# Infinitesimal (BiHom-)bialgebras of any weight (I): Basic definitions and
properties

Tianshui Ma  School of Mathematics and Information Science, Henan Normal
University, Xinxiang 453007, China<EMAIL_ADDRESS>and  Abdenacer
Makhlouf* Université de Haute Alsace, IRIMAS-Département de Mathématiques,18
rue des Frères Lumière F-68093 Mulhouse, France<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is to introduce and study $\lambda$-infinitesimal
BiHom-bialgebras (abbr. $\lambda$-infBH-bialgebra) and some related
structures. They can be seen as an extension of $\lambda$-infinitesimal
bialgebras considered by Ebrahimi-Fard, including Joni and Rota’s
infinitesimal bialgebras as well as Loday and Ronco’s infinitesimal
bialgebras, and including also infinitesimal BiHom-bialgebras introduced by
Liu, Makhlouf, Menini, Panaite. In this paper, we provide various relevant
constructions and new concepts. Two ways are provided for a unitary (resp.
counitary) algebra (coalgebra) to be a $\lambda$-infBH-bialgebra and the
notion of $\lambda$-infBH-Hopf module is introduced and discussed. It is
proved, in connexion with nonhomogeneous (co)associative BiHom-Yang-Baxter
equation, that every (left BiHom-)module (resp. comodule) over a
(anti-)quasitriangular (resp. (anti-)coquasitriangular) $\lambda$-infBH-
bialgebra carries a structure of $\lambda$-infBH-Hopf module. Moreover, two
approaches to construct BiHom-pre-Lie (co)algebras from $\lambda$-infBH-
bialgebras are presented.

###### Key words and phrases:

Infinitesimal bialgebra;infinitesimal Hopf module;quasitriangular
infinitesimal bialgebra

###### 2020 Mathematics Subject Classification:

17B61,17D30,17B38,17A30,16T10.

*Corresponding author

###### Contents

  1. 1 Introduction and Preliminaries
  2. 2 $\lambda$-infinitesimal BiHom-bialgebras
    1. 2.1 Definitions
    2. 2.2 Two tensor product structures
  3. 3 $\lambda$-infBH-Hopf modules
    1. 3.1 Definition and examples
    2. 3.2 Modules over (anti)quasitriangular $\lambda$-infBH-bialgebras
      1. 3.2.1 First approach
      2. 3.2.2 Second approach
    3. 3.3 BiHom-pre-Lie algebras from $\lambda$-infBH-bialgebras
    4. 3.4 Comodules over (anti-)coquasitriangular $\lambda$-infBH-bialgebra
    5. 3.5 BiHom-pre-Lie coalgebras from $\lambda$-infBH-bialgebras
  4. 4 Further research

##  1\. Introduction and Preliminaries

A triple $(A,\mu,\Delta)$ is called an infinitesimal bialgebra if $(A,\mu)$ is
an associative algebra, $(A,\Delta)$ is a coassociative coalgebra such that
$\Delta$ is a derivation, i.e., for all $a,b\in A$,
$\Delta(ab)=a\Delta(b)+\Delta(a)b$. This concept was introduced by Joni and
Rota in connection with the calculus of divided differences [12]. In [2],
Aguiar developed the basic theory of infinitesimal Hopf (bialgebra) algebras.
Infinitesimal versions of bicrossproduct, quasitriangular bialgebra, Yang-
Baxter equation and Drinfeld double are defined. It is shown that the path
algebra of an arbitrary quiver admits a canonical structure of infinitesimal
Hopf algebra. Hom-version of infinitesimal bialgebra was considered in [26],
BiHom-version in [15] and braided version in [25]. Other studies and results
related to infinitesimal bialgebras can be found in [3, 4, 9, 21, 18, 22]. The
notion of infinitesimal bialgebra of weight $-1$ was introduced by Loday and
Ronco [16] in order to obtain a structure theorem of cofree Hopf algebras,
which is a part of a 2-associative bialgebra. An example can be constructed
through polynomial algebra $K[x]$ with the coproduct
$\Delta(x^{n})=\sum_{p=1}^{n}x^{p}\otimes x^{n-p}$.

A generalization including both cases was provided by Ebrahimi-Fard in [6],
where infinitesimal bialgebra of weight $\lambda$ (abbr. $\lambda$-inf
bialgebra) are defined. More precisely, a $\lambda$-inf bialgebra is a triple
$(A,\mu,\Delta)$ consisting of an algebra $(A,\mu)$ (possibly without a unit)
and a coalgebra $(A,\Delta)$ (possibly without a counit) such that for all
$a,b\in A,\Delta(ab)=a\Delta(b)+\Delta(a)b+\lambda a\otimes b$. A solution of
the nonhomogeneous associative Yang-Baxter equation [24, 6] can produce a
$\lambda$-inf bialgebra. Further results about $\lambda$-inf bialgebras can be
found in[6, 8, 7, 16, 27, 28, 30]. It is worth mentioning that a unitary and
counitary $0$-inf bialgebra $H$ is trivial, i.e., $H=0$ and $\varepsilon(1)=0$
[2], but for non-zero weight cases, this result does not hold. For example, if
the weight $\lambda$ of infinitesimal bialgebra $H$ is $-1$ and suppose that
$H$ is unitary and counitary, then the counit $\varepsilon$ is an algebra map.
Examples of (co)unitary $\lambda$-inf bialgebras can be constructed on the
space of decorated planar rooted forests [29, 27].

Zhang and Guo introduced in [28] the concept of $\lambda$-infinitesimal Hopf
modules and proved that any module over a unitary quasitriangular
$\lambda$-inf bialgebra owns a structure of $\lambda$–infinitesimal Hopf
module. They also derived two pre-Lie algebras from a $\lambda$-inf bialgebra
and commutative $\lambda$-inf bialgebra. Also in [28], an example shows that
any unitary algebra $(A,\mu,1)$ is a unitary $\lambda$-inf bialgebra by taking
$\Delta(a)=-\lambda(a\otimes 1)$. A very interesting phenomenon that we find
is that if we define a coproduct on a unitary algebra $(A,\mu,1)$ by
$\overline{\Delta}(a)=-\lambda(1\otimes a)$ (just exchanging the positions of
a and $1$), then $(A,\mu,1,\overline{\Delta})$ is also a unitary $\lambda$-inf
bialgebra. Enlightened by this phenomenon, we introduce the notion of anti-
quasitriangular $\lambda$-inf bialgebra (see Definition 3.16) which is
different from the quasitriangular case based on their equivalent
characterizations (see Propositions 3.10 and 3.17). When $\lambda=-1$, the
equivalent characterizations for an anti-quasitriangular $\lambda$-inf
bialgebra in Proposition 3.17 are consistent with [5, Lemma 3.19]. Furthermore
we show that any module over a unitary anti-quasitriangular $\lambda$-inf
bialgebra still has a structure of $\lambda$-infinitesimal Hopf module and its
comodule action depends on the weight $\lambda$ (see Theorem 3.19). Anti-
quasitriangular $0$-inf bialgebra and quasitriangular $0$-inf bialgebra are
consistent. One purpose of this paper is to explore this new phenomenon.

In [10], Graziani, Makhlouf, Menini and Panaite introduced algebras in a group
Hom-category, which are called BiHom-associative algebras and involve two
commuting multiplicative linear maps. BiHom-type algebras can be seen as an
extension of Hom-type algebras, which first arose in quasi-deformations of Lie
algebras of vector fields and lead to the concept of Hom-Lie algebras
introduced by Hartwig, Larsson and Silvestrov in [11], see also [23]. The
examples of BiHom-type of algebras can be provided by the “Yau twist
principle”. In 2020, Liu, Makhlouf, Menini and Panaite proposed and studied
(quasitriangular) Joni-Rota’s infinitesimal BiHom-bialgebras [15] generalizing
the Hom-case studied by Yau in [26]. In [21], Ma and Yang presented the
Drinfeld double for infinitesimal BiHom-bialgebra and Ma, Li and Yang studied
the properties of infinitesimal BiHom-bialgebra from BiHom-coderivations [18].
However, the weight of infinitesimal BiHom-bialgebras studied above is zero.
The other purpose of this paper is to extend these studies and consider the
case of infinitesimal BiHom-bialgebra of any weight $\lambda$.

The layout of the paper is as follows. In Section 2, we extend Ebrahimi-Fard’s
$\lambda$-inf bialgebras [6] to the BiHom-case, which includes Joni and Rota’s
infinitesimal bialgebras [12], Loday and Ronco’s infinitesimal bialgebra [16]
and Liu, Makhlouf, Menini, Panaite’s infinitesimal BiHom-bialgebras [15]. We
prove that every unitary BiHom-associative algebra $(A,\mu,1,\alpha,\beta)$
(resp. counitary BiHom-coassociative coalgebra $(A,\Delta,\psi,\omega))$
possesses $\lambda$-infBH-bialgebra structures with the comultiplication
$\Delta:A\longrightarrow A\otimes A$ by $\Delta(a)=-\lambda(\omega(a)\otimes
1)$ or $\Delta(a)=-\lambda(1\otimes\psi(a))\big{)}$ (Example 2.4) (resp.
multiplication $\mu:A\otimes A\longrightarrow A$ by $\mu(a\otimes
b)=-\lambda\alpha(a)\varepsilon(b)$ or $\mu(a\otimes
b)=-\lambda\varepsilon(a)\beta(b)$ (Example 2.5)). These two examples arouse
the new constructions of $\lambda$-infBH-bialgebra by an element $r\in
A\otimes A$ (Definitions 3.9 and 3.16) (resp. $\sigma\in(A\otimes A)^{*}$
(Definition 3.34)). Let
$(A,\mu_{A},\Delta_{A},\varepsilon_{A},\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$
(resp.
$(C,\mu_{C},\Delta_{C},1_{C},\alpha_{C},\beta_{C},\psi_{C},\omega_{C})$) be a
counitary (resp. unitary) $\lambda$-infBH-bialgebra. We provide a new tensor
product algebra (resp. coalgebra) structure of two algebras (resp. coalgebras)
such that $\Delta_{A}:(A,\mu_{A},\alpha_{A},\beta_{A})\longrightarrow(A\otimes
A,\cdot_{\varepsilon},\alpha_{A}\otimes\alpha_{A},\beta_{A}\otimes\beta_{A})$
(resp. $\mu_{C}:(C\otimes
C,\Delta_{\eta},\psi_{C}\otimes\psi_{C},\omega_{C}\otimes\omega_{C})\longrightarrow(C,\Delta_{C},\psi_{C},\omega_{C})$)
is a morphism of BiHom-associative algebras (resp. coalgebras) (see Theorem
2.21).

In Section 3 we investigate $\lambda$-infBH-Hopf modules. First, we provide
some relevant and interesting examples. Then considering a unitary BiHom-
algebra $(A,\mu,1,\alpha,\beta)$, an element $r\in A\otimes A$ and
$\psi,\omega:A\longrightarrow A$ two linear maps, we define a linear map
$\Delta_{r}:A\longrightarrow A\otimes A$ by
$\Delta_{r}(a)=\alpha^{-1}(a)\triangleright
r-r\triangleleft\beta^{-1}(a)-\lambda(\omega(a)\otimes
1),~{}~{}~{}\forall~{}~{}a\in A$ (Eq.(3.2)) and prove that $\Delta_{r}$ is a
$\lambda$-BiHom-derivation and also find the equivalent condition for
$\Delta_{r}$ is BiHom-coassociative (Theorem 3.6). Then we introduce the
notions of $\lambda$-associative BiHom-Yang-Baxter equation (abbr.
$\lambda$-abhYBe) and quasitriangular unitary $\lambda$-infBH-bialgebra.
Therefore we investigate equivalent characterizations of quasitriangular
unitary $\lambda$-infBH-bialgebras (see Proposition 3.10) and obtain that
modules over quasitriangular unitary $\lambda$-infBH-bialgebras carry
structures of $\lambda$-infBH-Hopf modules (Theorem 3.12). Moreover, inspired
by Example 2.4, we define a new comultiplication for $\lambda$-infBH-bialgebra
by replacing $\lambda(\omega(a)\otimes 1)$ in $\Delta_{r}$ above with
$\lambda(1\otimes\psi(a))$, then a new interesting phenomenon appears. We
notice that $\widetilde{\Delta}_{r}$ defined by Eq.(3.8) is still a
$\lambda$-BiHom-derivation (Proposition 3.13). It turns out that the weight of
the induced associative BiHom-Yang-Baxter equation is $-\lambda$ by the
coassociativity of $\widetilde{\Delta}_{r}$ (Theorem 3.14). We call this
unitary $\lambda$-infBH-bialgebra determined by $\lambda$-abhYBe “anti-
quasitriangular” (Definition 3.16). Comparing the equivalent characterizations
of quasitriangular case in Proposition 3.10 and anti-quasitriangular case in
Proposition 3.17, we find the two cases are essentially different for the non-
zero weight. While for zero weight, they are same. Furthermore,
$\lambda$-infBH-Hopf modules also can be derived from the modules over anti-
quasitriangular unitary $\lambda$-infBH-bialgebras. We notice here that the
comodule coaction depends on the weight $\lambda$ (Theorem 3.19). Also in this
section, we study $\lambda$-infBH-bialgebras and $\lambda$-infBH-Hopf modules
from the perspective of (BiHom-)coderivation, which covers and at the same
time extends the results in [20, 18]. More precisely, corresponding to the
cases of unitary “quasitriangular” and “anti-quasitriangular” above, for a
counitary BiHom-coalgebra $(C,\Delta,\varepsilon,\psi,\omega)$ and an element
$\sigma\in(C\otimes C)^{\ast}$, we define the multiplications $\mu_{\sigma}$
(Eq.3.16) (resp. $\widetilde{\mu_{\sigma}}$ (Eq.3.17)), then we get the
concept of $\pm\lambda$-coassociative BiHom-Yang-Baxter equation (abbr.
$\pm\lambda$-coabhYBe) (Definition 3.32). We show that $\lambda$-infBH-Hopf
modules can be derived from every comodule over (anti-)coquasitriangular
counitary $\lambda$-infBH-bialgebra (Theorem 3.36). Lastly, we propose two
ways to construct BiHom-pre-Lie algebras from $\lambda$-infBH-bialgebras. The
structure in Theorem 3.26 makes the following diagram commutative.

$\textstyle{\text{Quasitriangular}\atop\text{$\lambda$-infBH
Bialgebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Lemma
3.20Corollary 3.8$\textstyle{\text{$\lambda$-Rota Baxter}\atop\text{BiHom-
algebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Lemma
3.21$\textstyle{\text{BiHom-
Dendriform}\atop\text{algebras}}$$\textstyle{\text{$\lambda$-infBH}\atop\text{Bialgebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Theorem
3.26$\textstyle{\text{BiHom-Pre-
Lie}\atop\text{algebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$[13,
Proposition 3.6]

We also put forward two methods to construct BiHom-pre-Lie coalgebras from
$\lambda$-infBH-bialgebras, one of which can also provide a commutative
diagram corresponding to the above one.

Throughout this paper, $K$ will be a field, and all vector spaces, tensor
products, and homomorphisms are over $K$. We use Sweedler’s notation for
terminology on coalgebras. For a coalgebra $C$ with a comultiplication
$\Delta$, we write $\Delta(c)=c_{1}\otimes c_{2}$, for all $c\in C$. We denote
by ${\rm id}_{M}$ the identity map from $M$ to $M$ and by $\tau:M\otimes
N\rightarrow N\otimes M$ the flip map. We abbreviate ”infinitesimal BiHom-” to
”infBH-”.

Let us recall from [10, 17] the following basic definitions and structures.

######  Definition 1.1.

A BiHom-associative algebra is a 4-tuple $(A,\mu,\alpha,\beta)$, where $A$ is
a linear space, $\alpha,\beta:A\longrightarrow A$ and $\mu:A\otimes
A\longrightarrow A$ (write $\mu(a\otimes b)=ab$) are linear maps satisfying
the following conditions, for all $a,b,c\in A$ :

$\displaystyle\alpha\circ\beta=\beta\circ\alpha,\quad\alpha(ab)=\alpha(a)\alpha(b),\quad\beta(ab)=\beta(a)\beta(b),$
(1.1) $\displaystyle\alpha(a)(bc)=(ab)\beta(c).$ (1.2)

A BiHom-associative algebra $(A,\mu,\alpha,\beta)$ is called unitary if there
exists an element $1_{A}\in A$ (called a unit) such that

$\displaystyle\alpha(1_{A})=1_{A},\quad\beta(1_{A})=1_{A},\quad
a1_{A}=\alpha(a),\quad 1_{A}a=\beta(a),\quad\forall a\in A.$ (1.3)

A morphism
$f:(A,\mu_{A},\alpha_{A},\beta_{A})\longrightarrow(B,\mu_{B},\alpha_{B},\beta_{B})$
of BiHom-associative algebras is a linear map $f:A\longrightarrow B$ such that
$\alpha_{B}\circ f=f\circ\alpha_{A}$, $\beta_{B}\circ f=f\circ\beta_{A}$ and
$f\circ\mu_{A}=\mu_{B}\circ(f\otimes f)$.

######  Remark 1.2.

“Yau twist”: Let $(A,\mu)$ be an associative algebra,
$\alpha,\beta:A\longrightarrow A$ two linear maps satisfying Eqs.(1.1). Then
$(A,\mu\circ(\alpha\otimes\beta),\alpha,\beta)$ is a BiHom-associative
algebra.

######  Definition 1.3.

A BiHom-coassociative coalgebra is a 4-tuple $(C,\Delta,\psi,\omega)$, in
which $C$ is a linear space, $\psi,\omega:C\longrightarrow C$ and
$\Delta:C\longrightarrow C\otimes C$ are linear maps, such that

$\displaystyle\psi\circ\omega=\omega\circ\psi,~{}~{}~{}(\psi\otimes\psi)\circ\Delta=\Delta\circ\psi,~{}~{}(\omega\otimes\omega)\circ\Delta=\Delta\circ\omega,$
(1.4)
$\displaystyle(\Delta\otimes\psi)\circ\Delta=(\omega\otimes\Delta)\circ\Delta.$
(1.5)

A BiHom-coassociative coalgebra $(C,\Delta,\psi,\omega)$ is called counitary
if there exists a linear map $\varepsilon:C\longrightarrow K$ (called a
counit) such that

$\displaystyle\varepsilon\circ\psi=\varepsilon,\quad\varepsilon\circ\omega=\varepsilon,\quad({\rm
id}_{C}\otimes\varepsilon)\circ\Delta=\omega,\quad(\varepsilon\otimes{\rm
id}_{C})\circ\Delta=\psi.$ (1.6)

A morphism
$g:(C,\Delta_{C},\psi_{C},\omega_{C})\longrightarrow(D,\Delta_{D},\psi_{D},\omega_{D})$
of BiHom-coassociative coalgebras is a linear map $g:C\longrightarrow D$ such
that $\omega_{D}\circ g=g\circ\omega_{C}$, $\psi_{D}\circ g=g\circ\psi_{C}$
and $\Delta_{D}\circ g=(g\otimes g)\circ\Delta_{C}$.

