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O-operators on associative algebras and dendriform algebras

An O-operator is a relative version of a Rota-Baxter operator and, in the Lie algebra context, is originated from the operator form of the classical Yang-Baxter equation. We generalize the well-known construction of dendriform dialgebras and trialgebras from Rota-Baxter algebras to a construction from O-operators. We then show that this construction from O-operators gives all dendriform dialgebras and trialgebras. Furthermore there are bijections between certain equivalence classes of invertible O-operators and certain equivalence classes of dendriform dialgebras and trialgebras.


Introduction
This paper shows that there is a close tie between two seemingly unrelated objects, namely Ooperators and dendriform dialgebras and trialgebras, generalizing and strengthening a previously established connection from Rota-Baxter algebras to dendriform algebras [1,2,13].
To fix notations, we let k denote a commutative unitary ring in this paper. By a k-algebra we mean an associative (not necessarily unitary) k-algebra, unless otherwise stated. Definition 1.1. Let R be a k-algebra and let λ ∈ k be given. If a k-linear map P : R → R satisfies the Rota-Baxter relation: (1) P(x)P(y) = P(P(x)y) + P(xP(y)) + λP(xy), ∀x, y ∈ R, then P is called a Rota-Baxter operator of weight λ and (R, P) is called a Rota-Baxter algebra of weight λ.
On the other hand, with motivation from periodicity of algebraic K-theory and operads, dendriform dialgebras were introduced by Loday [23] in the 1990s.

Definition 1.2.
A dendriform dialgebra is a triple (R, ≺, ≻) consisting of a k-module R and two bilinear operations ≺ and ≻ on R such that for all x, y, z ∈ R. Here x ⋆ y = x ≺ y + x ≻ y.
Date: March 18, 2010. This defines a functor from the category of Rota-Baxter algebras of weight 0 to the category of dendriform dialgebras. This work has inspired quite a few subsequent studies [3,4,5,9,13,14,17] that generalized and further clarified the relationship between Rota-Baxter algebras and dendriform dialgebras and trialgebras of Loday and Ronco [24], including the adjoint functor of the above functor, the related Poincare-Birkhoff-Witt theorem and Gröbner-Shirshov basis.
These studies further suggested that there should be a close relationship between Rota-Baxter algebras and dendriform dialgebras. Then it is natural to ask whether every dendriform dialgebra and trialgebra could be derived from a Rota-Baxter algebra by a construction like Eq. (3). As later examples show, this is quite far from being true.
Our main purpose of this paper is to show that there is a generalization of the concept of a Rota-Baxter operator that could derive all the dendriform dialgebras and trialgebras. It is given by the concept of an O-operator on a k-module and or a k-algebra. Such a concept was first introduced in the context of Lie algebras [4,10,21] to study the classical Yang-Baxter equations and integrable systems, and was recently generalized and applied to the study of Lax pairs and PostLie algebras [6]. In the associative algebra context, O-operators have been applied to study associative analogues of the classical Yang-Baxter equation [7].
For simplicity, we only define O-operators on modules in the introduction, referring the reader to later sections for the more case of O-operators on algebras.
When V is taken to be the A-bimodule (A, L, R) associated to the algebra A, an O-operator on the module is just a Rota-Baxter operator of weight zero.
Then as in the case of Rota-Baxter operators, we obtain a dendriform dialgebra (V, ≺ α , ≻ α ). We also define an O-operator on an algebra that generalizes a Rota-Baxter operator with a non-zero weight and show that an O-operator on an algebra gives a dendriform trialgebra. We prove in Section 2.3 that every dendriform dialgebras and trialgebra can be recovered from an O-operator in this way, in contrary to the case of a Rota-Baxter operator.
In Section 3 we further show that the dendriform dialgebra or trialgebra structure on V from an O-operator α : V → A transports to a dendriform dialgebra or trialgebra structure on A through α under a natural condition. To distinguish the two dendriform dialgebras and trialgebras from an O-operator α : V → A, we call them the dendriform dialgebras and trialgebras on the domain and the dendriform dialgebras and trialgebras on the range of α respectively.
By considering the multiplication on the range A, we show that, the correspondence from O-operators to dendriform dialgebras and trialgebras on the domain V implies a more refined correspondence from O-operators to dendriform dialgebras and trialgebra on the range A that are compatible with A in the sense that the dialgebra and trialgebra multiplications give a splitting (or decomposition) of the associative product of A. We finally quantify this refined correspondence by providing bijections between certain equivalent classes of O-operators with range in A and equivalent classes of compatible dendriform dialgebra and trialgebra structures on A.
NSF grant DMS 0505445 for support, and thanks the Chern Institute of Mathematics at Nankai University and the Center of Mathematics at Zhejiang University for their hospitalities.

