key: cord-0467719-qkx07w21
authors: Das, Apurba
title: Generalized Rota-Baxter systems
date: 2020-07-27
journal: nan
DOI: nan
sha: f6a4904c912ba63239b2810bb5d28da0763b617a
doc_id: 467719
cord_uid: qkx07w21

Rota-Baxter systems of T. Brzezi'{n}ski are a generalization of Rota-Baxter operators that are related to dendriform structures, associative Yang-Baxter pairs and covariant bialgebras. In this paper, we consider Rota-Baxter systems in the presence of bimodule, which we call generalized Rota-Baxter systems. We define a graded Lie algebra whose Maurer-Cartan elements are generalized Rota-Baxter systems. This allows us to define a cohomology theory for a generalized Rota-Baxter system. Formal one-parameter deformations of generalized Rota-Baxter systems are discussed from cohomological points of view. We further study Rota-Baxter systems, associative Yang-Baxter pairs, covariant bialgebras and introduce generalized averaging systems that are related to associative dialgebras. Next, we define generalized Rota-Baxter systems in the homotopy context and find relations with homotopy dendriform algebras. The paper ends by considering commuting Rota-Baxter systems and their relation with quadri-algebras.

The notion of Rota-Baxter operators was first introduced by Baxter [7] in the context of differential operators on commutative Banach algebras and further studied by Rota [32] in connections with probability and combinatorics. There are important applications of Rota-Baxter operators in the Connes-Kreimer's algebraic approach of the renormalization in quantum field theory [11] . In [2] Aguiar showed that Rota-Baxter operators produce dendriform structures. Aguiar [1, 3] also introduced an associative analogue of Yang-Baxter equation and extensively study infinitesimal bialgebras that were first appeared in a paper by Joni and Rota [23] . An associative Yang-Baxter solution gives rise to a Rota-Baxter operator and an infinitesimal bialgebra. In [35] Uchino introduced a generalization of Rota-Baxter operators that can also be considered as an associative analogue of Poisson structures. See [22] for more details on Rota-Baxter operators. Recently, deformations of associative Rota-Baxter operators and their governing cohomology has been studied in [13] .

In [9] Brzeziński introduced Rota-Baxter systems as a generalization of a Rota-Baxter operator. In a Rota-Baxter system, two operators are acting on the algebra and satisfying some Rota-Baxter type identities. More precisely, a pair (R, S) of linear maps on an algebra A is said to be a Rota-Baxter system if they satisfy R(a)R(b) = R(R(a)b + aS(b)), S(a)S(b) = S(R(a)b + aS(b)), for all a, b ∈ A. He also introduced associative Yang-Baxter pairs (as a generalization of Yang-Baxter solutions), covariant bialgebras (as a generalization of infinitesimal bialgebras) and find various relations among them that generalize the results of Aguiar. See [30] for the free construction of Rota-Baxter system and related structures.

Our main objective in this paper is Brzeziński's Rota-Baxter systems in the presence of A-bimodule M . Motivated by Uchino's [35] terminology of generalized Rota-Baxter operator, we call Rota-Baxter systems in the presence of bimodule as generalized Rota-Baxter systems. They were called O-operator system in [26] . Thus, Rota-Baxter systems on an associative algebra A are generalized Rota-Baxter systems on the adjoint A-bimodule A. A generalized Rota-Baxter system induces a dendriform structure on M , hence an associative structure on M . We give some new characterizations of (generalized) Rota-Baxter systems. Motivated from gauge transformations [33] and reductions [27] of Poisson structures, we define some new constructions of generalized Rota-Baxter systems.

Next, given an associative algebra A and an A-bimodule M , we construct a graded Lie algebra whose Maurer-Cartan elements are given by generalized Rota-Baxter systems. This characterization of a Rota-Baxter system allows us to define a cohomology theory for a generalized Rota-Baxter system. We further show that this cohomology is isomorphic to the Hochschild cohomology of the induced associative algebra M with coefficients in a suitable bimodule structure on the direct sum A ⊕ A. We also find a morphism from the cohomology of a generalized Rota-Baxter system to the cohomology of the induced dendriform algebra.

The classical deformation theory of Gerstenhaber [19] for associative algebras has been extended to various other algebraic structures including Lie algebras [28] and Leibniz algebras [5] . More generally, Balavoine [6] considered deformations of algebras over binary quadratic operads. Deformations of morphisms have also been considered in many articles [18, 20] . In [13] the author study deformations of Rota-Baxter operators from cohomological perspectives. Here we consider deformations of generalized Rota-Baxter systems. We show that the linear term in a formal deformation of a given generalized Rota-Baxter system is a 1-cocycle in the above-defined cohomology of the generalized Rota-Baxter system. Finally, given a finite order deformation, we construct a 2-cocycle in the cohomology. The corresponding cohomology class is called the obstruction class. When the obstruction class vanishes, the given deformation extends to next order.

We also study some further results on Rota-Baxter systems, associative Yang-Baxter pairs and covariant bialgebras that are introduced by Brzeziński [9] . We show that a pair (r, s) of skew-symmetric elements of A ⊗2 is an associative Yang-Baxter pair if and only if (r ♯ , s ♯ ) is a generalized Rota-Baxter system on the coadjoint A-bimodule A * . Following the perturbations of quasi-Hopf algebraà la Drinfel'd, we study perturbations of the coproduct in a covariant bialgebra. We find a necessary and sufficient condition for the perturbed object to be a new covariant bialgebra. A suitably compatible covariant bialgebra also induces a pre-Lie algebra structure that generalizes a result of Aguiar [3] for infinitesimal bialgebras. Finally, we introduce a notion of generalized averaging system as a generalization of an averaging operator [29, 31] . A generalized averaging system induces a dialgebra structure in the sense of Loday [24] . We observed that a generalized averaging system can be seen as a particular case of generalized Rota-Baxter system. Therefore, all cohomological results and deformations of generalized Rota-Baxter systems can be applied to generalized averaging systems.

The notion of strongly homotopy associative algebras or A ∞ -algebras were introduced by Stasheff in [34] . We introduce generalized Rota-Baxter systems on a bimodule over an A ∞ -algebra. We show that a generalized Rota-Baxter system induces a Dend ∞ -algebra structure on the underlying bimodule space. This generalizes a result of Brzeziński in the homotopy context. Finally, we consider Rota-Baxter systems on a dendriform algebra and show how they induce quadrialgebra structures. As a consequence, we obtain a quadri-algebra structure from two commuting Rota-Baxter systems on an associative algebra. The homotopy analogue of these results is also described.

The paper is organized as follows. In Section 2, we introduce generalized Rota-Baxter systems and give some characterizations and new constructions. In Section 3, we mainly study cohomology of a generalized Rota-Baxter system and find their relationship with the dendriform algebra cohomology. Deformations of generalized Rota-Baxter systems are considered in Section 4. Applications and some further properties of Rota-Baxter systems, associative Yang-Baxter pairs, covariant bialgebras and generalized averaging systems are given in Section 5. In Section 6, we define generalized Rota-Baxter systems in homotopy context and find their relations with Dend ∞ -algebras. Finally, in Section 7, we consider commuting Rota-Baxter systems and obtain quadri-algebras.

