Proof.
Forx,y,z∈A and α,β,ω∈Ω, by Eq. (26), we get x,xω=0 and
(27)px,y,zαβ=px,y·αz−z·αyβ=px·βy·αz−z·αy−y·αz−z·αy·βpx=px·βy·αz−px·βz·αy−y·αz·βpx+z·αy·βpx.
Similarly, we have(28)py,z,xαβ=py·βz·αx−py·βx·αz−z·αx·βpy+x·αz·βpy,pz,x,yαβ=pz·βx·αy−pz·βy·αx−x·αy·βpz+y·αx·βpz,px,y,zβα=px·αy·βz−px·αz·βy−y·βz·αpx+z·βy·αpx,py,z,xβα=py·αz·βx−py·αx·βz−z·βx·αpy+x·βz·αpy,pz,z,yβα=pz·αx·βy−pz·αy·βx−x·βy·αpz+y·βx·αpz.
By Eq. (6), we get
(29)px,y,zαβ+py,z,xαβ+pz,x,yαβ+px,y,zβα+py,z,xβα+pz,x,yβα=0.
Hence,A,,Ω,p is a compatible Hom-Lie algebra.Now, we give the definition of matching Hom-preLie algebras.Definition 12.
A matching Hom-preLie algebra is a k-module A together with a family of binary operations ∗ω:A⊗A⟶A,ω∈Ω and a linear map p:A⟶A such that
(30)px∗αy∗βz−x∗αy∗βpz=py∗βx∗αz−y∗βx∗αpzfor all x,y,z∈A and α,β∈Ω.
Now, we give the relationship between matching Hom-preLie algebras and compatible Hom-Lie algebras.Proposition 13.
LetA,∗Ω,p be a matching Hom-preLie algebra. Then A,,Ω,p is a compatible Hom-Lie algebra, where
(31),ω:A⊗A⟶A,x,yω≔x∗ωy−y∗ωx,forallx,y∈Aandω∈Ω.Proof.
Forx,y,z∈A and α,β∈Ω, by Eq. (31), we have x,xω=0 and
(32)px,y,zαβ=px,y∗αz−z∗αyβ=px∗βy∗αz−z∗αy−y∗αz−z∗αy∗βpx=px∗βy∗αz−px∗βz∗αy−y∗αz∗βpx+z∗αy∗βpx.
Similarly, we have(33)py,z,xαβ=py∗βz∗αx−py∗βx∗αz−z∗αx∗βpy+x∗αz∗βpy,pz,x,yαβ=pz∗βx∗αy−pz∗βy∗αx−x∗αy∗βpz+y∗αx∗βpz,px,y,zβα=px∗αy∗βz−px∗αz∗βy−y∗βz∗αpx+z∗βy∗αpx,py,z,xβα=py∗αz∗βx−py∗αx∗βz−z∗βx∗αpy+x∗βz∗αpy,pz,x,yβα=pz∗αx∗βy−pz∗αy∗βx−x∗βy∗αpz+y∗βx∗βpz.
Then, by Eq. (30), we get
(34)px,y,zαβ+py,z,xαβ+pz,x,yαβ+px,y,zβα+py,z,xβα+pz,x,yβα=0.
Hence,A,,Ω,p is a compatible Hom-Lie algebra.


## 3. Matching Rota-Baxter Algebras and Hom-Associative Algebras

In this section, we extend the notion of matching Rota-Baxter algebras to the Hom-associative algebra setting.Definition 14 [38].
LetλΩ≔λωω∈Ω⊆k be a set of scalars indexed by Ω. A matching Rota-Baxter algebra of weight λΩ is an associative algebra A equipped with a family PΩ≔Pωω∈Ω of linear operators Pω:R⟶R,ω∈Ω, that satisfy the matching Rota-Baxter equation
(35)Pαx·Pβy=Pαx·Pβy+PβPαx·y+λβPαx·y,forallx,y∈Aandα,β∈Ω.Definition 15.
A matching Hom-associative Rota-Baxter algebra is a quadruples A,·,PΩ,p, where A,PΩ is a matching Rota-Baxter algebra and A,·,p is a Hom-associative algebra.Takingp=id, we recover to matching Rota-Baxter associative algebras and denote it by A,·,PΩ. If Ω is a singleton, a matching Hom-associative Rota-Baxter algebra becomes a Hom-associative Rota-Baxter algebra given in [37].A Hom-associative Rota-Baxter algebra can be induced from an associative Rota-Baxter algebra with a particular algebra endomorphism [37]. The following result generalizes it to the matching Rota-Baxter case.Theorem 16.