O-operators and dendriform algebras on the domains
In this section we study the relationship between O-operators and dendriform dialgebras and trialgebras on the domains of these operators. The related concepts and notations are introduced in Section 2.1 and 2.2. Then we show that O-operators recover all dendriform dialgebras and trialgebras on the domains of the operators.

2.1.
A-bimodule k-algebras and O-operators. We start with a generalization of the well-known concept of bimodules. Definition 2.1. Let (A, * ) be a k-algebra with multiplication * .
(a) Let (R, •) be a k-algebra with multiplication •. Let ℓ, r : A → End k (R) be two linear maps. We call (R, •, ℓ, r) or simply R an A-bimodule k-algebra if (R, ℓ, r) is an A-bimodule that is compatible with the multiplication • on R. More precisely, we have is a k-linear map g : R 1 → R 2 that is both an A-bimodule homomorphism and a k-algebra homomorphism.
An A-bimodule (V, ℓ, r) becomes an A-bimodule k-algebra if we equip V with the zero multiplication.
For a k-algebra (A, * ) and x ∈ A, define the left and right actions Further define As is well-known, (A, L, R) is an A-bimodule. Moreover, (A, * , L, R) is an A-bimodule k-algebra.
Note that an A-bimodule k-algebra needs not be a left or right A-algebra. For example, the Abimodule k-algebra (A, * , L, R) is an A-algebra if and only if A is a commutative k-algebra.
We can now define our generalization [7] of Rota-Baxter operators.
(b) Let (R, •, ℓ, r) be an A-bimodule k-algebra and λ ∈ k. A linear map α : R → A is called an O-operator on the algebra R of weight λ if α satisfies

Remark 2.3.
(a) Obviously, for the A-bimodule k-algebra (A, * , L, R), an O-operator α : (A, * , L, R) → A of weight λ is just a Rota-Baxter operator on (A, * ) of the same weight. An O-operator can be viewed as a relative version of a Rota-Baxter operator in the sense that the domain and range of an O-operator might be different. (b) The construction of O-operators of λ = 0 has been defined by Uchino [26] under the name of a generalized Rota-Baxter operator who also obtained Theorem 2.7.(b).
We note the following simple relationship between O-operators on modules and O-operators on algebras of weight zero. Lemma 2.4. Let A be a k-algebra. If α : R → A is an O-operator on a k-algebra (R, •) of weight zero, then α is an O-operator on the underlying k-module of (R, •). Conversely, let α : V → A be an O-operator on a k-module V. Equip V with an associative multiplication (say the zero multiplication) •. Then α is an O-operator on the algebra (V, •) of weight zero.
Thus we have natural maps between O-operators on an algebra of weight zero and O-operators on a module. But the map from O-operators on a module to O-operators on an algebra of weight zero is not canonical in the sense that it depends on a choice of a multiplication on the module which will play a subtle role later in the paper (See the remark before Theorem 2.8). Thus we would like to distinguish these two kinds of O-operators.

Rota-Baxter algebras and dendriform algebras.
Generalizing the concept of a dendriform dialgebra of Loday defined in Section 1, the concept of a dendriform trialgebra was introduced by Loday and Ronco [24].
We summarize Proposition 2.6 by saying that dendriform dialgebra (resp. trialgebra) gives a splitting of the associative multiplication ⋆.