Note that given an associative algebra A and an A-bimodule M , the sum A ⊕ A ⊕ M carries an associative algebra structure with product

This is exactly the semi-direct product if we consider the direct sum associative algebra structure on A ⊕ A and the bimodule structure on M given by (a 1 , a 2 ) · m = a 1 · m and m · (a 1 , a 2 ) = m · a 2 .

2.4. Proposition. A pair (R, S) of maps from M to A is a generalized Rota-Baxter system if and only if the pair of maps

Hence (R, S) is a generalized Rota-Baxter system if and only if ( R, S) is a Rota-Baxter system.

Proof. For any u, v ∈ M , we have

Hence Gr((R, S)) is a subalgebra if and only if (R, S) is a generalized Rota-Baxter system.

Nijenhuis operators on associative algebras [10] are associative analog of classical Nijenhuis operator on Lie algebras. More precisely, a linear map N : A → A on an algebra A is said to be a Nijenhuis operator if it satisfies

The following result relates to generalized Rota-Baxter systems and Nijenhuis operators.

Proof. By a simple calculation, we have

and

It follows from (3) and (4) that N (R,S) is a Nijenhuis tensor if and only if (R, S) is a generalized Rota-Baxter system.

Let A be an associative algebra and M an A-bimodule. We assume that dimA = dimM . A pair (θ 0 , θ 1 ) of invertible linear maps from A to M is said to be an invertible 1-cocycle system if they satisfy

for all x, y ∈ A. It follows from the above definition that (θ, θ) is an invertible 1-cocycle system if and only if θ : A → M is an invertible 1-cocycle.

2.7. Proposition. Let A be an associative algebra and M an A-bimodule (dimA = dimM ). A pair (R, S) of invertible linear maps from M to A is a generalized Rota-Baxter system if and only if (R −1 , S −1 ) is an invertible 1-cocycle system.

Proof. It is easy to see that the identity (1) is equivalent to (by taking R(u) = x, R(v) = y)

Similarly, the identity (2) is equivalent to (by taking

It follows that (R, S) is a generalized Rota-Baxter system if and only if (R −1 , S −1 ) is an invertible 1-cocycle system.

A dendriform algebra is a triple (D, ≺, ≻) consisting of a vector space D together with bilinear operations ≺, ≻:

It follows that in a dendriform algebra (D, ≺, ≻), the sum operation

Let (D, ≺, ≻) and (D ′ , ≺ ′ , ≻ ′ ) be two dendriform algebras. A morphism between them is given by a linear map f :

The following result relates to generalized Rota-Baxter systems and dendriform algebras [9] .

2.9. Proposition. Let (R, S) be a generalized Rota-Baxter system on M over the algebra A. Then M carries a dendriform algebra structure with

Consequently, there is an associative algebra structure on M given by u * v = u ≺ v + u ≻ v for which both R and S are morphisms of associative algebras.

Next, we define morphisms between generalized Rota-Baxter systems. Let (R, S) be a generalized Rota-Baxter system on an A-bimodule M and (R ′ , S ′ ) be a generalized Rota-Baxter system on an A ′ -bimodule M ′ .

2.10. Definition. A morphism of generalized Rota-Baxter systems from (R, S) to (R ′ , S ′ ) consists of a triple (φ, ϕ, ψ) of algebra maps φ, ϕ : A → A ′ and a linear map ψ : M → M ′ satisfying

It is called an isomorphism if φ, ϕ and ψ are all linear isomorphisms.

2.11. Proposition. Let (φ, ϕ, ψ) be a morphism of generalized Rota-Baxter systems from (R, S) to (R ′ , S ′ ) as in the above definition. Then ψ : M → M ′ is a morphism between induced dendriform structures.

Proof. For any u, v ∈ M , we have

Similarly,

Hence the proof.

2.12. Definition. A (left) pre-Lie algebra is a vector space L together with a bilinear operation ⋄ :

The following result is well-known [2, 3] .

2.13. Proposition. (i) Let (L, ⋄) be a pre-Lie algebra. Then the vector space L is equipped with the bracket [a, b] = a ⋄ b − b ⋄ a is a Lie algebra, called the subadjacent Lie algebra.

(ii) Let (D, ≺, ≻) be a dendriform algebra.

Thus, it follows that if (R, S) is a generalized Rota-Baxter system on M over the algebra A, then there is a pre-Lie structure on M given by

2.1. Gauge transformations. Gauge transformations of Poisson structures by suitable closed 2-forms was introduced byŠevera and Weinstein [33] . Motivated from the fact that generalized Rota-Baxter operators (more generally, generalized Rota-Baxter systems) are the associative analogue of Poisson structures, we define here gauge transformations of generalized Rota-Baxter systems.

Let L ⊂ A ⊕ A ⊕ M be a subalgebra of the semi-direct product. For any linear map B : A ⊕ A → M , we define a subspace τ B (L) := {(a 1 , a 2 , u + B(a 1 , a 2 ))| (a 1 , a 2 , u) ∈ L}.

Then we have the following. Proof. For any (a 1 , a 2 , u),

In other words, B is a 1-cocycle.

Next, let (R, S) be a generalized Rota-Baxter system on M over the algebra A. 

In such a case, we call B a (R, S)-admissible 1-cocycle. Therefore, in this case, by Proposition 2.5, the pair of linear maps R • (id + B • (R, S)) −1 , S • (id + B • (R, S)) −1 is a generalized Rota-Baxter system on M over the algebra A. This generalized Rota-Baxter system is called the gauge transformation of (R, S) associated with the (R, S)-admissible 1-cocycle B. 

Hence the proof.

Reductions. Let A be an associative algebra and M an A-bimodule. Let (R, S) be a generalized Rota-Baxter system on M over the algebra A. Consider a subalgebra B ⊂ A and a vector subspace E ⊂ A with the property that the quotient B/E ∩ B is an associative algebra and the projection π :

Then it is easy to see that (E ∩ B) 0 N is an B/E ∩ B-bimodule with |b| · u = b · u and u · |b| = u · b, for |b| ∈ B/E ∩ B and u ∈ (E ∩ B) 0 N . Observe that b · u and u · b is in (E ∩ B) 0 N . This follows as for any a ∈ E ∩ B, we have a · (b · u) = (ab) · u = 0 and (b · u) · a = b · (u · a) = 0.

2.17. Definition. Let (R, S) be a generalized Rota-Baxter system on M over the algebra A. A triple (B, E, N ) as above is said to be reducible if there is a generalized Rota-Baxter system (R, S) :

The Marsden-Ratiu reduction theorem for generalized Rota-Baxter system is given by the following. Then we have

On the other hand,

. Similarly, we can show that S(u)S(v) = S(R(u) · v + u · S(v)). Hence (R, S) is a generalized Rota-Baxter system on (E ∩ B) 0 N over the algebra B/E ∩ B. Moreover, the condition (5) holds. Hence (B, E, N ) is reducible.