######  Definition 1.4.

Let $(A,\mu,\alpha_{A},\beta_{A})$ be a BiHom-associative algebra. A left
$(A,\mu,\alpha_{A},\beta_{A})$-module is a 4-tuple
$(M,\gamma,\alpha_{M},\beta_{M})$, where $M$ is a linear space,
$\alpha_{M},\beta_{M}:M\longrightarrow M$ and $\gamma:A\otimes
M\longrightarrow M$ (write $\gamma(a\otimes m)=a\triangleright m$) are linear
maps such that, for all $a,a^{\prime}\in A,m\in M$,

$\displaystyle\alpha_{M}\circ\beta_{M}=\beta_{M}\circ\alpha_{M},~{}~{}\alpha_{M}(a\triangleright
m)=\alpha_{A}(a)\triangleright\alpha_{M}(m),~{}~{}\beta_{M}(a\triangleright
m)=\beta_{A}(a)\triangleright\beta_{M}(m),$ (1.7)
$\displaystyle\alpha_{A}(a)\triangleright(a^{\prime}\triangleright
m)=(aa^{\prime})\triangleright\beta_{M}(m).$ (1.8)

Likewise, we can get the right version of
$(A,\mu,\alpha_{A},\beta_{A})$-module.

If $(M,\gamma,\alpha_{M},\beta_{M})$ is a left
$(A,\mu,\alpha_{A},\beta_{A})$-module and at the same time
$(M,\nu,\alpha_{M},\beta_{M})$ is a right
$(A,\mu,\alpha_{A},\beta_{A})$-module (write $\nu(m\otimes a)=m\triangleleft
a$), then $(M,\gamma,\nu,\alpha_{M},\beta_{M})$ is an
$(A,\mu,\alpha_{A},\beta_{A})$-bimodule if

$\displaystyle\alpha_{A}(a)\triangleright(m\triangleleft
a^{\prime})=(a\triangleright m)\triangleleft\beta_{A}(a^{\prime}).$ (1.9)

######  Proposition 1.5.

Let $(A,\mu,\alpha_{A},\beta_{A})$ be a BiHom-associative algebra and
$(M,\gamma_{M},\nu_{M},\alpha_{M},\beta_{M})$, $(N,\gamma_{N},\nu_{N}$,
$\alpha_{N},\beta_{N})$, $(V,\gamma_{V},\nu_{V},\alpha_{V},\beta_{V})$ be
$(A,\mu,\alpha_{A},\beta_{A})$-bimodules.
$\psi_{A},\omega_{A}:A\longrightarrow A$ are linear maps such that
$\psi_{A}(aa^{\prime})=\psi_{A}(a)\psi_{A}(a^{\prime})$,
$\omega_{A}(aa^{\prime})=\omega_{A}(a)\omega_{A}(a^{\prime})$ and any two of
the maps $\alpha_{A}$, $\beta_{A}$, $\psi_{A}$, $\omega_{A}$ commute. We
consider the following left and right actions of $A$ on $M\otimes N\otimes V$,
for all $a\in A,m\in M,n\in N,v\in V$

$\displaystyle a\triangleright(m\otimes n\otimes
v)=\omega_{A}(a)\triangleright m\otimes\beta_{N}(n)\otimes\beta_{V}(v),$
$\displaystyle(m\otimes n\otimes v)\triangleleft
a=\alpha_{M}(m)\otimes\alpha_{N}(n)\otimes v\triangleleft\psi_{A}(a).$

Then $(M\otimes N\otimes
V,\triangleright,\triangleleft,\alpha_{M}\otimes\alpha_{N}\otimes\alpha_{V},\beta_{M}\otimes\beta_{N}\otimes\beta_{V})$
is an $(A,\mu,\alpha_{A},\beta_{A})$-bimodule.

######  Remark 1.6.

(1) As special cases of Proposition 1.5, for all $a,x,y,z\in A$, we have the
following actions of $A$ on $A\otimes A$ and $A\otimes A\otimes A$ given by

$\displaystyle a\triangleright(x\otimes y)=\omega(a)x\otimes\beta(y),$ (1.10)
$\displaystyle(x\otimes y)\triangleleft a=\alpha(x)\otimes y\psi(a),$ (1.11)

and

$\displaystyle a\triangleright(x\otimes y\otimes
z)=\omega(a)x\otimes\beta(y)\otimes\beta(z),$ (1.12) $\displaystyle(x\otimes
y\otimes z)\triangleleft a=\alpha(x)\otimes\alpha(y)\otimes z\psi(a).$ (1.13)

respectively.

(2) The actions defined in Eqs.(1.10) and (1.11) coincide with the actions in
[15, Lemma 4.2].

##  2\. $\lambda$-infinitesimal BiHom-bialgebras

In this section, we introduce the notion of $\lambda$-infinitesimal BiHom-
bialgebra which is the BiHom-version of Ebrahimi-Fard’s
$\lambda$-infinitesimal bialgebra [6] including Joni and Rota’s infinitesimal
bialgebra [12], Loday and Ronco’s infinitesimal bialgebra [16] and Liu,
Makhlouf, Menini, Panaite’s infBH-bialgebra [15] as special cases.

###  2.1. Definitions

######  Definition 2.1.

Let $\lambda$ be a given element of $K$. (1) A $\lambda$-infinitesimal BiHom-
bialgebra (abbr. $\lambda$-infBH-bialgebra) is a 7-tuple
$(A,\mu,\Delta,\alpha,\beta,\psi,\omega)$ such that $(A,\mu,\alpha,\beta)$ is
a BiHom-associative algebra and $(A,\Delta,\psi,\omega)$ is a BiHom-
coassociative coalgebra satisfying the following conditions

$\displaystyle\alpha\circ\psi=\psi\circ\alpha,\
\alpha\circ\omega=\omega\circ\alpha,\ \beta\circ\psi=\psi\circ\beta,\
\beta\circ\omega=\omega\circ\beta,$ (2.1)
$\displaystyle(\alpha\otimes\alpha)\circ\Delta=\Delta\circ\alpha,\
(\beta\otimes\beta)\circ\Delta=\Delta\circ\beta,$ (2.2)
$\displaystyle\psi\circ\mu=\mu\circ(\psi\otimes\psi),\
\omega\circ\mu=\mu\circ(\omega\otimes\omega),$ (2.3)
$\displaystyle\Delta\circ\mu=(\mu\otimes\beta)\circ(\omega\otimes\Delta)+(\alpha\otimes\mu)\circ(\Delta\otimes\psi)+\lambda(\alpha\omega\otimes\beta\psi).$
(2.4)

(2) If further $(A,\mu,1,\alpha,\beta)$ is a unitary BiHom-associative
algebra, then the 8-tuple $(A,\mu,1,\Delta,$ $\alpha,\beta,\psi,\omega)$ is
called a unitary $\lambda$-infBH-bialgebra if

$\displaystyle\psi(1_{A})=1_{A},\ \omega(1_{A})=1_{A}.$ (2.5)

(3) If further $(A,\Delta,\varepsilon,\psi,\omega)$ is a counitary BiHom-
coassociative coalgebra, then the 8-tuple
$(A,\mu,\Delta,\varepsilon,\alpha,\beta,\psi,\omega)$ is called a counitary
$\lambda$-infBH-bialgebra if

$\displaystyle\varepsilon\circ\alpha=\varepsilon,\
\varepsilon\circ\beta=\varepsilon.$ (2.6)

######  Remark 2.2.

(1) If $\lambda=0$ in Definition 2.1, then we get infBH-bialgebras introduced
by Liu, Makhlouf, Menini, Panaite in [15, Definition 4.1] and also studied in
[17, 18, 21]. If further $\alpha=\beta=\psi=\omega={\rm id}$, then one can
obtain Joni and Rota’s infinitesimal bialgebras [12].

(2) If $\lambda=-1$ in Definition 2.1, then we have the BiHom-version of Loday
and Ronco’s infinitesimal bialgebra [16].

(3) Definition 2.1 is the BiHom-version of Ebrahimi-Fard’s
$\lambda$-infinitesimal bialgebras [6] studied in [28].

(4) A morphism between $\lambda$-infBH-bialgebras is a linear map that
commutes with the structure maps $\alpha,\beta,\psi,\omega$, the
multiplication $\mu$, and the comultiplication $\Delta$.

The following theorem can provide examples of $\lambda$-infBH-bialgebras from
$\lambda$-inf bialgebras.

######  Theorem 2.3.

Let $(A,\mu,\Delta)$ be a $\lambda$-infinitesimal bialgebra [6] and
$\alpha,\beta,\psi,\omega:A\longrightarrow A$ be morphisms of algebras and
coalgebras such that any two of them commute. Then
$A_{(\alpha,\beta,\psi,\omega)}:=(A,\mu_{(\alpha,\beta)}:=\mu\circ(\alpha\otimes\beta),\Delta_{(\psi,\omega)}:=(\omega\otimes\psi)\circ\Delta,\alpha,\beta,\psi,\omega)$
is a $\lambda$-infBH-bialgebra, called the Yau twist of $(A,\mu,\Delta)$.

###### Proof.

The fact that $(A,\mu_{(\alpha,\beta)},\alpha,\beta)$ is a BiHom-associative
algebra and $(A,\Delta_{(\psi,\omega)},\psi,\omega)$ is a BiHom-coassociative
coalgebra is known from [10]. By the assumption, we have Eqs.(2.1)-(2.3). Next
we only need to prove Eq.(2.4). For simplicity, we denote
$\mu_{(\alpha,\beta)}(a\otimes b)=a\ast b=\alpha(a)\cdot\beta(b)$ and
$\Delta_{(\psi,\omega)}(a)=a_{[1]}\otimes
a_{[2]}=\omega(a_{1})\otimes\psi(a_{2})$, for all $a,b\in A$, we compute:

$\displaystyle\omega(a)\ast
b_{[1]}\otimes\beta(b_{[2]})+\alpha(a_{[1]})\otimes
a_{[2]}\ast\psi(b)+\lambda\alpha\omega(a)\otimes\beta\psi(b)$ $\displaystyle=$
$\displaystyle\alpha\omega(a)\beta\omega(b_{1})\otimes\beta\psi(b_{2})+\alpha\omega(a_{1})\otimes\alpha\psi(a_{2})\beta\psi(b)+\lambda\alpha\omega(a)\otimes\beta\psi(b)$
$\displaystyle=$
$\displaystyle\omega(\alpha(a)\beta(b_{1}))\otimes\beta\psi(b_{2})+\alpha\omega(a_{1})\otimes\psi(\alpha(a_{2})\beta(b))+\lambda\alpha\omega(a)\otimes\beta\psi(b)$
$\displaystyle=$
$\displaystyle(\omega\otimes\psi)(\alpha(a)\beta(b_{1})\otimes\beta(b_{2})+\alpha(a_{1})\otimes\alpha(a_{2})\beta(b)+\lambda\alpha(a)\otimes\beta(b))$
$\displaystyle\stackrel{{\scriptstyle}}{{=}}$
$\displaystyle(\omega\otimes\psi)\Delta(\alpha(a)\beta(b))$ $\displaystyle=$
$\displaystyle\Delta_{(\psi,\omega)}(a\ast b),$

finishing the proof. ∎

The following two examples show that every (co)unitary BiHom-(co)associative
(co)algebra possesses $\lambda$-infBH-bialgebra structures.

######  Example 2.4.

Let $(A,\mu,1,\alpha,\beta)$ be a unitary BiHom-associative algebra,
$\psi,\omega:A\longrightarrow A$ be linear maps such that Eqs.(2.1), (2.3) and
(2.5) hold. Define the comultiplication $\Delta:A\longrightarrow A\otimes A$
by

$\displaystyle\Delta(a)=-\lambda(\omega(a)\otimes 1)\ \ \big{(}\hbox{resp.}\
\Delta(a)=-\lambda(1\otimes\psi(a))\big{)}$

for all $a\in A$. Then $(A,\mu,1,\Delta,\alpha,\beta,\psi,\omega)$ is a
unitary $\lambda$-infBH-bialgebra.

###### Proof.

We only check the case of $\Delta(a)=-\lambda(\omega(a)\otimes 1)$, another
case is similar. For all $a,b\in A$, we calculate

$\displaystyle(\Delta\otimes\psi)\circ\Delta(a)$ $\displaystyle=$
$\displaystyle-\lambda(\Delta\otimes\psi)(\omega(a)\otimes
1)=\lambda^{2}(\omega^{2}(a)\otimes 1\otimes 1)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.30})}}{{=}}$
$\displaystyle\lambda^{2}(\omega^{2}(a)\otimes\omega(1)\otimes
1)=-\lambda(\omega\otimes\Delta)(\omega(a)\otimes 1)$ $\displaystyle=$
$\displaystyle(\omega\otimes\Delta)\circ\Delta(a).$

Therefore, BiHom-coassociativity of $(A,\Delta,\psi,\omega)$ holds. Next, we
have

$\displaystyle\omega(a)b_{1}\otimes\beta(b_{2})+\alpha(a_{1})\otimes
a_{2}\psi(b)+\lambda(\alpha\omega(a)\otimes\beta\psi(b))$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{eq:1.5})}}{{=}}-\lambda\omega(a)\omega(b)\otimes
1-\lambda\alpha\omega(a)\otimes\beta\psi(b)+\lambda(\alpha\omega(a)\otimes\beta\psi(b))$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{eq:12.3})}}{{=}}-\lambda\omega(ab)\otimes
1=\Delta(ab),$

finishing the proof. ∎

######  Example 2.5.

Let $(A,\Delta,\varepsilon,\psi,\omega)$ be a counitary BiHom-coassociative
coalgebra and $\alpha,\beta:A\longrightarrow A$ be linear maps satisfying
Eqs.(2.1), (2.2) and (2.6). If we define the multiplication $\mu:A\otimes
A\longrightarrow A$ by

$\displaystyle\mu(a\otimes b)=-\lambda\alpha(a)\varepsilon(b)\ \
\big{(}\hbox{resp.}\ \mu(a\otimes b)=-\lambda\varepsilon(a)\beta(b)\big{)}$

for all $a,b\in A$, then $(A,\mu,\Delta,\varepsilon,\alpha,\beta,\psi,\omega)$
is a counitary $\lambda$-infBH-bialgebra.

###### Proof.

Here we prove the case of $\mu(a\otimes b)=-\lambda\varepsilon(a)\beta(b)$,
the other case is left to the reader. For all $a,b\in A$, we calculate

$\displaystyle\alpha(a)(bc)$ $\displaystyle=$
$\displaystyle-\lambda\alpha(a)(\varepsilon(b)\beta(c))\stackrel{{\scriptstyle(\ref{eq:12.31})}}{{=}}\lambda^{2}\varepsilon(a)\varepsilon(b)\beta^{2}(c)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.31})}}{{=}}$
$\displaystyle-\lambda(\varepsilon(a)\beta(b))\beta(c)=(ab)\beta(c).$

So BiHom-associativity of $(A,\mu,\alpha,\beta)$ holds. The compatibility
condition can be verified as follows.

$\displaystyle\omega(a)b_{1}\otimes\beta(b_{2})+\alpha(a_{1})\otimes
a_{2}\psi(b)+\lambda(\alpha\omega(a)\otimes\beta\psi(b))$
$\displaystyle\qquad=-\lambda\varepsilon\omega(a)\beta(b_{1})\otimes\beta(b_{2})-\lambda\alpha(a_{1})\otimes\varepsilon(a_{2})\beta\psi(b)+\lambda(\alpha\omega(a)\otimes\beta\psi(b))$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{eq:1.11})}}{{=}}-\lambda\varepsilon(a)\beta(b_{1})\otimes\beta(b_{2})-\lambda\alpha\omega(a)\otimes\beta\psi(b)+\lambda(\alpha\omega(a)\otimes\beta\psi(b))$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{eq:12.3})}}{{=}}-\lambda\varepsilon(a)\beta(b_{1})\otimes\beta(b_{2})=\Delta(ab),$

as desired. ∎

######  Remark 2.6.

We remark here that there are two cases containing in Example 2.4 (resp. 2.5),
which motivates the construction of anti-quasitriangular (resp. anti-
coquasitriangular) infBH-bialgebras in Section 3.2 (resp. 3.4).

In order to illustrate the self-duality of $\lambda$-infBH-bialgebras, we
introduce the following concepts.

######  Definition 2.7.

Let $\lambda$ be a given element of $K$, $(A,\mu,\alpha,\beta)$ be a BiHom-
associative algebra, $\psi,\omega:A\longrightarrow A$ be two linear maps such
that Eq.(2.1) holds. Then $\delta:A\longrightarrow A\otimes A$ (write
$\delta(a)=a_{[1]}\otimes a_{[2]}$) is a $\lambda$-BiHom-derivation if it
satisfies

$\displaystyle(\alpha\otimes\alpha)\circ\delta=\delta\circ\alpha,\
(\beta\otimes\beta)\circ\delta=\delta\circ\beta,\
(\psi\otimes\psi)\circ\delta=\delta\circ\psi,\
(\omega\otimes\omega)\circ\delta=\delta\circ\omega,$ (2.7)
$\displaystyle\delta(ab)=a\triangleright\delta(b)+\delta(a)\triangleleft
b+\lambda\alpha\omega(a)\otimes\beta\psi(b).$ (2.8)

By using the module action in Proposition 1.5, Eq.(2.8) is exactly

$\delta(ab)=\omega(a)b_{[1]}\otimes\beta(b_{[2]})+\alpha(a_{[1]})\otimes
a_{[2]}\psi(b)+\lambda\alpha\omega(a)\otimes\beta\psi(b),~{}\forall a,b\in A.$

Dually, we have

######  Definition 2.8.

Let $\lambda$ be a given element of $K$, $(C,\Delta,\psi,\omega)$ be a BiHom-
coassociative coalgebra, $\alpha,\beta:C\longrightarrow C$ be two linear maps
such that Eq.(2.1) holds. Then $\vartheta:C\otimes C\longrightarrow C$ (write
$\vartheta(c\otimes d)=c\diamond d$) is a $\lambda$-BiHom-coderivation if it
satisfies

$\displaystyle\vartheta\circ(\omega\otimes\omega)=\omega\circ\vartheta,\
\vartheta\circ(\psi\otimes\psi)=\psi\circ\vartheta,\
\vartheta\circ(\alpha\otimes\alpha)=\alpha\circ\vartheta,\
\vartheta\circ(\beta\otimes\beta)=\beta\circ\vartheta,$ (2.9)
$\displaystyle(c\diamond d)_{1}\otimes(c\diamond d)_{2}=(\omega(c)\diamond
d_{1})\otimes\beta(d_{2})+\alpha(c_{1})\otimes(c_{2}\diamond\psi(d))+\lambda\alpha\omega(c)\otimes\beta\psi(d),$
(2.10)

for all $c,d\in C$.