2.3.
From O-operators to dendriform algebras on the domains. We first show that the procedure of deriving dendriform dialgebras and trialgebras from Rota-Baxter operators can be generalized to O-operators.
Similar arguments can be applied to verify the other axioms for a dendriform trialgebra in Eq. (12).
The second statement follows from Proposition 2.6 and the definition of α: where the disjoint union runs through all pairs (R, A) where A is a k-algebra and R is an Abimodule k-algebra such that |R| = V. Then from the map Φ alg λ,V,A in Eq. (24) we obtain Similarly, for a k-module V and k-algebra A, denote By Theorem 2.7.(b), we obtain a map Let O mod (V, −) denote the set of O-operators on the module V. In other words, where A runs through all the k-algebras. Then we have is the disjoint union of multiple copies of O mod (V, −), one copy for each associative multiplication on V. Therefore, the surjectivity of Φ mod V is a stronger property than the surjectivity of Φ alg 0,V .
By this theorem, all dendriform dialgebra (resp. trialgebra) structures on V could be recovered from O-operators on the module (resp. on the algebra).
. This proves the surjectivity of Φ mod V .

O-operators and dendriform algebras on the ranges
We next study another kind of relationship between O-operators and dendriform dialgebras and trialgebras by focusing on the algebra (A, * ) in an O-operator α : R → A. We first show that, under a natural condition, an O-operator α : R → A on the module (resp. on the algebra) gives a dendriform dialgebra (resp. trialgebra) structure on A that gives a splitting of * in the sense of Proposition 2.6 (see the remark thereafter). We then show that the O-operators α : R → A, as the k-module (resp. k-algebra) R varies, recover all dendriform dialgebra or trialgebra structures on (A, * ) with the splitting property. We in fact give bijections between suitable equivalence classes of these O-operators and (equivalent classes of) dendriform dialgebras and trialgebras.
3.1. From O-operators to dendriform algebras on the ranges. We first give the following consequence of Theorem 2.7, providing a dendriform dialgebra or a trialgebra on the range of an O-operator.
(a) Let (R, •, ℓ, r) be an A-bimodule k-algebra. Let α : R → A be an O-operator on the algebra of weight λ. If ker α is an ideal of (R, •), then there is a dendriform trialgebra structure on α(R) given by Proof. (a) We first prove that the multiplications in Eq. (31) are well-defined. More precisely, for with the first term on the right hand side vanishing. The third term also vanishes since ker α is an ideal of (R, •). Thus the second term also vanishes and (u − u ′ )r(α(v)) is in ker α. We then find that is in ker α. This verifies the first equation in Eq. (34). The other two equations are verified similarly. Then the axioms in Eq. (12) for (α(R), ≺ α,A , ≻ α,A , · α,A ) to be a dendriform trialgebra follows from the axioms for (R, ≺ α , ≻ α , · α ) to be a dendriform trialgebra.
Since α is an O-operator, we have This proves the second statement in Item (a). Then the last statement follows as a direct consequence.
Proposition 3.1 motivate us to introduce the following notations.
be the maps defined by Proposition 3.1.

Bijective correspondences.
Instead of proving just the surjectivities of the maps Ψ alg A and Ψ mod A defined by Eq. (35) and Eq. (36), we give a more quantitative description of these maps. We first prove a lemma that justifies the concepts that will be introduced next. Lemma 3.3. Let (A, * ) be a k-algebra and let (R, •, ℓ, r) be an A-bimodule k-algebra. Let α :
We can now define equivalence relations among O-operators and dendriform algebras.

Introduction
This paper shows that there is a close tie between two seemingly unrelated objects, namely Ooperators and dendriform dialgebras and trialgebras, generalizing and strengthening a previously established connection from Rota-Baxter algebras to dendriform algebras [1,2,13].
To fix notations, we let k denote a commutative unitary ring in this paper. By a k-algebra we mean an associative (not necessarily unitary) k-algebra, unless otherwise stated. Definition 1.1. Let R be a k-algebra and let λ ∈ k be given. If a k-linear map P : R → R satisfies the Rota-Baxter relation: (1) P(x)P(y) = P(P(x)y) + P(xP(y)) + λP(xy), ∀x, y ∈ R, then P is called a Rota-Baxter operator of weight λ and (R, P) is called a Rota-Baxter algebra of weight λ.
On the other hand, with motivation from periodicity of algebraic K-theory and operads, dendriform dialgebras were introduced by Loday [23] in the 1990s.