As a consequence, we have the following. 

In this section, we construct a graded Lie algebra whose Maurer-Cartan elements are precisely generalized Rota-Baxter systems. The graded Lie algebra is obtained from Voronov's derived bracket construction.

We first recall the Gerstenhaber bracket on multilinear maps on a vector space. Let V be a vector space. For any f ∈ Hom(V ⊗m , V ) and g ∈ Hom(V ⊗n , V ), the Gerstenhaber bracket

Then the graded space ⊕ n Hom(V ⊗n , V ) with the Gerstenhaber bracket [ , ] is a degree −1 graded Lie algebra.

Let 

3.1. Proposition. With the above notations, µ defines an associative product on A and (l, r) gives rise to an A-bimodule structure on M if and only if µ

On the other hand,

Hence (µ + µ + l 1 + r 2 ) • (µ + µ + l 1 + r 2 ) = 0 if and only if µ is associative and (l, r) defines an A-bimodule structure on M .

Thus it follows from the above Proposition that the graded vector space Hom((A⊕A⊕M ) ⊗2 , A⊕A⊕M ) with the differential d µ+µ+l1+r2 := [µ + µ + l 1 + r 2 , ] is a dgLa. Moreover, it follows from the definition of the bracket that n≥1 Hom(M ⊗n , A ⊕ A) is an abelian subalgebra. Therefore, by the derived bracket construction of Voronov [36] yields a graded Lie algebra bracket on n≥1 Hom(M ⊗n , A ⊕ A) given by

. Let pr 1 , pr 2 : A ⊕ A → A denote the projection maps onto the first and second factor, respectively. Explicitly, the bracket (6) is given by

We can extend this bracket to the graded space n≥0 Hom(M ⊗n , A ⊕ A) by the following Proof. The first part follows from previous discussions. For the second part, it follows from (7) and (8) 

Thus, generalized Rota-Baxter systems can be characterized as Maurer-Cartan elements in a gLa. It follows from the above theorem that if (R, S) is a generalized Rota-Baxter system, then d (R,S) := (R, S), is a differential on C * (M, A) and makes the gLa (C • (M, A), , ) into a dgLa.

The cohomology of the cochain complex (C • (M, A), d (R,S) ) is called the cohomology of the generalized Rota-Baxter system (R, S). We denote the corresponding cohomology groups simply by H • (M, A). In the next subsection, we view this cohomology as the Hochschild cohomology of (M, * ) with coefficients in a suitable representation on A ⊕ A.

The proof the following result is standard in the study of Maurer-Cartan elements in a gLa.

Theorem. Let (R, S) be a generalized Rota-Baxter system on M over the algebra A. Then for any pair

3.1. Hochschild cohomology. Let (R, S) be a generalized Rota-Baxter system on M over the algebra A.

Consider the associative algebra (M,

The following result generalizes [35, Lemma 2.11].

3.4. Proposition. The associative algebra (M, * ) has a bimodule representation on A ⊕ A given by

Proof. The result follows from a straightforward calculation.

It follows from the above lemma that we can consider the Hochschild cohomology of the associative algebra (M, * ) with coefficients in the bimodule A ⊕ A. More precisely, the Hochschild complex is given by

3.5. Proposition. Let (R, S) be a generalized Rota-Baxter system on M over the algebra A. Then

Proof. For (f, g) ∈ C n Hoch (M, A ⊕ A) and u 1 , . . . , u n+1 ∈ M , we have from (7) that

Similarly, we can show that pr 2 (R, S),

It follows from the above proposition that the cohomology of a generalized Rota-Baxter system induced from the Maurer-Cartan element coincides with the above Hochschild cohomology.

Relation with dendriform algebra cohomology. Cohomology of dendriform algebras was first introduced by Loday [24] for trivial coefficients. Later on, an operadic approach of the cohomology was given in the book by Loday and Vallette [25] . The explicit description of the cohomology can be found in [12] . In this subsection, we find the relationship between the cohomology of a generalized Rota-Baxter system and the cohomology of the corresponding dendriform algebra.

For each n ≥ 1, let C n be the set of first n natural numbers. We denote the elements of C n by { [1] , [2] , . . . , [n]} for convenience. Then it has been shown in [12] that for any vector space D, the collection of spaces

for f ∈ O(m), g ∈ O(n), 1 ≤ i ≤ m and [r] ∈ C m+n−1 . Therefore, by a result of Gerstenhaber and Voronov [21] , the graded space O(• + 1) = ⊕ n≥1 O(n + 1) carries a graded Lie bracket

for f ∈ O(m + 1) and g ∈ O(n + 1). If (D, ≺, ≻) is a dendriform algebra, then the element π ∈ O(2) defined by

is a Maurer-Cartan element in the graded Lie algebra (O(• + 1), , ). Therefore, π defines a differential δ π (f ) = (−1) n−1 π, f , for f ∈ O(n). The corresponding cohomology groups are called the cohomology of the dendriform algebra (D, ≺, ≻) with coefficients in itself. We denote the cohomology groups by H • dend (D, D). Let (R, S) be a generalized Rota-Baxter system on an A-bimodule M . Consider the corresponding dendriform algebra (M, ≺, ≻) given in Proposition 2.9. Let π ∈ Hom(K[C 2 ] ⊗ M ⊗2 , M ) denotes the corresponding Maurer-Cartan element, i.e.

We define a collection of maps

We have the following lemma which is crucial for the next theorem.

3.6. Lemma. The collection {Θ n } of maps preserve the respective graded Lie brackets, i.e.

Proof. For u 0 , u 1 , . . . , u m+n ∈ M , we have from (13) that

Similarly, for u 1 , . . . , u m+n+1 ∈ M ,

For 2 ≤ r ≤ m + n, it follows from (13) and partial compositions (12) that

Hence the result follows. Proof. It follows from (14) that Θ 1 (R, S) = π M . Therefore, by the above lemma, the following diagram commutes

Hence the result follows.

Formal deformation theory of associative Rota-Baxter operators was studied by the author in [13] along the line of Gerstenhaber [19] . In this section, we study formal one-parameter deformations of generalized Rota-Baxter systems.

Let 

Thus the followings are hold

By expanding these equations and comparing coefficients of various powers of t, we obtain for n ≥ 0,

Both of these identities hold for n = 0 as (R, S) is a generalized Rota-Baxter system. However, for n = 1, we obtain

These identities are equivalent to (R, S), (R 1 , S 1 ) = 0.

In other words, the pair (R 1 , S 1 ) is a 1-cocycle in the cohomology of the generalized Rota-Baxter system (R, S). It is called the infinitesimal of the deformation. In general, if (R 1 , S 1 ) = · · · = (R k−1 , S k−1 ) = 0 then (R k , S k ) is a 1-cocycle in the cohomology of (R, S).