######  Remark 2.9.

$0$-BiHom-(co)derivation was introduced in [21] ([18]). If further,
$\alpha=\beta=\psi=\omega={\rm id}$, we can cover the classical case.

######  Proposition 2.10.

Let $(A,\mu,\alpha,\beta)$ be a BiHom-associative algebra and
$(A,\Delta,\psi,\omega)$ a BiHom- coassociative coalgebra such that Eqs.(2.1),
(2.2) and (2.3) hold. Then the following statements are equivalent:

(1) $\Delta:A\longrightarrow A\otimes A$ is a $\lambda$-BiHom-derivation.

(2) $\mu:A\otimes A\longrightarrow A$ is a $\lambda$-BiHom-coderivation.

(3) Eq.(2.4) holds.

###### Proof.

Note that the compatibility condition Eq.(2.4) can be written as

$\displaystyle\Delta(ab)=\omega(a)b_{1}\otimes\beta(b_{2})+\alpha(a_{1})\otimes
a_{2}\psi(b)+\lambda(\alpha\omega(a)\otimes\beta\psi(b)),$

which implies that $\Delta:A\longrightarrow A\otimes A$ is a $\lambda$-BiHom-
derivation in the BiHom-associative algebra $(A,\mu,\alpha,\beta)$, in other
words, $\mu:A\otimes A\longrightarrow A$ is a $\lambda$-BiHom-coderivation in
the BiHom-coassociative coalgebra $(A,\Delta,\psi,\omega)$. ∎

######  Theorem 2.11.

Let $A$ be a finite dimensional vector space. Then
$(A,\mu,\Delta,\alpha,\beta,\psi,\omega)$ is a $\lambda$-infBH-bialgebra if
and only if
$(A^{\ast},\Delta^{\ast},\mu^{\ast},\omega^{\ast},\psi^{\ast},\beta^{\ast},\alpha^{\ast})$
is a $\lambda$-infBH-bialgebra with the multiplication

$\displaystyle A^{\ast}\otimes A^{\ast}\cong(A\otimes
A)^{\ast}\stackrel{{\scriptstyle\Delta^{\ast}}}{{\longrightarrow}}A^{\ast},$

and the coproduct

$\displaystyle
A^{\ast}\stackrel{{\scriptstyle\mu^{\ast}}}{{\longrightarrow}}(A\otimes
A)^{\ast}\cong A^{\ast}\otimes A^{\ast}.$

###### Proof.

It can be proved by Proposition 2.10 and [10, Theorem 5.5 and 5.6]. ∎

###  2.2. Two tensor product structures

######  Lemma 2.12.

(1) Let $(A,\mu,1,\Delta,\alpha,\beta,\psi,\omega)$ be a unitary
$\lambda$-infBH-bialgebra. Then

$\displaystyle\Delta(1)=-\lambda(1\otimes 1).$

(2) Let $(A,\mu,\Delta,\varepsilon,\alpha,\beta,\psi,\omega)$ be a counitary
$\lambda$-infBH-bialgebra. Then

$\displaystyle\varepsilon(ab)=-\lambda\varepsilon(a)\varepsilon(b),$

for all $a,b\in A$.

###### Proof.

(1) can be checked below.

$\displaystyle\Delta(1)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.4})}}{{=}}$
$\displaystyle\omega(1)1_{1}\otimes\beta(1_{2})+\alpha(1_{1})\otimes
1_{2}\psi(1)+\lambda(\alpha\omega(1)\otimes\beta\psi(1))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.30})(\ref{eq:1.5})}}{{=}}$
$\displaystyle\beta(1_{1})\otimes\beta(1_{2})+\alpha(1_{1})\otimes\alpha(1_{2})+\lambda(1\otimes
1)$ $\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.2})}}{{=}}$
$\displaystyle\Delta\circ\beta(1)+\Delta\circ\alpha(1)+\lambda(1\otimes 1)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.5})}}{{=}}$ $\displaystyle
2\Delta(1)+\lambda(1\otimes 1).$

(2) Dual to (1). ∎

######  Remark 2.13.

The counit $\varepsilon$ of counitary $(-1)$-infBH-bialgebra is an algebra
morphism. In this case, if the BiHom-associative algebra
$(A,\mu,\alpha,\beta)$ has a unit $1$, then $\varepsilon(1)=1_{K}$.

Motivated by Lemma 2.12 (2), we can get an extended version below, which
generalizes the augmented algebras introduced by Aguiar [4].

######  Definition 2.14.

Let $\lambda$ be a given element of $K$. A $\lambda$-augmented BiHom-
associative algebra is a 5-tuple $(A,\mu,\chi,\alpha,\beta)$ consisting of a
BiHom-associative algebra $(A,\mu,\alpha,\beta)$ (possibly without unit) and a
linear augmentation map $\chi:A\longrightarrow K$ of weight $\lambda$
satisfying $\chi\circ\alpha=\chi$, $\chi\circ\beta=\chi$ and

$\displaystyle\chi(ab)=-\lambda\chi(a)\chi(b),$ (2.11)

for all $a,b\in A$.

Let $(A,\mu_{A},\chi_{A},\alpha_{A},\beta_{A})$ and
$(B,\mu_{B},\chi_{B},\alpha_{B},\beta_{B})$ be two $\lambda$-augmented BiHom-
associative algebras. An algebra map $f:A\longrightarrow B$ is said to be
augmented if it satisfies $\alpha_{B}\circ f=f\circ\alpha_{A}$,
$\beta_{B}\circ f=f\circ\beta_{A}$ and $\chi_{B}\circ f=\chi_{A}$.

######  Remark 2.15.

By Lemma 2.12, a counitary $\lambda$-infBH-bialgebra is a $\lambda$-augmented
BiHom-associative algebra.

######  Proposition 2.16.

Let $(A,\mu_{A},\chi_{A},\alpha_{A},\beta_{A})$ and
$(B,\mu_{B},\chi_{B},\alpha_{B},\beta_{B})$ be two $\lambda$-augmented BiHom-
associative algebras. Then

(1) $(A\otimes
B,\cdot_{\chi},\alpha_{A}\otimes\alpha_{B},\beta_{A}\otimes\beta_{B})$ is a
BiHom-associative algebra with the multiplication $\cdot_{\chi}$ defined by

$\displaystyle(a\otimes b)\cdot_{\chi}(a^{\prime}\otimes b^{\prime})$
$\displaystyle\qquad=\chi_{B}(b)aa^{\prime}\otimes\beta_{B}(b^{\prime})+\chi_{A}(a^{\prime})\alpha_{A}(a)\otimes
bb^{\prime}+\lambda\chi_{A}(a^{\prime})\chi_{B}(b)\alpha_{A}(a)\otimes\beta_{B}(b^{\prime}),$
(2.12)

for all $a,a^{\prime}\in A$ and $b,b^{\prime}\in B$.

(2) Furthermore, $(A\otimes B,\cdot_{\chi},\chi_{A\otimes
B},\alpha_{A}\otimes\alpha_{B},\beta_{A}\otimes\beta_{B})$ is a
$\lambda$-augmented BiHom-associative algebra with the augmentation map given
by

$\displaystyle\chi_{A\otimes B}=\chi_{A}\otimes\chi_{B}.$ (2.13)

###### Proof.

(1) We only check the BiHom-associativity as follows. For all
$a,a^{\prime},a^{\prime\prime}\in A$ and $b,b^{\prime},b^{\prime\prime}\in B$,
on one hand,

$\displaystyle((a\otimes b)\cdot_{\chi}(a^{\prime}\otimes
b^{\prime}))\cdot_{\chi}(\beta_{A}(a^{\prime\prime})\otimes\beta_{B}(b^{\prime\prime}))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.6})}}{{=}}$
$\displaystyle\chi_{B}(b)(\chi_{B}(b^{\prime})(aa^{\prime})\beta_{A}(a^{\prime\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})+\chi_{A}(a^{\prime\prime})\alpha_{A}(aa^{\prime})\otimes\beta_{B}(b^{\prime})\beta_{B}(b^{\prime\prime})$
$\displaystyle+\lambda\chi_{A}(a^{\prime\prime})\chi_{B}(b^{\prime})\alpha_{A}(aa^{\prime})\otimes\beta^{2}_{B}(b^{\prime\prime}))+\chi_{A}(a^{\prime})(\chi_{B}(bb^{\prime})\alpha_{A}(a)\beta_{A}(a^{\prime\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})$
$\displaystyle+\chi_{A}(a^{\prime\prime})\alpha^{2}_{A}(a)\otimes(bb^{\prime})\beta_{B}(b^{\prime\prime})+\lambda\chi_{A}(a^{\prime\prime})\chi_{B}(bb^{\prime})\alpha^{2}_{A}(a)\otimes\beta^{2}_{B}(b^{\prime\prime}))$
$\displaystyle+\lambda\chi_{A}(a^{\prime})\chi_{B}(b)(\chi_{B}(b^{\prime})\alpha_{A}(a)\beta_{A}(a^{\prime\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})+\chi_{A}(a^{\prime\prime})\alpha^{2}_{A}(a)\otimes\beta_{B}(b^{\prime})\beta_{B}(b^{\prime\prime})$
$\displaystyle+\lambda\chi_{A}(a^{\prime\prime})\chi_{B}(b^{\prime})\alpha^{2}_{A}(a)\otimes\beta^{2}_{B}(b^{\prime\prime}))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.3})(\ref{eq:12.5})}}{{=}}$
$\displaystyle\chi_{B}(b)\chi_{B}(b^{\prime})\alpha_{A}(a)(a^{\prime}a^{\prime\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})+\chi_{B}(b)\chi_{A}(a^{\prime\prime})\alpha_{A}(a)\alpha_{A}(a^{\prime})\otimes\beta_{B}(b^{\prime})\beta_{B}(b^{\prime\prime})$
$\displaystyle+\lambda\chi_{B}(b)\chi_{A}(a^{\prime\prime})\chi_{B}(b^{\prime})\alpha_{A}(a)\alpha_{A}(a^{\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})-\lambda\chi_{A}(a^{\prime})\chi_{B}(b)\chi_{B}(b^{\prime})\alpha_{A}(a)\beta_{A}(a^{\prime\prime})$
$\displaystyle\otimes\beta^{2}_{B}(b^{\prime\prime})+\chi_{A}(a^{\prime})\chi_{A}(a^{\prime\prime})\alpha^{2}_{A}(a)\otimes\alpha_{B}(b)(b^{\prime}b^{\prime\prime})-\lambda^{2}\chi_{A}(a^{\prime})\chi_{A}(a^{\prime\prime})\chi_{B}(b)\chi_{B}(b^{\prime})\alpha^{2}_{A}(a)$
$\displaystyle\otimes\beta^{2}_{B}(b^{\prime\prime})+\lambda\chi_{A}(a^{\prime})\chi_{B}(b)\chi_{B}(b^{\prime})\alpha_{A}(a)\beta_{A}(a^{\prime\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})+\lambda\chi_{A}(a^{\prime})\chi_{B}(b)\chi_{A}(a^{\prime\prime})\alpha^{2}_{A}(a)$
$\displaystyle\otimes\beta_{B}(b^{\prime})\beta_{B}(b^{\prime\prime})+\lambda^{2}\chi_{A}(a^{\prime})\chi_{B}(b)\chi_{A}(a^{\prime\prime})\chi_{B}(b^{\prime})\alpha^{2}_{A}(a)\otimes\beta^{2}_{B}(b^{\prime\prime})$
$\displaystyle=$
$\displaystyle\chi_{B}(b)\chi_{B}(b^{\prime})\alpha_{A}(a)(a^{\prime}a^{\prime\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})+\chi_{B}(b)\chi_{A}(a^{\prime\prime})\alpha_{A}(a)\alpha_{A}(a^{\prime})\otimes\beta_{B}(b^{\prime})\beta_{B}(b^{\prime\prime})$
$\displaystyle+\lambda\chi_{B}(b)\chi_{A}(a^{\prime\prime})\chi_{B}(b^{\prime})\alpha_{A}(a)\alpha_{A}(a^{\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})+\chi_{A}(a^{\prime})\chi_{A}(a^{\prime\prime})\alpha^{2}_{A}(a)\otimes\alpha_{B}(b)(b^{\prime}b^{\prime\prime})$
$\displaystyle+\lambda\chi_{A}(a^{\prime})\chi_{B}(b)\chi_{A}(a^{\prime\prime})\alpha^{2}_{A}(a)\otimes\beta_{B}(b^{\prime})\beta_{B}(b^{\prime\prime})\stackrel{{\scriptstyle\bigtriangleup}}{{=}}I.$

On the other hand,

$\displaystyle(\alpha_{A}(a)\otimes\alpha_{B}(b))\cdot_{\chi}((a^{\prime}\otimes
b^{\prime})\cdot_{\chi}(a^{\prime\prime}\otimes b^{\prime\prime}))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.6})}}{{=}}$
$\displaystyle\chi_{B}(b^{\prime})(\chi_{B}(b)\alpha_{A}(a)(a^{\prime}a^{\prime\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})+\chi_{A}(a^{\prime}a^{\prime\prime})\alpha^{2}_{A}(a)\otimes\alpha_{B}(b)\beta_{B}(b^{\prime\prime})$
$\displaystyle+\lambda\chi_{A}(a^{\prime}a^{\prime\prime})\chi_{B}(b)\alpha^{2}_{A}(a)\otimes\beta^{2}_{B}(b^{\prime\prime}))+\chi_{A}(a^{\prime\prime})(\chi_{B}(b)\alpha_{A}(a)\alpha_{A}(a^{\prime})\otimes\beta_{B}(b^{\prime}b^{\prime\prime})$
$\displaystyle+\chi_{A}(a^{\prime})\alpha^{2}_{A}(a)\otimes\alpha_{B}(b)(b^{\prime}b^{\prime\prime})+\lambda\chi_{A}(a^{\prime})\chi_{B}(b)\alpha^{2}_{A}(a)\otimes\beta_{B}(b^{\prime}b^{\prime\prime}))$
$\displaystyle+\lambda\chi_{A}(a^{\prime\prime})\chi_{B}(b^{\prime})(\chi_{B}(b)\alpha_{A}(a)\alpha_{A}(a^{\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})+\chi_{A}(a^{\prime})\alpha^{2}_{A}(a)\otimes\alpha_{B}(b)\beta_{B}(b^{\prime\prime})$
$\displaystyle+\lambda\chi_{A}(a^{\prime})\chi_{B}(b)\alpha^{2}_{A}(a)\otimes\beta^{2}_{B}(b^{\prime\prime}))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.5})}}{{=}}$
$\displaystyle\chi_{B}(b^{\prime})\chi_{B}(b)\alpha_{A}(a)(a^{\prime}a^{\prime\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})-\lambda\chi_{B}(b^{\prime})\chi_{A}(a^{\prime})\chi_{A}(a^{\prime\prime})\alpha^{2}_{A}(a)\otimes\alpha_{B}(b)\beta_{B}(b^{\prime\prime})$
$\displaystyle-\lambda^{2}\chi_{B}(b^{\prime})\chi_{A}(a^{\prime})\chi_{A}(a^{\prime\prime})\chi_{B}(b)\alpha^{2}_{A}(a)\otimes\beta^{2}_{B}(b^{\prime\prime})+\chi_{A}(a^{\prime\prime})\chi_{B}(b)\alpha_{A}(a)\alpha_{A}(a^{\prime})\otimes\beta_{B}(b^{\prime})\beta_{B}(b^{\prime\prime})$
$\displaystyle+\chi_{A}(a^{\prime\prime})\chi_{A}(a^{\prime})\alpha^{2}_{A}(a)\otimes\alpha_{B}(b)(b^{\prime}b^{\prime\prime})+\lambda\chi_{A}(a^{\prime\prime})\chi_{A}(a^{\prime})\chi_{B}(b)\alpha^{2}_{A}(a)\otimes\beta_{B}(b^{\prime})\beta_{B}(b^{\prime})$
$\displaystyle+\lambda\chi_{A}(a^{\prime\prime})\chi_{B}(b^{\prime})\chi_{B}(b)\alpha_{A}(a)\alpha_{A}(a^{\prime})\otimes\beta^{2}_{B}(b^{\prime\prime})+\lambda\chi_{A}(a^{\prime\prime})\chi_{B}(b^{\prime})\chi_{A}(a^{\prime})\alpha^{2}_{A}(a)\otimes\alpha_{B}(b)\beta(b^{\prime\prime})$
$\displaystyle+\lambda^{2}\chi_{A}(a^{\prime\prime})\chi_{B}(b^{\prime})\chi_{A}(a^{\prime})\chi_{B}(b)\alpha^{2}_{A}(a)\otimes\beta^{2}_{B}(b^{\prime\prime})=I,$

as desired.

(2) For all $a,a^{\prime}\in A$ and $b,b^{\prime}\in B$,

$\displaystyle\chi_{A\otimes B}((a\otimes b)\cdot_{\chi}(a^{\prime}\otimes
b^{\prime}))$ $\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.6})}}{{=}}$
$\displaystyle\chi_{A\otimes
B}(\chi_{B}(b)aa^{\prime}\otimes\beta_{B}(b^{\prime})+\chi_{A}(a^{\prime})\alpha_{A}(a)\otimes
bb^{\prime}+\lambda\chi_{A}(a^{\prime})\chi_{B}(b)\alpha_{A}(a)\otimes\beta_{B}(b^{\prime}))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.7})}}{{=}}$
$\displaystyle\chi_{B}(b)\chi_{A}(aa^{\prime})\chi_{B}(b^{\prime})+\chi_{A}(a^{\prime})\chi_{A}(a)\chi_{B}(bb^{\prime})+\lambda\chi_{A}(a^{\prime})\chi_{B}(b)\chi_{A}(a)\chi_{B}(b^{\prime})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.5})}}{{=}}$
$\displaystyle-\lambda\chi_{A}(a)\chi_{B}(b)\chi_{A}(a^{\prime})\chi_{B}(b^{\prime})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.7})}}{{=}}$
$\displaystyle-\lambda\chi_{A\otimes B}(a\otimes b)\chi_{A\otimes
B}(a^{\prime}\otimes b^{\prime}),$

finishing the proof. ∎

Dual to Definition 2.14, we have

######  Definition 2.17.

Let $\lambda$ be a given element of $K$. A $\lambda$-coaugmented BiHom-
coassociative coalgebra is a 5-tuple $(A,\Delta,\zeta,\psi,\omega)$ consisting
of a BiHom-coassociative coalgebra $(A,\Delta,\psi,\omega)$ (possibly without
counit) and a linear coaugmentation map $\zeta:K\longrightarrow A$ of weight
$\lambda$ satisfying $\omega\circ\zeta=\zeta$, $\psi\circ\zeta=\zeta$ and

$\displaystyle\Delta\circ\zeta=-\lambda\zeta\otimes\zeta.$ (2.14)

######  Remark 2.18.

By Lemma 2.12, a unitary $\lambda$-infBH-bialgebra is a $\lambda$-coaugmented
BiHom-coassociative coalgebra with the coaugmentation map
$\eta:K\longrightarrow A$ ($\eta(1_{K})=1_{A}$).