LetA,·,PΩ be a matching Rota-Baxter algebra and p:A⟶A be an algebra endomorphism which commutes with Pω for all ω∈Ω. Then A,·p,PΩ,p, where x·py≔px·y, is a matching Hom-associative Rota-Baxter algebra.Proof.
The Hom-associative structure of the algebra follows from Yau’s Theorem in [52]. We only need to show that the matching Rota-Baxter equation holds. For x,y∈A and α,β∈Ω,
(36)Pαx·pPβy=pPαx·Pβybythedefinitionof·p=pPαx·Pβy+PβPαx·y+λβPαx·ybyEq.10=pPαx·Pβy+pPβPαx·y+λβpPαx·y=Pαpx·Pβy+PβpPαx·y+λPαpx·ybyp∘Pω=Pω∘p=Pαx·pPβy+PβPαx·py+λPαx·py,as required.Given a matching Hom-associative Rota-Baxter algebraA,·,PΩ,p, it is natural to wonder that whether this matching Hom-associative Rota-Baxter algebra is induced by an ordinary associative matching Rota-Baxter algebra A,·′,PΩ, i.e., p is an algebra endomorphism with respect to ·′ and ·=p∘·′.LetA,·,p be a multiplicative Hom-associative algebra, i.e., pa·b=pa·pb for all a,b∈A. It was proved in [53] that in case p is invertible, A,p−1∘·is an associative algebra. It is generalized to the multiplicative Hom-associative Rota-Baxter algebras in [37], and the following result generalizes it to the multiplicative matching Hom-associative Rota-Baxter algebras.Proposition 17.
LetA,·,PΩ,p be a multiplicative matching Hom-assoicative Rota-Baxter algebra, where p is invertible and p∘Pω=Pω∘p for each ω∈Ω. Then, A,·′≔p−1∘·,PΩ is an associative matching Rota-Baxter algebra.Proof.
Forx,y,z∈A, we have
(37)x·′y·′z−x·′y·′z=p−1p−1x·y·z−p−1x·p−1y·zby·′=p−1∘·=p−2x·y·pz−px·y·zbypx·py=px·y=0.
Hence, the associativity condition holds. Forα,β∈Ω, we have
(38)Pαx·′Pβy=p−1Pαx·Pβy=p−1Pαx·Pβy+PβPαx·y+λβPαx·y=Pαp−1x·Pβy+Pβp−1Pαx·y+λβPαp−1x·y=Pαx·′Pβy+PβPαx·′y+λβPαx·′y.Hence, the matching Rota-Baxter equation holds for the new multiplication, andA,·′,PΩ is an associative matching Rota-Baxter algebra.There are two new ways of constructing Hom-associative algebras from a given multiplicative Hom-associative algebra [37, 54].Definition 18. ([37, 54]).
LetA,·,p be a multiplicative Hom-algebra and n≥0. Then, the following two algebras are also Hom-associative algebras:
(a)
then-th derived Hom-algebra of type 1 of A defined by
(39)An=A,·n=pn∘·,pn+1,(b)
then-th derived Hom-algebra of type 2 of A defined by
(40)An=A,·n=p2n−1∘·,p2n.Now, we show that then-th derived Hom-algebra of type 1 and 2 of a multiplicative matching Hom-associative Rota-Baxter algebra is also a matching Hom-associative Rota-Baxter algebra generalizing the Rota-Baxter case in [37].Theorem 19.
LetA,·,PΩ,p be a multiplicative matching Hom-associative Rota-Baxter algebra such that p∘Pω=Pω∘p for all ω∈Ω. Then,
(a)
then-th derived Hom-algebra of type 1A,·n=pn∘·,pn+1 is a matching Hom-associative Rota-Baxter algebra(b)
then-th derived Hom-algebra of type 2 A,·n=p2n−1∘·,p2n is a matching Hom-associative Rota-Baxter algebraProof.
