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

Chen, Dan; Peng, Xiao-Song; Zargeh, Chia; Zhang, Yi

doi:https://doi.org/10.1155/2020/9792726

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2020 | Article ID 9792726 | https://doi.org/10.1155/2020/9792726

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

Dan Chen,¹Xiao-Song Peng,¹Chia Zargeh,²and Yi Zhang^1,3

Academic Editor: Yao-Zhong Zhang

Received11 Feb 2020

Revised10 May 2020

Accepted01 Jun 2020

Published10 Jul 2020

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.

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 consisting of a k-module , a bilinear skew-symmetric bracket and an algebra endomorphism satisfying the following Hom-Jacobi identity:

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], -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 -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.

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: where and is a unitary associative algebra. The corresponding equation was 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.

The 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 Figure 3.

We 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 section 2, 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 section 3, 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.

Section 4 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 section 5, 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, let 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 is a nonempty set. We denote by the collection of operations , , where 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-module together with a collection of binary operations and a linear map such that A matching Hom-associative algebra is called totally compatible if it satisfies

More generally,

Definition 2. A compatible Hom-associative algebra is a k-module together with a collection of binary operations and a linear map such that for all and . For simplicity, we denote it by .

Remark 3. (a)Any matching Hom-associative algebra or totally compatible Hom-associative algebra is a compatible Hom-associative algebra(b)By taking , we recover to the definition of matching associative algebras, totally compatible associative algebra and compatible associative algebra given in [38](c)If is a singleton and the characteristic of 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-module equipped with a collection of binary operations , and a linear map such that for all and .

Remark 5. A totally compatible Hom-associative algebra has a natural matching Hom-Lie algebra structure with the Lie bracket defined by

The matching Hom-Lie algebra has a close relationship with Hom-Lie algebras. We first record a lemma for a preparation.

Lemma 6. Let be a matching Hom-Lie algebra. Consider linear combinations where for with finite supports. Then

Proof. By Eq. (10), for , we have Similarly, we also have Since is a matching Hom-Lie algebra, then Thus as desired.

Proposition 7. Let be a matching Hom-Lie algebra. Consider linear combinations with a finite support. Then, is a Hom-Lie algebra.

Proof. It follows from Lemma 6 by taking .

More generally, we propose

Definition 8. A compatible Hom-Lie algebra is a k-module together with a set of binary operations and a linear map such that for all and .

Remark 9. (a)Every matching Hom-Lie algebra is a compatible Hom-Lie algebra.(b)Given two Hom-Lie algebras and . Define a new bracket as follows:Clearly, this new bracket is both skew symmetric and bilinear. Then, is further a Hom-Lie algebra if satisfies the Hom-Jacobi identity By a direct calculation, we get that this condition is equivalent to Eq. (18).

Proposition 10. Let be a matching Hom-Lie algebra. Then for and , we have

Proof. Since Eq. (8) holds for any and , we get Eqs. (8) and (22) result in By the arbitrariness of , we have and so

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. Let be a compatible Hom-associative algebra. Then is a compatible Hom-Lie algebra, where

Proof. For and , by Eq. (26), we get and Similarly, we have By Eq. (6), we get Hence, 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 together with a family of binary operations and a linear map such that for all and .
Now, we give the relationship between matching Hom-preLie algebras and compatible Hom-Lie algebras.

Proposition 13. Let be a matching Hom-preLie algebra. Then is a compatible Hom-Lie algebra, where

Proof. For and , by Eq. (31), we have and Similarly, we have Then, by Eq. (30), we get Hence, 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]. Let be a set of scalars indexed by . A matching Rota-Baxter algebra of weight is an associative algebra equipped with a family of linear operators that satisfy the matching Rota-Baxter equation

Definition 15. A matching Hom-associative Rota-Baxter algebra is a quadruples , where is a matching Rota-Baxter algebra and is a Hom-associative algebra.

Taking , we recover to matching Rota-Baxter associative algebras and denote it by . If 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. Let be a matching Rota-Baxter algebra and be an algebra endomorphism which commutes with for all . Then , where , 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 and , as required.