Definition. (Equivalence) Let (R t , S t ) and (R ′ t , S ′ t ) be two formal one-parameter deformations of a generalized Rota-Baxter system (R, S). They are said to be equivalent if there exists a, b ∈ A; linear maps φ i , ϕ i ∈ End(A) and ψ i ∈ End(M ) for i ≥ 2 such that

is a morphism of generalized Rota-Baxter systems from (R t , S t ) to (R ′ t , S ′ t ).

Thus it follows that the conditions of Definition 2.10 hold. In particular, we have

Comparing coefficients of t in both identities, we get

. Hoch (a, b) )(u). In summary, we obtain the following. 4.3. Theorem. The infinitesimal of a deformation of a generalized Rota-Baxter system is a 1-cocycle and the corresponding cohomology class depends only on the equivalence class of the deformation.

The following result relates the deformation of a generalized Rota-Baxter system and the deformation of the dendriform structure in the sense of [12] .

Proposition. Let (R t , S t ) be a formal one-parameter deformation of a generalized Rota-Baxter system (R, S). Then (≺ t , ≻ t ) defines a deformation of the induced dendriform algebra (M, ≺, ≻) given in Proposition 

Thus in a deformation of order N , the identities (15) and (16) hold for n = 0, 1, . . . , N . These identities can be equivalently expressed as

4.6. Definition. A deformation (R t , S t ) of order N is said to be extensible if there exists a pair (R N +1 , S N +1 ) of linear maps from M to A such that (

) is a deformation of order N + 1.

In such a case, one more deformation needs to be satisfied, namely,

Note that on the right hand side of the above equation does not involve R N +1 or S N +1 . Hence it depends only on the order N deformation (R t , S t ). It is called the 'obstruction' to extend the deformation (R t , S t ) and is denoted by Ob (Rt,St) .

Proposition. The obstruction Ob (Rt,St) is a 2-cocycle in the cohomology of the generalized Rota-Baxter system (R, S).

Proof. We have

Hence the proof.

This shows that the obstruction gives rise to a second cohomology class [Ob (Rt,St) ] ∈ H 2 (M, A), called the obstruction class.

As a consequence of (17) we obtain the following. 

In this section, we give some further study of Rota-Baxter systems, associative Yang-Baxter pairs and covariant bialgebras introduced by Brzeziński [9] . Some applications of the results of previous sections are given to these structures. Finally, we introduce generalized averaging systems and observe that they are particular cases of generalized Rota-Baxter systems. A generalized averaging system induces an associative dialgebra structure in the sense of Loday [24] .

It has been shown in Section 2 that a Rota-Baxter system on an associative algebra A is a generalized Rota-Baxter system on the adjoint bimodule over the algebra A. Thus, the results of the previous sections can be applied to Rota-Baxter systems by considering the adjoint bimodule.

Combining Theorem 3.2 and Theorem 3.3 for the adjoint bimodule, we get the following.

A be an associative algebra. Then there is a graded Lie bracket , defined by (7) The Hochschild cohomology of the associative algebra (A, * ) with coefficients in the above bimodule A⊕ A is isomorphic to the cohomology induced from the Maurer-Cartan element (R, S) in the graded Lie algebra (C • (A, A) , , ). This cohomology is called the cohomology of the Rota-Baxter system (R, S) on the algebra A.

Like the previous section, we may define deformations of a Rota-Baxter system (R, S) on an algebra A. Such deformations can be studied in terms of the cohomology of (R, S).

Associative Yang-Baxter pairs. Let A be an associative algebra. In [9] the author introduced associative Yang-Baxter pairs as generalization of associative Yang-Baxter solutions [1] . Let us introduce the following notations first. For r = r (1) ⊗ r (2) and s = s (1) ⊗ s (2) ∈ A ⊗2 , we define r 13 s 12 = r (1) s (1) ⊗ s (2) ⊗ r (2) , r 12 s 23 = r (1) ⊗ r (2) s (1) ⊗ s (2) , r 23 s 13 = s (1) ⊗ r (1) ⊗ r (2) s (2) be elements of A ⊗3 . A pair (r, s) of elements of A ⊗2 is called an associative Yang-Baxter pair if they satisfy    r 13 r 12 − r 12 r 23 + s 23 r 13 = 0, s 13 r 12 − s 12 s 23 + s 23 s 13 = 0.

In [9] Brzeziński showed that if (r = r (1) ⊗ r (2) , s = s (1) ⊗ s (2) ) is an associative Yang-Baxter pair, then the pair (R, S) of linear maps on A defined by R(a) = r (1) ar (2) and S(a) = s (1) as (2) 

is a Rota-Baxter system. In the following, we associate a generalized Rota-Baxter system to any skewsymmetric associative Yang-Baxter pair.

For r = r (1) ⊗ r (2) and s = s (1) ⊗ s (2) , we define linear maps r ♯ , s ♯ : A * → A by r ♯ (α) = α(r (2) )r (1) and s ♯ (α) = α(s (2) )s (1) , for α ∈ A * . With these notations, we have the following.

Proposition. Let r, s be two skew-symmetric elements of A ⊗2 . Then (r, s) is an associative Yang-Baxter pair if and only if (r ♯ , s ♯ ) is a generalized Rota-Baxter system on the coadjoint module A * over the algebra A.

Proof. For α, β ∈ A * , we have r ♯ (α)r ♯ (β) = α(r (2) )β(r (2) )r (1)r (1) . On the other hand,

We consider an auxiliary map (β ⊗ α) ♯ :

Note that (r 13 r 12 − r 12 r 23 + s 23 r 13 ) = 0 holds if and only if (β ⊗ α) ♯ (r 13 r 12 − r 12 r 23 + s 23 r 13 ) = 0, for all α, β ∈ A * , or equivalently, β(r (2) )α(r (2) )r (1)r(1) − β(r (2) r (1) )α(r (2) )r (1) + β(s (1) )α(s (2) r (2) )r (1) = 0. This is equivalent to r ♯ (α)r ♯ (β) = r ♯ (r ♯ (α) · β + α · s ♯ (β)), for all α, β ∈ A * .

Similarly, we have s ♯ (α)s ♯ (β) = α(s (1) )β(s (1) )s (2)s(2) and s ♯ (r ♯ (α) · β + α · s ♯ (β)) = s ♯ α(r (2) )(r (1) · β) − β(s (1) )(α · s (2) ) = − α(r (2) )((r (1) · β)(s (1) ))s (2) + β(s (1) )((α · s (2) )(s (1) ))s (2) = − α(r (2) )β(s (1) r (1) )s (2) + β(s (1) )α(s (2)s(1) )s (2) .

. Then s 13 r 12 − s 12 s 23 + s 23 s 13 = 0 if and only if ♯ (β ⊗ α)(s 13 r 12 − s 12 s 23 + s 23 s 13 ) = 0 , for all α, β ∈ A * . This is equivalent to β(s (1) r (1) )α(r (2) )s (2) − β(s (1) )α(s (2)s(1) )s (2) + β(s (1) )α(s (1) )s (2) s (2) = 0, or equivalently, s ♯ (α)s ♯ (β) = s ♯ (r ♯ (α) · β + α · s ♯ (β)), for all α, β ∈ A * . Hence the proof.