(a) By [54], A,·n,pn+1 is a Hom-associative algebra. Now, we show the matching Rota-Baxter equation holds. For x,y,z∈A and α,β∈Ω, we have
(41)Pαx·nPβy=pnPαx·Pβy=pnPαx·Pβy+PβPαx·y+λβPαx·y=Pαpnx·Pβy+PβpnPαx·y+λβPαpnx·y=Pαx·nPβy+PβPαx·ny+λβPαx·ny.
Thus, the matching Rota-Baxter equation holds for the new multiplication.
(b) By [54], A,·n=p2n−1∘·,p2n is also a Hom-associative algebra. For x,y,z∈A and α,β∈Ω, we have
(42)Pαx·nPβy=p2n−1Pαx·Pβy=p2n−1Pαx·Pβy+PβPαx·y+λβPαx·y=Pαp2n−1x·Pβy+Pβp2n−1Pαx·y+λβPαp2n−1x·y=Pαx·nPβy+PβPαx·ny+λβPαx·ny.
This completes the proof.LetA,· be an associative algebra. The centroid of A is defined by
(43)CentA≔p∈EndA∣px·y=px·y=x·pyforallx,y∈A.The same definition of the centroid is assumed for Hom-associative algebras.In [4], Benayadi and Makhlouf gave the construction of Hom-algebras using elements of the centroid for Lie algebras. In [37], the construction was extended to Rota-Baxter algebras. Now, we generalize it to the matching Rota-Baxter case.Proposition 20.
LetA,·,PΩ be an associative matching Rota-Baxter algebra. For p∈CentA and x,y∈A, define
(44)x·p1y≔px·yandx·p2y≔px·py.
Ifp∘Pω=Pω∘p for all ω∈Ω, then A,·p1,PΩ,p and A,·p2,PΩ,p are matching Hom-associative Rota-Baxter algebras.Proof By [37].
A,·p1,p and A,·p2,p are Hom-associative algebras. Now, we show that they are also matching Rota-Baxter algebras. For x,y∈A and α,β∈Ω, we have
(45)Pαx·p1Pβy=pPαx·Pβy=Pαpx·Pβy=Pαpx·Pβy+PβPαpx·y+λβPαpx·y=Pαx·p1Pβy+PβPαx·p1y+λβPαx·p1y
and(46)Pαx·p2Pβy=pPαx·pPβy=Pαpx·Pβpy=Pαpx·Pβpy+PβPαpx·py+λβPαpx·py=Pαpx·pPβy+PβpPαx·py+λβPαpx·py=Pαx·p2Pβy+PβPαx·p2y+λβPαx·p2y.
This completes the proof.


## 4. Matching Hom-Dendriform Algebras and Matching Hom-Tridendriform Algebras

In this section, we introduce the notions of matching Hom-dendriform algebras and matching Hom-tridendriform algebras generalizing the definitions of matching dendriform algebras and matching tridendriform algebras given in [38].Definition 21.
A matching Hom-dendriform algebra is a k-moduleD together with a family of binary operations ⊙ω:D⊗D⟶D, where⊙∈≺,≻ and ω∈Ω, and a linear map p:D⟶D such that for all x,y,z∈D and α,β∈Ω,
(47)x≺αy≺βpz=px≺αy≺βz+px≺βy≻αz,x≻αy≺βpz=px≻αy≺βz,x≺βy≻αpz+x≻αy≻βpz=px≻αy≻βz.
For simplicity, we denote it byD,≺Ω,≻Ω,p.Definition 22.
A matching Hom-tridendriform algebra is a k-moduleD together with a family of binary operations ⊙ω:D⊗D⟶D, where⊙e∈≺,•,≻ and ω∈Ω , and a linear map p:D⟶D such that for all x,y,z∈D and α,β∈Ω,
(48)x≺αy≺βpz=px≺αy≺βz+px≺βy≻αz+px≺αy•βz,(49)x≻αy≺βpz=px≻αy≺βz,(50)px≻αy≻βz=x≺βy≻αpz+x≻αy≻βpz+x•βy≻αpz,(51)x≻αy•βpz=px≻αy•βz,(52)x≺αy•βpz=px•βy≻αz,(53)x•αy≺βpz=px•αy≺βz,(54)x•αy•βpz=px•αy•βz.Definition 23.
