On quadri-bialgebras

TL;DR

This paper introduces quadri-bialgebras, a new algebraic structure linked to quadri-algebras, and explores their connections to Manin triples, nondegenerate cocycles, and solutions to the Q-equations, expanding the algebraic framework.

Contribution

It defines quadri-bialgebras, establishes their equivalence to Manin triples, and connects them to classical Yang-Baxter variations and operator constructions.

Findings

Quadri-bialgebras are equivalent to Manin triples of dendriform and quadri-algebras.

They relate to nondegenerate 2-cocycles and invariant bilinear forms.

They originate from a variation of the classical Yang-Baxter equation called Q-equations.

Abstract

We introduce the notion of a quadri-bialgebra, which gives a bialgebra theory for the quadri-algebra introduced by Aguiar and Loday. We show that a quadri-bialgebra is equivalent to a Manin triple of dendriform algebras associated to a nondegenerate 2-cocycle, and to a Manin triple of quadri-algebras associated to a nondegenerate invariant bilinear form. Quadri-bialgebras also come from a variation of the classical Yang-Baxter equation, called the $Q$-equations. Moreover, quadri-bialgebras fit into the framework of construction of Rota-Baxter operators and Nijenhuis operators on the double spaces of quadri-algebras.

Equations

$$W^{\perp }=\{x\in V|\mathfrak{B}(x,y)=0,\forall y\in W\}.$$

$$r_{12}:=\sum _i{a}_i\otimes b_i\otimes 1,r_{13}:=\sum _i{a}_i\otimes 1\otimes b_i,r_{23}:=\sum _{i}1\otimes a_i\otimes b_i,$$

$$r_{12}\diamond r_{13}=\sum _{i,j}{a}_i\diamond a_j\otimes b_i\otimes b_j,r_{13}\diamond r_{23}=\sum _{i,j}{a}_i\otimes a_j\otimes b_i\diamond b_j,r_{23}\diamond r_{12}=\sum _{i,j}{a}_j\otimes a_i\diamond b_j\otimes b_i.$$

$$\tau (u\otimes v)=v\otimes u,\forall u,v\in V.$$

$$⟨u^{\ast }\otimes v^{\ast },r⟩=⟨u^{\ast },T_r(v^{\ast })⟩,\forall u^{\ast },v^{\ast }\in V^{\ast },$$

$$⟨v_1,T^{\ast }(v_2^{\ast })⟩=⟨T(v_1),v_2^{\ast }⟩,\forall v_1\in V_1,v_2^{\ast }\in V_2^{\ast }.$$

$$⟨{\rho }^{\ast }(x)v^{\ast },u⟩=⟨v^{\ast },\rho (x)u⟩,\forall x\in A,u\in V,v^{\ast }\in V^{\ast }.$$

$$(x\prec y)\prec z=x\prec (y\star z),(x\succ y)\prec z=x\succ (y\prec z),(x\star y)\succ z=x\succ (y\succ z),$$

$$x\star y:=x\prec y+x\succ y,\forall x,y\in A,$$

$$r_{\prec }(y)r_{\prec }(x)=r_{\prec }(x\star y),r_{\prec }(y)l_{\prec }(x)=l_{\prec }(x)r_{\star }(y),l_{\prec }(x\prec y)=l_{\prec }(x)l_{\star }(y),$$

$$r_{\prec }(y)r_{\succ }(x)=r_{\succ }(x\prec y),r_{\prec }(y)l_{\succ }(x)=l_{\succ }(x)r_{\prec }(y),l_{\prec }(x\succ y)=l_{\succ }(x)l_{\prec }(y),$$

$$r_{\succ }(y)r_{\star }(x)=r_{\succ }(x\succ y),r_{\succ }(y)l_{\star }(x)=l_{\succ }(x)r_{\succ }(y),l_{\succ }(x\star y)=l_{\succ }(x)l_{\succ }(y).$$

$$T(u)\prec T(v)=T(l_{\prec }(T(u))v+r_{\prec }(T(v))u),T(u)\succ T(u)=T(l_{\succ }(T(u))v+r_{\succ }(T(v))u),\forall u,v\in V.$$

$$(x\nwarrow y)\nwarrow z=x\nwarrow (y\star z),(x\nearrow y)\nwarrow z=x\nearrow (y\prec z),(x\wedge y)\nearrow z=x\nearrow (y\succ z),$$

$$(x\swarrow y)\nwarrow z=x\swarrow (y\wedge z),(x\searrow y)\nwarrow z=x\searrow (y\nwarrow z),(x\vee y)\nearrow z=x\searrow (y\nearrow z),$$

$$(x\prec y)\swarrow z=x\swarrow (y\vee z),(x\succ y)\swarrow z=x\searrow (y\swarrow z),(x\star y)\searrow z=x\searrow (y\searrow z),$$

$$x\succ y:=x\nearrow y+x\searrow y,x\prec y:=x\nwarrow y+x\swarrow y,$$

$$x\vee y:=x\swarrow y+x\searrow y,x\wedge y:=x\nwarrow y+x\nearrow y,$$

$$x\star y:=x\searrow y+x\nearrow y+x\nwarrow y+x\swarrow y=x\succ y+x\prec y=x\vee y+x\wedge y.$$

$$\mathfrak{B}(x\star y,z)=\mathfrak{B}(y,z\wedge x)+\mathfrak{B}(x,y\vee z),\forall x,y,z\in A.$$

$$\mathfrak{B}(x\nwarrow y,z)=\mathfrak{B}(x,y\star z),\mathfrak{B}(x\nearrow y,z)=-\mathfrak{B}(y,z\vee x),$$

$$\mathfrak{B}(x\swarrow y,z)=-\mathfrak{B}(x,y\wedge z),\mathfrak{B}(x\searrow y,z)=\mathfrak{B}(y,z\star x),\forall x,y,z\in A.$$

$$\phi (A^{+})\subset B^{+},\phi (A^{-})\subset B^{-},{\mathfrak{B}}_A(x,y)={\mathfrak{B}}_B(\phi (x),\phi (y)),\forall x,y\in A.$$

$${\mathfrak{B}}_S(x+a^{\ast },y+b^{\ast }):=⟨a^{\ast },y⟩+⟨x,b^{\ast }⟩,\forall x,y\in A;a^{\ast },b^{\ast }\in A^{\ast },$$

$$(l_{{\wedge }_B}(a)x){\wedge }_{A}y+l_{{\wedge }_B}(r_{{\wedge }_A}(x)a)y=l_{{\wedge }_B}(a)(x{\star }_{A}y),$$

$$l_{{\wedge }_B}(l_{{\wedge }_A}(x)a)y+(r_{{\wedge }_B}(a)x){\wedge }_{A}y=x{\wedge }_A(l_{{\star }_B}(a)y)+r_{{\wedge }_B}(r_{{\star }_A}(y)a)x,$$

$$r_{{\wedge }_B}(a)(x{\wedge }_{A}y)=r_{{\wedge }_B}(l_{{\star }_A}(y)a)x+x{\wedge }_A(r_{{\star }_B}(a)y),$$

$$(l_{{\vee }_B}(a)x){\wedge }_{A}y+l_{{\wedge }_B}(r_{{\vee }_A}(x)a)y=l_{{\vee }_B}(a)(x{\wedge }_{A}y),$$

$$l_{{\wedge }_B}(l_{{\vee }_A}(x)a)y+(r_{{\vee }_B}(a)x){\wedge }_{A}y=x{\vee }_A(l_{{\wedge }_B}(a)y)+r_{{\vee }_B}(r_{{\wedge }_A}(y)a)x,$$

$$r_{{\wedge }_B}(a)(x{\vee }_{A}y)=r_{{\vee }_B}(l_{{\wedge }_A}(y)a)x+x{\vee }_A(r_{{\wedge }_B}(a)y),$$

Full text

1. 1 Introduction
2. 2 Notations and backgrounds
3. 3 Manin triples of dendriform algebras associated to a nondegenerate 2-cocycle and matched pairs of dendriform algebras
4. 4 Bimodules and matched pairs of quadri-algebras
5. 5 Quadri-bialgebras
6. 6 Coboundary quadri-bialgebras
7. 7 The $Q$-equations
8. 8 Construction of linear operators on double spaces of quadri-algebras

1. Introduction

Motivated by the study of periodicity in algebraic K-theory, J.-L. Loday introduced the notion of a dendriform algebra [23]. It has attracted quite much interest because of its connections with various fields in mathematics and physics (see [20] and the references therein).

There is a remarkable fact that a Rota-Baxter operator (of weight zero), which first arose in probability theory [9] and later became popular in combinatorics and other areas [21, 26, 27], on an associative algebra naturally gives a dendriform algebra structure on the underlying vector space of the associative algebra [1, 2, 14]. Such unexpected relationship between dendriform algebras in the field of operads and algebraic topology and Rota-Baxter operators in the field of combinatorics and probability theory was extended to several other structures. In order to determine the algebraic structures behind a pair of commuting Rota-Baxter operators (on an associative algebra), which appeared, for example, in the space of the linear endomorphisms of an infinitesimal bialgebra, Aguiar and Loday introduced the notion of a quadri-algebra [4], which is a vector space equipped with four binary operations satisfying nine axioms. Moreover, quadri-algebras have deep relationships with combinatorics and the theory of Hopf algebras [4, 16]. A quadri-algebra is also regarded as the underlying algebra structure of a dendriform algebra with a nondegenerate 2-cocycle [7] and is one of the first examples of operad splitting [8, 16, 24].