The above proposition shows that the dual space A * carries a dendriform structure given by α ≺ β = β(s (2) )(α · s (1) ) and α ≻ β = α(r (2) )(r (1) · β). Hence the corresponding associative product on A * is given by α * β = r ♯ (α) · β + α · s ♯ (β) = α(r (2) )(r (1) · β) + β(s (2) )(α · s (1) ). (19) Moreover, the maps r ♯ , s ♯ : A * → A are both associative algebra morphisms.

Let (r, s) and (r ′ , s ′ ) be two (skew-symmetric) associative Yang-Baxter pairs on A. They are said to be equivalent if there exists an algebra morphism φ : A → A satisfying (φ ⊗ φ)(r) = r ′ and (φ ⊗ φ)(s) = s ′ .

Next, we consider certain morphisms between skew-symmetric associative Yang-Baxter pairs that induce morphisms between corresponding generalized Rota-Baxter systems.

Definition. Let (r, s) and (r ′ , s ′ ) be two skew-symmetric associative Yang-Baxter pairs. A weak morphism from (r, s) to (r ′ , s ′ ) is a triple (φ, ϕ, ψ) consist of two algebra morphisms φ, ϕ : A → A and a linear map ψ :

A weak morphism (φ, ϕ, ψ) is called a weak isomorphism if φ, ϕ, ψ are all linear isomorphisms. It follows that two skew-symmetric associative Yang-Baxter pairs (r, s) and (r ′ , s ′ ) are equivalent if and only if there is an algebra morphism φ : A → A such that (φ, φ, φ −1 ) is a weak morphism from (r, s) to (r ′ , s ′ ). Proposition. Let (r, s) and (r ′ , s ′ ) be two skew-symmetric associative Yang-Baxter pairs. Then (φ, ϕ, ψ) is a weak (iso)morphism from (r, s) to (r ′ , s ′ ) if and only if (φ, ϕ, ψ * ) is a (iso)morphism of generalized Rota-Baxter systems from (r ♯ , s ♯ ) to (r ′♯ , s ′♯ ). 2) . Suppose (φ, ϕ, ψ * ) is a morphism of generalized Rota-Baxter systems from (r ♯ , s ♯ ) to (r ′♯ , s ′♯ ). Then by definition,

for all a ∈ A, ξ ∈ A * . Note that r ♯ (ξ) = −ξ(r (1) )r (2) and r ′♯ (ξ) = −ξ(r ′ (1) )r ′ (2) . Therefore, for any ξ, η ∈ A * ,

Thus it follows from the first condition of (20) 

Thus, we have ψ(aφ(b)) = ψ(a)b. Similarly, we get ψ(ϕ(a)b) = aψ(b). Therefore, (φ, ϕ, ψ) becomes a weak morphism from (r, s) to (r ′ , s ′ ). The converse part is similar and we omit the details.

Let (r, s) be an associative Yang-Baxter pair on an associative algebra A. A deformation of (r, s) is a pair of formal sums (r t = t i r i , s t = t i s i ) of elements of A ⊗2 with r 0 = r, s 0 = s and such that (r t , s t ) is an associative Yang-Baxter pair. If (r, s) is a skew-symmetric pair, then we assume that each r i and s i are also skew-symmetric. In this case, we say that (r t , s t ) is a skew-symmetric deformation of (r, s).

is a deformation of the Rota-Baxter system (R, S) given in (18) , where R i (a) = r i(1) ar i (2) and S i (a) = s i(1) as i (2) . On the other hand, if (r t = t i r i , s t = t i s i ) is a skew-symmetric deformation of a skew-symmetric Yang-Baxter pair (r, s), then (

is a deformation of the generalized Rota-Baxter system (r ♯ , s ♯ ).

as an extension of infinitesimal bialgebras. In this subsection, we study perturbations of the coproduct in a covariant bialgebra and associate a pre-Lie algebra structure to any compatible covariant bialgebra. Some remarks about deformations are also mentioned. Let A be an associative algebra. Then A ⊗ A can be given an A-bimodule structure by a · (b ⊗ c) = ab ⊗ c

Then a linear map △ : A → A ⊗ A is said to be a covariant derivation with respect to (δ 1 , δ 2 ) if

5.6. Definition. A covariant bialgebra consists of a tuple (A, µ, △, δ 1 , δ 2 ) in which (A, µ) is an associative algebra, (A, △) is an coassociative coalgebra, the maps δ 1 , δ 2 : A → A ⊗ A are derivations on A with values in the bimodule A ⊗ A such that △ is a covariant derivation with respect to (δ 1 , δ 2 ).

The following result constructs a covariant bialgebra from an associative Yang-Baxter pair [9] Let (A, µ) be an associative algebra and (r, s) be an associative Yang-Baxter pair. Define maps △, δ r , δ s :

Then (A, µ, △, δ r , δ s ) is a covariant bialgebra, called quasitriangular covariant bialgebra.

In [16, 17] Drinfel'd consider the perturbations of the quasi-Hopf algebra structure. In the same spirit, we study here perturbations of the coproduct in a covariant bialgebra. Before that, we introduce some additional notations. Note that, the tensor product A ⊗3 also carries an A-bimodule structure by

We will use this A-bimodule in the next theorem. 

Proof. It is easy to see that δ r and δ s are derivations, so the sums δ 1 + δ r and δ 2 + δ s . We now show that △ + △ is a covariant derivation with respect to (δ 1 + δ r , δ 2 + δ s ). We have

Similarly, we can show that (△ + △)(ab) = a · (△ + △)(b) + (δ 2 + δ s )(a) · b. Thus we proved our claim. We are now left with the coassociativity of △ + △. We show that △ + △ is coassociative if and only if the condition (22) holds. Note that

We have (id ⊗ △) • △ = (△ ⊗ id) • △ as △ is coassociative. Therefore, it follows from (23) and (24) 

= a · (r 13 r 12 − r 12 r 23 + s 23 r 13 ) − (s 13 r 12 − s 12 s 23 + s 23 s 13 ) · a.

Here the right-hand side follows from [9, Proposition 3.15] . For r = r (1) ⊗ r (2) and s = s (1) ⊗ s (2) , we have

On the other hand, by letting △(a) = a (1) ⊗ a (2) , we get

Therefore, we get from (25) and (26) that

The right-hand side is same as a · (r 13 r 12 − r 12 r 23 + s 23 r 13 ) − (s 13 r 12 − s 12 s 23 + s 23 s 13 ) · a (equivalently, (△ + △) is coassociative) if and only if (22) holds.