(a)
LetD,≺Ω,≻Ω,p and D′,≺Ω′,≻Ω′,p′ be two matching Hom-dendriform algebras. A linear map f:D⟶D′ is called a matching Hom-dendriform algebra morphism if for all ω∈Ω(55)≺ω′∘f⊗f=f∘≺ω,≻ω∘f⊗f=f∘≻ωandp′∘f=f∘p.(b)
LetD,≺Ω,•Ω,≻Ω,p and D′,≺Ω′,•Ω,≻Ω′,p′ be two matching Hom-tridendriform algebras. A linear map f:D⟶D′ is called a matching Hom-tridendriform algebra morphism if for all ω∈Ω(56)≺ω′∘f⊗f=f∘≺ω,•ω′∘f⊗f=f∘•ω′,≻ω∘f⊗f=f∘≻ωandp′∘f=f∘p.The following results show that we can construct a matching Hom-(tri)dendriform algebra from a matching (tri)dendriform algebra, generalizing the (tri)dendriform case in [37].Theorem 24.
(a)
LetD,≺Ω,≻Ω be a matching dendriform algebra and p:D⟶D be a matching dendriform algebra endomorphism. Then, Ap=A,≺p,Ω,≻p,Ω,p, where ≺p,ω≔p∘≺ω and ≻p,ω≔p∘≻ω for each ω∈Ω, is a matching Hom-dendriform algebra. Moreover, suppose that A′,≺Ω′,≻Ω′ is another matching dendriform algebra and p′:A′⟶A′ is a matching dendriform algebra endomorphism. If f:A⟶A′ is a matching dendriform algebra morphism that satisfies f∘p=p′∘f, then
(57)f:D,≺p,Ω,≻p,Ω,p⟶D′,≺′p,Ω,≻′p,Ω,p′is a morphism of matching Hom-dendriform algebras.(b)
LetD,≺Ω,•Ω,≻Ω be a matching tridendriform algebra and p:D⟶D be a matching tridendriform algebra endomorphism. Then, Ap=A,≺p,Ω,•p,Ω,≻p,Ω,p, where ≺p,ω≔p∘≺ω, •p,ω≔p∘•ω and ≻p,ω=p∘≻ω for each ω∈Ω, is a matching Hom-tridendriform algebra. Moreover, suppose that A′,≺Ω′,•Ω′,≻Ω′ is another matching tridendriform algebra and p′:A′⟶A′ is a matching tridendriform algebra endomorphism. If f:A⟶A′ is a matching tridendriform algebra morphism that satisfies f∘p=p′∘f, then
(58)f:D,≺p,Ω,•p,Ω,≻p,Ω,p⟶D′,≺p,Ω′,•p,Ω′,≻p,Ω′,p′is a morphism of matching Hom-tridendriform algebras.Proof.
We just prove Item (b) and Item (a) can be proved similarly. For anyx,y,z∈A and α,β∈Ω, we have
(59)x≺p,αy≺p,βpz=ppx≺αy≺βpz=p2x≺αy≺βz;px≺p,αy≺p,βz=ppx≺αpy≺βz=p2x≺αy≺βz;px≺p,βy≻p,αz=ppx≺βpy≻αz=p2x≺βy≻αz;px≺p,αy•p,βz=ppx≺αpy•βz=p2x≺αy•βz.
Hence,(60)x≺p,αy≺p,βpz=px≺p,αy≺p,βz+px≺p,βy≻p,αz+px≺p,αy•p,βz,that is Eq. (48) holds for A,≺p,Ω,•p,Ω,≻p,Ω,p. Similarly, Eqs. (49), (50), (51), (52), (53), (54) hold. Hence, A,≺p,Ω,•p,Ω,≻p,Ω,p is a matching Hom-tridendriform algebra. And
(61)fx≺p′,αfy=p′fx≺αfy=p′∘fx≺αy=f∘px≺αy=fx≺p,αy;fx≻p′,αfy=p′fx≻αfy=p′∘fx≻αy=f∘px≻αy=fx≻p,αy;fx•p′,αfy=p′fx•αfy=p′∘fx•αy=f∘px•y=fx•p,αy.
Hence,f:D,≺p,Ω,•p,Ω,≻p,Ω,p⟶D′,≺p,Ω′,•p,Ω′,≻p,Ω′,p′ is a morphism of matching Hom-tridendriform algebras.Now, we show that any linear combinations of the operations of a matching Hom-dendriform algebra still result in a matching Hom-dendriform algebra, generalizing the matching dendriform case in [38].Proposition 25.