######  Proposition 2.19.

Let $(C,\Delta_{C},\zeta_{C},\psi_{C},\omega_{C})$ and
$(D,\Delta_{D},\zeta_{D},\psi_{D},\omega_{D})$ be two $\lambda$-coaugmented
BiHom-coassociative coalgebras. Then

(1) $(C\otimes
D,\Delta_{\zeta},\psi_{C}\otimes\psi_{D},\omega_{C}\otimes\omega_{D})$ is a
BiHom-coassociative coalgebra with the coproduct defined by

$\displaystyle\Delta_{\zeta}(c\otimes
d)=(c_{1}\otimes\mathcal{I}_{D})\otimes(c_{2}\otimes\psi_{D}(d))+(\omega_{C}(c)\otimes
d_{1})\otimes(\mathcal{I}_{C}\otimes d_{2})$
$\displaystyle\qquad\qquad\qquad+\lambda(\omega_{C}(c)\otimes\mathcal{I}_{D})\otimes(\mathcal{I}_{C}\otimes\psi_{D}(d)).$
(2.15)

for all $c\in C$, $d\in D$ and
$\zeta_{C}(1_{K})=\mathcal{I}_{C},\zeta_{D}(1_{K})=\mathcal{I}_{D}$.

(2) Furthermore, $(C\otimes D,\Delta,\zeta_{C\otimes
D},\psi_{C}\otimes\psi_{D},\omega_{C}\otimes\omega_{D})$ is a
$\lambda$-coaugmented BiHom-coassociative coalgebra with the coaugmentation
map given by

$\displaystyle\zeta_{C\otimes D}=\zeta_{C}\otimes\zeta_{D}.$ (2.16)

###### Proof.

Dual to the proof of Proposition 2.16. ∎

######  Remark 2.20.

Proposition 2.16 (2.19) shows that the tensor product of two
$\lambda$-(co)augmented BiHom-(co)associative (co)algebras is closed, which
implies that the category of $\lambda$-(co)augmented BiHom-(co)associative
(co)algebras is a tensor category.

Based on the new (co)algebra structures on the tensor product of two
(co)algebras in Proposition 2.16 (2.19), we have the following results but the
usual tensor product (co)algebra structures do not work.

######  Theorem 2.21.

(1) Let $(A,\mu,\Delta,\varepsilon,\alpha,\beta,\psi,\omega)$ be a counitary
$\lambda$-infBH-bialgebra and view $(A\otimes
A,\cdot_{\varepsilon},\alpha\otimes\alpha,\beta\otimes\beta)$ as a BiHom-
associative algebra as in Proposition 2.16. Then
$\Delta:(A,\mu,\alpha,\beta)\longrightarrow(A\otimes
A,\cdot_{\varepsilon},\alpha\otimes\alpha,\beta\otimes\beta)$ is a morphism of
BiHom-associative algebras.

(2) Let $(C,\mu,\Delta,1,\alpha,\beta,\psi,\omega)$ be a unitary
$\lambda$-infBH-bialgebra and view $(C\otimes
C,\Delta_{\eta},\psi\otimes\psi,\omega\otimes\omega)$ as a BiHom-coassociative
coalgebra as in Proposition 2.19. Then $\mu:(C\otimes
C,\Delta_{\eta},\psi\otimes\psi,\omega\otimes\omega)\longrightarrow(C,\Delta,\psi,\omega)$
is a morphism of BiHom-coassociative coalgebras.

###### Proof.

(1) We only need to prove that
$\Delta(aa^{\prime})=\Delta(a)\cdot_{\varepsilon}\Delta(a^{\prime})$ for all
$a,a^{\prime}\in A$. In fact, we have

$\displaystyle\Delta(a)\cdot_{\varepsilon}\Delta(a^{\prime})$ $\displaystyle=$
$\displaystyle(a_{1}\otimes a_{2})\cdot_{\varepsilon}(a^{\prime}_{1}\otimes
a^{\prime}_{2})$ $\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.6})}}{{=}}$
$\displaystyle\varepsilon(a_{2})a_{1}a^{\prime}_{1}\otimes\beta(a^{\prime}_{2})+\varepsilon(a^{\prime}_{1})\alpha(a_{1})\otimes
a_{2}a^{\prime}_{2}+\lambda\varepsilon(a^{\prime}_{1})\varepsilon(a_{2})\alpha(a_{1})\otimes\beta(a^{\prime}_{2})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.11})}}{{=}}$
$\displaystyle\omega(a)a^{\prime}_{1}\otimes\beta(a^{\prime}_{2})+\alpha(a_{1})\otimes
a_{2}\psi(a^{\prime})+\lambda\alpha\omega(a)\otimes\beta\psi(a^{\prime})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.4})}}{{=}}$
$\displaystyle\Delta(aa^{\prime}).$

(2) Dual to (1). ∎

##  3\. $\lambda$-infBH-Hopf modules

In order to study representations of $\lambda$-infBH-bialgebras, In this
section, we introduce the notion of $\lambda$-infBH-Hopf modules and prove
that (co)modules of (co)quasitriangular $\lambda$-infBH-bialgebras can induce
structures of $\lambda$-infBH-Hopf modules, which are similar to the classical
Hopf algebra theory.

###  3.1. Definition and examples

Based on the notion of $\lambda$-infBH-bialgebra, we get the following natural
definition.

######  Definition 3.1.

Let $\lambda$ be a given element of $K$ and
$(A,\mu,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$ be a
$\lambda$-infBH-bialgebra. A (left) $\lambda$-infBH-Hopf module is a 7-tuple
$(M,\gamma,\rho,\alpha_{M},\beta_{M},$ $\psi_{M},\omega_{M})$, where
$(M,\gamma,\alpha_{M}$, $\beta_{M})$ is a left
$(A,\mu,\alpha_{A},\beta_{A})$-module and $(M,\rho,\psi_{M},\omega_{M})$ is a
left $(A,\Delta,\psi_{A},\omega_{A})$-comodule, such that any two maps of
$\alpha_{M},\beta_{M},\psi_{M},\omega_{M}$ commute and

$\displaystyle\rho\gamma=(\mu\otimes\beta_{M})(\omega_{A}\otimes\rho)+(\alpha_{A}\otimes\gamma)(\Delta\otimes\psi_{M})+\lambda\alpha_{A}\omega_{A}\otimes\beta_{M}\psi_{M}.$
(3.1)

If further $(A,\mu,1,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$
($(A,\mu,\varepsilon,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$) is a
(co)unitary $\lambda$-infBH-bialgebra, then $(M,\gamma,\rho,$
$\alpha_{M},\beta_{M},\psi_{M},\omega_{M})$ is called a (co)unitary (left)
$\lambda$-infBH-Hopf module.

######  Remark 3.2.

(1) In terms of above notations, the compatibility condition of left
$\lambda$-infBH-Hopf module given in Eq.(3.1) may be written as

$\displaystyle(a\triangleright m)_{-1}\otimes(a\triangleright
m)_{0}=\omega_{A}(a)m_{-1}\otimes\beta_{M}(m_{0})+\alpha_{A}(a_{1})\otimes
a_{2}\triangleright\psi_{M}(m)+\lambda\alpha_{A}\omega_{A}(a)\otimes\beta_{M}\psi_{M}(m),$

here we write $\gamma(a\otimes m)=a\triangleright m$ and
$\rho(m)=m_{-1}\otimes m_{0}$.

(2) The right version can be given similarly, which will be used in the
forthcoming paper [19] to construct a $\lambda$-infBH-Hopf bimodule over a
$\lambda$-infBH-bialgebra. In this paper, a $\lambda$-infBH-Hopf module means
the left one.

We provide in the following some relevant examples.

######  Example 3.3.

Let $\lambda$ be a given element of $K$. (1) Let
$(A,\mu,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$ be a
$\lambda$-infBH-bialgebra. Obviously,
$(A,\mu,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$ itself is a
$\lambda$-infBH-Hopf module by taking $\gamma=\mu$ and $\rho=\Delta$
($\nu=\mu$ and $\varphi=\Delta$).

(2) Let $(A,\mu,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$ be a
$\lambda$-infBH-bialgebra, $V$ be a vector space and
$\alpha_{V},\beta_{V},\psi_{V},\omega_{V}:V\longrightarrow V$ be four linear
maps such that any two of them commute. Then $(A\otimes
V,\gamma,\rho,\alpha_{A}\otimes\alpha_{V},\beta_{A}\otimes\beta_{V},\psi_{A}\otimes\psi_{V},\omega_{A}\otimes\omega_{V})$
is a $\lambda$-infBH-Hopf module with the structure maps

$\displaystyle\gamma=\mu\otimes\beta_{V}:A\otimes A\otimes V\longrightarrow
A\otimes V,\ \rho=\Delta\otimes\psi_{V}:A\otimes V\longrightarrow A\otimes
A\otimes V.$

We have the following two more general versions than the above example.

(3) Let $(A,\mu,1,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$ be a
unitary $\lambda$-infBH-bialgebra, $V$ be a vector space, $\alpha_{V}$,
$\beta_{V}$, $\psi_{V}$, $\omega_{V}:V\longrightarrow V$ be four linear maps
such that any two of them commute. Then $(A\otimes
V,\gamma,\rho,\alpha_{A}\otimes\alpha_{V},\beta_{A}\otimes\beta_{V},\psi_{A}\otimes\psi_{V},\omega_{A}\otimes\omega_{V})$
is a $\lambda$-infBH-Hopf module with the structure maps

$\displaystyle\gamma:A\otimes(A\otimes V)\longrightarrow A\otimes V$
$\displaystyle~{}~{}~{}~{}\quad a\otimes(b\otimes n)\longmapsto
ab\otimes\beta_{V}(v)$

and

$\displaystyle\rho:A\otimes V\longrightarrow A\otimes(A\otimes V)$
$\displaystyle~{}~{}~{}\quad a\otimes
v\longmapsto\Delta(a)\otimes\psi_{V}(v)+\lambda\omega_{A}(a)\otimes
1_{A}\otimes\psi_{V}(v)$

for all $a,b\in A$ and $v\in V$.

###### Proof.

In fact, for all $a,b,c\in A$ and $v\in V$, we have

$\displaystyle\gamma(\alpha_{A}\otimes\gamma)(a\otimes b\otimes c\otimes v)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.3})}}{{=}}$
$\displaystyle(ab)\beta_{A}(c)\otimes\beta^{2}_{V}(v)$ $\displaystyle=$
$\displaystyle\gamma(\mu\otimes\beta_{A\otimes V})(a\otimes b\otimes c\otimes
v),$ $\displaystyle(\omega_{A}\otimes\rho)\rho(a\otimes v)$ $\displaystyle=$
$\displaystyle\omega_{A}(a_{1})\otimes a_{21}\otimes
a_{22}\otimes\psi_{V}^{2}(v)+\lambda\omega_{A}(a_{1})\otimes\omega_{A}(a_{2})\otimes
1_{A}\otimes\psi_{V}^{2}(v)$ $\displaystyle+\lambda\omega_{A}^{2}(a)\otimes
1_{A1}\otimes
1_{A2}\otimes\psi_{V}^{2}(v)+\lambda^{2}\omega_{A}^{2}(a)\otimes\omega_{A}(1_{A})\otimes
1_{A}\otimes\psi_{V}^{2}(v)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.9})(\ref{eq:12.30})}}{{=}}$
$\displaystyle a_{11}\otimes
a_{12}\otimes\psi_{A}(a_{2})\otimes\psi_{V}^{2}(v)+\lambda\omega_{A}(a_{1})\otimes\omega_{A}(a_{2})\otimes
1_{A}\otimes\psi_{V}^{2}(v)$ $\displaystyle\hskip 170.71652pt\quad(\hbox{also
by Lemma \ref{lem:12.4}})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.7})(\ref{eq:12.30})}}{{=}}$
$\displaystyle(\Delta\otimes\psi_{A\otimes V})\rho(a\otimes v).$

Thus $(A\otimes
V,\gamma,\alpha_{A}\otimes\alpha_{V},\beta_{A}\otimes\beta_{V})$ is a left
$(A,\mu,\alpha_{A},\beta_{A})$-module and $(A\otimes
V,\rho,\psi_{A}\otimes\psi_{V},\omega_{A}\otimes\omega_{V})$ is a left
$(A,\Delta,\psi,\omega)$-comodule. The compatibility condition can be checked
as follows:

$\displaystyle\rho\gamma(a\otimes(b\otimes v))$ $\displaystyle=$
$\displaystyle(ab)_{1}\otimes(ab)_{2}\otimes\psi_{V}\beta_{V}(v)+\lambda\omega_{A}(ab)\otimes
1_{A}\otimes\psi_{V}\beta_{V}(v)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.4})(\ref{eq:12.3})}}{{=}}$
$\displaystyle\omega_{A}(a)b_{1}\otimes\beta_{A}(b_{2})\otimes\psi_{V}\beta_{V}(v)+\lambda\omega_{A}(a)\omega_{A}(b)\otimes
1_{A}\otimes\psi_{V}\beta_{V}(v)$ $\displaystyle+\alpha_{A}(a_{1})\otimes
a_{2}\psi_{A}(b)\otimes\psi_{V}\beta_{V}(v)+\lambda\alpha_{A}\omega_{A}(a)\otimes\beta_{A}\psi_{A}(b)\otimes\psi_{V}\beta_{V}(v)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.5})}}{{=}}$
$\displaystyle(\mu\otimes\beta_{A\otimes
V})(\omega_{A}\otimes\rho)(a\otimes(b\otimes
v))+(\alpha_{A}\otimes\gamma)(\Delta\otimes\psi_{A\otimes
V})(a\otimes(b\otimes v)),$

completing the proof. ∎

(4) Let $(A,\mu,\Delta,\varepsilon,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$
be a counitary $\lambda$-infBH-bialgebra, $V$ be a vector space, and
$\alpha_{V},\beta_{V},\psi_{V},\omega_{V}:V\longrightarrow V$ be four linear
maps such that any two of them commute. Then $(A\otimes
V,\gamma,\rho,\alpha_{A}\otimes\alpha_{V},\beta_{A}\otimes\beta_{V},\psi_{A}\otimes\psi_{V},\omega_{A}\otimes\omega_{V})$
is a $\lambda$-infBH-Hopf module with the structure maps

$\displaystyle\gamma:A\otimes(A\otimes V)\longrightarrow A\otimes V$
$\displaystyle~{}~{}~{}\quad a\otimes(b\otimes v)\longmapsto
ab\otimes\beta_{V}(v)+\lambda\varepsilon(b)\alpha_{A}(a)\otimes\beta_{V}(v)$

and

$\displaystyle\rho:A\otimes V\longrightarrow A\otimes(A\otimes V)$
$\displaystyle~{}~{}~{}\quad a\otimes v\longmapsto a_{1}\otimes
a_{2}\otimes\psi_{V}(v)$

for all $a,b\in A$ and $v\in V$.

###### Proof.

By the proof of Example 3.3 (3), we know that $(A\otimes
V,\rho,\psi_{A}\otimes\psi_{V},\omega_{A}\otimes\omega_{V})$ is a left
$(A,\Delta,\psi_{A},\omega_{A})$-comodule. For all $a,b,c\in A$ and $v\in V$,
we have

$\displaystyle\alpha_{A}(a)\triangleright(b\triangleright(c\otimes v))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.31})}}{{=}}$
$\displaystyle\alpha_{A}(a)(bc)\otimes\beta^{2}_{V}(v)+\lambda\varepsilon(bc)\alpha_{A}^{2}\otimes\beta_{V}^{2}(v)$
$\displaystyle+\lambda\varepsilon(c)\alpha_{A}(a)\alpha_{A}(b)\otimes\beta_{V}^{2}(v)+\lambda^{2}\varepsilon(b)\varepsilon(c)\alpha_{A}^{2}(a)\otimes\beta_{V}^{2}(v)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.3})}}{{=}}$
$\displaystyle(ab)\beta_{A}(c)\otimes\beta^{2}_{V}(v)-\lambda^{2}\varepsilon(b)\varepsilon(c)\alpha_{A}^{2}(a)\otimes\beta_{V}^{2}(v)$
$\displaystyle+\lambda\varepsilon(c)\alpha_{A}(a)\alpha_{A}(b)\otimes\beta_{V}^{2}(v)+\lambda^{2}\varepsilon(b)\varepsilon(c)\alpha_{A}^{2}(a)\otimes\beta_{V}^{2}(v)~{}(\hbox{by
Lemma \ref{lem:12.4}})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.2})}}{{=}}$
$\displaystyle(ab)\beta_{A}(c)\otimes\beta^{2}_{V}(v)+\lambda\varepsilon(c)\alpha_{A}(ab)\otimes\beta_{V}^{2}(v)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.31})}}{{=}}$ $\displaystyle
ab\triangleright\beta_{A\otimes V}(c\otimes v).$

Thus $(A\otimes
V,\gamma,\alpha_{A}\otimes\alpha_{V},\beta_{A}\otimes\beta_{V})$ is a left
$(A,\mu,\alpha_{A},\beta_{A})$-module. Then it remains to check the
compatibility condition of the left $\lambda$-infBH-Hopf module.