Definition 1.2.
A dendriform dialgebra is a triple (R, ≺, ≻) consisting of a k-module R and two bilinear operations ≺ and ≻ on R such that for all x, y, z ∈ R. Here x ⋆ y = x ≺ y + x ≻ y.
Aguiar [1] first established the following connection from Rota-Baxter algebras to dendriform dialgebras.
Date: March 18, 2010. This defines a functor from the category of Rota-Baxter algebras of weight 0 to the category of dendriform dialgebras. This work has inspired quite a few subsequent studies [3,4,5,9,13,14,17] that generalized and further clarified the relationship between Rota-Baxter algebras and dendriform dialgebras and trialgebras of Loday and Ronco [24], including the adjoint functor of the above functor, the related Poincare-Birkhoff-Witt theorem and Gröbner-Shirshov basis.
These studies further suggested that there should be a close relationship between Rota-Baxter algebras and dendriform dialgebras. Then it is natural to ask whether every dendriform dialgebra and trialgebra could be derived from a Rota-Baxter algebra by a construction like Eq. (3). As later examples show, this is quite far from being true.
Our main purpose of this paper is to show that there is a generalization of the concept of a Rota-Baxter operator that could derive all the dendriform dialgebras and trialgebras. It is given by the concept of an O-operator on a k-module and or a k-algebra. Such a concept was first introduced in the context of Lie algebras [4,10,21] to study the classical Yang-Baxter equations and integrable systems, and was recently generalized and applied to the study of Lax pairs and PostLie algebras [6]. In the associative algebra context, O-operators have been applied to study associative analogues of the classical Yang-Baxter equation [7].
For simplicity, we only define O-operators on modules in the introduction, referring the reader to later sections for the more case of O-operators on algebras.
Let (A, * ) be a k-algebra. Let (V, ℓ, r) be an A-bimodule, consisting of a compatible pair of a left A-module (V, ℓ) given by ℓ : A → End(V) and a right A-module (V, r) given by r : When V is taken to be the A-bimodule (A, L, R) associated to the algebra A, an O-operator on the module is just a Rota-Baxter operator of weight zero.
Then as in the case of Rota-Baxter operators, we obtain a dendriform dialgebra (V, ≺ α , ≻ α ). We also define an O-operator on an algebra that generalizes a Rota-Baxter operator with a non-zero weight and show that an O-operator on an algebra gives a dendriform trialgebra. We prove in Section 2.3 that every dendriform dialgebras and trialgebra can be recovered from an O-operator in this way, in contrary to the case of a Rota-Baxter operator.
In Section 3 we further show that the dendriform dialgebra or trialgebra structure on V from an O-operator α : V → A transports to a dendriform dialgebra or trialgebra structure on A through α under a natural condition. To distinguish the two dendriform dialgebras and trialgebras from an O-operator α : V → A, we call them the dendriform dialgebras and trialgebras on the domain and the dendriform dialgebras and trialgebras on the range of α respectively.
By considering the multiplication on the range A, we show that, the correspondence from O-operators to dendriform dialgebras and trialgebras on the domain V implies a more refined correspondence from O-operators to dendriform dialgebras and trialgebra on the range A that are compatible with A in the sense that the dialgebra and trialgebra multiplications give a splitting (or decomposition) of the associative product of A. We finally quantify this refined correspondence by providing bijections between certain equivalent classes of O-operators with range in A and equivalent classes of compatible dendriform dialgebra and trialgebra structures on A.
NSF grant DMS 0505445 for support, and thanks the Chern Institute of Mathematics at Nankai University and the Center of Mathematics at Zhejiang University for their hospitalities.

O-operators and dendriform algebras on the domains
In this section we study the relationship between O-operators and dendriform dialgebras and trialgebras on the domains of these operators. The related concepts and notations are introduced in Section 2.1 and 2.2. Then we show that O-operators recover all dendriform dialgebras and trialgebras on the domains of the operators.

A-bimodule k-algebras and O-operators.
We start with a generalization of the well-known concept of bimodules. (a) Let (R, •) be a k-algebra with multiplication •. Let ℓ, r : A → End k (R) be two linear maps. We call (R, •, ℓ, r) or simply R an A-bimodule k-algebra if (R, ℓ, r) is an A-bimodule that is compatible with the multiplication • on R. More precisely, we have is a k-linear map g : R 1 → R 2 that is both an A-bimodule homomorphism and a k-algebra homomorphism.
An A-bimodule (V, ℓ, r) becomes an A-bimodule k-algebra if we equip V with the zero multiplication.
For a k-algebra (A, * ) and x ∈ A, define the left and right actions Further define As is well-known, (A, L, R) is an A-bimodule. Moreover, (A, * , L, R) is an A-bimodule k-algebra.