In the following, we construct a pre-Lie algebra from a suitable covariant bialgebra generalizing a result of Aguiar [3] . We need the following definition: A covariant bialgebra (A, µ, △, δ 1 , δ 2 ) is said to be a compatible covariant bialgebra if

Any infinitesimal bialgebra is by definition a compatible covariant bialgebra. Another example of a compatible covariant bialgebra can be given by the following. Let (A, µ) be an associative algebra and a, b ∈ A satisfying a 2 = b 2 = ba = 0. Then (r = a ⊗ a, s = b ⊗ b) is an associative Yang-Baxter pair. The corresponding quasitriangular covariant bialgebra (A, µ, △ ′ , δ r , δ s ) is a compatible covariant bialgebra.

The following result relates to compatible covariant bialgebras and pre-Lie algebras.

5.8. Proposition. Let (A, µ, △, δ 1 , δ 2 ) be a compatible covariant bialgebra. Then (A, ⋄) is a pre-Lie algebra, (2) ).

To expand it, we observe that

Here we have used the notations δ 1 (a) = a 1 (1) ⊗ a 1 (2) and δ 2 (a) = a 2 (1) ⊗ a 2 (2) . Hence, we have (2) .

Therefore,

By interchanging a and b, we get

Finally, since (A, µ, △, δ 1 , δ 2 ) is a compatible covariant algebra, we have (id ⊗ δ 1 ) • △(c) = (δ 2 ⊗ id) • △(c), or equivalently,

Thus, it follows from (27) and (28) ⋄ c) holds. Hence the proof.

Let A be an associative algebra and (r, s) be a skew-symmetric associative Yang-Baxter pair. Consider the associative algebra structure on A * given in (19) induced from the generalized Rota-Baxter system (r ♯ , s ♯ ). On the other hand, we can consider the quasitriangular covariant bialgebra (A, µ, △, δ r , δ s ) given in (21) . Since △ : A → A ⊗ A, △(a) = a · r − s · a is coassociative, its dual defines an associative multiplication on A * . The comparison between the above two associative structures on A * is given by the following. 5.9. Proposition. Let (r, s) be a skew-symmetric associative Yang-Baxter pair on an associative algebra A. Then the associative structure on A * induced from the generalized Rota-Baxter system (r ♯ , s ♯ ) coincides with the dual of the coassociative coproduct △ :

Proof. Note that the dual of △ that defines an associative product on A * is given by the following composition

Here Ξ :

= α(r (2) )β(ar (1) ) − α(s (2) a)β(s (1) ).

On the other hand, r ♯ (α) · β + α · s ♯ (β), a = α(r (2) ) r (1) · β, a − β(s (1) ) α · s (2) , a = α(r (2) )β(ar (1) ) − β(s (1) )α(s (2) a).

Thus it follows that △ * (Ξ(α ⊗ β)) = r ♯ (α) · β + α · s ♯ (β). Hence the proof.

Next, we introduce a notion of weak morphism between covariant bialgebras and relate them with weak morphism between associative Yang-Baxter pairs. Let (A, µ, △, δ 1 , δ 2 ) and (A ′ , µ ′ , △ ′ , δ ′ 1 , δ ′ 2 ) be two covariant bialgebras. A morphism between them is given by an algebra map φ : A → A ′ that is also a coalgebra map satisfying additionally (φ ⊗ φ)

It is called an isomorphism of covariant bialgebras if φ is a linear isomorphism.

A weak morphism of covariant bialgebras from (A, µ, △, δ 1 , δ 2 ) to (A, µ, △ ′ , δ ′ 1 , δ ′ 2 ) consists of a triple (φ, ϕ, ψ) in which φ, ϕ : A → A are algebra maps, ψ :

Let (A, µ, △, δ 1 , δ 2 ) and (A, µ, △ ′ , δ ′ 1 , δ ′ 2 ) be two covariant bialgebras. Then it is easy to see that φ : A → A is an isomorphism of covariant bialgebras if and only if (φ, φ, φ −1 ) is a weak isomorphism of covariant bialgebras. 5.11. Proposition. Let (A, µ) be an associative algebra and (r, s), (r ′ , s ′ ) be two skew-symmetric associative Yang-Baxter pairs. If (φ, ϕ, ψ) is a weak morphism (resp. weak isomorphism) from (r, s) to (r ′ , s ′ ), then (φ, ϕ, ψ) is a weak morphism (resp. weak isomorphism) of covariant bialgebras from (A, µ, △, δ r , δ s ) to (A, µ, △ ′ , δ r ′ , δ s ′ ).

Proof. To prove that (φ, ϕ, ψ) is a weak morphism of covariant bialgebras, it remains to show that ψ : △) is a coalgebra map and satisfying (ψ ⊗ ψ) • δ r ′ = δ r • ψ, (ψ ⊗ ψ) • δ s ′ = δ s • ψ. Note that ψ is a coalgebra map if and only if ψ * is an algebra map. For any α, β ∈ A * and a ∈ A, we have ψ * (α * (r,s) β), a = r ♯ (α) · β + α · s ♯ (β), ψ(a) = β, ψ(a)r ♯ (α) + α, s ♯ (β)ψ(a) = β, ψ(aφr ♯ (α)) + α, ψ(ϕs ♯ (β)a) = β, ψ(ar ′♯ ψ * (α)) + α, ψ(s ′♯ ψ * (β)a)

This proves that ψ * is an algebra map. Moreover, the condition (ψ ⊗ ψ) • δ r ′ = δ r • ψ is equivalent to ψ * (r ♯ (α) · β + α · r ♯ (β)) = r ′♯ ψ * (α) · ψ * (β) + ψ * (α) · r ′♯ ψ * (β), the proof of which is similar to the above calculation (replacing s by r). Similarly, for the condition (ψ ⊗ ψ) • δ s ′ = δ s • ψ. Hence the proof.

Generalized averaging systems. In this subsection, we consider a notion of the generalized averaging system as a generalization of averaging operator [29] in the presence of bimodules. We will see that they are particular cases of generalized Rota-Baxter systems. Let A be an associative algebra and M be an A-bimodule.

A generalized left (resp. right) averaging system consists of a pair (R, S) of linear

A generalized averaging system is a pair (R, S) which is both a left averaging system and a right averaging system.

Definition. [24] An associative dialgebra is a vector space D together with bilinear maps ⊣, ⊢: D ⊗ D → D satisfying the following identities

5.14. Proposition. Let (R, S) be generalized averaging system on M over the algebra A. Then M carries an associative dialgebra structure with products

Proof. For any u, v, w ∈ M , we have

Hence the proof.

Proposition. Let r = r (1) ⊗ r (2) , s = s (1) ⊗ s (2) be two elements of A ⊗2 such that r 13 r 12 = r 12 r 23 and s 13 r 12 = s 12 s 23 . Then the pair (R, S) of linear maps on A defined by R(a) = r (1) ar (2) and S(a) = s (1) as (2) is a left averaging system on A.

Proof. From the hypothesis, we have

In the above two identities, replacing the first tensor product by a and the second product by b, and using the definition of R and S, we get respectively R(R(a)b) = R(a)R(b) and S(R(a)b) = S(a)S(b). Hence the proof.