LetI be an nonempty set. For each i∈I, let Ai:Ω⟶k be a map with finite supports, identified with finite set Ai=ai,ωω∈Ω,ai,ω∈k.
(a)
LetD,≺Ω,≻Ω,p be a matching Hom-dendriform algebra. Define the following binary operations:
(62)⊙i≔∑ω∈Ωai,ω⊙,where⊙∈≺,≻andi∈I.Then,D,≺I,≻I,p is also a matching Hom-dendriform algebra.(b)
LetT,≺Ω,•Ω,≻Ω,p be a matching Hom-tridendriform algebra. Define the following binary operations:
(63)⊙i≔∑ω∈Ωai,ω⊙ω,where⊙∈≺,•,≻andi∈I.Then,T,≺I,•I,≻I,p is also a matching Hom-tridendriform algebra.Proof.
We just prove Item (b) and Item (a) can be proved similarly. Forx,y,z∈D and i,j∈I, we have
(64)x≺iy≺jpz=∑β∈Ωbj,β∑α∈Ωai,αx≺αy≺βpz=∑α∈Ω∑β∈Ωai,αbj,βx≺αy≺βpz=∑α∈Ω∑β∈Ωai,αbj,βpx≺αy≺βz+px≺βy≻αz+px≺αy•βz=∑α∈Ωai,αpx≺α∑β∈Ωbj,βy≺βz+∑β∈Ωbj,βpx≺β∑α∈Ωai,αy≻αz+∑α∈Ωai,αpx≺α∑β∈Ωbj,βy•βz=∑α∈Ωai,αpx≺αy≺jz+∑β∈Ωbj,βpx≺βy≻iz+∑α∈Ωai,αpx≺αy•jz=px≺iy≺jz+px≺jy≻iz+px≺iy•jz.
Hence, Eq. (48) holds. Similarly, Eqs. (49), (50), (51), (52), (53), (54) hold. Hence, T,≺I,•I,≻I,p is a matching Hom-tridendriform algebra.The following results establish the connections between matching Hom-(tri)dendriform algebras and compatible Hom-associative algebras, generalizing the well-known result that a (tri) dendriform algebra has an associative algebraic structure.Theorem 26.
(a)
LetA,≺Ω,≻Ω,p be a matching Hom-dendriform algebra. Then A,·Ω,p is a compatible Hom-associative algebra, where
(65)·ω:A⊗A⟶A,x·ωy≔x≺ωy+x≻ωyforx,y∈Aandω∈Ω.(b)
LetA,≺Ω,•Ω,≻Ω,p be a matching Hom-tridendriform algebra. Then, A,·Ω,p is a compatible Hom-associative algebra, where
(66)·ω:A⊗A⟶A,x·ωy≔x≺ωy+x•ωy+x≻ωyforx,y∈Aandω∈Ω.Proof.
We only prove Item (b) and Item (a) can be proved similarly. Forx,y,z∈A and α,β∈Ω, we have
(67)x·αy·βpz+x·βy·αpz=x≺αy+x•αy+x≻αy·βpz+x≺βy+x•βy+x≻βy·αpz=x≺αy≺βpz+x•αy≺βpz+x≻αy≺βpz+x≺αy•βpz+x•αy•βpz+x≻αy•βpz+x≺αy≻βpz+x•αy≻βpz+x≻αy≻βpz+x≺βy≺αpz+x≻βy≺αpz+x•βy≺αpz+x≺βy•αpz+x≻βy•αpz+x•βy•αpz+x≺βy≻αpz+x•βy≻αpz+x≻βy≻αpz,px·αy·βz+px·βy·αz=px·αy≺βz+y•βz+y≻βz+px·βy≺αz+y•αz+y≻αz=px≺αy≺βz+px≺αy•βz+px≺αy≻βz+px•αy≺βz+px•αy•βz+px•αy≻βz+px≻αy≺βz+px≻αy•βz+px≻αy≻βz+px≺βy≺αz+px≺βy•αz+px≺βy≻αz+px•βy≺αz+px•βy•αz+px•βy≻αz+px≻βy≺αz+px≻βy•αz+px≻βy≻αz.
By Eqs (48), (49), (50), (51), (52), (53), (54), we get
(68)x·αy·βpz+x·βy·αpz=px·αy·βz+px·βy·αz.Hence,A,·Ω,p is a compatible Hom-associative algebra.Now, we explore the relationship between matching Hom-dendriform algebras and matching Hom-preLie algebras.Theorem 27.