$\displaystyle\rho\gamma(a\otimes(b\otimes v))$ $\displaystyle=$
$\displaystyle\Delta(ab)\otimes\beta_{V}\psi_{V}(v)+\lambda\varepsilon(b)\alpha_{A}(a)_{1}\otimes\alpha_{A}(a)_{2}\otimes\beta_{V}\psi_{V}(v)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.4})(\ref{eq:12.2})}}{{=}}$
$\displaystyle\omega_{A}(a)b_{1}\otimes\beta_{A}(b_{2})\otimes\beta_{V}\psi_{V}(v)+\alpha_{A}(a_{1})\otimes
a_{2}\psi_{A}(b)\otimes\beta_{V}\psi_{V}(v)$
$\displaystyle+\lambda\varepsilon(b)\alpha_{A}(a_{1})\otimes\alpha_{A}(a_{2})\otimes\psi_{V}\beta_{V}(v)+\lambda\alpha_{A}\omega_{A}(a)\otimes\beta_{A}\psi_{A}(b)\otimes\beta_{V}\psi_{V}(v)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.11})}}{{=}}$
$\displaystyle(\mu\otimes\beta_{A\otimes
V})(\omega_{A}\otimes\rho)(a\otimes(b\otimes
v))+(\alpha_{A}\otimes\gamma)(\Delta\otimes\psi_{A\otimes
V})(a\otimes(b\otimes v)).$

These finish the proof since other conditions are obviously satisfied. ∎

(5) Let $(A,\mu,\Delta,1_{A},\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$ be a
unitary $0$-infBH-bialgebra such that $\alpha_{A}$ is invertible and
$(N,\rho,\psi_{N},\omega_{N})$ be a left
$(A,\Delta,\psi_{A},\omega_{A})$-comodule,
$\alpha_{N},\beta_{N}:N\longrightarrow N$ be linear maps such that any two
maps of $\alpha_{N},\beta_{N},\psi_{N},\omega_{N}$ commute. Then the space
$A\otimes N$ is a $0$-infBH-Hopf module with the structure maps

$\displaystyle\gamma:A\otimes(A\otimes N)\longrightarrow A\otimes N$
$\displaystyle~{}~{}~{}~{}\quad a\otimes(b\otimes n)\longmapsto
ab\otimes\beta_{N}(n)$

and

$\displaystyle\rho:A\otimes N\longrightarrow A\otimes(A\otimes N)$
$\displaystyle~{}~{}~{}\quad a\otimes
n\longmapsto\Delta(a)\otimes\psi_{N}(n)+\omega_{A}\alpha^{-1}_{A}(a)n_{-1}\otimes
1_{A}\otimes n_{0}$

for all $a,b\in A$ and $n\in N$.

###### Proof.

For all $a,b\in A$, $n\in N$, we have

$\displaystyle(\omega_{A}\otimes\rho)\rho(a\otimes n)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.5})}}{{=}}$
$\displaystyle\omega_{A}(a_{1})\otimes a_{21}\otimes
a_{22}\otimes\psi_{N}^{2}(n)+\omega_{A}(a_{1})\otimes\omega_{A}\alpha_{A}^{-1}(a_{2})\psi_{N}(n)_{-1}\otimes
1_{A}\otimes\psi_{N}(n)_{0}$
$\displaystyle+\omega_{A}^{2}\alpha_{A}^{-1}(a)\omega_{A}(n_{-1})\otimes
1_{A1}\otimes
1_{A2}\otimes\psi_{N}(n_{0})+\omega_{A}^{2}\alpha_{A}^{-1}(a)\omega_{A}(n_{-1})$
$\displaystyle\otimes\omega_{A}\alpha_{A}^{-1}(1_{A})n_{0-1}\otimes
1_{A}\otimes n_{00}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.9})}}{{=}}$ $\displaystyle
a_{11}\otimes
a_{12}\otimes\psi_{A}(a_{2})\otimes\psi_{N}^{2}(n)+\omega_{A}^{2}\alpha_{A}^{-1}(a)n_{-11}\otimes\beta_{A}(n_{-12})\otimes
1_{A}\otimes\psi_{N}(n_{0})$
$\displaystyle+\omega_{A}(a_{1})\otimes\omega_{A}\alpha_{A}^{-1}(a_{2})\psi_{A}(n_{-1})\otimes
1_{A}\otimes\psi_{N}(n_{0})\quad(\hbox{by Lemma \ref{lem:12.4}})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.4})}}{{=}}$
$\displaystyle(\Delta\otimes\psi_{A\otimes N})\rho(a\otimes n),$

and

$\displaystyle\rho\gamma(a\otimes(b\otimes n))$ $\displaystyle=$
$\displaystyle(ab)_{1}\otimes(ab)_{2}\otimes\psi_{N}\beta_{N}(n)+\omega_{A}\alpha_{A}^{-1}(ab)\beta_{A}(n_{-1})\otimes
1_{A}\otimes\beta_{N}(n_{0})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.4})}}{{=}}$
$\displaystyle\omega_{A}(a)b_{1}\otimes\beta_{A}(b_{2})\otimes\psi_{N}\beta_{N}(n)+\alpha_{A}(a_{1})\otimes
a_{2}\psi_{A}(b)\otimes\psi_{N}\beta_{N}(n)$
$\displaystyle+\omega_{A}\alpha_{A}^{-1}(ab)\beta_{A}(n_{-1})\otimes
1_{A}\otimes\beta_{N}(n_{0})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.3})}}{{=}}$
$\displaystyle\omega_{A}(a)b_{1}\otimes\beta_{A}(b_{2})\otimes\psi_{N}\beta_{N}(n)+\omega_{A}(a)(\omega_{A}\alpha_{A}^{-1}(b)n_{-1})\otimes
1_{A}\otimes\beta_{N}(n_{0})$ $\displaystyle+\alpha_{A}(a_{1})\otimes
a_{2}\psi_{A}(b)\otimes\psi_{N}\beta_{N}(n)$ $\displaystyle=$
$\displaystyle(\mu\otimes\beta_{A\otimes
N})(\omega_{A}\otimes\rho)(a\otimes(b\otimes
n))+(\alpha_{A}\otimes\gamma)(\Delta\otimes\psi_{A\otimes
N})(a\otimes(b\otimes n)).$

The rest is obvious by Example 3.3 (3). These complete the proof. ∎

(6) Let $(A,\mu,\Delta,\varepsilon,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$
be a counitary $0$-infBH-bialgebra such that $\omega_{A}$ is invertible and
$(N,\triangleright,\alpha_{N},\beta_{N})$ be a left
$(A,\mu,\alpha_{A},\beta_{A})$-module, $\psi_{N},\omega_{N}:N\longrightarrow
N$ be linear maps such that any two maps of
$\alpha_{N},\beta_{N},\psi_{N},\omega_{N}$ commute. Then the space $A\otimes
N$ is a $0$-infBH-Hopf module with the structure maps

$\displaystyle\gamma:A\otimes(A\otimes N)\longrightarrow A\otimes N$
$\displaystyle~{}~{}~{}\quad a\otimes(b\otimes n)\longmapsto
ab\otimes\beta_{N}(n)+\varepsilon(b)\alpha_{A}\omega^{-1}_{A}(a_{1})\otimes(a_{2}\triangleright
n)$

and

$\displaystyle\rho:A\otimes N\longrightarrow A\otimes(A\otimes N)$
$\displaystyle~{}~{}~{}\quad a\otimes n\longmapsto a_{1}\otimes
a_{2}\otimes\psi_{N}(n)$

for all $a,b\in A$ and $n\in N$.

###### Proof.

By Example 3.3 (4), we know that $(A\otimes
N,\rho,\psi_{A}\otimes\psi_{N},\omega_{A}\otimes\omega_{N})$ is a left
$(A,\Delta,\psi_{A},\omega_{A})$-comodule. For any $a,b,c\in A$ and $n\in N$,
we have

$\displaystyle\alpha_{A\otimes N}(a\triangleright(b\otimes n))$
$\displaystyle=$
$\displaystyle\alpha_{A}(ab)\otimes\alpha_{N}\beta_{N}(n)+\varepsilon(b)\alpha^{2}_{A}\omega^{-1}_{A}(a_{1})\otimes\alpha_{N}(a_{2}\triangleright
n)$ $\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.2})(\ref{eq:1.13})}}{{=}}$
$\displaystyle\alpha_{A}(a)\alpha_{A}(b)\otimes\alpha_{N}\beta_{N}(n)+\varepsilon(b)\alpha^{2}_{A}\omega^{-1}_{A}(a_{1})\otimes(\alpha_{A}(a_{2})\triangleright\alpha_{N}(n))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.31})}}{{=}}$
$\displaystyle\alpha_{A}(a)\triangleright\alpha_{A\otimes N}(b\otimes n).$

Similarly, $\beta_{A\otimes N}(a\triangleright(b\otimes
n))=\beta_{A}(a)\triangleright\beta_{A\otimes N}(b\otimes n)$.

$\displaystyle\alpha_{A}(a)\triangleright(b\triangleright(c\otimes n))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.31})}}{{=}}$
$\displaystyle\alpha_{A}(a)(bc)\otimes\beta^{2}_{N}(n)+\varepsilon(bc)\alpha_{A}\omega^{-1}_{A}\alpha_{A}(a)_{1}\otimes(\alpha_{A}(a)_{2}\triangleright\beta_{N}(n))$
$\displaystyle+\varepsilon(c)\alpha_{A}(a)\alpha_{A}\omega^{-1}_{A}(b_{1})\otimes\beta_{N}(b_{2}\triangleright
n)+\varepsilon(c)\varepsilon(b_{1})\alpha_{A}\omega^{-1}_{A}\alpha_{A}(a)_{1}$
$\displaystyle\otimes(\alpha_{A}(a)_{2}\triangleright(b_{2}\triangleright n))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.11})}}{{=}}$
$\displaystyle\alpha_{A}(a)(bc)\otimes\beta^{2}_{N}(n)+\varepsilon(c)\alpha_{A}(a)\alpha_{A}\omega^{-1}_{A}(b_{1})\otimes\beta_{N}(b_{2}\triangleright
n)$
$\displaystyle+\varepsilon(c)\alpha^{2}_{A}\omega^{-1}_{A}(a_{1})\otimes(\alpha_{A}(a_{2})\triangleright(\psi_{A}(b)\triangleright
n))\quad(\hbox{by Lemma \ref{lem:12.4}})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.15})(\ref{eq:1.13})}}{{=}}$
$\displaystyle\alpha_{A}(a)(bc)\otimes\beta^{2}_{N}(n)+\varepsilon(c)\alpha_{A}(a)\alpha_{A}\omega^{-1}_{A}(b_{1})\otimes(\beta_{A}(b_{2})\triangleright\beta_{N}(n))$
$\displaystyle+\varepsilon(c)\alpha^{2}_{A}\omega^{-1}_{A}(a_{1})\otimes(a_{2}\psi_{A}(b)\triangleright\beta_{N}(n))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.3})(\ref{eq:12.4})}}{{=}}$
$\displaystyle(ab)\beta_{A}(c)\otimes\beta^{2}_{N}(n)+\varepsilon(c)\alpha_{A}\omega^{-1}_{A}(ab)_{1}\otimes((ab)_{2}\triangleright\beta_{N}(n))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.31})}}{{=}}$ $\displaystyle
ab\triangleright\beta_{A\otimes N}(c\otimes n).$

Thus $(A\otimes
V,\gamma,\alpha_{A}\otimes\alpha_{V},\beta_{A}\otimes\beta_{V})$ is a left
$(A,\mu,\alpha_{A},\beta_{A})$-module. Then it remains to check the
compatibility condition of the left $0$-infBH-Hopf module.

$\displaystyle\rho\gamma(a\otimes(b\otimes n))$ $\displaystyle=$
$\displaystyle\Delta(ab)\otimes\beta_{N}\psi_{N}(n)+\varepsilon(b)\Delta(\alpha_{A}\omega^{-1}_{A}(a_{1}))\otimes\psi_{N}(a_{2}\triangleright
n)$ $\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.4})}}{{=}}$
$\displaystyle\omega_{A}(a)b_{1}\otimes\beta_{A}(b_{2})\otimes\beta_{N}\psi_{N}(n)+\alpha_{A}(a_{1})\otimes
a_{2}\psi_{A}(b)\otimes\beta_{N}\psi_{N}(n)$
$\displaystyle+\varepsilon(b)\alpha_{A}\omega^{-1}_{A}(a_{11})\otimes\alpha_{A}\omega^{-1}_{A}(a_{12})\otimes\psi_{N}(a_{2}\triangleright
n)$ $\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.9})}}{{=}}$
$\displaystyle\omega_{A}(a)b_{1}\otimes\beta_{A}(b_{2})\otimes\beta_{N}\psi_{N}(n)+\alpha_{A}(a_{1})\otimes
a_{2}\psi_{A}(b)\otimes\beta_{N}\psi_{N}(n)$
$\displaystyle+\varepsilon(b)\alpha_{A}(a_{1})\otimes\alpha_{A}\omega^{-1}_{A}(a_{21})\otimes(a_{22}\triangleright\psi_{N}(n))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.31})}}{{=}}$
$\displaystyle(\mu\otimes\beta_{A\otimes
N})(\omega_{A}(a)\otimes\Delta(b)\otimes\psi_{N}(n))+(\alpha_{A}\otimes\gamma)(\Delta(a)\otimes\psi_{A}(b)\otimes\psi_{N}(n))$
$\displaystyle=$ $\displaystyle(\mu\otimes\beta_{A\otimes
N})(\omega_{A}\otimes\rho)(a\otimes(b\otimes
n))+(\alpha_{A}\otimes\gamma)(\Delta\otimes\psi_{A\otimes
N})(a\otimes(b\otimes n)).$

These complete the proof. ∎

######  Remark 3.4.

If $\lambda=0$ in Example 3.3 (3) and (4), then we can get Example 3.3 (2).

###  3.2. Modules over (anti)quasitriangular $\lambda$-infBH-bialgebras

In this subsection, we prove that every module over (anti)quasitriangular
$\lambda$-infBH-bialgebra can induce a $\lambda$-infBH-Hopf module.

####  3.2.1. First approach

First we provide a characterization of $\lambda$-infBH-bialgebra by an element
$r\in A\otimes A$.

Let $(A,\mu,1,\alpha,\beta)$ be a unitary BiHom-algebra,
$\psi,\omega:A\longrightarrow A$ be two linear maps such that Eqs.(2.1), (2.3)
and (2.5) hold, $r\in A\otimes A$ be an $\alpha,\beta,\psi,\omega$-invariant
element. We mean an element $r\in A\otimes A$ is $f$-invariant if $(f\otimes
f)(r)=r$, where $f:A\longrightarrow A$ is a linear map. Define a linear map
$\Delta_{r}:A\longrightarrow A\otimes A$ by

$\displaystyle\Delta_{r}(a)=\alpha^{-1}(a)\triangleright
r-r\triangleleft\beta^{-1}(a)-\lambda(\omega(a)\otimes
1),~{}~{}~{}\forall~{}~{}a\in A,$ (3.2)

i.e.,

$\displaystyle\Delta_{r}(a)=\omega\alpha^{-1}(a)r^{1}\otimes\beta(r^{2})-\alpha(r^{1})\otimes
r^{2}\psi\beta^{-1}(a)-\lambda(\omega(a)\otimes 1).$

######  Lemma 3.5.

The map $\Delta_{r}$ defined by Eq.(3.2) is a $\lambda$-BiHom-derivation.

###### Proof.

We only check Eq.(2.8) for $\Delta_{r}$ as follows. For all $a,b\in A$,

$\displaystyle a\triangleright\Delta_{r}(b)+\Delta_{r}(a)\triangleleft
b+\lambda\alpha\omega(a)\otimes\beta\psi(b)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:14.1})(\ref{eq:14.2})(\ref{eq:1.5})}}{{=}}$
$\displaystyle\omega(a)(\omega\alpha^{-1}(b)r^{1})\otimes\beta^{2}(r^{2})-\omega(a)\alpha(r^{1})\otimes\beta(r^{2})\psi(b)-\lambda\omega(a)\omega(b)\otimes
1$
$\displaystyle+\omega(a)\alpha(r^{1})\otimes\beta(r^{2})\psi(b)-\alpha^{2}(r^{1})\otimes(r^{2}\psi\beta^{-1}(a))\psi(b)-\lambda\alpha\omega(a)\otimes\beta\psi(b)$
$\displaystyle+\lambda\alpha\omega(a)\otimes\beta\psi(b)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.3})}}{{=}}$
$\displaystyle(\omega\alpha^{-1}(a)\omega\alpha^{-1}(b))r^{1}\otimes\beta(r^{2})-\omega(a)\alpha(r^{1})\otimes\beta(r^{2})\psi(b)-\lambda\omega(a)\omega(b)\otimes
1$
$\displaystyle+\omega(a)\alpha(r^{1})\otimes\beta(r^{2})\psi(b)-\alpha(r^{1})\otimes
r^{2}(\psi\beta^{-1}(a)\psi\beta^{-1}(b))$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.3})(\ref{eq:1.2})}}{{=}}$
$\displaystyle\omega\alpha^{-1}(ab)r^{1}\otimes\beta(r^{2})-\alpha(r^{1})\otimes
r^{2}\psi\beta^{-1}(ab)-\lambda\omega(ab)\otimes 1$ $\displaystyle=$
$\displaystyle\Delta_{r}(ab),$

as desired. ∎

For convenience, we follow the notations in [15] or [21]. Let
$(A,\mu,\alpha,\beta)$ be a unitary BiHom-associative algebra,
$\psi,\omega:A\longrightarrow A$ be linear maps, $r\in A\otimes A$. We define
the following elements in $A\otimes A\otimes A$:

$\displaystyle r_{12}r_{23}=\alpha(r^{1})\otimes
r^{2}\bar{r}^{1}\otimes\beta(\bar{r}^{2}),~{}~{}r_{13}r_{12}=\omega(r^{1})\bar{r}^{1}\otimes\beta(\bar{r}^{2})\otimes\alpha\psi(r^{2}),$
$\displaystyle
r_{23}r_{13}=\beta\omega(r^{1})\otimes\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\psi(r^{2}),\qquad
r_{13}=\omega(r^{1})\otimes 1\otimes\psi(r^{2}),$ $\displaystyle
r_{12}=r\otimes 1,\qquad r_{23}=1\otimes r.$

######  Theorem 3.6.

Let $(A,\mu,\alpha,\beta)$ be a unitary BiHom-associative algebra such that
$\alpha,\beta$ are bijective, $\psi,\omega:A\longrightarrow A$ be linear maps,
$r=r^{1}\otimes r^{2}\in A\otimes A$ be $\alpha,\beta,\psi,\omega$-invariant
and moreover Eqs.(2.1), (2.3) and (2.5) hold. Then the $\lambda$-BiHom-
derivation $\Delta_{r}$ defined by Eq.(3.2) is BiHom-coassociative if and only
if

$\displaystyle\omega\alpha^{-1}(a)\triangleright(r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}-\lambda
r_{13})=(r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}-\lambda
r_{13})\triangleleft\psi\beta^{-1}(a).$ (3.3)