Note that an A-bimodule k-algebra needs not be a left or right A-algebra. For example, the Abimodule k-algebra (A, * , L, R) is an A-algebra if and only if A is a commutative k-algebra.
We can now define our generalization [7] of Rota-Baxter operators.
(b) Let (R, •, ℓ, r) be an A-bimodule k-algebra and λ ∈ k. A linear map α : R → A is called an O-operator on the algebra R of weight λ if α satisfies Thus we have natural maps between O-operators on an algebra of weight zero and O-operators on a module. But the map from O-operators on a module to O-operators on an algebra of weight zero is not canonical in the sense that it depends on a choice of a multiplication on the module which will play a subtle role later in the paper (See the remark before Theorem 2.8). Thus we would like to distinguish these two kinds of O-operators.

Rota-Baxter algebras and dendriform algebras.
Generalizing the concept of a dendriform dialgebra of Loday defined in Section 1, the concept of a dendriform trialgebra was introduced by Loday and Ronco [24].
We summarize Proposition 2.6 by saying that dendriform dialgebra (resp. trialgebra) gives a splitting of the associative multiplication ⋆.

From O-operators to dendriform algebras on the domains.
We first show that the procedure of deriving dendriform dialgebras and trialgebras from Rota-Baxter operators can be generalized to O-operators.
Similar arguments can be applied to verify the other axioms for a dendriform trialgebra in Eq. (12).
The second statement follows from Proposition 2.6 and the definition of α: By this theorem, all dendriform dialgebra (resp. trialgebra) structures on V could be recovered from O-operators on the module (resp. on the algebra).

O-operators and dendriform algebras on the ranges
We next study another kind of relationship between O-operators and dendriform dialgebras and trialgebras by focusing on the algebra (A, * ) in an O-operator α : R → A. We first show that, under a natural condition, an O-operator α : R → A on the module (resp. on the algebra) gives a dendriform dialgebra (resp. trialgebra) structure on A that gives a splitting of * in the sense of Proposition 2.6 (see the remark thereafter). We then show that the O-operators α : R → A, as the k-module (resp. k-algebra) R varies, recover all dendriform dialgebra or trialgebra structures on (A, * ) with the splitting property. We in fact give bijections between suitable equivalence classes of these O-operators and (equivalent classes of) dendriform dialgebras and trialgebras.
Proof. (a) We first prove that the multiplications in Eq. (31) are well-defined. More precisely, for u, u ′ , v, v ′ ∈ R such that α(u) = α(u ′ ) and α(v) = α(v ′ ), we check that with the first term on the right hand side vanishing. The third term also vanishes since ker α is an ideal of (R, •). Thus the second term also vanishes and (u − u ′ )r(α(v)) is in ker α. We then find that is in ker α. This verifies the first equation in Eq. (34). The other two equations are verified similarly. Then the axioms in Eq. (12) for (α(R), ≺ α,A , ≻ α,A , · α,A ) to be a dendriform trialgebra follows from the axioms for (R, ≺ α , ≻ α , · α ) to be a dendriform trialgebra.
Since α is an O-operator, we have This proves the second statement in Item (a). Then the last statement follows as a direct consequence.
(b). Let (V, ℓ, r) be an A-bimodule and let α : V → A be an O-operator on the module V. Then when V is equipped with an associative multiplication • (say • ≡ 0), α becomes an O-operator on the algebra (V, •) of weight zero. Then by Item (a), (V, Proposition 3.1 motivate us to introduce the following notations.
Thus α • g is an O-operator of weight λ.
We can now define equivalence relations among O-operators and dendriform algebras.
By the same argument, we have g((ℓ 1 F −1 )(x)v) = ℓ 2 (x)g(v), ∀x ∈ A, v ∈ R and g(u • 1 v) = g(u) • 2 g(v), ∀u, v ∈ R. Since g is also bijective, we have proved that g is the isomorphism of A-bimodule k-algebras that we want. This completes the proof.