LetA,≺Ω,≻Ω,p be a matching Hom-dendriform algebra. Then A,∗Ω,p is a matching Hom-preLie algebra, where
(69)∗ω:A⊗A⟶A,x∗ωy≔x≻ωy−y≺ωxforx,y∈Aandω∈Ω.Proof.
Forx,y,z∈A and α,β∈Ω, we have
(70)px∗αy∗βz−x∗αy∗βpz=px∗αy≻βz−z≺βy−x≻αy−y≺αx∗βpz=px≻αy≻βz−px≻αz≺βy−y≻βz≺αpx+z≺βy≺αpx−x≻αy≻βpz+y≺αx≻βpz+pz≺βx≻αy−pz≺βy≺αxand
(71)py∗βx∗αz−y∗βx∗αpz=py∗βx≻αz−z≺αx−y≻βx−x≺βy∗αpz=py≻βx≻αz−py≻βz≺αx−x≻αz≺βpy+z≺αx≺βpy−y≻βx≻αpz+x≺βy≻αpz+pz≺αy≻βx−pz≺αx≺βy.
By Eqs (48), (49), (50), (51), (52), (53), (54), we get
(72)px∗αy∗βz−x∗αy∗βpz=py∗βx∗αz−y∗βx∗αpz.Hence,A,∗Ω,p is a matching Hom-preLie algebra.A matching Rota-Baxter algebraA,·,PΩ is of weight 0 if the set λΩ=0. The connections between Rota-Baxter algebras and (tri)dendriform algebras are given in [36, 41] and extended to matching Rota-Baxter algebras. Now, we generalize it to matching Hom-associative Rota-Baxter algebra.Proposition 28.
(a)
LetA,·,PΩ,p be a matching Hom-associative Rota-Baxter algebra of weight 0. Assume that p∘Pω=Pω∘p for each ω∈Ω. Define the operations ≺ω and ≻ω for ω∈Ω by
(73)x≺ωy≔x·Pωyandx≻ωy=Pωx·y,forx,y∈A.ThenA,≺Ω,≻Ω,p is a matching Hom-dendriform algebra.(b)
LetA,·,PΩ,p be a matching Hom-associative Rota-Baxter algebra. Assume that p∘Pω=Pω∘p for each ω∈Ω. Define the operations ≺ω, ≻ω, ω∈Ω by
(74)x≺ωy≔x·Pωy+λωx·yandx≻ωy=Pωx·y,forx,y∈A.Then,A,≺Ω,≻Ω,p is a matching Hom-dendriform algebra.Proof.
Since Item (a) can be seen as a special case of Item (b) by takingλΩ=0, we only prove Item (b). For x,y,z∈A and α,β∈Ω, we have
(75)px≺αy≺βz+px≺βy≻αz=px≺αy·Pβz+λβy·z+px≺βPαy·z=px·Pαy·Pβz+λβy·z+λαpx·y·Pβz+λβy·z+px·PβPαy·z+λβpx·Pαy·z=pxPαy·Pβz+λαpx·y·Pβz+λαλβpx·y·z+λβpx·Pαy·z=x·Pαy+λαx·y·Pβpz+λβx·Pαy+λαx·y·pz=x·Pαy+λαx·y≺βpz=x≺αy≺βpz.
Also,(76)x≻αy≺βpz=Pαx·y≺βpz=Pαx·y·Pβpz+λβPαx·y·pz=Pαpx·y·Pβz+λβPαpx·y·z=Pαpx·y·Pβz+λβy·z=Pαpx·y≺βz=px≻αy≺βz
and(77)x≺βy≻αpz+x≻αy≻βpz=x·Pβy+λβx·y≻αpz+Pαx·y≻βpz=Pαx·Pβy+λβx·y·pz+PβPαx·y·pz=Pαx·Pβy+PβPαx·y+λβPαx·y·pz=Pαx·Pβy·pz=Pαpx·Pβy·z=px≻αy≻βz.Hence,A,≺Ω,≻Ω,p is a matching Hom-dendriform algebra.Proposition 29.