###### Proof.

For all $a\in A$ and $\bar{r}=r$, on one hand,

$\displaystyle(\Delta_{r}\otimes\psi)\circ\Delta_{r}(a)$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{eq:12.30})}}{{=}}(\omega^{2}\alpha^{-2}(a)\omega\alpha^{-1}(r^{1}))\bar{r}^{1}\otimes\beta(\bar{r}^{2})\otimes\psi\beta(r^{2})-\alpha(\bar{r}^{1})\otimes\bar{r}^{2}(\alpha^{-1}\beta^{-1}\psi\omega(a)\psi\beta^{-1}(r^{1}))\otimes\psi\beta(r^{2})$
$\displaystyle\qquad\qquad-\lambda\omega^{2}\alpha^{-1}(a)\omega(r^{1})\otimes
1\otimes\psi\beta(r^{2})-\omega(r^{1})\bar{r}^{1}\otimes\beta(\bar{r}^{2})\otimes\psi(r^{2})\psi^{2}\beta^{-1}(a)$
$\displaystyle\qquad\qquad+\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\alpha\beta^{-1}\psi(r^{1})\otimes\psi(r^{2})\psi^{2}\beta^{-1}(a)+\lambda\alpha\omega(r^{1})\otimes
1\otimes\psi(r^{2})\psi^{2}\beta^{-1}(a)$
$\displaystyle\qquad\qquad-\lambda\alpha^{-1}\omega^{2}(a)r^{1}\otimes\beta(r^{2})\otimes
1+\lambda\alpha(r^{1})\otimes r^{2}\psi\beta^{-1}\omega(a)\otimes
1+\lambda^{2}\omega^{2}(a)\otimes 1\otimes 1$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{eq:1.3})}}{{=}}\omega^{2}\alpha^{-1}(a)(\omega\alpha^{-1}(r^{1})\beta^{-1}(\bar{r}^{1}))\otimes\beta(\bar{r}^{2})\otimes\psi\beta(r^{2})$
$\displaystyle\qquad\qquad-\alpha(\bar{r}^{1})\otimes\bar{r}^{2}(\alpha^{-1}\beta^{-1}\psi\omega(a)\psi\beta^{-1}(r^{1}))\otimes\psi\beta(r^{2})$
$\displaystyle\qquad\qquad-\lambda\omega^{2}\alpha^{-1}(a)\omega(r^{1})\otimes
1\otimes\psi\beta(r^{2})-\omega(r^{1})\bar{r}^{1}\otimes\beta(\bar{r}^{2})\otimes\psi(r^{2})\psi^{2}\beta^{-1}(a)$
$\displaystyle\qquad\qquad+\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\alpha\beta^{-1}\psi(r^{1})\otimes\psi(r^{2})\psi^{2}\beta^{-1}(a)+\lambda\alpha\omega(r^{1})\otimes
1\otimes\psi(r^{2})\psi^{2}\beta^{-1}(a)$
$\displaystyle\qquad\qquad-\lambda\alpha^{-1}\omega^{2}(a)r^{1}\otimes\beta(r^{2})\otimes
1+\lambda\alpha(r^{1})\otimes r^{2}\psi\beta^{-1}\omega(a)\otimes
1+\lambda^{2}\omega^{2}(a)\otimes 1\otimes 1$
$\displaystyle\quad\stackrel{{\scriptstyle(\ref{eq:1.5})(\ref{eq:14.3})(\ref{eq:14.4})}}{{=}}\omega\alpha^{-1}(a)\triangleright(\omega\alpha^{-1}(r^{1})\beta^{-1}(\bar{r}^{1})\otimes\bar{r}^{2}\otimes\psi(r^{2}))$
$\displaystyle\qquad\qquad-\alpha(\bar{r}^{1})\otimes\bar{r}^{2}(\alpha^{-1}\beta^{-1}\psi\omega(a)\psi\beta^{-1}(r^{1}))\otimes\psi\beta(r^{2})$
$\displaystyle\qquad\qquad-\lambda\omega\alpha^{-1}(a)\triangleright(\omega(r^{1})\otimes
1\otimes\psi(r^{2}))-(\omega\alpha^{-1}(r^{1})\alpha^{-1}(\bar{r}^{1})\otimes\alpha^{-1}\beta(\bar{r}^{2})\otimes\psi(r^{2}))\triangleleft\psi\beta^{-1}(a)$
$\displaystyle\qquad\qquad+(\bar{r}^{1}\otimes\alpha^{-1}(\bar{r}^{2})\beta^{-1}\psi(r^{1})\otimes\psi(r^{2}))\triangleleft\psi\beta^{-1}(a)+\lambda(\omega(r^{1})\otimes
1\otimes\psi(r^{2}))\triangleleft\psi\beta^{-1}(a)$
$\displaystyle\qquad\qquad-\lambda\omega\alpha^{-1}(a)\triangleright(r^{1}\otimes
r^{2}\otimes 1)+\lambda\alpha(r^{1})\otimes
r^{2}\psi\beta^{-1}\omega(a)\otimes 1+\lambda^{2}\omega^{2}(a)\otimes 1\otimes
1$ $\displaystyle\qquad\quad=\omega\alpha^{-1}(a)\triangleright
r_{13}r_{12}-\alpha(\bar{r}^{1})\otimes\bar{r}^{2}(\alpha^{-1}\beta^{-1}\psi\omega(a)\beta^{-1}(r^{1}))\otimes\beta(r^{2})$
$\displaystyle\qquad\qquad-\lambda\omega\alpha^{-1}(a)\triangleright
r_{13}-r_{13}r_{12}\triangleleft\psi\beta^{-1}(a)+r_{12}r_{23}\triangleleft\psi\beta^{-1}(a)+\lambda
r_{13}\triangleleft\psi\beta^{-1}(a)$
$\displaystyle\qquad\qquad-\lambda\omega\alpha^{-1}(a)\triangleright
r_{12}+\lambda\alpha(r^{1})\otimes r^{2}\psi\beta^{-1}\omega(a)\otimes
1+\lambda^{2}\omega^{2}(a)\otimes 1\otimes 1.$

On the other hand,

$\displaystyle(\omega\otimes\Delta_{r})\circ\Delta_{r}(a)$
$\displaystyle\quad\stackrel{{\scriptstyle(\ref{eq:12.30})}}{{=}}\omega^{2}\alpha^{-1}(a)\omega(r^{1})\otimes(\omega\alpha^{-1}\beta(r^{2})\bar{r}^{1}\otimes\beta(\bar{r}^{2})-\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\psi\beta^{-1}\beta(r^{2})-\lambda\omega\beta(r^{2})\otimes
1)$
$\displaystyle\qquad-\omega\alpha(r^{1})\otimes(\omega\alpha^{-1}(r^{2}\psi\beta^{-1}(a))\bar{r}^{1}\otimes\beta(\bar{r}^{2})-\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\psi\beta^{-1}(r^{2}\psi\beta^{-1}(a))$
$\displaystyle\qquad-\lambda\omega(r^{2}\psi\beta^{-1}(a))\otimes
1)-\lambda\omega^{2}(a)\otimes(1\cdot
r^{1}\otimes\beta(r^{2})-\alpha(r^{1})\otimes r^{2}\cdot 1-\lambda 1\otimes
1)$
$\displaystyle\quad\stackrel{{\scriptstyle(\ref{eq:1.5})}}{{=}}\omega^{2}\alpha^{-1}(a)\omega(r^{1})\otimes\omega\alpha^{-1}\beta(r^{2})\bar{r}^{1}\otimes\beta(\bar{r}^{2})-\omega^{2}\alpha^{-1}(a)\omega(r^{1})\otimes\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\psi(r^{2})$
$\displaystyle\qquad-\lambda\omega^{2}\alpha^{-1}(a)\omega(r^{1})\otimes\omega\beta(r^{2})\otimes
1-\omega\alpha(r^{1})\otimes(\omega\alpha^{-1}(r^{2})\alpha^{-1}\beta^{-1}\psi\omega(a))\bar{r}^{1}\otimes\beta(\bar{r}^{2})$
$\displaystyle\qquad+\omega\alpha(r^{1})\otimes\alpha(\bar{r}^{1})\otimes\bar{r}^{2}(\psi\beta^{-1}(r^{2})\psi^{2}\beta^{-2}(a))+\lambda\omega\alpha(r^{1})\otimes\omega(r^{2})\omega\psi\beta^{-1}(a)\otimes
1$ $\displaystyle\qquad-\lambda\omega^{2}(a)\otimes r^{1}\otimes
r^{2}+\lambda\omega^{2}(a)\otimes r^{1}\otimes
r^{2}+\lambda^{2}\omega^{2}(a)\otimes 1\otimes 1$
$\displaystyle\quad\stackrel{{\scriptstyle(\ref{eq:1.3})}}{{=}}\omega^{2}\alpha^{-1}(a)\omega(r^{1})\otimes\omega\alpha^{-1}\beta(r^{2})\bar{r}^{1}\otimes\beta(\bar{r}^{2})-\omega^{2}\alpha^{-1}(a)\omega(r^{1})\otimes\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\psi(r^{2})$
$\displaystyle\qquad-\lambda\omega^{2}\alpha^{-1}(a)\omega(r^{1})\otimes\omega\beta(r^{2})\otimes
1-\omega\alpha(r^{1})\otimes\omega(r^{2})(\alpha^{-1}\beta^{-1}\psi\omega(a)\beta^{-1}(\bar{r}^{1}))\otimes\beta(\bar{r}^{2})$
$\displaystyle\qquad+\omega\alpha(r^{1})\otimes\alpha(\bar{r}^{1})\otimes(\alpha^{-1}(\bar{r}^{2})\psi\beta^{-1}(r^{2}))\psi^{2}\beta^{-1}(a))+\lambda\omega\alpha(r^{1})\otimes\omega(r^{2})\omega\psi\beta^{-1}(a)\otimes
1$ $\displaystyle\qquad+\lambda^{2}\omega^{2}(a)\otimes 1\otimes 1$
$\displaystyle~{}\stackrel{{\scriptstyle(\ref{eq:14.3})(\ref{eq:14.4})}}{{=}}\omega\alpha^{-1}(a)\triangleright(\omega(r^{1})\otimes\omega\alpha^{-1}(r^{2})\beta^{-1}(\bar{r}^{1})\otimes\bar{r}^{2})$
$\displaystyle\qquad-\omega\alpha^{-1}(a)\triangleright(\omega(r^{1})\otimes\alpha\beta^{-1}(\bar{r}^{1})\otimes\beta^{-1}(\bar{r}^{2})\psi\beta^{-1}(r^{2}))$
$\displaystyle\qquad-\lambda\omega\alpha^{-1}(a)\triangleright(\omega(r^{1})\otimes\omega(r^{2})\otimes
1)-\omega\alpha(r^{1})\otimes\omega(r^{2})(\alpha^{-1}\beta^{-1}\psi\omega(a)\beta^{-1}(\bar{r}^{1}))\otimes\beta(\bar{r}^{2})$
$\displaystyle\qquad+(\omega(r^{1})\otimes\bar{r}^{1}\otimes\alpha^{-1}(\bar{r}^{2})\psi\beta^{-1}(r^{2}))\triangleleft\psi\beta^{-1}(a))+\lambda\omega\alpha(r^{1})\otimes\omega(r^{2})\omega\psi\beta^{-1}(a)\otimes
1$ $\displaystyle\qquad+\lambda^{2}\omega^{2}(a)\otimes 1\otimes 1$
$\displaystyle\quad=\omega\alpha^{-1}(a)\triangleright
r_{12}r_{23}-\omega\alpha^{-1}(a)\triangleright
r_{23}r_{13}-\lambda\omega\alpha^{-1}(a)\triangleright r_{12}$
$\displaystyle\qquad-\alpha(r^{1})\otimes
r^{2}(\alpha^{-1}\beta^{-1}\psi\omega(a)\beta^{-1}(\bar{r}^{1}))\otimes\beta(\bar{r}^{2})+r_{23}r_{13}\triangleleft\psi\beta^{-1}(a)$
$\displaystyle\qquad+\lambda\alpha(r^{1})\otimes
r^{2}\omega\psi\beta^{-1}(a)\otimes 1+\lambda^{2}\omega^{2}(a)\otimes 1\otimes
1.$

Therefore $\Delta_{r}$ is BiHom-coassociative if and only if Eq.(3.3) holds,
finishing the proof. ∎

Based on the above result, we introduce the notion of nonhomogeneous
associative BiHom-Yang-Baxter equation.

######  Definition 3.7.

Let $(A,\mu,1,\alpha,\beta)$ be a unitary BiHom-associative algebra,
$\psi,\omega:A\longrightarrow A$ be linear maps and $r\in A\otimes A$. We call

$\displaystyle r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}=\lambda r_{13}$ (3.4)

the $\lambda$-associative BiHom-Yang-Baxter equation (abbr. $\lambda$-abhYBe)
in $(A,\mu,1,\alpha,\beta)$ where $\lambda$ is a given element in $K$.

By Lemma 3.5 and Theorem 3.6, we have

######  Corollary 3.8.

Let $(A,\mu,1,\alpha,\beta)$ be a unitary BiHom-associative algebra such that
$\alpha,\beta$ are bijective, $\psi,\omega:A\longrightarrow A$ be linear maps,
$r=r^{1}\otimes r^{2}\in A\otimes A$ be $\alpha,\beta,\psi,\omega$-invariant
and moreover Eqs.(2.1), (2.3) and (2.5) hold. If $r$ is a solution of the
$\lambda$-abhYBe, then $(A,\mu,\Delta_{r},1,\alpha,\beta,\psi,\omega)$ is a
$\lambda$-infBH-bialgebra, where $\Delta_{r}$ is defined by Eq.(3.2).

######  Definition 3.9.

Under the assumptions of Corollary 3.8, a quasitriangular unitary
$\lambda$-infBH-bialgebra is a 8-tuple $(A,\mu,1,\alpha,\beta,\psi,\omega,r)$
consisting of a unitary BiHom-associative algebra $(A,\mu,1,\alpha,\beta)$ and
a solution $r\in A\otimes A$ of a $\lambda$-abhYBe.

######  Proposition 3.10.

Under the assumptions of Corollary 3.8,
$(A,\mu,\Delta=\Delta_{r},1,\alpha,\beta,\psi,\omega)$, where $\Delta_{r}$ is
defined by Eq.(3.2), is a quasitriangular unitary $\lambda$-infBH-bialgebra if
and only if

$\displaystyle(\Delta\otimes\psi)(r)=-r_{23}r_{13}$ (3.5)

or

$\displaystyle(\omega\otimes\Delta)(r)=r_{13}r_{12}-\lambda(r_{13}+r_{12}).$
(3.6)

holds.

###### Proof.

It is sufficient to prove that Eq.(3.4) is equivalent to Eq.(3.5) or Eq.(3.6).
While

$\displaystyle(\Delta\otimes\psi)(r)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:14.5})}}{{=}}$
$\displaystyle\omega\alpha^{-1}(r^{1})\bar{r}^{1}\otimes\beta(\bar{r}^{2})\otimes\psi(r^{2})-\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\psi\beta^{-1}(r^{1})\otimes\psi(r^{2})$
$\displaystyle-\lambda\omega(r^{1})\otimes 1\otimes\psi(r^{2})$
$\displaystyle=$ $\displaystyle r_{13}r_{12}-r_{12}r_{23}-\lambda r_{13},$

and

$\displaystyle(\omega\otimes\Delta)(r)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:14.5})}}{{=}}$
$\displaystyle\omega(r^{1})\otimes\omega\alpha^{-1}(r^{2})\bar{r}^{1}\otimes\beta(\bar{r}^{2})-\omega(r^{1})\otimes\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\psi\beta^{-1}(r^{2})$
$\displaystyle-\lambda\omega(r^{1})\otimes\omega(r^{2})\otimes 1$
$\displaystyle=$ $\displaystyle r_{12}r_{23}-r_{23}r_{13}-\lambda r_{12},$

as desired. ∎

######  Remark 3.11.

If $\alpha=\beta=\psi=\omega={\rm id}$ and $\lambda=0$ in Proposition 3.10,
then we can obtain [2, Proposition 5.5]. We notice here that any one of
Eqs.(3.5) and (3.6) is equivalent to Eq.(3.4).

$\lambda$-infBH-Hopf modules can be obtained from the modules over
quasitriangular unitary $\lambda$-infBH-bialgebra by the following procedure.

######  Theorem 3.12.

Let $(A,\mu,1,\alpha_{A},\beta_{A},\psi_{A},\omega_{A},r)$ be a
quasitriangular unitary $\lambda$-infBH-bialgebra and
$(M,\gamma,\alpha_{M},\beta_{M})$ be a left
$(A,\mu,\alpha_{A},\beta_{A})$-module, $\psi_{M},\omega_{M}:M\longrightarrow
M$ be linear maps such that $\beta_{M}\circ\psi_{M}=\psi_{M}\circ\beta_{M}$,
$\psi_{M}\circ\gamma=\gamma\circ(\psi_{A}\otimes\psi_{M})$. Then
$(M,\gamma,\rho,\alpha_{M},\beta_{M},\psi_{M},\omega_{M})$ becomes a
$\lambda$-infBH-Hopf module with the coaction $\rho:M\longrightarrow A\otimes
M$ given by

$\displaystyle\rho(m):=-\alpha_{A}(r^{1})\otimes
r^{2}\triangleright\psi_{M}\beta_{M}^{-1}(m),\forall~{}m\in M.$ (3.7)

###### Proof.

We first prove that $(M,\rho,\psi_{M},\omega_{M})$ is a left
$(A,\Delta_{r},\psi_{A},\omega_{A})$-comodule as follows. For all $m\in M$, we
have

$\displaystyle(\Delta_{r}\otimes\psi_{M})\rho(m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:14.8})}}{{=}}$
$\displaystyle\beta_{A}\omega_{A}(\bar{r}^{1})\otimes\alpha_{A}(r^{1})\otimes\alpha_{A}^{-1}(r^{2})\alpha_{A}^{-1}\psi_{A}(\bar{r}^{2})\triangleright\psi_{M}^{2}\beta_{M}^{-1}(m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.15})}}{{=}}$
$\displaystyle\alpha_{A}\omega_{A}(\bar{r}^{1})\otimes\alpha_{A}(r^{1})\otimes
r^{2}\triangleright(\psi_{A}\beta_{A}^{-1}(\bar{r}^{2})\triangleright\psi_{M}^{2}\beta_{M}^{-2}(m))$
$\displaystyle=$ $\displaystyle(\omega_{A}\otimes\rho)\rho(m).$

Next we then check the compatibility condition. For all $a\in A$ and $m\in M$,
we have

$\displaystyle\omega_{A}(a)m_{-1}\otimes\beta_{M}(m_{0})+\alpha_{A}(a_{1})\otimes
a_{2}\triangleright\psi_{M}(m)+\lambda\alpha_{A}\omega_{A}(a)\otimes\beta_{M}\psi_{M}(m)$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{eq:14.5})}}{{=}}-\omega_{A}(a)\alpha_{A}(r^{1})\otimes\beta_{A}(r^{2})\triangleright\psi_{M}(m)+\alpha_{A}(\omega_{A}\alpha_{A}^{-1}(a)r^{1})\otimes\beta_{A}(r^{2})\triangleright\psi_{M}(m)$
$\displaystyle\qquad\quad\qquad-\alpha_{A}^{2}(r^{1})\otimes
r^{2}\psi_{A}\beta_{A}^{-1}(a)\triangleright\psi_{M}(m)-\lambda\alpha_{A}\omega_{A}(a)\otimes
1\triangleright\psi_{M}(m)$
$\displaystyle\qquad\quad\qquad+\lambda\alpha_{A}\omega_{A}(a)\otimes\beta_{M}\psi_{M}(m)$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{eq:1.2})(\ref{eq:1.5})}}{{=}}-\alpha_{A}(r^{1})\otimes\alpha_{A}^{-1}(r^{2})\psi_{A}\beta_{A}^{-1}(a)\triangleright\psi_{M}(m)$
$\displaystyle\hskip
5.69054pt\qquad\stackrel{{\scriptstyle(\ref{eq:1.15})}}{{=}}-\alpha_{A}(r^{1})\otimes
r^{2}(\psi_{A}\beta_{A}^{-1}(a)\triangleright\psi_{M}\beta_{M}^{-1}(m))$
$\displaystyle\qquad\quad=(a\triangleright m)_{-1}\otimes(a\triangleright
m)_{0},$

completing the proof. ∎

####  3.2.2. Second approach

Inspired by Example 2.4, we can define a new comultiplication for
$\lambda$-infBH-bialgebra by replacing $\lambda(\omega(a)\otimes 1)$ in
Eq.(3.2) by $\lambda(1\otimes\psi(a))$, i.e.,

$\displaystyle\widetilde{\Delta}_{r}(a)=\alpha^{-1}(a)\triangleright
r-r\triangleleft\beta^{-1}(a)-\lambda(1\otimes\psi(a)).$ (3.8)

In this case, we only list the parallel results and omit the partial proofs.