This paper establishes a bialgebra theory for quadri-algebras, utilizing methods from the theory of Lie bialgebras [12, 13]. Explicitly, in the finite-dimensional case, we consider an analogue of a Manin triple of Lie algebras which is equivalent to a Lie bialgebra, namely, a Manin triple of dendriform algebras associated to a nondegenerate 2-cocycle. We find that it is in fact equivalent to certain bialgebra structure of the underlying quadri-algebra, which leads to the notion of quadri-bialgebra. Furthermore, it is interesting to find that such a structure is also equivalent to another Manin triple on the level of quadri-algebras, that is, a Manin triple of quadri-algebras with a nondegenerate invariant bilinear form. Quadri-bialgebras have certain similar properties of Lie bialgebras. For example, there are the so-called coboundary quadri-bialgebras which lead to a construction from an analogue of the classical Yang-Baxter equation and there also exists a “Drinfeld double” construction for a finite-dimensional quadri-bialgebra. We would like to point out that the finite-dimensional restriction is imposed to motivated the definition of a quadri-bialgebra in its equivalence to Manin triples and matched pairs. The notion of a quadri-bialgebra itself, once established (in Definition 5.2), can be applied with no dimensional restrictions.

Moreover, we find that quadri-bialgebras fit into the framework of constructions of certain linear operators, such as Rota-Baxter operators and Nijenhuis operators in combinatorics [9, 14, 15, 26, 27], renormalization of perturbative quantum field theory (pQFT) [11, 17, 18, 19] and quantum physics [10], on the double spaces.

The paper is organized as follows. In Section 2, we recall some notions and basic facts on dendriform algebras and quadri-algebras. In Section 3, we introduce the notion of a Manin triple of dendriform algebras associated to a nondegenerate 2-cocycle and then interpret it in terms of matched pairs of dendriform algebras. In Section 4, we consider the notion of a Manin triple of quadri-algebras associated to a nondegenerate invariant bilinear form and then interpret it in terms of matched pairs of quadri-algebras. We also show the equivalence between the two Manin triples introduced in the two sections. In Section 5, we define a quadri-bialgebra as a bialgebra structure equivalent to the aforementioned Manin triples by assuming that there is a quadri-algebra structure on the dual space. In Section 6, we study the coboundary case which leads to a construction from certain algebraic equations, which could be regarded as analogues of the classical Yang-Baxter equation. In Section 7, we recall some results in [7] which reduce the aforementioned algebraic equations to a simple form, namely, the $Q$-equations referring to a set of two equations. We list some properties of the $Q$-equations including the ones given in [7] from another point of view. In Section 8, we construct families of Nijenhuis operators and Rota-Baxter operators on certain double spaces of quadri-algebras, including the Drinfeld $Q$-doubles obtained from quadri-bialgebras.

2. Notations and backgrounds

Throughout this paper, all algebras and vector spaces are finite-dimensional over a fixed base field $\mathbb{F}$. We put together notations and conventions to be used throughout the paper.

(a) Let $V$ be a vector space. Let $\mathfrak{B}:V\otimes V\rightarrow\mathbb{F}$ be a symmetric or skew-symmetric bilinear form on a vector space $V$. A subspace $W$ is called isotropic if $W\subset W^{\perp}$, where

[TABLE]

A subspace $W$ is called Lagrangian if $W=W^{\perp}$.

(b) Let $(A,\diamond)$ be a vector space with a bilinear operation $\diamond:A\otimes A\rightarrow A$. Let $L_{\diamond}(x)$ and $R_{\diamond}(x)$ denote the left and right multiplication operators respectively, that is, $L_{\diamond}(x)y=R_{\diamond}(y)x=x\diamond y$ for all $x,y\in A$. We also simply denote them by $L(x)$ and $R(x)$ respectively if the meaning of $\diamond$ is clear from the context. Moreover, let $L_{\diamond},R_{\diamond}:A\rightarrow\mathfrak{gl}(A)$ be the linear maps defined by $x\mapsto L_{\diamond}(x)$ and $x\mapsto R_{\diamond}(x)$ respectively.

(c) Let $V$ be a vector space and let $r=\sum_{i}{a_{i}\otimes b_{i}}\in V\otimes V$. Set

[TABLE]

where $1$ is a symbol playing a similar role of unit. If in addition, there exists a bilinear operation $\diamond:V\otimes V\rightarrow V$ on $V$, then the operation between two $r$s is in an obvious way. For example,

[TABLE]

Note that Eq. (3) is independent of the existence of the unit.

(d) Let $V$ be a vector space. Let $\tau:V\otimes V\to V\otimes V$ be the flip map defined as

[TABLE]

(e) Let $V$ be a vector space. Any $r\in V\otimes V$ is identified with the linear map $T_{r}:V^{*}\rightarrow V$ defined by

[TABLE]

where $\langle,\rangle$ is the canonical paring between $V$ and $V^{*}$. $r\in V\otimes V$ is called nondegenerate if the above induced linear map $T_{r}$ is invertible.

(f) Let $V_{1},V_{2}$ be two vector spaces and let $T:V_{1}\rightarrow V_{2}$ be a linear map. Denote the dual (linear) map by $T^{*}:V_{2}^{*}\rightarrow V_{1}^{*}$ defined by

[TABLE]

(g) Let $A$ be a vector space with a set of bilinear operations and let $V$ be a vector space. For any linear map $\rho:A\rightarrow\mathfrak{gl}(V)$, define a linear map $\rho^{*}:A\rightarrow\mathfrak{gl}(V^{*})$ by

[TABLE]

Note that in this case, $\rho^{*}$ is different from the one given by Eq. (6) which regards $\mathfrak{gl}(V)$ as a vector space also.

(h) Let $V$ be a vector space, we sometimes use 1 to denote the identity transformation on $V$.

Here are the notions that motivated our study.

Definition 2.1**.**

[23] A dendriform algebra $(A,\prec,\succ)$ is a vector space $A$ together with two bilinear operations $\prec,\succ:A\otimes A\rightarrow A$ such that (for all $x,y,z\in A$)

[TABLE]

where $x\star y=x\prec y+x\succ y$. Moreover, a homomorphism between two dendriform algebras is defined to be a linear map (between the two underlying vector spaces) which preserves the operations respectively. **

Proposition 2.2**.**

[23]* Let $(A,\prec,\succ)$ be a dendriform algebra. Then the bilinear operation given by*

[TABLE]

defines an associative algebra, which is called the associated associative algebra and denoted by $(As(A),\star)$.

Definition 2.3**.**

[3, 6] Let $(A,\prec,\succ)$ be a dendriform algebra and let $V$ be a vector space. Let $l_{\prec},l_{\succ},r_{\prec},r_{\succ}:A\rightarrow\mathfrak{gl}(V)$ be four linear maps. Then $V$ or $(V,l_{\prec},r_{\prec},l_{\succ},r_{\succ})$ is called a bimodule of $A$ if the following equations hold (for all $x,y\in A$):**

[TABLE]

[TABLE]

[TABLE]

Proposition 2.4**.**

[6]* Let $(V,l_{\prec},r_{\prec},l_{\succ},r_{\succ})$ be a bimodule of a dendriform algebra $(A,\prec$, $\succ)$. Then $(V^{*},-r_{\succ}^{*},l_{\succ}^{*}+l_{\prec}^{*},r_{\succ}^{*}+r_{\prec}^{*},-l_{\prec}^{*})$ is a bimodule of $(A,\prec,\succ)$.*

Definition 2.5**.**

[6, 22] Let $(V,l_{\prec},r_{\prec},l_{\succ},r_{\succ})$ be a bimodule of a dendriform algebra $(A,\prec$, $\succ)$. A linear map $T:V\rightarrow A$ is called an $\mathcal{O}$-operator associated to $(V,l_{\prec},r_{\prec},l_{\succ},r_{\succ})$ if $T$ satisfies

[TABLE]

Definition 2.6**.**

[4]** A quadri-algebra $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ is a vector space $A$ together with four bilinear operations $\nwarrow,\nearrow,\swarrow$ and $\searrow:A\otimes A\rightarrow A$ satisfying the axioms below (for all $x,y,z\in A$)

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

A homomorphism between two quadri-algebras is defined as a linear map (between the two underlying vector spaces) which preserves the operations respectively.

Proposition 2.7**.**

[4]* Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra.*

1. (a)

The bilinear operation given by Eq. (17) defines a dendriform algebra $A_{h}:=(A,\prec,\succ)$, which is called the associated horizontal dendriform algebra. 2. (b)

The bilinear operation given by Eq. (18) defines a dendriform algebra $A_{v}:=(A,\wedge,\vee)$, which is called the associated vertical dendriform algebra. 3. (c)

The bilinear operation given by Eq. (19) defines an associative algebra $(A,\star)$, which is called the associated associative algebra, which is the associated associative algebra of both the dendriform algebras $(A,\prec,\succ)$ and $(A,\wedge,\vee)$.