Given a matching Hom-associative Rota-Baxter algebra , it is natural to wonder that whether this matching Hom-associative Rota-Baxter algebra is induced by an ordinary associative matching Rota-Baxter algebra , i.e., is an algebra endomorphism with respect to and .

Let be a multiplicative Hom-associative algebra, i.e., for all . It was proved in [53] that in case is invertible, 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. Let be a multiplicative matching Hom-assoicative Rota-Baxter algebra, where is invertible and for each . Then, is an associative matching Rota-Baxter algebra.

Proof. For , we have Hence, the associativity condition holds. For , we have

Hence, the matching Rota-Baxter equation holds for the new multiplication, and 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]). Let be a multiplicative Hom-algebra and . Then, the following two algebras are also Hom-associative algebras: (a)the -th derived Hom-algebra of type of defined by (b)the -th derived Hom-algebra of type of defined by

Now, we show that the -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. Let be a multiplicative matching Hom-associative Rota-Baxter algebra such that for all . Then, (a)the -th derived Hom-algebra of type is a matching Hom-associative Rota-Baxter algebra(b)the -th derived Hom-algebra of type 2 is a matching Hom-associative Rota-Baxter algebra

Proof. (a) By [54], is a Hom-associative algebra. Now, we show the matching Rota-Baxter equation holds. For and , we have Thus, the matching Rota-Baxter equation holds for the new multiplication.
(b) By [54], is also a Hom-associative algebra. For and , we have This completes the proof.

Let be an associative algebra. The centroid of A is defined by

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. Let be an associative matching Rota-Baxter algebra. For and , define If for all , then and are matching Hom-associative Rota-Baxter algebras.

Proof By [37]. and are Hom-associative algebras. Now, we show that they are also matching Rota-Baxter algebras. For and , we have and 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-module together with a family of binary operations , where and , and a linear map such that for all and , For simplicity, we denote it by .

Definition 22. A matching Hom-tridendriform algebra is a k-module together with a family of binary operations , where and , and a linear map such that for all and ,

Definition 23. (a)Let and be two matching Hom-dendriform algebras. A linear map is called a matching Hom-dendriform algebra morphism if for all (b)Let and be two matching Hom-tridendriform algebras. A linear map is called a matching Hom-tridendriform algebra morphism if for all

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)Let be a matching dendriform algebra and be a matching dendriform algebra endomorphism. Then, , where and for each , is a matching Hom-dendriform algebra. Moreover, suppose that is another matching dendriform algebra and is a matching dendriform algebra endomorphism. If is a matching dendriform algebra morphism that satisfies , then is a morphism of matching Hom-dendriform algebras.(b)Let be a matching tridendriform algebra and be a matching tridendriform algebra endomorphism. Then, , where , and for each , is a matching Hom-tridendriform algebra. Moreover, suppose that is another matching tridendriform algebra and is a matching tridendriform algebra endomorphism. If is a matching tridendriform algebra morphism that satisfies , then is a morphism of matching Hom-tridendriform algebras.

Proof. We just prove Item (b) and Item (a) can be proved similarly. For any and , we have Hence, that is Eq. (48) holds for . Similarly, Eqs. (49), (50), (51), (52), (53), (54) hold. Hence, is a matching Hom-tridendriform algebra. And Hence, 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. Let be an nonempty set. For each , let be a map with finite supports, identified with finite set . (a)Let be a matching Hom-dendriform algebra. Define the following binary operations: Then, is also a matching Hom-dendriform algebra.(b)Let be a matching Hom-tridendriform algebra. Define the following binary operations: Then, is also a matching Hom-tridendriform algebra.

Proof. We just prove Item (b) and Item (a) can be proved similarly. For and , we have Hence, Eq. (48) holds. Similarly, Eqs. (49), (50), (51), (52), (53), (54) hold. Hence, 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)Let be a matching Hom-dendriform algebra. Then is a compatible Hom-associative algebra, where (b)Let be a matching Hom-tridendriform algebra. Then, is a compatible Hom-associative algebra, where

Proof. We only prove Item (b) and Item (a) can be proved similarly. For and , we have By Eqs (48), (49), (50), (51), (52), (53), (54), we get

Hence, is a compatible Hom-associative algebra.