LetA,·,PΩ,p be a matching Hom-associative Rota-Baxter algebra. Assume that p∘Pω=Pω∘p for each ω∈Ω. Define the operations ≺ω, ≻ω and •ω for ω∈Ω by
(78)x≺ωy≔x·Pωy,x≻ωy=Pωx·yandx•ωy=λωx·y,forx,y∈A.Then,A,≺Ω,•Ω,≻Ω,p is a matching Hom-tridendriform algebra.Proof.
Forx,y,z∈A and α,β∈Ω, we have
(79)x≺αy≺βpz=x·Pαy·Pβpz=px·Pαy·Pβz=px·Pαy·Pβz+PβPαy·z+λβPαy·z=px≺αy≺βz+px≺βy≻αz+x≺αy•βz,x≻αy≺βpz=Pαx·y·Pβpz=Pαpx·y·Pβz=px≻αy≺βz,px≻αy≻βz=Pαpx·Pβy·z=Pαx·Pβy·pz=Pαx·Pβy+PβPαx·y+λβPαx·y·pz=x≺βy≻αpz+x≻αy≻βpz+x•βy≻αpz,x≻αy•βpz=λβPαx·y·pz=λβPαpx·y·z=px≻αy•βz,x≺αy•βpz=λβx·Pαy·pz=λβpx·Pαy·z=px•βy≻αz,x•αy≺βpz=λαx·y·Pβpz=λαpx·y·Pβz=px•αy≺βz,x•αy•βpz=λαλβx·y·pz=λαλβpx·y·z=px•αy•βz,as required.Corollary 30.
(a)
LetA,·,PΩ,p be a matching Hom-associative Rota-Baxter algebra of weight 0. Then, A,∗Ω is a matching Hom-preLie algebra, where
(80)x∗ωy≔Pωx·y−y·Pωxforx,y∈Aandω∈Ω.(b)
LetA,·,PΩ,p be a matching Hom-associative Rota-Baxter algebra. Then, A,∗Ω is a matching Hom-preLie algebra, where
(81)x∗ωy≔Pωx·y−y·Pωx−λωy·xforx,y∈Aandω∈Ω.Proof.
(a) It follows from Theorem27 and Proposition 28 (a). (b) It follows from Theorem 27 and Proposition 28 (b).


## 5. Matching Rta-Baxter Operators and Hom-Nonassociative Algebras

Rota-Baxter Lie algebras were introduced independently by Belavin and Drinfeld and Semenov-Tian-Shansky in [51, 55] and were related to solutions of the (modified) Yang-Baxter equation. Makhlouf extended Rota-Baxter operators to the context of Hom-Lie algebras. Now, we generalize it to the matching Rota-Baxter case.Definition 31.
LetλΩ≔λωω∈Ω⊆k be a family indexed by Ω. A matching Hom-Lie Rota-Baxter algebra is a Hom-Lie algebra g,,,p endowed with a set of linear maps Pω:g⟶g, where ω∈Ω, subject to the relation
(82)Pαx,Pβy=Pαx,Pβy+PβPαx,y+λβPαx,y,
for allx,y∈g and α,β∈Ω. For simplicity, we denote it by g,,,PΩ,p.Theorem 32.
Letg,,,PΩ be a matching Lie Rota-Baxter algebra and p:g⟶g be a Lie algebra endomorphism such that p∘Pω=Pω∘p for each ω∈Ω. Then, g,,p,PΩ,p, where ,p≔p∘,, is a matching Hom-Lie Rota-Baxter algebra.Proof.
Sincepx,y,zpp=ppx,py,z=p2x,y,z, the Hom-Jacobi identity for g,,p,p follows from the Jacobi identity of g,,. The skew-symmetry of g,,p,p holds from the skew-symmetry of g,,;, hence, g,,p,p is a Hom-Lie algebra.
Forx,y∈g and α,β∈Ω, we have
(83)Pαx,Pβyp=pPαx,Pβy=pPαx,Pβy+PβPαx,y+λβPαx,y=Pαpx,Pβy+PβpPαx,y+λβPαpx,y=Pαx,Pβyp+PβPαx,yp+λβPαx,yp,as required.Proposition 33.
Letg,,,PΩ,p be a matching Hom-Lie Rota-Baxter algebra such that p∘Pω=Pω∘p for each ω∈Ω. Then g,,p−1≔p−1∘,,PΩ is a matching Lie Rota-Baxter algebra.Proof.