Proposition 2.8**.**

[7]* Let $A$ be a vector space with four bilinear operations denoted by $\nwarrow,\nearrow$, $\swarrow$ and $\searrow:A\otimes A\rightarrow A$. Then the following conditions are equivalent:*

1. (a)

$(A,\nwarrow,\nearrow,\swarrow,\searrow)$* is a quadri-algebra;* 2. (b)

$(A,\prec,\succ)$* defined by Eq. (17) is a dendriform algebra and $(A,L_{\swarrow},R_{\nwarrow},L_{\searrow},R_{\nearrow})$ is a bimodule.* 3. (c)

$(A,\wedge,\vee)$* defined by Eq.  (18) is a dendriform algebra and $(A,L_{\nearrow},R_{\nwarrow},L_{\searrow},R_{\swarrow})$ is a bimodule.*

Corollary 2.9**.**

[7]* Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra. Then $(A^{*},-R_{\nearrow}^{*},L_{\vee}^{*},R_{\wedge}^{*}$, $-L_{\swarrow}^{*})$ is a bimodule of the associated horizontal dendriform algebra $(A,\prec,\succ)$ and $(A^{*},-R_{\swarrow}^{*}$, $L_{\succ}^{*}$, $R_{\prec}^{*}$, $-L_{\nearrow}^{*})$ is a bimodule of the associated vertical dendriform algebra $(A,\wedge,\vee)$.*

In the following, we will focus on the study of the associated vertical dendriform algebras of quadri-algebras in this paper. The corresponding study on the associated horizontal dendriform algebras is completely similar.

Definition 2.10**.**

[6] Let $(A,\wedge,\vee)$ be a dendriform algebra and let $(As(A),\star)$ be the associated associative algebra. Suppose that $\mathfrak{B}:A\otimes A\rightarrow\mathbb{F}$ is a symmetric bilinear form. $\mathfrak{B}$ is called a $2$-cocycle on $A$ if $\mathfrak{B}$ satisfies

[TABLE]

Proposition 2.11**.**

[7]* Let $(A,\wedge,\vee)$ be a dendriform algebra equipped with a nondegenerate $2$-cocycle $\mathfrak{B}$. Define four bilinear operations $\nwarrow,\nearrow,\swarrow,\searrow:A\otimes A\rightarrow A$ by*

[TABLE]

[TABLE]

Then $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ is a quadri-algebra such that $(A,\wedge,\vee)$ is the associated vertical dendriform algebra.

3. Manin triples of

dendriform algebras associated to a nondegenerate 2-cocycle and matched pairs of dendriform algebras

This section studies Manin triples of dendriform algebras and the closely related matched pairs of dendriform algebras.

Definition 3.1**.**

A Manin triple of dendriform algebras associated to a nondegenerate 2-cocycle is a triple of dendriform algebras $(A,A^{+},A^{-})$ together with a nondegenerate $2$-cocycle $\mathfrak{B}$ on $A$, such that:

1. (a)

$A^{+}$* and $A^{-}$ are subalgebras of $A$;* 2. (b)

$A=A^{+}\oplus A^{-}$* as vector spaces;* 3. (c)

$A^{+}$* and $A^{-}$ are isotropic with respect to $\mathfrak{B}$.*

It is denoted by $(A,A^{+},A^{-},\mathfrak{B})$. A homomorphism between two Manin triples $(A,A^{+},A^{-},\mathfrak{B}_{A})$ and $(B,B^{+},B^{-},\mathfrak{B}_{B})$ of dendriform algebras associated to a nondegenerate 2-cocycle is a homomorphism of dendriform algebras $\varphi:A\rightarrow B$ such that

[TABLE]

Definition 3.2**.**

Let $(A,\wedge,\vee)$ be a dendriform algebra. Suppose that there is a dendriform algebra structure on the dual space $A^{*}$. If there is a dendriform algebra structure on the direct sum of the underlying vector spaces of $A$ and $A^{*}$ such that $A$ and $A^{*}$ are subalgebras and the natural symmetric bilinear form on $A\oplus A^{*}$ given by

[TABLE]

is a $2$-cocycle, then $(A\oplus A^{*},A,A^{*},\mathfrak{B}_{S})$ is called a standard Manin triple of dendriform algebras associated to $\mathfrak{B}_{S}$.

Obviously, a standard Manin triple of dendriform algebras is a Manin triple of dendriform algebras. Conversely, we have

Proposition 3.3**.**

Every Manin triple of dendriform algebras associated to a nondegenerate 2-cocycle is isomorphic to a standard one.

Proof.

Since in this case $A^{-}$ and $(A^{+})^{*}$ are identified by the nondegenerate $2$-cocycle, the dendriform algebra structure on $A^{-}$ is transferred to $(A^{+})^{*}$. Hence the dendriform algebra structure on $A^{+}\oplus A^{-}$ is transferred to $A^{+}\oplus(A^{+})^{*}$. Then the conclusion follows. ∎

Proposition 3.4**.**

[6]* Let $(A,\wedge_{A},\vee_{A})$ and $(B,\wedge_{B},\vee_{B})$ be two dendriform algebras. Suppose that there are linear maps $l_{\wedge_{A}},r_{\wedge_{A}},l_{\vee_{A}},r_{\vee_{A}}:A\rightarrow\mathfrak{gl}(B)$ and $l_{\wedge_{B}},r_{\wedge_{B}},l_{\vee_{B}},r_{\vee_{B}}:B\rightarrow\mathfrak{gl}(A)$ such that $(l_{\wedge_{A}},r_{\wedge_{A}},l_{\vee_{A}},r_{\vee_{A}})$ is a bimodule of $A$ and $(l_{\wedge_{B}},r_{\wedge_{B}},l_{\vee_{B}},r_{\vee_{B}})$ is a bimodule of $B$, and they satisfy the following conditions:*

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for all $x,y\in A,a,b\in B$. Then there is a dendriform algebra structure on the vector space $A\oplus B$ given by

[TABLE]

[TABLE]

for all $x,y\in A,a,b\in B.$ This dendriform algebra, given by the data $(A,B$, $l_{\wedge_{A}},r_{\wedge_{A}}$, $l_{\vee_{A}},r_{\vee_{A}},l_{\wedge_{B}}$, $r_{\wedge_{B}}$, $l_{\vee_{B}}$, $r_{\vee_{B}})$ which is called a matched pair of dendriform algebras, is denoted by $A\bowtie_{l_{\wedge_{A}},r_{\wedge_{A}},l_{\vee_{A}},r_{\vee_{A}}}^{l_{\wedge_{B}},r_{\wedge_{B}},l_{\vee_{B}},r_{\vee_{B}}}B$ or simply $A\bowtie B$. On the other hand, every dendriform algebra which is the direct sum of the underlying vector spaces of two dendriform subalgebras can be obtained from the above way.

Proposition 3.5**.**

Let $(A,\nwarrow_{A},\nearrow_{A},\swarrow_{A},\searrow_{A})$ be a quadri-algebra. Suppose that there is a quadri-algebra structure $\nwarrow_{A^{*}},\nearrow_{A^{*}},\swarrow_{A^{*}},\searrow_{A^{*}}$ on the dual space $A^{*}$. Let $A_{v}$ and $(A^{*})_{v}$ be the associated vertical dendriform algebras respectively. Then there exists a dendriform algebra structure on the vector space $A\oplus A^{*}$ such that $A_{v}$ and $(A^{*})_{v}$ are isotropic subalgebras associated to the $2$-cocycle $\mathfrak{B}_{S}$ given by Eq. (24), that is, $(A_{v}\oplus(A^{*})_{v},A_{v},(A^{*})_{v},\mathfrak{B}_{S})$ is a standard Manin triple of dendriform algebras associated to the 2-cocycle $\mathfrak{B}_{S}$, if and only if $(A_{v},(A^{*})_{v},-R_{\swarrow_{A}}^{*},L_{\succ_{A}}^{*},R_{\prec_{A}}^{*},-L_{\nearrow_{A}}^{*},-R_{\swarrow_{A^{*}}}^{*}$, $L_{\succ_{A^{*}}}^{*}$, $R_{\prec_{A^{*}}}^{*}$, $-L_{\nearrow_{A^{*}}}^{*})$ is a matched pair of dendriform algebras.

Proof.

If $(A_{v},(A^{*})_{v},-R_{\swarrow_{A}}^{*},L_{\succ_{A}}^{*},R_{\prec_{A}}^{*},-L_{\nearrow_{A}}^{*},-R_{\swarrow_{A^{*}}}^{*},L_{\succ_{A^{*}}}^{*},R_{\prec_{A^{*}}}^{*},-L_{\nearrow_{A^{*}}}^{*})$ is a matched pair of dendriform algebras, then it is straightforward to check that the bilinear form $\mathfrak{B}_{S}$ given by Eq. (24) on $A_{v}\bowtie_{-R_{\swarrow_{A}}^{*},L_{\succ_{A}}^{*},R_{\prec_{A}}^{*},-L_{\nearrow_{A}}^{*}}^{-R_{\swarrow_{A^{*}}}^{*},L_{\succ_{A^{*}}}^{*},R_{\prec_{A^{*}}}^{*},-L_{\nearrow_{A^{*}}}^{*}}(A^{*})_{v}$ is a 2-cocycle. So $(A_{v}\oplus(A_{v})^{*},A_{v},(A^{*})_{v},\mathfrak{B}_{S})$ is a standard Manin triple of dendriform algebras associated to $\mathfrak{B}_{S}$.

Conversely, if $(A_{v}\oplus(A_{v})^{*},A_{v},(A_{v})^{*},\mathfrak{B}_{S})$ is a standard Manin triple of dendriform algebras associated to the 2-cocycle $\mathfrak{B}_{S}$, then for all $x,y\in A,a^{*},b^{*}\in A^{*}$, we have

[TABLE]

[TABLE]

[TABLE]

So $x\wedge a^{*}=-R_{\swarrow_{A}}^{*}(x)a^{*}+L_{\succ_{A^{*}}}^{*}(a^{*})x$. Similarly, we show that

[TABLE]

for all $x\in A,a^{*}\in A^{*}$. Hence $(A_{v},(A^{*})_{v},-R_{\swarrow_{A}}^{*},L_{\succ_{A}}^{*},R_{\prec_{A}}^{*},-L_{\nearrow_{A}}^{*},-R_{\swarrow_{A^{*}}}^{*},L_{\succ_{A^{*}}}^{*},R_{\prec_{A^{*}}}^{*},-L_{\nearrow_{A^{*}}}^{*})$ is a matched pair of dendriform algebras.∎

4. Bimodules and matched pairs of quadri-algebras

In this section we introduce the notion of a Manin triple of quadri-algebras associated to a nondegenerate invariant bilinear form and establish its equivalence with a matched pair of quadri-algebras on the one hand and with a Manin triple of dendriform algebras associated to a nondegenerate 2-cocycle on the other hand.