Now, we explore the relationship between matching Hom-dendriform algebras and matching Hom-preLie algebras.

Theorem 27. Let be a matching Hom-dendriform algebra. Then is a matching Hom-preLie algebra, where

Proof. For and , we have and By Eqs (48), (49), (50), (51), (52), (53), (54), we get

Hence, is a matching Hom-preLie algebra.

A matching Rota-Baxter algebra is of weight 0 if the set . 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)Let be a matching Hom-associative Rota-Baxter algebra of weight . Assume that for each . Define the operations and for by Then is a matching Hom-dendriform algebra.(b)Let be a matching Hom-associative Rota-Baxter algebra. Assume that for each . Define the operations , , by Then, is a matching Hom-dendriform algebra.

Proof. Since Item (a) can be seen as a special case of Item (b) by taking , we only prove Item (b). For and , we have Also, and

Hence, is a matching Hom-dendriform algebra.

Proposition 29. Let be a matching Hom-associative Rota-Baxter algebra. Assume that for each . Define the operations , and for by

Then, is a matching Hom-tridendriform algebra.

Proof. For and , we have as required.

Corollary 30. (a)Let be a matching Hom-associative Rota-Baxter algebra of weight 0. Then, is a matching Hom-preLie algebra, where (b)Let be a matching Hom-associative Rota-Baxter algebra. Then, is a matching Hom-preLie algebra, where

Proof. (a) It follows from Theorem 27 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. Let be a family indexed by . A matching Hom-Lie Rota-Baxter algebra is a Hom-Lie algebra endowed with a set of linear maps , where , subject to the relation for all and . For simplicity, we denote it by .

Theorem 32. Let be a matching Lie Rota-Baxter algebra and be a Lie algebra endomorphism such that for each . Then, , where , is a matching Hom-Lie Rota-Baxter algebra.

Proof. Since , the Hom-Jacobi identity for follows from the Jacobi identity of . The skew-symmetry of holds from the skew-symmetry of ;, hence, is a Hom-Lie algebra.
For and , we have as required.

Proposition 33. Let be a matching Hom-Lie Rota-Baxter algebra such that for each . Then is a matching Lie Rota-Baxter algebra.

Proof. Since , the Jacobi identity of holds from the Hom-Jacobi identity of . The skew-symmetry of holds from skew symmetry of ; hence, is a Lie algebra.
Since , . Then, as required.

Definition 34. Let be a multiplicative Hom-Lie algebra and . The th derived Hom-algebra of is defined by

Theorem 35. Let be a multiplicative matching Hom-Lie Rota-Baxter algebra and assume that for each . Then its th derived Hom-algebra is a matching Hom-Lie Rota-Baxter algebra.

Proof. Following [54], the -th derived Hom-algebra is a Hom-Lie algebra. For and , as required.

In the following, we construct matching Hom-Lie Rota-Baxter algebras involving elements of the centroid of matching Lie Rota-Baxter algebras. Let be a matching Lie Rota-Baxter algebra. The centroid is defined by

Proposition 36. Let be a matching Lie Rota-Baxter algebra. Let and set for Assume that for each . Then, and are matching Hom-Lie Rota-Baxter algebras

Proof. Following Proposition 1.12 of [4], and are Hom-Lie algebras. Also, and This completes the proof.

Proposition 37. Let be a matching Hom-Lie Rota-Baxter algebra of weight zero (i.e. for all ). Assume that for each . Then, is a matching Hom-pre-Lie algebra, where

Proof. For and , we have This completes the proof.

Data Availability

No data were used to support this study.

Disclosure

No potential conflict of interest was reported by the authors.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments which improved the paper. This work was supported by the National Natural Science Foundation of China (grant no. 11771191) and the Fundamental Research Funds for the Central Universities (grant no. lzujbky-2020-12). The work of Chia Zargeh was partially supported by CNPq grant.

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Copyright © 2020 Dan Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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