######  Proposition 3.13.

The $\widetilde{\Delta}_{r}$ defined by Eq.(3.8) is a $\lambda$-BiHom-
derivation.

######  Theorem 3.14.

Let $(A,\mu,\alpha,\beta)$ be a unitary BiHom-associative algebra such that
$\alpha,\beta$ are bijective, $\psi,\omega:A\longrightarrow A$ be linear maps,
$r=r^{1}\otimes r^{2}\in A\otimes A$ be $\alpha,\beta,\psi,\omega$-invariant
and moreover Eqs.(2.1), (2.3) and (2.5) hold. Then the $\lambda$-BiHom-
derivation $\widetilde{\Delta}_{r}$ defined by Eq.(3.8) is BiHom-coassociative
if and only if

$\displaystyle\omega\alpha^{-1}(a)\triangleright(r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}+\lambda
r_{13})=(r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}+\lambda
r_{13})\triangleleft\psi\beta^{-1}(a).$ (3.9)

######  Remark 3.15.

If we substitute $\lambda$ in Eq.(3.3) by $-\lambda$, then we obtain Eq.(3.9).

######  Definition 3.16.

Under the assumptions of Corollary 3.8, an anti-quasitriangular unitary
$\lambda$-infBH-bialgebra is a 8-tuple $(A,\mu,1,\alpha,\beta,\psi,\omega,r)$
consisting of a unitary BiHom-associative algebra $(A,\mu,1,\alpha,\beta)$ and
a solution $r\in A\otimes A$ of a $(-\lambda)$-abhYBe.

######  Proposition 3.17.

Under the assumption of Corollary 3.8,
$(A,\mu,\Delta=\widetilde{\Delta}_{r},1,\alpha,\beta,\psi,\omega)$, where
$\widetilde{\Delta}_{r}$ is defined by Eq.(3.8), is an anti-quasitriangular
unitary $\lambda$-infBH-bialgebra if and only if

$\displaystyle(\Delta\otimes\psi)(r)=-r_{23}r_{13}-\lambda(r_{23}+r_{13})$
(3.10)

or

$\displaystyle(\omega\otimes\Delta)(r)=r_{13}r_{12}.$ (3.11)

###### Proof.

The result can be proved by the following equalities:

$\displaystyle(\Delta\otimes\psi)(r)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:14.25})}}{{=}}$
$\displaystyle\omega\alpha^{-1}(r^{1})\bar{r}^{1}\otimes\beta(\bar{r}^{2})\otimes\psi(r^{2})-\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\psi\beta^{-1}(r^{1})\otimes\psi(r^{2})$
$\displaystyle-\lambda(1\otimes\psi(r^{1}))\otimes\psi(r^{2})$
$\displaystyle=$ $\displaystyle r_{13}r_{12}-r_{12}r_{23}-\lambda r_{23},$

and

$\displaystyle(\omega\otimes\Delta)(r)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:14.25})}}{{=}}$
$\displaystyle\omega(r^{1})\otimes\omega\alpha^{-1}(r^{2})\bar{r}^{1}\otimes\beta(\bar{r}^{2})-\omega(r^{1})\otimes\alpha(\bar{r}^{1})\otimes\bar{r}^{2}\psi\beta^{-1}(r^{2})$
$\displaystyle-\lambda\omega(r^{1})\otimes 1\otimes\psi(r^{2})$
$\displaystyle=$ $\displaystyle r_{12}r_{23}-r_{23}r_{13}-\lambda r_{13},$

as desired. ∎

######  Remark 3.18.

(1) If $\lambda=0$, then the conditions in Proposition 3.10 and the ones in
Proposition 3.17 are consistent, and in this case Proposition 3.10 and 3.17
are the BiHom-version of [2, Proposition 5.5].

(2) If $\lambda\neq 0$, then the conditions in Proposition 3.10 and the ones
in Proposition 3.17 are different which shows that there are essential
differences between quasitriangular unitary $\lambda$-infBH-bialgebras and the
corresponding anti-case.

(3) When $\lambda=-1$ in Proposition 3.17, we obtain [17, Corollary 2.32]. If
further, the structure maps $\alpha=\beta=\psi=\omega={\rm id}$, Proposition
3.17 is consistent with [5, Lemma 3.19].

$\lambda$-infBH-Hopf modules also can be constructed from the modules over
anti-quasitriangular unitary $\lambda$-infBH-bialgebras, which is different
from Theorem 3.12 for the quasitriangular case.

######  Theorem 3.19.

Let $(A,\mu,1,\alpha_{A},\beta_{A},\psi_{A},\omega_{A},r)$ be an anti-
quasitriangular unitary $\lambda$-infBH-bialgebra and
$(M,\widetilde{\gamma},\alpha_{M},\beta_{M})$ a left
$(A,\mu,\alpha_{A},\beta_{A})$-module, $\psi_{M},\omega_{M}:M\longrightarrow
M$ be linear maps such that $\beta_{M}\circ\psi_{M}=\psi_{M}\circ\beta_{M}$,
$\psi_{M}\circ\widetilde{\gamma}=\widetilde{\gamma}\circ(\psi_{A}\otimes\psi_{M})$.
Then
$(M,\widetilde{\gamma},\widetilde{\rho},\alpha_{M},\beta_{M},\psi_{M},\omega_{M})$
becomes a left $\lambda$-infBH-Hopf module with the coaction
$\widetilde{\rho}:M\longrightarrow A\otimes M$ given by

$\displaystyle\widetilde{\rho}(m):=-\alpha_{A}(r^{1})\otimes
r^{2}\triangleright\psi_{M}\beta_{M}^{-1}(m)-\lambda
1\otimes\psi_{M}(m),\forall~{}m\in M.$ (3.12)

###### Proof.

We first prove that $(M,\widetilde{\rho},\psi_{M},\omega_{M})$ is a left
$(A,\widetilde{\Delta_{r}},\psi_{A},\omega_{A})$-comodule as follows. For all
$m\in M$, we have

$\displaystyle(\widetilde{\Delta_{r}}\otimes\psi_{M})\widetilde{\rho}(m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:14.28})}}{{=}}$
$\displaystyle\lambda\omega_{A}(r^{1})\otimes
1\otimes\alpha_{A}^{-1}\psi_{A}(r^{2})\triangleright\psi_{M}^{2}\beta_{M}^{-1}(m)+\lambda
1\otimes
r^{1}\otimes\alpha_{A}^{-1}(r^{2})\triangleright\psi_{M}^{2}\beta_{M}^{-1}(m)$
$\displaystyle+\beta_{A}\omega_{A}(r^{1})\otimes\alpha_{A}(\bar{r}^{1})\otimes\alpha_{A}^{-1}(\bar{r}^{2})\alpha_{A}^{-1}\psi_{A}(r^{2})\triangleright\psi_{M}^{2}\beta_{M}^{-1}(m)+\lambda^{2}1\otimes
1\otimes\psi_{M}^{2}(m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.15})}}{{=}}$
$\displaystyle\omega_{A}\alpha_{A}(r^{1})\otimes\alpha_{A}(\bar{r}^{1})\otimes\bar{r}^{2}\triangleright(\psi_{A}\beta_{A}^{-1}(r^{2})\triangleright\psi_{M}^{2}\beta_{M}^{-2}(m))+\lambda^{2}1\otimes
1\otimes\psi^{2}_{M}(m)$
$\displaystyle+\lambda\omega_{A}\alpha_{A}(r^{1})\otimes
1\otimes\psi_{A}(r^{2})\triangleright\psi^{2}_{M}\beta^{-1}_{M}(m)+\lambda
1\otimes\alpha_{A}(r^{1})\otimes
r^{2}\triangleright\psi^{2}_{M}\beta^{-1}_{M}(m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.30})}}{{=}}$
$\displaystyle(\omega_{A}\otimes\widetilde{\rho})\widetilde{\rho}(m).$

The compatibility condition can be checked as follows. For all $a\in A$ and
$m\in M$,

$\displaystyle\omega_{A}(a)m_{-1}\otimes\beta_{M}(m_{0})+\alpha_{A}(a_{1})\otimes
a_{2}\triangleright\psi_{M}(m)+\lambda\alpha_{A}\omega_{A}(a)\otimes\beta_{M}\psi_{M}(m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:14.25})}}{{=}}$
$\displaystyle-\omega_{A}(a)\alpha_{A}(r^{1})\otimes\beta_{A}(r^{2})\triangleright\psi_{M}(m)+\lambda\omega_{A}(a)\cdot
1\otimes\beta_{M}\psi_{M}(m)+\omega_{A}(a)\alpha_{A}(r^{1})\otimes\beta_{A}(r^{2})\triangleright\psi_{M}(m)$
$\displaystyle-\alpha_{A}^{2}(r^{1})\otimes
r^{2}\psi_{A}\beta_{A}^{-1}(a)\triangleright\psi_{M}(m)-\lambda
1\otimes\psi_{A}(a)\triangleright\psi_{M}(m)+\lambda\alpha_{A}\omega_{A}(a)\otimes\beta_{M}\psi_{M}(m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.5})}}{{=}}$
$\displaystyle-\alpha_{A}(r^{1})\otimes\alpha^{-1}_{A}(r^{2})\psi_{A}\beta_{A}^{-1}(a)\triangleright\psi_{M}(m)-\lambda
1\otimes\psi_{A}(a)\triangleright\psi_{M}(m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.13})}}{{=}}$
$\displaystyle-\alpha_{A}(r^{1})\otimes
r^{2}\triangleright\psi_{M}\beta^{-1}_{M}(a\triangleright m)-\lambda
1\otimes\psi_{M}(a\triangleright m)=(a\triangleright
m)_{-1}\otimes(a\triangleright m)_{0},$

as desired. ∎

###  3.3. BiHom-pre-Lie algebras from $\lambda$-infBH-bialgebras

In this subsection, we provide two approaches to construct BiHom-pre-Lie
algebras from $\lambda$-infBH-bialgebras.

Rota-Baxter operators on a BiHom-associative algebras (here we call this
structure Rota-Baxter BiHom-associative algebra) were studied in [14,
Definition 2.1] or [22, Definition 2.1].

######  Lemma 3.20.

([17, Corollary 2.6]) Let $(A,\mu,1,\alpha,\beta)$ be a unitary BiHom-
associative algebra, $\psi,\omega:A\longrightarrow A$ be linear maps such that
$\alpha,\beta,\psi,\omega$ are bijective and Eqs.(2.1), (2.3) hold. Assume
that $r$ is a solution of $(\pm\lambda)$-abhYBe in $(A,\mu,\alpha,\beta)$ and
$r$ is $\alpha,\beta,\psi,\omega$-invariant. Define $R:A\longrightarrow A$ by

$\displaystyle
R(a)=\mp\beta^{2}\psi(r^{1})(\alpha^{-1}\beta^{-1}(a)\alpha\omega(r^{2})),\forall
a\in A.$

Then $(A,\mu,R,\alpha,\beta)$ is a Rota-Baxter BiHom-associative algebra of
weight $\lambda$.

The BiHom-version of Loday’s dendriform algebra was introduced in [14,
Definition 3.1].

######  Lemma 3.21.

Let $(A,R,\alpha,\beta)$ be a Rota-Baxter BiHom-associative algebra of weight
$\lambda$ and $\prec,\succ:A\otimes A\longrightarrow A$ be linear maps defined
by

$\displaystyle a\prec b=aR(b)+\lambda ab,\quad a\succ b=R(a)b$

(resp.

$\displaystyle a\prec b=aR(b),\quad a\succ b=R(a)b+\lambda ab$
$\displaystyle)$

for all $a,b\in A$. Then $(A,\prec,\succ,\alpha,\beta)$ is a BiHom-dendriform
algebra.

###### Proof.

It can be derived by [22, Lemma 2.3, Theorem 2.5]. ∎

######  Remark 3.22.

When $\lambda=0$, Lemma 3.21 recovers [14, Corollary 4.4].

######  Corollary 3.23.

Let $(A,\mu,1,\alpha,\beta,\psi,\omega,r)$ be a quasitriangular unitary
$\lambda$-infBH-bialgebra. Define two binary operations $\prec,\succ$ on $A$
by

$\displaystyle a\succ
b=-(\beta^{2}\psi(r^{1})(\alpha^{-1}\beta^{-1}(a)\alpha\omega(r^{2})))b,\
a\prec
b=-a(\beta^{2}\psi(r^{1})(\alpha^{-1}\beta^{-1}(b)\alpha\omega(r^{2})))+\lambda
ab$

or

$\displaystyle a\succ
b=-(\beta^{2}\psi(r^{1})(\alpha^{-1}\beta^{-1}(a)\alpha\omega(r^{2})))b+\lambda
ab,\ a\prec
b=-a(\beta^{2}\psi(r^{1})(\alpha^{-1}\beta^{-1}(b)\alpha\omega(r^{2}))).$

Then the 5-tuple $(A,\prec,\succ,\alpha,\beta)$ is a BiHom-dendriform algebra.

###### Proof.

The result follows from Lemma 3.20 and Lemma 3.21. ∎

######  Remark 3.24.

If we delete the minus signs “$-$” in Corollary 3.23, then the corresponding
results for the case of anti-quasitriangular unitary $\lambda$-infBH-bialgebra
can be obtained.

######  Definition 3.25.

([13, Definition 3.1]) A (left) BiHom-pre-Lie algebra $(A,\cdot,\alpha,\beta)$
is a 4-tuple in which $A$ is a vector space and $\cdot:A\otimes
A\longrightarrow A$, $\alpha,\beta:A\longrightarrow A$ are linear maps
satisfying $\alpha\circ\beta=\beta\circ\alpha$, $\alpha(a\cdot
b)=\alpha(a)\cdot\alpha(b)$, $\beta(a\cdot b)=\beta(a)\cdot\beta(b)$ and

$\displaystyle\alpha\beta(a)\cdot(\alpha(b)\cdot
c)-(\beta(a)\cdot\alpha(b))\cdot\beta(c)=\alpha\beta(b)\cdot(\alpha(a)\cdot
c)-(\beta(b)\cdot\alpha(a))\cdot\beta(c),$ (3.13)

for all $a,b,c\in A$.

Now we get a new construction of BiHom-pre-Lie algebra which ensures that the
following diagram is commutative.