Definition 4.1**.**

[7] Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $V$ be a vector space. Let $l_{\circ},r_{\circ}:A\rightarrow\mathfrak{gl}(V)$ be eight linear maps, where $\circ\in\{\nwarrow,\nearrow,\swarrow,\searrow\}$. Then $V$ or $(V,l_{\nwarrow},r_{\nwarrow},l_{\nearrow},r_{\nearrow}$, $l_{\swarrow}$, $r_{\swarrow}$, $l_{\searrow},r_{\searrow})$) is called a bimodule of $A$ if for all $x,y\in A$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In fact, $(V,l_{\nwarrow},r_{\nwarrow},l_{\nearrow},r_{\nearrow},l_{\swarrow},r_{\swarrow},l_{\searrow},r_{\searrow})$ is a bimodule of a quadri-algebra $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ if and only if the direct sum $A\oplus V$ of the underlying vector spaces of $A$ and $V$ is turned into a quadri-algebra (the semidirect product) by defining multiplications in $A\oplus V$ by (we still denote them by $\nwarrow,\nearrow,\swarrow,\searrow$):

[TABLE]

We denote it by $A\ltimes_{l_{\nwarrow},r_{\nwarrow},l_{\nearrow},r_{\nearrow},l_{\swarrow},r_{\swarrow},l_{\searrow},r_{\searrow}}V$ or simply $A\ltimes V$.

Lemma 4.2**.**

[7]* Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra. If $(V,l_{\nwarrow},r_{\nwarrow},l_{\nearrow},r_{\nearrow},l_{\swarrow},r_{\swarrow},l_{\searrow}$, $r_{\searrow})$ is a bimodule of $A$, then $(V^{*},r_{\searrow}^{*},l_{\star}^{*},-r_{\vee}^{*},-l_{\prec}^{*},-r_{\succ}^{*},-l_{\wedge}^{*},r_{\star}^{*},l_{\nwarrow}^{*})$ is a bimodule of $A$.*

Definition 4.3**.**

Let $(A,\nwarrow_{A},\nearrow_{A},\swarrow_{A},\searrow_{A})$ and $(B,\nwarrow_{B},\nearrow_{B},\swarrow_{B},\searrow_{B})$ be two quadri-algebras. Suppose that there exist linear maps $l_{\nwarrow_{A}},r_{\nwarrow_{A}},l_{\nearrow_{A}},r_{\nearrow_{A}},l_{\swarrow_{A}},r_{\swarrow_{A}},l_{\searrow_{A}},r_{\searrow_{A}}:A\rightarrow\mathfrak{gl}(B)$ and $l_{\nwarrow_{B}},r_{\nwarrow_{B}}$, $l_{\nearrow_{B}},r_{\nearrow_{B}}$, $l_{\swarrow_{B}},r_{\swarrow_{B}}$, $l_{\searrow_{B}},r_{\searrow_{B}}:B\rightarrow\mathfrak{gl}(A)$ such that for all $x,y\in A,a,b\in B$,

[TABLE]

define a quadri-algebra structure on $A\oplus B$. Then $(A,B,l_{\nwarrow_{A}},r_{\nwarrow_{A}}$, $l_{\nearrow_{A}},r_{\nearrow_{A}}$, $l_{\swarrow_{A}}$, $r_{\swarrow_{A}},l_{\searrow_{A}}$, $r_{\searrow_{A}},l_{\nwarrow_{B}}$, $r_{\nwarrow_{B}},l_{\nearrow_{B}}$, $r_{\nearrow_{B}},l_{\swarrow_{B}},r_{\swarrow_{B}},l_{\searrow_{B}},r_{\searrow_{B}})$ is called a matched pair of quadri-algebras and the quadri-algebra structure on $A\oplus B$ is denoted by $A\bowtie_{l_{\nwarrow_{A}},r_{\nwarrow_{A}},l_{\nearrow_{A}},r_{\nearrow_{A}},l_{\swarrow_{A}},r_{\swarrow_{A}},l_{\searrow_{A}},r_{\searrow_{A}}}^{l_{\nwarrow_{B}},r_{\nwarrow_{B}},l_{\nearrow_{B}},r_{\nearrow_{B}},l_{\swarrow_{B}},r_{\swarrow_{B}},l_{\searrow_{B}},r_{\searrow_{B}}}B$ or simply $A\bowtie B$.**

Remark 4.4**.**

Similar to Proposition 3.4, one can also write down the necessary and sufficient conditions that the above linear maps make $A\oplus B$ into a quadri-algebra. We omit the details since we will not use such a conclusion in this paper. In particular, in this case, $A$ and $B$ are bimodules of $B$ and $A$ respectively.**

Proposition 4.5**.**

Let $(A,B,l_{\nwarrow_{A}},r_{\nwarrow_{A}},l_{\nearrow_{A}},r_{\nearrow_{A}},l_{\swarrow_{A}},r_{\swarrow_{A}},l_{\searrow_{A}}$, $r_{\searrow_{A}},l_{\nwarrow_{B}},r_{\nwarrow_{B}}$, $l_{\nearrow_{B}}$, $r_{\nearrow_{B}}$, $l_{\swarrow_{B}}$, $r_{\swarrow_{B}}$, $l_{\searrow_{B}}$, $r_{\searrow_{B}})$ be a matched pair of quadri-algebras. Then $((A)_{v},(B)_{v},l_{\nwarrow_{A}}+l_{\nearrow_{A}},r_{\nwarrow_{A}}+r_{\nearrow_{A}},l_{\swarrow_{A}}$ $+l_{\searrow_{A}},r_{\swarrow_{A}}+r_{\searrow_{A}},l_{\nwarrow_{B}}+l_{\nearrow_{B}},r_{\nwarrow_{B}}+r_{\nearrow_{B}},l_{\swarrow_{B}}+l_{\searrow_{B}},r_{\swarrow_{B}}+r_{\searrow_{B}})$ is a matched pair of dendriform algebras.

Proof.

It is straightforward.∎

Proposition 4.6**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra. Suppose that there is a quadri-algebra structure $(\nwarrow_{*},\nearrow_{*},\swarrow_{*},\searrow_{*})$ on the dual space $A^{*}$. Then $(A_{v},(A^{*})_{v},-R_{\swarrow}^{*},L_{\succ}^{*},R_{\prec}^{*},-L_{\nearrow}^{*}$, $-R_{\swarrow_{*}}^{*},L_{\succ_{*}}^{*},R_{\prec_{*}}^{*},-L_{\nearrow_{*}}^{*})$ is a matched pair of dendriform algebras if and only if $(A,A^{*},R_{\searrow}^{*},L_{\star}^{*}$, $-R_{\vee}^{*},-L_{\prec}^{*},-R_{\succ}^{*},-L_{\wedge}^{*},R_{\star}^{*},L_{\nwarrow}^{*},R_{\searrow_{*}}^{*},L_{\star_{*}}^{*},-R_{\vee_{*}}^{*},-L_{\prec_{*}}^{*},-R_{\succ_{*}}^{*},-L_{\wedge_{*}}^{*},R_{\star_{*}}^{*},L_{\nwarrow_{*}}^{*})$ is a matched pair of quadri-algebras.

Proof.

By Proposition 4.5, we only need to prove the “only if” part of the conclusion. In fact, if $(A_{v},(A^{*})_{v},-R_{\swarrow}^{*},L_{\succ}^{*},R_{\prec}^{*},-L_{\nearrow}^{*},-R_{\swarrow_{*}}^{*},L_{\succ_{*}}^{*},R_{\prec_{*}}^{*},-L_{\nearrow_{*}}^{*})$ is a matched pair of dendriform algebras, then by Proposition 3.5, we find that $(A_{v}\oplus(A^{*})_{v},A_{v},(A^{*})_{v},\mathfrak{B}_{S})$ is a standard Manin triple of dendriform algebras associated to the 2-cocycle $\mathfrak{B}_{S}$. Hence there exists a quadri-algebra structure on $A_{v}\bowtie_{-R_{\swarrow}^{*},L_{\succ}^{*},R_{\prec}^{*},-L_{\nearrow}^{*}}^{-R_{\swarrow_{*}}^{*},L_{\succ_{*}}^{*},R_{\prec_{*}}^{*},-L_{\nearrow_{*}}^{*}}(A^{*})_{v}$ given by Proposition 2.11 which we denote by $(\nwarrow_{\bullet},\nearrow_{\bullet},\swarrow_{\bullet},\searrow_{\bullet}$). Moreover, $A$ and $A^{*}$ are the quadri-subalgebras. For all $x,y\in A,a^{*},b^{*}\in A^{*}$, we have

[TABLE]

So $x\nwarrow_{\bullet}a^{*}=R_{\searrow}^{*}(x)a^{*}+L_{\star_{*}}^{*}(a^{*})x$. Similarly, we show that

[TABLE]

and $a^{*}\searrow_{\bullet}x=R_{\star_{*}}^{*}(a^{*})x+L_{\nwarrow}^{*}(x)a^{*}.$ Therefore $(A,A^{*},R_{\searrow}^{*},L_{\star}^{*},-R_{\vee}^{*},-L_{\prec}^{*},-R_{\succ}^{*},-L_{\wedge}^{*},R_{\star}^{*},L_{\nwarrow}^{*}$, $R_{\searrow_{*}}^{*}$, $L_{\star_{*}}^{*},-R_{\vee_{*}}^{*},-L_{\prec_{*}}^{*},-R_{\succ_{*}}^{*},-L_{\wedge_{*}}^{*},R_{\star_{*}}^{*},L_{\nwarrow_{*}}^{*})$ is a matched pair of quadri-algebras.∎

In fact, the equivalence in Proposition 4.6 between two matched pairs can be interpreted in terms of their corresponding Manin triples as follows, in Corollary 4.10.