Sincex,y,zp−1p−1=p−1x,p−1y,z, the Jacobi identity of g,,p−1 holds from the Hom-Jacobi identity of g,,,p. The skew-symmetry of g,,p−1 holds from skew symmetry of g,,,p; hence, g,,p−1 is a Lie algebra.
Sincep∘Pω=Pω∘p, p−1∘Pω=Pω∘p−1. Then,
(84)Pαx,Pβyp−1=p−1Pαx,Pβy=p−1Pαx,Pβy+PβPαx,y+λβPαx,y=Pαp−1x,Pβy+Pβp−1Pαx,y+λβPαp−1x,y=Pαx,Pβyp−1+PβPαx,yp−1+λβPαx,yp−1,as required.Definition 34.
Letg,,,p be a multiplicative Hom-Lie algebra and n≥0. The n th derived Hom-algebra of g is defined by
(85)gn=g,,n=pn∘,,pn+1.Theorem 35.
Letg,,,PΩ,p be a multiplicative matching Hom-Lie Rota-Baxter algebra and assume that p∘Pω=Pω∘p for each ω∈Ω. Then its n th derived Hom-algebra is a matching Hom-Lie Rota-Baxter algebra.Proof.
Following [54], the n-th derived Hom-algebra is a Hom-Lie algebra. For x,y∈g and α,β∈Ω,
(86)Pαx,Pβyn=pnPαx,Pβy=pnPαx,Pβy+PβPαx,y+λβPαx,y=Pαpnx,Pβy+PβpnPαx,y+λβPαpnx,y=Pαx,Pβyn+PβPαx,yn+λβPαx,yn,as required.In the following, we construct matching Hom-Lie Rota-Baxter algebras involving elements of the centroid of matching Lie Rota-Baxter algebras. Letg,,,Ω,R be a matching Lie Rota-Baxter algebra. The centroid is defined by
(87)Centg≔p∈Endg:px,y=px,y,∀x,y∈g.Proposition 36.
Letg,,,PΩ be a matching Lie Rota-Baxter algebra. Let p∈Centg and set for x,y∈g(88)x,yp1≔px,yandx,yp2≔px,py.
Assume thatp∘Pω=Pω∘p for each ω∈Ω. Then, g,,p1,PΩ,p and g,,p2,PΩ,p are matching Hom-Lie Rota-Baxter algebrasProof.
Following Proposition 1.12 of [4], g,,p1,p and g,,p2,p are Hom-Lie algebras. Also,
(89)Pαx,Pβyp1=pPαx,Pβy=pPαx,Pβy=pPαx,Pβy+PβPαx,y+λβPαx,y=Pαpx,Pβy+PβpPαx,y+λβPαpx,y=Pαx,Pβyp1+PβPαx,yp1+λβPαx,yp1and
(90)Pαx,Pβyp2=pPαx,pPβy=pPαx,pPβy=−p2Pβy,Pαx=p2Pαx,Pβy=p2Pαx,Pβy+PβPαx,y+λβPαx,y=Pαpx,pPβy+PβpPαx,py+λβPαpx,py=Pαx,Pβyp2+PβPαx,yp2+λβPαx,yp2.
This completes the proof.Proposition 37.
LetA,,,PΩ,p be a matching Hom-Lie Rota-Baxter algebra of weight zero (i.e. λω=0 for all ω∈Ω). Assume that p∘Pω=Pω∘p for each ω∈Ω. Then, A,∗ω∣ω∈Ω,p is a matching Hom-pre-Lie algebra, where
(91)x∗ωy=Pωx,yforx,y∈Aandω∈Ω.Proof.
Forx,y,z∈g and α,β∈Ω, we have
(92)px∗αy∗βz−x∗αy∗βz=Pαpx,Pβy,z−PβPαx,y,pzbyEq.91=Pαpx,Pβy,z−Pαx,Pβy,pz+Pαx,Pβy,pzbyEq.82=pPαx,Pβy,z+pz,Pαx,Pβy−PαPβy,x,pzbyp∘Pα=Pα∘p=−pPβy,z,Pαx−PαPβy,x,pzbyHom−Jacobiidentity=Pβpy,Pαx,z−PαPβy,x,pz=py∗βx∗αz−y∗βx∗αpz.
This completes the proof.



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*Source: 9792726-2020-07-10.xml*