Remark. Similar to the proof of above proposition, if r, s satisfies r 12 r 23 = s 23 r 13 and s 12 s 23 = s 23 s 13 , then the above-defined (R, S) is a right averaging system. Therefore, if r, s satisfies r 13 r 12 = r 12 r 23 = s 23 r 13 and s 13 r 12 = s 12 s 23 = s 23 s 13 ,

then (R, S) is a Rota-Baxter system. We call the above system (29) of equations as the Frobenius-separability system in view of [8] .

In the rest of this subsection, we will consider the generalized left averaging system. Note that if (M, l, r) is an A-bimodule then (M, l, 0) and (M, 0, r) are both A-bimodules. Then it follows that any generalized left (resp. right) averaging system is a generalized Rota-Baxter system on the A-bimodule (M, l, 0) (resp. (M, 0, r)). Therefore, we get the following results.

5.17. Theorem. Let A be an associative algebra and M be an A-bimodule. Then there is a graded Lie algebra structure on the graded vector space n≥0 Hom(M ⊗n , A ⊕ A) whose Maurer-Cartan elements are given by generalized left (resp. right) averaging systems.

The induced cohomology groups are the cohomology of the generalized left (resp. right) averaging system. One may define deformations of a generalized left (resp. right) averaging system which are governed by the above cohomology.

In this section, we introduce generalized Rota-Baxter systems on a bimodule over an A ∞ -algebra. They are a generalization of Rota-Baxter operators on A ∞ -algebras considered in [12] . We show that a generalized Rota-Baxter system induces a Dend ∞ -algebra structure. We first recall the notion of A ∞ -algebras from [34] .

Definition. An A ∞ -algebra consists of a graded vector space A = ⊕A i together with a collection of multilinear maps {µ k : A ⊗k → A| deg(µ k ) = k − 2} k≥1 satisfying the following set of identities: for any n ≥ 1, i+j=n+1 j λ=1 (−1) λ(i+1)+i(|a1|+···+|a λ−1 |) µ j a 1 , . . . , a λ−1 , µ i (a λ , . . . , a λ+i−1 ), a λ+i , . . . , a n = 0, (30) for a i ∈ A |ai| , 1 ≤ i ≤ n.

When µ k = 0 for k ≥ 3, we get differential graded associative algebras. If further µ 1 = 0, one obtains graded associative algebras. An A ∞ -algebra whose underlying graded vector space A is concentrated in degree 0 is nothing but an associative algebra.

Let (A, µ k ) be an A ∞ -algebra. An A ∞ -bimodule over it consists of a graded vector space M = ⊕M i together with a collection of multilinear maps η k :

These multilinear maps are supposed to satisfy the identities (30) with exactly one of a 1 , . . . , a n is from M and the corresponding multilinear operation µ is replaced by η.

It follows that any A ∞ -algebra (A, µ k ) is an A ∞ -bimodule over itself. 

for each k ≥ 1 and u 1 , . . . , u k ∈ M.

When R = S, we call R a generalized Rota-Baxter operator. On the other hand if the A ∞ -bimodule is taken to be A itself, we call (R, S) a Rota-Baxter system on the A ∞ -algebra (A, µ k ). Any Rota-Baxter system on an associative algebra can be seen as a Rota-Baxter system on an A ∞ -algebra whose underlying graded vector space is concentrated in degree 0.

It is known that an A ∞ -algebra structure on A is equivalent to a square-zero coderivation (of degree −1) on the cofree coassociative coalgebra T c (sA). It could be interesting to find the interpretation of a Rota-Baxter operator (more generally, a Rota-Baxter system) on an A ∞ -algebra in terms of the compatibility with the coderivation on T c (sA).

The above definition of a (generalized) Rota-Baxter system is more justified by the following example and Theorem 6.5. 

In this case, we first define an A ∞ -algebra structure on the 2-term complex M µ1=d −−−→ A ⊕ N with structure maps given by µ 2 ((a, n 1 ), (b, n 2 )) = (ab, a · n 2 + n 1 · b), µ 2 ((a, n 1 ), u) = a · u, µ 2 (u, (a, n 1 )) = u · a, µ k = 0, for k ≥ 3.

Then it can be checked that R = ((R ⊕ R N ), R M ) and S = ((S ⊕ S N ), S M ) constitute a Rota-Baxter system on the above A ∞ -algebra.

Next, we recall the notion of Dend ∞ -algebras introduced in [25] . Here we will consider the equivalent definition given in [12] . First recall from [12] that there are certain maps R 0 (m; a λ+i , . . . , a n = 0, for a 1 , . . . , a n ∈ A.

A Dend ∞ -algebra (A, µ k,[r] ) whose underlying graded vector space A is concentrated in degree 0 is a dendriform algebra with ≺ = µ 2, [1] and ≻ = µ 2, [2] . It has been proved in [12, 25] that if (A, µ k, [r] ) is a Dend ∞ -algebra then (A, µ k ) is an A ∞ -algebra, where µ k = µ k, [1] + · · · + µ k,[k] , for 1 ≤ k < ∞.

The following theorem is the homotopy version of Proposition 2.9.

6.5. Theorem. Let (A, µ k ) be an A ∞ -algebra and (M, η k ) be an A ∞ -bimodule. If (R, S) is a generalized Rota-Baxter system, then (M, µ k, [r] ) is a Dend ∞ -algebra where µ k,[r] (u 1 , . . . , u k ) = η k R(u 1 ), . . . , R(u r−1 ), u r , S(u r+1 ), . . . , S(u k ) , for k ≥ 1 and 1 ≤ r ≤ k.

Proof. Since (R, S) is a generalized Rota-Baxter system, it follows from (31) and (32) that . . . , u k )) and µ k (S(u 1 ), . . . , S(u k )) = S(

On the other hand, the A ∞ condition on (A, µ k ) implies that i+j=n+1 j λ=1 ± µ j a 1 , . . . , a λ−1 , µ i (a λ , . . . , a λ+i−1 ), a λ+i , . . . , a n = 0,

for all n ≥ 1 and a 1 , . . . , a n ∈ A. The same identity holds if exactly one of (a 1 , . . . , a n ) is from M and the corresponding µ is replaced by η. Consider the elements R(u 1 ), . . . , R(u r−1 ), u r , S(u r+1 ), . . . , S(u n ) , for some fixed 1 ≤ r ≤ n. For any fixed i, j and λ, if r ≤ λ − 1, then the term inside the summation look Substitute these identities in (33), we get the identities of a Dend ∞ -algebra.

In this section, we consider Rota-Baxter systems on a dendriform algebra and commuting Rota-Baxter systems on an associative algebra. We show how these structures induce quadri-algebras introduced by Aguiar and Loday [4] . The homotopy version of these results is also described. 

for a, b ∈ D.