Definition 4.7**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $\mathfrak{B}$ be a symmetric bilinear form. If $\mathfrak{B}$ satisfies Eqs. (21)-(22), then $\mathfrak{B}$ is called invariant on $A$.**

Proposition 4.8**.**

[7]* Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $\mathfrak{B}$ be a symmetric bilinear form. If $\mathfrak{B}$ is invariant on $A$, then $\mathfrak{B}$ is a 2-cocycle of the associated vertical dendriform algebra $(A_{v},\wedge,\vee)$. Conversely, if $\mathfrak{B}$ is a nondegenerate 2-cocycle of a dendriform algebra, then $\mathfrak{B}$ is invariant on the quadri-algebra given by Eqs. (21)-(22).*

Definition 4.9**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra. Suppose that there is a quadri-algebra structure on $A^{*}$. If there is a quadri-algebra structure on the direct sum of the underlying vector space of $A$ and $A^{*}$ such that $A$ and $A^{*}$ are quadri-subalgebras and the bilinear form $\mathfrak{B}_{S}$ on $A\oplus A^{*}$ given by Eq. (24) is invariant, then $(A\bowtie A^{*},A,A^{*},\mathfrak{B}_{S})$ is called a (standard) Manin triple of quadri-algebras associated to the nondegenerate invariant bilinear form $\mathfrak{B}_{S}$.**

By Proposition 4.8, the following conclusion is obvious:

Corollary 4.10**.**

The quadruple $(A\bowtie A^{*},A,A^{*},\mathfrak{B}_{S})$ is a standard Manin triple of quadri-algebras associated to the nondegenerate invariant bilinear form $\mathfrak{B}_{S}$ if and only if $(A_{v}\bowtie(A_{v})^{*},A_{v},A_{v}^{*},{\mathfrak{B}}_{S})$ is a standard Manin triple of dendriform algebras associated to the nondegenerate 2-cocycle $\mathfrak{B}_{S}$.

Remark 4.11**.**

By Proposition 4.6, it is obvious that a standard Manin triple of quadri-algebras associated to the nondegenerate invariant bilinear form $\mathfrak{B}_{S}$ can be interpreted in terms of a matched pair of quadri-algebras (cf. Theorem 5.3).

5. Quadri-bialgebras

With the preparations in the previous sections, we now introduce the notion of a quadri-bialgebra and establish its equivalence with a Manin triple of dendriform algebras and of quadri-algebras, and a matched pair of dendriform algebras and of quadri-algebras.

Proposition 5.1**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-algebra equipped with four cooperations $\alpha,\beta,\tilde{\alpha},\tilde{\beta}:A\rightarrow A\otimes A$. Suppose that $\alpha^{*},\beta^{*},\tilde{\alpha}^{*},\tilde{\beta}^{*}:A^{*}\otimes A^{*}\subset(A\otimes A)^{*}\rightarrow A^{*}$ induce a quadri-algebra structure on $A^{*}$. Set $\nwarrow_{*}:=\alpha^{*},\nearrow_{*}:=\beta^{*},\swarrow_{*}:=\tilde{\alpha}^{*},\searrow_{*}:=\tilde{\beta}^{*}$. Then $(A_{v},(A^{*})_{v},-R_{\swarrow}^{*},L_{\succ}^{*},R_{\prec}^{*},-L_{\nearrow}^{*},-R_{\swarrow_{*}}^{*},L_{\succ_{*}}^{*},R_{\prec_{*}}^{*},-L_{\nearrow_{*}}^{*})$ is a matched pair of dendriform algebras if and only if the following equations hold:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for all $x,y\in A$.

Proof.

From Proposition 3.5, we only need to prove that each of Eqs. (56)-(73) is equivalent to the corresponding one in Eqs. (25)-(42) in taking $A=A_{v},B=(A^{*})_{v}$ and

[TABLE]

As an example, we give an explicit proof that

[TABLE]

holds if and only if Eq. (58) holds. The other equivalences are similar. In fact, let the left hand side of Eq. (74) act on an arbitrary element $b^{*}\in A^{*}$. Then we have

[TABLE]

So the conclusion follows. ∎

Definition 5.2**.**

1. (a)

Let $(A,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a vector space with four comultiplications $\alpha,\beta,\tilde{\alpha},\tilde{\beta}:A\rightarrow A\otimes A$. Let the dual operations be $\alpha^{*},\beta^{*},\tilde{\alpha}^{*},\tilde{\beta}^{*}:A^{*}\otimes A^{*}\subset(A\otimes A)^{*}\rightarrow A^{*}$. If $(A^{*},\alpha^{*},\beta^{*},\tilde{\alpha}^{*},\tilde{\beta}^{*})$ becomes a quadri-algebra, then we call $(A,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ a quadri-coalgebra. A homomorphism between two quadri-coalgebras is defined as a linear map (between the two underlying vector spaces) which preserves the corresponding cooperations. 2. (b)

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-algebra equipped with four cooperations $\alpha,\beta,\tilde{\alpha}$, $\tilde{\beta}:A\rightarrow A\otimes A$ such that $(A,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ is a quadri-coalgebra and $\alpha,\beta,\tilde{\alpha}$ and $\tilde{\beta}$ satisfy Eqs. (56)-(73). Then $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ is called a quadri-bialgebra. A homomorphism between two quadri-bialgebras is defined as a linear map (between the two underlying vector spaces) which is a homomorphisms of both quadri-algebras and quadri-coalgebras.

Combining Proposition 5.1 and the discussion in the previous sections, we have the following conclusion:

Theorem 5.3**.**

Let $(A,\nwarrow_{A},\nearrow_{A},\swarrow_{A},\searrow_{A},\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-algebra with four comultiplications $\alpha,\beta,\tilde{\alpha},\tilde{\beta}:A\rightarrow A\otimes A$, such that $(A,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ is a quadri-coalgebra. Then, with the notations $\nwarrow_{B}:=\alpha^{*},\nearrow_{B}:=\beta^{*},\swarrow_{B}:=\tilde{\alpha}^{*},\searrow_{B}:=\tilde{\beta}^{*}$, and with $\mathfrak{B}_{S}$ given by Eq. (24), the following conditions are equivalent

1. (a)

$(A_{v}\bowtie(A^{*})_{v},A_{v},(A^{*})_{v},\mathfrak{B}_{S})$* is a standard Manin triple of dendriform algebras associated to the nondegenerate 2-cocycle $\mathfrak{B}_{S}$;* 2. (b)

$(A\bowtie A^{*},A,A^{*},\mathfrak{B}_{S})$* is a standard Manin triple of quadri-algebras associated to the nondegenerate invariant bilinear form $\mathfrak{B}_{S}$.* 3. (c)

$(A_{v},(A^{*})_{v},-R_{\swarrow_{A}}^{*},L_{\succ_{A}}^{*},R_{\prec_{A}}^{*},-L_{\nearrow_{A}}^{*},-R_{\swarrow_{A^{*}}}^{*},L_{\succ_{A^{*}}}^{*},R_{\prec_{A^{*}}}^{*},-L_{\nearrow_{A^{*}}}^{*})$* is a matched pair of dendriform algebras;* 4. (d)

$(A,A^{*},R_{\searrow_{A}}^{*},L_{\star_{A}}^{*},-R_{\vee_{A}}^{*},-L_{\prec_{A}}^{*},-R_{\succ_{A}}^{*},-L_{\wedge_{A}}^{*},R_{\star_{A}}^{*},L_{\nwarrow_{A}}^{*},R_{\searrow_{A^{*}}}^{*},L_{\star_{A^{*}}}^{*},-R_{\vee_{A^{*}}}^{*},-L_{\prec_{A^{*}}}^{*}$, $-R_{\succ_{A^{*}}}^{*}$, $-L_{\wedge_{A^{*}}}^{*}$, $R_{\star_{A^{*}}}^{*},L_{\nwarrow_{A^{*}}}^{*})$ is a matched pair of quadri-algebras; 5. (e)

$(A,\nwarrow_{A},\nearrow_{A},\swarrow_{A},\searrow_{A},\alpha,\beta,\tilde{\alpha},\tilde{\beta})$* is a quadri-bialgebra.*

By a standard argument (cf. [6], Proposition 2.2.10), we get the following result:

Proposition 5.4**.**

Two Manin triples of dendriform algebras associated to a nondegenerate 2-cocycle are isomorphic if and only if their corresponding quadri-bialgebras are isomorphic.

Remark 5.5**.**

It is obvious that for a quadri-bialgebra $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$, the dual $(A^{*}$, $\alpha^{*}$, $\beta^{*}$, $\tilde{\alpha}^{*},\tilde{\beta}^{*},\gamma,\delta,\tilde{\gamma}$, $\tilde{\delta})$ is also a quadri-bialgebra, where $\gamma^{*}=\nwarrow,\delta^{*}=\nearrow,\tilde{\gamma}^{*}=\swarrow$, $\tilde{\delta}^{*}=\searrow$. **

6. Coboundary quadri-bialgebras

We now give a more detailed study of a class of quadri-bialgebras with certain coboundary conditions which can be obtained as solutions of a system of equations.

Definition 6.1**.**

A quadri-bialgebra $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ is called coboundary if $\alpha,\beta,\tilde{\alpha}$ and $\tilde{\beta}$ satisfy

[TABLE]

where $r_{\nwarrow},r_{\nearrow},r_{\swarrow},r_{\searrow}\in A\otimes A$ and $x\in A$.**

Proposition 6.2**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-algebra with four comultiplications $\alpha,\beta,\tilde{\alpha},\tilde{\beta}$ defined by Eqs. (75)-(78) respectively. If $r_{\nwarrow}=r_{\nearrow}=r_{\swarrow}=r_{\searrow}=r\in A\otimes A$ and $r$ is skew-symmetric, then $\alpha,\beta,\tilde{\alpha}$ and $\tilde{\beta}$ satisfy Eqs. (56)-(73).

Proof.

Straightforward. ∎

Lemma 6.3**.**

Let $A$ be a vector space and $\alpha,\beta,\tilde{\alpha},\tilde{\beta}:A\rightarrow A\otimes A$ be four cooperations. Then $(A,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ is a quadri-coalgebra if and only if the linear maps $R_{i}:A\rightarrow A\otimes A\otimes A$ $(i\in\{1,...,9\})$ defined by the following equations are zero:

[TABLE]

for all $x\in A$.

Proof.

It follows from the definition of a quadri-algebra.∎

Definition 6.4**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $r\in A\otimes A$. The following equations are called the $Q_{i}^{j}$-equations for $i=1,2,3$ and $j=1,2$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proposition 6.5**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $r\in A\otimes A$. Define $\alpha,\beta,\tilde{\alpha}$ and $\tilde{\beta}$ by Eqs.  (75)-(78) respectively, where $r_{\nwarrow}=r_{\nearrow}=r_{\swarrow}=r_{\searrow}=r$. Then $(A,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ becomes a quadri-coalgebra if and only if the following equations hold (for all $x\in A$):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

Let $x\in A$. We only need to prove that $R_{i}(x)=0,1\leq i\leq 9$, if and only if the corresponding equation in Eqs. (94)-(102) holds. Since all the proofs are similar, we only give a detailed proof for that $R_{2}(x)=0$ if and only if Eq. (95) holds. In fact, after rearranging the terms suitably, we divide $R_{2}(x)$ into three parts: $R_{2}(x)=(R1)+(R2)+(R3),$ where

[TABLE]

So the conclusion follows.∎

From the above discussion, we arrive at the following conclusion.

Theorem 6.6**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $r\in A\otimes A$ be skew-symmetric. Then the comultiplications $\alpha,\beta,\tilde{\alpha}$ and $\tilde{\beta}$ defined by Eqs. (75)-(78) with $r_{\nwarrow}=r_{\nearrow}=r_{\swarrow}=r_{\searrow}=r$ respectively make $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ into a quadri-bialgebra if and only if Eqs. (94)-(102) are satisfied.

Moreover, we have the following “Drinfeld double” construction of a quadri-bialgebra [12].

Theorem 6.7**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-bialgebra. Then there exists a canonical quadri-bialgebra structure on $A\oplus A^{*}$ such that the inclusion $i_{1}:A\rightarrow A\oplus A^{*}$ is a homomorphism of quadri-bialgebras, which is from $(A,\nwarrow,\nearrow,\swarrow,\searrow,-\alpha,-\beta,-\tilde{\alpha},-\tilde{\beta})$ to $A\oplus A^{*}$, and the inclusion $i_{2}:A^{*}\rightarrow A\oplus A^{*}$ is also a homomorphism of quadri-bialgebras from the quadri-bialgebra given in Remark 5.5 to $A\oplus A^{*}$.

Proof.

Let $r=\sum_{i}e_{i}\otimes e_{i}^{*}$ correspond to the identity map ${\rm id}:A\rightarrow A$, where $\{e_{i},...,e_{s}\}$ is a basis of $A$ and $\{e_{1}^{*},...e_{s}^{*}\}$ is the dual basis. Suppose the quadri-algebra structure $(\nwarrow_{\bullet},\nearrow_{\bullet},\swarrow_{\bullet},\searrow_{\bullet})$ on $A\oplus A^{*}$ is given by $\mathcal{QD}(A):=A\bowtie_{R_{\searrow}^{*},L_{\star}^{*},-R_{\vee}^{*},-L_{\prec}^{*},-R_{\succ}^{*},-L_{\wedge}^{*},R_{\star}^{*},L_{\nwarrow}^{*}}^{R_{\searrow_{*}}^{*},L_{\star_{*}}^{*},-R_{\vee_{*}}^{*},-L_{\prec_{*}}^{*},-R_{\succ_{*}}^{*},-L_{\wedge_{*}}^{*},R_{\star_{*}}^{*},L_{\nwarrow_{*}}^{*}}A^{*}$, where the subscript $\ast$ is used to denote the quadri-algebra structure on $A^{*}$. Then for all $x,y\in A,a^{*},b^{*}\in A^{*}$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We prove that $r$ satisfies Eqs. (56)-(73). We provide a detailed proof that $r$ satisfies Eq. (68) as an example. The proofs of the other cases follow from the same argument.

for all $\mu,\nu\in\mathcal{QD}(A)$, Eq. (68) is equivalent to

[TABLE]

The proof of the equation is divided into the following four cases: (I). $\mu,\nu\in A$; (II). $\mu,\nu\in A^{*}$; (III). $\mu\in A,\nu\in A^{*}$; (IV). $\mu\in A^{*},\nu\in A$. We proceed to prove the last case as the proofs of the other cases are similar. Let $\mu=e_{i}^{*},\nu=e_{j}$. Then for all $m$ and $n$, the coefficient of $e_{m}^{*}\otimes e_{n}$ is

[TABLE]

Similarly, the coefficients of $e_{m}\otimes e_{n},e_{m}\otimes e_{n}^{*}$ and $e_{m}^{*}\otimes e_{n}^{*}$ are zero.

Furthermore, we have

[TABLE]

So $Q_{1}^{1}=0$. Similarly, $r$ satisfies Eqs. (89)-(93). So the cooperations (for all $x\in A\oplus A^{*}$)

[TABLE]

[TABLE]

induce a quadri-bialgebra structure on $A\oplus A^{*}$.

On the other hand, for $e_{i}\in A$, we have:

[TABLE]

Similarly, we have $\beta_{\mathcal{QD}}(e_{i})=-\beta(e_{i}),\tilde{\alpha}_{\mathcal{QD}}(e_{i})=-\tilde{\alpha}(e_{i})$ and $\tilde{\beta}_{\mathcal{QD}}(e_{i})=-\tilde{\beta}(e_{i})$. So $i_{1}$ is a homomorphism of quadri-bialgebras. Similarly, we prove that $i_{2}$ is also a homomorphism of quadri-bialgebras.∎

Definition 6.8**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-bialgebra. With the quadri-bialgebra structure given in Theorem 6.7, $A\oplus A^{*}$ is called the Drinfeld $Q$-double of $A$ and we denote it by $\mathcal{QD}(A)$. On the other hand, due to the symmetry between $A$ and $A^{*}$ (Remark 5.5), $\tilde{r}:=\sum_{i}e_{i}^{*}\otimes e_{i}$ also induces a (coboundary) quadri-bialgebra structure on $A\oplus A^{*}$, and we denote it by $\tilde{\mathcal{QD}}(A)$.**

Proposition 6.9**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-bialgebra. Suppose that $\alpha,\beta,\tilde{\alpha}$ and $\tilde{\beta}$ are defined by Eqs. (75)-(78) with $r_{\nwarrow}=r_{\nearrow}=r_{\swarrow}=r_{\searrow}=r$ respectively.

1. (a)

If $r$ satisfies Eqs. (88)-(93), then $T_{r}$ is a homomorphism of quadri-bialgebras which is from the quadri-bialgebra given in Remark 5.5 to $(A,\nwarrow,\nearrow$, $\swarrow,\searrow,-\alpha,-\beta$, $-\tilde{\alpha}$, $-\tilde{\beta})$. 2. (b)

If $r$ satisfies Eqs. (88)-(93) and $r$ is skew-symmetric, then

[TABLE]

is a homomorphism of quadri-bialgebras from $\tilde{\mathcal{QD}}(A)$ to $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$.

Proof.

(a). Note that $(1\otimes\alpha)r=r_{12}\nwarrow r_{13},(\alpha\otimes 1)r=-r_{13}\nwarrow r_{23}$. Set $\nwarrow_{*}:=\alpha^{*}$. Then for all $a^{*},b^{*}\in A^{*}$ we have

[TABLE]

Here we use the Sweedler’s notation: $\gamma(a^{*})=a_{(1)}^{*}\otimes a_{(2)}^{*}$. Similarly $T_{r}$ also preserves the other operations and cooperations. So the conclusion follows.