A quadri-algebra is a vector space A together with four binary operations տ (northwest), ր (north-east), ւ (south-west) and ց (south-east) satisfying the following 3 identities

for a, b, c ∈ A. Here we use the following notations

It turns out that in a quadri-algebra as above, (A, ❁, ❂) and (A, ∧, ∨) are both dendriform algebras. Thus, (A, * ) is an associative algebra. Quadri-algebras are studied from multiplicative operadic points of view in [14] . 7.3. Proposition. Let (D, ≺, ≻) be a dendriform algebra and (R, S) be a Rota-Baxter system on D. Then (D, տ, ր, ւ, ց) is a quadri-algebra where

Proof. We will verify the first 3 identities of a quadri-algebra. The verification of remaining 6 identities are similar. For the first identity, we have Here we have verified the first two identities of (34) . The other two identities are similar to verify.

Combining Propositions 7.3 and 7.4, we get the following. 7.6. Example. Let A be an associative algebra and (r = r (1) ⊗ r (2) , s = s (1) ⊗ s (2) ) be an associative Yang-Baxter pair. Then it has been shown in [9, Proposition 3.4 ] that the pair (R, S) is a Rota-Baxter system on A, where R, S : A → A are given by R(a) = r (1) ar (2) and S(a) = s (1) as (2) .

Suppose (p = p (1) ⊗p (2) , q = q (1) ⊗q (2) ) is another associative Yang-Baxter pair (with associated Rota-Baxter system (P, Q)) such that p and q both commute with r and s as elements of A ⊗ A op . Then (P, Q) and (R, S) are commuting Rota-Baxter systems. 7.7. Example. Let A be an associative algebra and σ, τ be two commuting algebra maps. Suppose P is a σ-twisted Rota-Baxter operator and R is a τ -twisted Rota-Baxter operator (see Example 2.3) such that P • R = R • P , P • τ = τ • P and R • σ = σ • R. Then (P, Q = σ • P ) and (R, S = τ • R) are commuting Rota-Baxter systems.

The above results can be even extended in the homotopy context. Note that the notion of Quad ∞ -algebras (homotopy quadri-algebras) can be explicitly define using [14, 15] where various Loday-type algebras and homotopy Loday-type algebras are considered. Here we first recall the definition.

Let Q n = C n × C n , for n ≥ 1, where C n 's are given in Subsection 3.2. Therefore, Q n = {([r], [s])| 1 ≤ r, s ≤ n}. We define the structure maps R 0 (m; 1, . . . , n, . . . , 1) : Q m+n−1 → Q m and R i (m; 1, . . . , n, . . . , 1) : Q m+n−1 → K[C n ] × K[C n ] as the product of the structure maps defined by the table in Section 6, i.e. 

a λ+i , . . . , a n = 0.

The following result is the homotopy version of the fact that a quadri-algebra induce two dendriform structures on the underlying vector space. (R(a 1 ) , . . . , R(a s−1 ), a s , S(a s+1 ), . . . , S(a k )), for 1 ≤ k < ∞, 1 ≤ r, s ≤ k.

Proof. The proof is similar to the approach of Theorem 6.5, and hence we omit the details.

Let (A, µ k ) be an A ∞ -algebra. Two Rota-Baxter systems (P, Q) and (R, S) are said to be compatible if P • R = R • P , P • S = S • P , Q • R = R • Q and Q • S = S • Q. 7.12. Proposition. Let (P, Q) and (R, S) be two commuting Rota-Baxter systems on an A ∞ -algebra (A, µ k ). Then (R, S) is a Rota-Baxter system on the Dend ∞ -algebra (A, µ k, [r] ) induced from the Rota-Baxter system (P, Q) by Theorem 6.5. Consequently, there is a Quad ∞ -algebra (A, µ k,([r],[s]) ).

Proof. We have µ k,[r] (R(a 1 ), . . . , R(a k )) = µ k (P R(a 1 ), . . . , P R(a r−1 ), Ra r , QR(a r+1 ), . . . , QR(a k )) = µ k (RP (a 1 ), . . . , RP (a r−1 ), Ra r , RQ(a r+1 ), . . . , RQ(a k )) = R r−1 i=1 µ k RP (a 1 ), . . . , RP (a i−1 ), P a i , SP (a i+1 ), . . . , SP (a r−1 ), Sa r , SQ(a r+1 ), . . . , SQ(a k ) + µ k RP (a 1 ), . . . , RP (a r−1 ), a r , SQ(a r+1 ), . . . , SQ(a k ) + k i=r+1 µ k RP (a 1 ), . . . , RP (a r−1 ), Ra r , RQ(a r+1 ), . . . , RQ(a i−1 ), Qa i , SQ(a i+1 ), . . . , SQ(a k ) = R r−1 i=1 µ k P R(a 1 ), . . . , P R(a i−1 ), P a i , P S(a i+1 ), . . . , P S(a r−1 ), Sa r , QS(a r+1 ), . . . , QS(a k ) + µ k P R(a 1 ), . . . , P R(a r−1 ), a r , QS(a r+1 ), . . . , QS(a k ) + k i=r+1 µ k P R(a 1 ), . . . , P R(a r−1 ), Ra r , QR(a r+1 ), . . . , QR(a i−1 ), Qa i , QS(a i+1 ), . . . , QS(a k ) = R k i=1 µ k,[r] (R(a 1 ), . . . , R(a i−1 ), a i , S(a i+1 ), . . . , S(a k )) .

Thus, we have proved the identity (36) . The proof of (37) is similar.

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Déformations des algébres de Leibniz

Deformations of algebras over a quadratic operad

An analytic problem whose solution follows from a simple algebraic identity

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Rota-Baxter systems, dendriform algebras and covariant bialgebras

Quantum Bi-Hamiltonian Systems

Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem

Cohomology and deformations of dendriform algebras, and Dend∞-algebras

Deformations of associative Rota-Baxter operators

Deformations of Loday-type algebras and their morphisms

Involutive and oriented dendriform algebras

Drinfel'd, On quasitriangular quasi-Hopf algebras and a certain group closely connected with Gal

Drinfel'd, On the structure of quasitriangular quasi-Hopf algebras

The L∞-deformation complex of diagrams of algebras

On the deformation of rings and algebras

On the deformation of algebra morphisms and diagrams

Homotopy G-algebras and moduli space operad

An introduction to Rota-Baxter algebra

Coalgebras and Bialgebras in combinatorics

Dialgebras, Dialgebras and related operads

Algebraic operads

Curved O-operator systems

Reduction of Poisson manifolds

Cohomology and deformations in graded Lie algebras

Averaging algebras, Schröder numbers, rooted trees and operads

Free Rota-Baxter systems and a Hopf algebra structure

On the dynamic theory of incompressible viscous fluids

Baxter algebras and combinatorial identities I, II

Poisson geometry with a 3-form background

Homotopy associativity of H-spaces II

Quantum analogy of Poisson geometry, related dendriform algebras and Rota-Baxter operators

Higher derived brackets and homotopy algebras

Acknowledgements. The research is supported by the postdoctoral fellowship of Indian Institute of Technology (IIT) Kanpur. The author thanks the Institute for support. This work is completely done at home during the lockdown period for COVID-19. He also wishes to thank his family members for support.