(103). We still denote the products of $\tilde{\mathcal{QD}}(A)$ by $\nwarrow,\nearrow,\swarrow,\searrow$. First we prove that $\tilde{T}_{r}$ is a homomorphism of quadri-algebras, that is, $\tilde{T}_{r}(\mu\diamond\nu)=\tilde{T}_{r}(\mu)\diamond\tilde{T}_{r}(\nu)$ for all $\mu,\nu\in A\oplus A^{*}$ and $\diamond\in\{\nwarrow,\nearrow,\swarrow,\searrow\}$. It is obvious when $\mu,\nu\in A$. Moreover, for all $x\in A,a^{*}\in A^{*}$ we have

[TABLE]

where we use the fact that $r$ is skew-symmetric. Similarly, $\tilde{T}_{r}(a^{*}\nwarrow x)=\tilde{T}_{r}(a^{*})\nwarrow\tilde{T}_{r}(x)$ and, for each $\diamond\in\{\nearrow,\swarrow,\searrow\}$, we have $\tilde{T}(\mu\diamond\nu)=\tilde{T}_{r}(\mu)\diamond\tilde{T}_{r}(\nu)$ for all $\mu\in A,\nu\in A^{*}$ or $\mu\in A^{*},\nu\in A$. On the other hand, by Item (a), for all $a^{*},b^{*}\in A^{*}$, we have

[TABLE]

where $\diamond\in\{\nwarrow,\nearrow,\swarrow,\searrow\}$. Therefore, $\tilde{T}_{r}$ is a homomorphism of quadri-algebras.

Furthermore, let $\{e_{1},...,e_{n}\}$ be a basis of $A$ and let $\{e_{1}^{*},...,e_{n}^{*}\}$ be the dual basis. Define $\tilde{r}:=\sum_{i}e_{i}^{*}\otimes e_{i}$. Then we have

[TABLE]

which implies that $\tilde{T}_{r}$ is a homomorphism of quadri-coalgebras. So the conclusion follows.∎

7. The $Q$-equations

We now show that the equations in Definition 93 boil down to a pair of equations, called the $Q$-equations which have a clear connection with ${\mathcal{O}}$-operators.

By Theorem 6.6, we have the following conclusion.

Proposition 7.1**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and $r\in A\otimes A$ be skew-symmetric. Then the comultiplications $\alpha,\beta,\tilde{\alpha}$ and $\tilde{\beta}$ defined by Eqs. (75)-(78) with $r_{\nwarrow}=r_{\nearrow}=r_{\swarrow}=r_{\searrow}=r$ respectively make $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ into a quadri-bialgebra if $r$ satisfies Eqs. (88)-(93).

As a related notion, we recall

Definition 7.2**.**

[7] Let $(A,\nwarrow,\nearrow$, $\swarrow,\searrow)$ be a quadri-algebra and let $(V,l_{\nwarrow},r_{\nwarrow},l_{\nearrow},r_{\nearrow}$, $l_{\swarrow},r_{\swarrow}$, $l_{\searrow},r_{\searrow})$ be a bimodule. An $\mathcal{O}$-operator of $A$ associated to the bimodule $V$ is a linear map $T$ from $V$ to $A$ such that for all $u,v\in V$,

[TABLE]

In fact, the following relation holds.

Proposition 7.3**.**

[7]* Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $r\in A\otimes A$ be skew-symmetric. Then the following conditions are equivalent:*

1. (a)

$r$* satisfies Eqs. (88)-(89);* 2. (b)

$r$* satisfies Eqs. (90)-(91);* 3. (c)

$r$* satisfies Eqs. (92)-(93);* 4. (d)

$T_{r}$* is an $\mathcal{O}$-operator of $A$ associated to the bimodule $(A^{*},R_{\searrow}^{*},L_{\star}^{*},-R_{\vee}^{*},-L_{\prec}^{*},-R_{\succ}^{*},-L_{\wedge}^{*},R_{\star}^{*}$, $L_{\nwarrow}^{*})$;* 5. (e)

$T_{r}$* is an $\mathcal{O}$-operator of $A_{v}$ associated to the bimodule $(A^{*},-R_{\swarrow}^{*},L_{\succ}^{*},R_{\prec}^{*},-L_{\nearrow}^{*})$.*

Definition 7.4**.**

[7] Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $r\in A\otimes A$. A set of equations (88) and (89) is called the $Q$-equations in $(A,\nwarrow,\nearrow,\swarrow,\searrow)$. **

Remark 7.5**.**

In the sense of Proposition 7.3 (in terms of $\mathcal{O}$-operators), the $Q$-equations in a quadri-algebra can be regarded as an analogue of the classical Yang-Baxter equation in a Lie algebra [5, 22], which led to the introduction of the $Q$-equations in [7]. **

The following conclusion is obvious:

Corollary 7.6**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $r\in A\otimes A$ be skew-symmetric. Then the comultiplications $\alpha,\beta,\tilde{\alpha}$ and $\tilde{\beta}$ defined by Eqs. (75)-(78) with $r_{\nwarrow}=r_{\nearrow}=r_{\swarrow}=r_{\searrow}=r$ respectively make $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ into a quadri-bialgebra if $r$ is a solution of the $Q$-equations.

Proposition 7.7**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $r$ be a skew-symmetric solution of the $Q$-equations. Then the quadri-algebra structure $(\nwarrow_{\bullet},\nearrow_{\bullet},\swarrow_{\bullet},\searrow_{\bullet})$ on the Drinfeld $Q$-double $\mathcal{QD}(A)$ in Theorem 6.7 can be given as follows (for all $x\in A,a^{*},b^{*}\in A^{*}$):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

Let $\{e_{1},...,e_{n}\}$ be a basis of $A$ and let $\{e_{1}^{*},...,e_{n}^{*}\}$ be the dual basis. Suppose that

[TABLE]

and $r=\sum_{i,j}a_{ij}e_{i}\otimes e_{j}$, where $a_{ij}=-a_{ji}$. Then $T_{r}(e_{i}^{*})=\sum_{k}a_{ki}e_{k}$. for all $k,l$, we have

[TABLE]

Hence for all $a^{*},b^{*}\in A^{*}$, we have that $a^{*}\nwarrow_{\bullet}b^{*}=R_{\searrow}^{*}(T_{r}(a^{*}))b^{*}+L_{\star}^{*}(T_{r}(b^{*}))a^{*}$. Similarly $a^{*}\nearrow_{\bullet}b^{*}=-R_{\vee}^{*}(T_{r}(a^{*}))b^{*}-L_{\prec}^{*}(T_{r}(b^{*}))a^{*}$. So Eq. (105) holds. Eq. (106) is proved in a similar way. On the other hand, we have

[TABLE]

Thus, $e_{k}^{*}\nwarrow_{\bullet}e_{l}=R_{\searrow}^{*}(e_{k}^{*})e_{l}+L_{\star}^{*}(e_{l})e_{k}^{*}=-T_{r}(L_{\star}^{*}(e_{l})e_{k}^{*})+T_{r}(e_{k}^{*})\nwarrow e_{l}+L_{\star}^{*}(e_{l})e_{k}^{*}$. Therefore Eq. (107) holds. Eqs. (108)-(114) are verified in the same way. ∎

Proposition 7.8**.**

[7]* Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra and let $r\in A\otimes A$. Suppose that $r$ is skew-symmetric and nondegenerate. Then $r$ is a solution of the $Q$-equations if and only if the inverse of the isomorphism $A^{*}\rightarrow A$ induced by $r$, regarded as a bilinear form $\omega$ on $A$, satisfies*

[TABLE]

[TABLE]

Definition 7.9**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra. A bilinear form $\omega:A\otimes A\rightarrow\mathbb{F}$ is called a $2$-cocycle on $A$ if it satisfies Eqs. (115) and (116). **

It is easy to show that if $\omega$ is a $2$-cocycle on a quadri-algebra $A$, then for all $x,y\in A$, $\mathfrak{B}(x,y):=\omega(x,y)+\omega(y,x)$ is a $2$-cocycle on the associated vertical dendriform algebra $A_{v}$.

Proposition 7.10**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-bialgebra obtained from a skew-symmetric solution of the $Q$-equations. Let $(A,A^{*},R_{\searrow}^{*},L_{\star}^{*},-R_{\vee}^{*},-L_{\prec}^{*}$, $-R_{\succ}^{*}$, $-L_{\wedge}^{*},R_{\star}^{*},L_{\nwarrow}^{*},R_{\searrow_{*}}^{*},L_{\star_{*}}^{*}$, $-R_{\vee_{*}}^{*},-L_{\prec_{*}}^{*},-R_{\succ_{*}}^{*},-L_{\wedge_{*}}^{*},R_{\star_{*}}^{*},L_{\nwarrow_{*}}^{*})$ be the corresponding matched pair of quadri-algebras, where the subscript $\ast$ denotes the quadri-algebra structure on $A^{*}$. Set

[TABLE]

1. (a)

$A\bowtie_{R_{\searrow}^{*},L_{\star}^{*},-R_{\vee}^{*},-L_{\prec}^{*},-R_{\succ}^{*},-L_{\wedge}^{*},R_{\star}^{*},L_{\nwarrow}^{*}}^{R_{\searrow_{*}}^{*},L_{\star_{*}}^{*},-R_{\vee_{*}}^{*},-L_{\prec_{*}}^{*},-R_{\succ_{*}}^{*},-L_{\wedge_{*}}^{*},R_{\star_{*}}^{*},L_{\nwarrow_{*}}^{*}}A^{*}$* is isomorphic to $A\ltimes A^{*}$ as quadri-algebras.* 2. (b)

The skew-symmetric solutions of the $Q$-equations in $A$ are in one-to-one correspondence with linear maps $T_{r}:A^{*}\rightarrow A$ whose graphs are the Lagrangian quadri-subalgebras of $A\ltimes A^{*}$ with respect to the bilinear form $\mathfrak{B}_{S}$ defined by Eq. (24). Here the graph of a linear map $T:A^{*}\rightarrow A$ is defined as $\mathrm{graph}(T):=\{(T(a^{*}),a^{*})|a^{*}\in A^{*}\}\subset A\ltimes A^{*}$.

Proof.

We denote the quadri-algebra structure on $A\ltimes A^{*}$ by $\nwarrow_{\dagger},\nearrow_{\dagger},\swarrow_{\dagger},\searrow_{\dagger}$. Define a linear map $\theta:A\bowtie_{R_{\searrow}^{*},L_{\star}^{*},-R_{\vee}^{*},-L_{\prec}^{*},-R_{\succ}^{*},-L_{\wedge}^{*},R_{\star}^{*},L_{\nwarrow}^{*}}^{R_{\searrow_{*}}^{*},L_{\star_{*}}^{*},-R_{\vee_{*}}^{*},-L_{\prec_{*}}^{*},-R_{\succ_{*}}^{*},-L_{\wedge_{*}}^{*},R_{\star_{*}}^{*},L_{\nwarrow_{*}}^{*}}A^{*}\rightarrow A\ltimes A^{*}$ by

[TABLE]

It is straightforward to check that $\theta$ is a homomorphism of quadri-algebras. Moreover, $\theta$ is bijective. So Item (a) holds.

Suppose that $r$ is a skew-symmetric solution of the $Q$-equations. Then by Item (a) we know that $\theta(A^{*})=\mathrm{graph}(T_{r})$ and $\theta(A)=A$ are isotropic complementary quadri-subalgebras of $A\ltimes A^{*}$ and dual to each others with respect to the bilinear form $\mathfrak{B}_{S}$. So $\mathrm{graph}(T_{r})$ is a Lagrangian quadri-subalgebra of $A\ltimes A^{*}$. Conversely, let $T_{r}:A^{*}\rightarrow A$ be a linear map whose graph $\mathrm{graph}(T_{r})$ is a Lagrangian quadri-subalgebra of $A\ltimes A^{*}$. Since $\mathrm{graph}(T_{r})$ is Lagrangian, $r$ is skew-symmetric. Moreover, let $a^{*},b^{*}\in A^{*}$. Since $\mathrm{graph}(T_{r})$ is a quadri-subalgebra of $A\ltimes A^{*}$, we have

[TABLE]

Therefore, $T_{r}(a^{*})\nwarrow T_{r}(b^{*})=T_{r}(R_{\searrow}^{*}(T_{r}(a^{*}))b^{*}+L_{\star}^{*}(T_{r}(b^{*}))a^{*})$. Similarly we have

[TABLE]

[TABLE]

[TABLE]

So by Proposition 7.3, we conclude that $r$ is a skew-symmetric solution of the $Q$-equations. ∎

8. Construction of linear operators on double spaces of quadri-algebras

An interesting (and important) feature of dendriform algebras, quadri-algebras and other similar algebra structures mentioned in [16] is that they have close relations with various linear operators in combinatorics [1, 2, 3, 4, 9, 14, 15, 16, 26, 27].

Definition 8.1**.**

Let $A$ be a vector space with a set of bilinear operations $\Omega:=\{\ast_{n}:A\otimes A\rightarrow A,n=1,...,m\}$. A linear operator $P$ on $A$ is called a Rota-Baxter operator of weight $\lambda$ ($\in\mathbb{F}$) if, for each $\ast\in\Omega$, we have

[TABLE]

A linear operator $N$ on $A$ is called a Nijenhuis operator if, for each $\ast\in\Omega$, we have

[TABLE]

We have the following relationship between Rota-Baxter operators of weight $\lambda$ and Nijenhuis operators.

Proposition 8.2**.**

[14]* With the notations in Definition 8.1, if $N:A\rightarrow A$ is a Nijenhuis operator on $A$ satisfying $N^{2}=\lambda^{2}{\rm id}$, then P:=$${-\lambda{\rm id}-N}\over 2 is a Rota-Baxter operator of weight $\lambda$ (on $A$).*

By a direct computation, we have

Proposition 8.3**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-bialgebra, which is induced from a skew-symmetric solution $r$ of the $Q$-equations. If, in addition, $r$ is nondegenerate, then for all $x\in A,a^{*}\in A^{*}$,

[TABLE]

is a Nijenhuis operator on the Drinfeld $Q$-double $\mathcal{QD}(A)$, where $\lambda_{i}\in\mathbb{F}$, $i=1,2,3,4$.

In the following we assume that the coefficient field $\mathbb{F}$ is the real number field $\mathbb{R}$ and fix a $\lambda\in\mathbb{R}$. With the conditions in Proposition 8.3, if we consider a Nijenhuis operator satisfying $N_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}^{2}=\lambda^{2}{\rm id}$ and apply Proposition 8.2, then we can get three families of Rota-Baxter operators of weight $\lambda$ on $\mathcal{QD}(A)$:

[TABLE]

for all $x\in A,a^{*}\in A^{*}$, where $k,\hat{k},k_{1},k_{2}\in\mathbb{R}$. Here we exclude the trivial case when $P=-\lambda{\rm id}$. Furthermore, it is easy to check that these Rota-Baxter operators are idempotents (i.e., $P^{2}=P$) if and only if $\lambda=-1$, that is, they are Rota-Baxter operators of weight $-1$. We record them explicitly as follows (see the discussion at the end of this section for their important roles in renormalization in quantum field theory):

[TABLE]

for all $x\in A,a^{*}\in A^{*}$, where $k,\hat{k},k_{1},k_{2}\in\mathbb{R}$.

On the other hand, the requirement that $r$ is nondegenerate can be dropped if the $\lambda_{3}$ appearing in the definition of $N_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}$ equals to zero. More precisely, we have the following immediate conclusion.

Proposition 8.4**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha,\beta,\tilde{\alpha},\tilde{\beta})$ be a quadri-bialgebra, which is induced from a skew-symmetric solution $r$ of the $Q$-equations. Then for all $x\in A,a^{*}\in A^{*}$,

[TABLE]

is a Nijenhuis operator on $\mathcal{QD}(A)$, where $\lambda_{i}\in\mathbb{F}$, $i=1,2,3$.

Remark 8.5**.**

Similarly, we consider the case when $N_{\lambda_{1},\lambda_{2},\lambda_{3}}=\lambda^{2}{\rm id}$ for $\lambda\in\mathbb{R}$ and then apply Proposition 8.2 to get certain families of Rota-Baxter operators (of weight $\lambda$) on $\mathcal{QD}(A)$. In fact, one can show that the Rota-Baxter operators (of weight $\lambda$) are given by (F2) and the Rota-Baxter operators which are idempotents are given by (G2), where $x\in A,a^{*}\in A^{*}$.**

Proposition 8.6**.**

Let $(A,\nwarrow,\nearrow,\swarrow,\searrow)$ be a quadri-algebra. Then

[TABLE]

is a Nijenhuis operator on $A_{v}\ltimes_{L_{\nearrow},R_{\nwarrow},L_{\searrow},R_{\swarrow}}A$, where $x,y\in A$ and $\lambda_{i}\in\mathbb{F}$, $i=1,2,3,4$. Moreover, if $(A_{v},A_{v},L_{\nearrow},R_{\nwarrow},L_{\searrow},R_{\swarrow},L_{\nearrow},R_{\nwarrow},L_{\searrow},R_{\swarrow})$ is a matched pair of dendriform algebras, then the linear operator defined by Eq. (122) is also a Nijenhuis operator on $A_{v}\bowtie_{L_{\nearrow},R_{\nwarrow},L_{\searrow},R_{\swarrow}}^{L_{\nearrow},R_{\nwarrow},L_{\searrow},R_{\swarrow}}A_{v}$. On the other hand,

[TABLE]

is an isomorphism of dendriform algebras from $A_{v}\bowtie_{L_{\nearrow},R_{\nwarrow},L_{\searrow},R_{\swarrow}}^{L_{\nearrow},R_{\nwarrow},L_{\searrow},R_{\swarrow}}A_{v}$ to $A_{v}\ltimes_{L_{\nearrow},R_{\nwarrow},L_{\searrow},R_{\swarrow}}A$.

Proof.

It is straightforward.∎

Remark 8.7**.**

Similarly, we can also consider the case that $N_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}=\lambda^{2}{\rm id}$ and then apply Proposition 8.2 to get certain families of Rota-Baxter operators of weight $\lambda$ on the double spaces given in Proposition 8.6.**

Remark 8.8**.**

One can use these Nijenhuis operators and Rota-Baxter operators (of weight $\lambda$) to construct $NS$-algebra and tridendriform algebra structures (see [16] and the references therein). Moreover, it is easy to show that these Nijenhuis operators also satisfy Eq. (119) on their associated associative algebras, that is, they are the so-called associative Nijenhuis tensors of the associated associative algebras in the sense of [10], where this notion was introduced in the study of Wigner problem in quantum physics.**

Remark 8.9**.**

Obviously, if $P$ is a Rota-Baxter operator of weight $\lambda$ on a quadri-algebra, then it is a Rota-Baxter operator of weight $\lambda$ on the associated associative algebra. Furthermore, Rota-Baxter operators on associative algebras which are idempotents play a key role in the algebraic Birkhoff decomposition in pQFT [11, 18, 19]. See [17] also for a nice survey of this topic. As discussed above, we have constructed certain families of Rota-Baxter operators on the double spaces of quadri-algebras.**

Acknowledgements. The authors are grateful to Professor J.-L. Loday for his inspiring suggestions and to L. Guo for helpful discussions. This work is supported by NSFC (11931009). C. Bai is also supported by the Fundamental Research Funds for the Central Universities and Nankai ZhiDe Foundation.