$\textstyle{\text{Quasitriangular}\atop\text{$\lambda$-infBH
Bialgebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Lemma
3.20Corollary 3.8$\textstyle{\text{$\lambda$-Rota Baxter}\atop\text{BiHom-
algebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Lemma
3.21$\textstyle{\text{BiHom-
Dendriform}\atop\text{algebras}}$$\textstyle{\text{$\lambda$-infBH}\atop\text{Bialgebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Theorem
3.26$\textstyle{\text{BiHom-Pre-
Lie}\atop\text{algebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$[13,
Proposition 3.6]

######  Theorem 3.26.

Let $(A,\mu,\Delta,\alpha,\beta,\psi,\omega)$ be a $\lambda$-infBH-bialgebra
such that $\alpha,\beta,\psi,\omega$ are invertible. Then
$(A,\star,\alpha,\beta)$ is a BiHom-pre-Lie algebra, where

$\displaystyle\star:A\otimes A\longrightarrow A,\ a\star
b=(\alpha^{-2}\beta\omega^{-1}(b_{1})\beta^{-1}(a))\psi^{-1}(b_{2}).$ (3.14)

###### Proof.

For all $a,b,c\in A$, we calculate

$\displaystyle\Delta(\alpha(b)\star c)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.7})(\ref{eq:12.3})(\ref{eq:12.4})(\ref{eq:1.2})}}{{=}}$
$\displaystyle(\alpha^{-2}\beta(c_{1})\alpha\beta^{-1}\omega(b))\psi^{-1}(c_{21})\otimes\beta\psi^{-1}(c_{22})+\alpha^{-1}\beta(c_{1})\alpha^{2}\beta^{-1}(b_{1})\otimes\alpha(b_{2})c_{2}$
$\displaystyle+\beta\omega^{-1}(c_{11})\otimes(\alpha^{-2}\beta\omega^{-1}(c_{12})\alpha\beta^{-1}\psi(b))c_{2}$
$\displaystyle+\lambda\beta(c_{1})\otimes\alpha\psi(b)c_{2}+\lambda\alpha^{-1}\beta(c_{1})\alpha^{2}\beta^{-1}\omega(b)\otimes\beta(c_{2})$

and

$\displaystyle\alpha\beta(a)\star(\alpha(b)\star c)$ $\displaystyle=$
$\displaystyle(((\alpha^{-4}\beta^{2}\omega^{-1}(c_{1})\alpha^{-1}(b))\alpha^{-2}\beta\psi^{-1}\omega^{-1}(c_{21}))\alpha(a))\beta\psi^{-2}(c_{22})$
$\displaystyle+((\alpha^{-3}\beta^{2}\omega^{-1}(c_{1})\omega^{-1}(b_{1}))\alpha(a))(\alpha\psi^{-1}(b_{2})\psi^{-1}(c_{2}))$
$\displaystyle+(\alpha^{-2}\beta^{2}\omega^{-2}(c_{11})\alpha(a))((\alpha^{-2}\beta\psi^{-1}\omega^{-1}(c_{12})\alpha\beta^{-1}(b))\psi^{-1}(c_{2}))$
$\displaystyle+\lambda(\alpha^{-2}\beta^{2}\omega^{-1}(c_{1})\alpha(a))(\alpha(b)\psi^{-1}(c_{2}))+\lambda((\alpha^{-3}\beta^{2}\omega^{-1}(c_{1})b)\alpha(a))\psi^{-1}\beta(c_{2}).$

Morever,

$\displaystyle(\beta(a)\star\alpha(b))\star\beta(c)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.2})}}{{=}}$
$\displaystyle(\alpha^{-2}\beta^{2}\omega^{-1}(c_{1})((\alpha^{-1}\omega^{-1}(b_{1})\beta^{-1}(a))\alpha\beta^{-1}\psi^{-1}(b_{2})))\psi^{-1}\beta(c_{2})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.3})}}{{=}}$
$\displaystyle((\alpha^{-3}\beta^{2}\omega^{-1}(c_{1})\omega^{-1}(b_{1}))\alpha(a))(\alpha\psi^{-1}(b_{2})\psi^{-1}(c_{2})).$

Hence,

$\displaystyle\alpha\beta(a)\star(\alpha(b)\star
c)-(\beta(a)\star\alpha(b))\star\beta(c)$ $\displaystyle=$
$\displaystyle(((\alpha^{-4}\beta^{2}\omega^{-1}(c_{1})\alpha^{-1}(b))\alpha^{-2}\beta\psi^{-1}\omega^{-1}(c_{21}))\alpha(a))\beta\psi^{-2}(c_{22})$
$\displaystyle+(\alpha^{-2}\beta^{2}\omega^{-2}(c_{11})\alpha(a))((\alpha^{-2}\beta\psi^{-1}\omega^{-1}(c_{12})\alpha\beta^{-1}(b))\psi^{-1}(c_{2}))$
$\displaystyle+\lambda(\alpha^{-2}\beta^{2}\omega^{-1}(c_{1})\alpha(a))(\alpha(b)\psi^{-1}(c_{2}))+\lambda((\alpha^{-3}\beta^{2}\omega^{-1}(c_{1})b)\alpha(a))\psi^{-1}\beta(c_{2})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.3})(\ref{eq:1.9})}}{{=}}$
$\displaystyle(((\alpha^{-4}\beta^{2}\omega^{-1}(c_{1})\alpha^{-1}(b))\alpha^{-2}\beta\psi^{-1}\omega^{-1}(c_{21}))\alpha(a))\beta\psi^{-2}(c_{22})$
$\displaystyle+(((\alpha^{-4}\beta^{2}\omega^{-1}(c_{1})\alpha^{-1}(a))\alpha^{-2}\beta\psi^{-1}\omega^{-1}(c_{21}))\alpha(b))\psi^{-2}\beta(c_{22})$
$\displaystyle+\lambda((\alpha^{-3}\beta^{2}\omega^{-1}(c_{1})a)\alpha(b))\psi^{-1}\beta(c_{2})+\lambda((\alpha^{-3}\beta^{2}\omega^{-1}(c_{1})b)\alpha(a))\psi^{-1}\beta(c_{2}),$

completing the proof since the positions of $a$ and $b$ are symmetric. ∎

We next provide a new way to construct BiHom-pre-Lie algebra from
$\lambda$-infBH-bialgebra in which the structure maps
$\alpha,\beta,\psi,\omega$ are not invertible. Here we omit the proof since
its proof is similar to Theorem 3.26.

######  Theorem 3.27.

Let $(A,\mu,\Delta,\alpha,\beta,\psi,\omega)$ be a $\lambda$-infBH-bialgebra.
Then $(A,\star,\alpha^{2}\beta,\alpha^{2}\beta^{2}\psi\omega)$ is a BiHom-pre-
Lie algebra, where

$\displaystyle\star:A\otimes A\longrightarrow A,\ a\star
b=(\beta^{2}\psi(b_{1})\alpha(a))\alpha^{2}\beta\omega(b_{2}).$ (3.15)

######  Remark 3.28.

[15, Theorem 4.6.] is the special case of Theorem 3.27 for $\lambda=0$.

######  Remark 3.29.

From the perspective of the commutative diagram below, Example 2.4 is very
interesting.

---
$\textstyle{\text{1(\text{resp.}-1)-infBH-}\atop\text{bialgebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Theorem
3.26 $\scriptstyle{\lambda=1(\text{resp.}-1)\text{ in Example}\
\ref{ex:12.3}\quad}$$\textstyle{\text{BiHom-pre-
Lie}\atop\text{algebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$[13,
Proposition 3.4]$\textstyle{\text{BiHom-
associative}\atop\text{algebras}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
[10, Proposition 3.15]$\textstyle{\text{BiHom-Lie}\atop\text{algebras}}$

###  3.4. Comodules over (anti-)coquasitriangular $\lambda$-infBH-bialgebra

Coquasitriangular infinitesimal bialgebras were introduced in [20] from the
mixed bialgebra and its BiHom-version was studied in [18]. In this subsection,
we prove that every comodule over (anti-)coquasitriangular $\lambda$-infBH-
bialgebra can induce a $\lambda$-infBH-Hopf module. Most of the conclusions in
this subsection are parallel to those in Section 3.2, we sketch the proof for
the convenience of reading.

Let $\lambda$ be a given element in $K$, $(C,\Delta,\varepsilon,\psi,\omega)$
be a counitary BiHom-coassociative coalgebra satisfying that $\psi,\omega$ are
bijective, $\alpha,\beta:C\longrightarrow C$ be two linear maps such that
Eqs.(2.1), (2.2) and (2.6) hold, and $\sigma\in(C\otimes C)^{\ast}$ be
$\alpha,\beta,\psi,\omega$-invariant. Here an element $\sigma\in(C\otimes
C)^{\ast}$ is $F$-invariant if $\sigma\circ(F\otimes F)=\sigma$, where
$F:C\longrightarrow C$ is a linear map. Define linear map
$\mu_{\sigma}~{}(\hbox{resp.}~{}\widetilde{\mu_{\sigma}}):C\otimes
C\longrightarrow C$ by

$\displaystyle\mu_{\sigma}(c\otimes
d)=\alpha\omega^{-1}(c_{1})\sigma(c_{2},\psi(d))-\sigma(\omega(c),d_{1})\beta\psi^{-1}(d_{2})-\lambda\alpha(c)\varepsilon(d)$
(3.16)

(resp.

$\displaystyle\widetilde{\mu_{\sigma}}(c\otimes
d)=\alpha\omega^{-1}(c_{1})\sigma(c_{2},\psi(d))-\sigma(\omega(c),d_{1})\beta\psi^{-1}(d_{2})-\lambda\varepsilon(c)\beta(d).)$
(3.17)

In what follows, we only prove the case corresponding to $\mu_{\sigma}$, the
other case for $\widetilde{\mu_{\sigma}}$ can be checked similarly.

######  Lemma 3.30.

The $\mu_{\sigma}$ (resp. $\widetilde{\mu_{\sigma}}$) defined by Eq.(3.16)
(resp. Eq.(3.17)) is a $\lambda$-BiHom-coderivation.

###### Proof.

By $\sigma\in(C\otimes C)^{\ast}$ is $\alpha,\beta,\psi,\omega$-invariant, we
know that Eq.(2.9) holds. For $a,b\in A$, Eq.(2.10) for $\mu_{\sigma}$ can be
proved below.

$\displaystyle\mu_{\sigma}(\omega(c)\otimes
d_{1})\otimes\beta(d_{2})+\alpha(c_{1})\otimes\mu_{\sigma}(c_{2}\otimes\psi(d))+\lambda\alpha\omega(c)\otimes\beta\psi(d)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.9})(\ref{eq:1.11})}}{{=}}$
$\displaystyle\alpha\omega^{-1}(c_{11})\otimes\alpha\omega^{-1}(c_{12})\sigma(\psi(c_{2}),\psi^{2}(d))-\sigma(\omega(c),d_{1})\beta\psi^{-1}(d_{21})\otimes\beta\psi^{-1}(d_{22})$
$\displaystyle-\lambda\alpha(c_{1})\otimes\alpha(c_{2})\varepsilon(d)$
$\displaystyle\stackrel{{\scriptstyle}}{{=}}$
$\displaystyle\Delta\circ\mu_{\sigma}(c\otimes d),$

as desired. ∎

######  Theorem 3.31.

Let $(C,\Delta,\psi,\omega)$ be a counitary BiHom-coassociative coalgebra such
that $\psi,\omega$ are bijective, $\alpha,\beta:C\rightarrow C$ be linear
maps, $\sigma\in(C\otimes C)^{\ast}$ be $\alpha,\beta,\psi,\omega$-invariant
and moreover Eqs.(2.1), (2.2) and (2.6) hold. Then the $\lambda$-BiHom-
coderivation $\mu_{\sigma}$ (resp. $\widetilde{\mu_{\sigma}}$) defined by
Eq.(3.16) (resp. Eq.(3.17)) is BiHom-associative if and only if for all
$c,d,e\in C$,

$\displaystyle\alpha^{2}\omega^{-1}(c_{1})(\sigma(\alpha\omega^{-1}(c_{21}),\beta\psi(e))\sigma(c_{22},\psi^{2}(d))-\sigma(\omega(c_{2}),\psi(d_{1}))\sigma(d_{2},\psi(e))$
$\displaystyle+\sigma(\alpha(c_{2}),\beta(e_{2}))\sigma(\omega(d),e_{1})-\lambda(\hbox{resp.}\
(-\lambda))\sigma(\alpha(c_{2}),\psi\beta(e))\varepsilon(d))$
$\displaystyle=(\sigma(\alpha(c_{1}),\beta(e_{1}))\sigma(c_{2},\psi(d))-\sigma(\omega(c),d_{1})\sigma(\omega(d_{2}),\psi(e_{1}))$
(3.18)
$\displaystyle+\sigma(\alpha\omega(c),\beta\psi^{-1}(e_{12}))\sigma(\omega^{2}(d),e_{11})-\lambda(\hbox{resp.}\
(-\lambda))\sigma(\alpha\omega(c),\beta(e_{1}))\varepsilon(d))\beta^{2}\psi^{-1}(e_{2}).$

###### Proof.

For all $c,d,e\in C$, based on Eqs.(1.5) and (1.6), one can get

$\displaystyle\alpha(c)(de)$ $\displaystyle=$
$\displaystyle\alpha^{2}\omega^{-1}(c_{1})\sigma(d_{2},\psi(e))\sigma(\alpha(c_{2}),\alpha\omega^{-1}\psi(d_{1}))-\alpha\beta\psi^{-1}\omega^{-1}(d_{21})\sigma(\psi^{-1}(d_{22}),\psi(e))\sigma(\omega(c),d_{1})$
$\displaystyle-\alpha^{2}\omega^{-1}(c_{1})\sigma(\omega(d),e_{1})\sigma(\alpha(c_{2}),\beta(e_{2}))+\sigma(\omega(d),\omega^{-1}(e_{11}))\sigma(\omega\alpha(c),\beta\psi^{-1}(e_{12}))\beta^{2}\psi^{-1}(e_{2})$
$\displaystyle-\lambda\varepsilon(e)\alpha^{2}\omega^{-1}(c_{1})\sigma(c_{2},\psi(d))+\lambda\sigma(\omega(c),d_{1})\varepsilon(e)\alpha\beta\psi^{-1}(d_{2})+\lambda^{2}\alpha^{2}(c)\varepsilon(d)\varepsilon(e)$

and

$\displaystyle(cd)\beta(e)$ $\displaystyle=$
$\displaystyle\alpha^{2}\omega^{-1}(c_{1})\sigma(\alpha\omega^{-1}(c_{21}),\psi\beta(e))\sigma(\psi^{-1}(c_{22}),\psi(d))-\sigma(c_{2},\psi(d))\sigma(\alpha(c_{1}),\beta(e_{1}))\beta^{2}\psi^{-1}(e_{2})$
$\displaystyle-\lambda\alpha^{2}\omega^{-1}(c_{1})\varepsilon(e)\sigma(c_{2},\psi(d))-\alpha\beta\psi^{-1}\omega^{-1}(d_{21})\sigma(\psi^{-1}(d_{22}),\psi(e))\sigma(\omega(c),d_{1})$
$\displaystyle+\sigma(\omega(c),d_{1})\sigma(\omega\psi^{-1}(d_{2}),e_{1})\beta^{2}\psi^{-1}(e_{2})+\lambda\alpha\beta\psi^{-1}(d_{2})\varepsilon(e)\sigma(\omega(c),d_{1})$
$\displaystyle-\lambda\alpha^{2}\omega^{-1}(c_{1})\varepsilon(d)\sigma(\alpha(c_{2}),\psi\beta(e))+\lambda\varepsilon(d)\sigma(\omega\alpha(c),\beta(e))\beta^{2}\psi^{-1}(e_{2})+\lambda^{2}\alpha^{2}(c)\varepsilon(d)\varepsilon(e).$

Therefore we can finish the proof. ∎

Now we introduce the nonhomogeneous type of coassociative BiHom-Yang-Baxter
equation.

######  Definition 3.32.

Let $(C,\Delta,\varepsilon,\psi,\omega)$ be a counitary BiHom-coassociative
coalgebra, $\alpha,\beta:C\longrightarrow C$ be linear maps and
$\sigma\in(C\otimes C)^{\ast}$. We call

$\displaystyle\sigma(\alpha(c_{1}),\beta\omega(e))\sigma(c_{2},\psi(d))-\sigma(\omega(c),d_{1})\sigma(d_{2},\psi(e))\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad+\sigma(\omega(d),e_{1})\sigma(\alpha\psi(c),\beta(e_{2}))=\lambda(\hbox{resp.}\
(-\lambda))\sigma(\alpha(c),\beta(e))\varepsilon(d)\qquad$ (3.19)

the $\lambda$(resp. ($-\lambda$))-coassociative BiHom-Yang-Baxter equation
(abbr. $\lambda$(resp. ($-\lambda$))-coabhYBe) in
$(C,\Delta,\varepsilon,\psi,\omega)$ where $\lambda$ is a given element in
$K$.

By Lemma 3.30 and Theorem 3.31, we have

######  Corollary 3.33.

Let $(C,\Delta,\varepsilon,\psi,\omega)$ be a counitary BiHom-coassociative
coalgebra such that $\psi,\omega$ are bijective,
$\alpha,\beta:A\longrightarrow A$ be linear maps, $\sigma\in(C\otimes
C)^{\ast}$ be $\alpha,\beta,\psi,\omega$-invariant and moreover Eqs.(2.1),
(2.3) and (2.6) hold. If $\sigma$ is a solution of the $\lambda$(resp.
($-\lambda$))-coabhYBe, then $(C,\mu=\mu_{\sigma}(\hbox{resp.}\
\widetilde{\mu_{\sigma}}),\Delta,\varepsilon,\alpha,\beta,\psi,\omega)$ is a
$\lambda$-infBH-bialgebra, where $\mu_{\sigma}$(resp.
$\widetilde{\mu_{\sigma}}$) is defined by Eq.(3.16)(resp. Eq.(3.17)).

######  Definition 3.34.

Under the assumption of Corollary 3.33, a coquasitriangular (resp. anti-
coquasi triangular) counitary $\lambda$-infBH-bialgebra is a 8-tuple
$(C,\Delta,\varepsilon,\alpha,\beta,\psi,\omega,\sigma)$ consisting of a
counitary BiHom-coassociative coalgebra $(C,\Delta,\varepsilon,\psi,\omega)$
and a solution $\sigma\in(C\otimes C)^{\ast}$ of a $\lambda$(resp.
$(-\lambda)$)-coabhYBe.

######  Proposition 3.35.

Under the assumption of Corollary 3.33, $(C,\mu=\mu_{\sigma}(\hbox{resp.}\
\widetilde{\mu_{\sigma}}),\Delta,\varepsilon,\alpha,\beta,\psi,\omega)$, where
$\mu_{\sigma}(\hbox{resp.}\ \widetilde{\mu_{\sigma}})$ is defined by
Eq.(3.16)(resp. Eq.(3.17)), is a coquasitriangular (resp. anti-
coquasitriangular) counitary $\lambda$-infBH-bialgebra if and only if

$\displaystyle\sigma(cd,\beta(e))=-\sigma(\omega(d),e_{1})\sigma(\alpha\psi(c),\beta(e_{2}))$
(3.20)

or

$\displaystyle\sigma(\alpha(c),de)=\sigma(\alpha(c_{1}),\beta\omega(e))\sigma(c_{2},\psi(d))-\lambda\sigma(\alpha(c),\beta(e))\varepsilon(d)-\lambda\sigma(c,d)\varepsilon(e)$
(3.21)

(resp.

$\displaystyle\sigma(cd,\beta(e))=-\sigma(\omega(d),e_{1})\sigma(\alpha\psi(c),\beta(e_{2}))-\lambda\sigma(\alpha(c),\beta(e))\varepsilon(d)-\lambda\sigma(d,e)\varepsilon(c)$
(3.22)

or

$\displaystyle\sigma(\alpha(c),de)=\sigma(\alpha(c_{1}),\beta\omega(e))\sigma(c_{2},\psi(d))$
$\displaystyle)$ (3.23)

hold for all $c,d,e\in C$.

###### Proof.

We only sketch the proof of the coquasitriangular case as follows. By the
definition of $\mu$ in Eq.(3.16) and invariant condition for $\sigma$, one
easily checks that

$\displaystyle\sigma(cd,\beta(e))\sigma(\alpha(c_{1}),\beta\omega(e))\sigma(c_{2},\psi(d))-\sigma(\omega(c),d_{1})\sigma(d_{2},\psi(e))-\lambda\sigma(\alpha(c),\beta(e))\varepsilon(d)$

and

$\displaystyle\sigma(\alpha(c),de)=\sigma(\omega(c),d_{1})\sigma(d_{2},\psi(e))-\sigma(\omega(d),e_{1})\sigma(\alpha\psi(c),\beta(e_{2}))-\lambda\varepsilon(e)\sigma(c,d).$

The rest is obvious. ∎

The following theorem provides constructions of $\lambda$-infBH-Hopf modules
from comodules of (anti-)coquasitriangular counitary $\lambda$-infBH-
bialgebras.

######  Theorem 3.36.

Let $(C,\Delta,\varepsilon,\alpha_{C},\beta_{C},\psi_{C},\omega_{C},\sigma)$
be a coquasitriangular (resp. anti-coquasitriangular) counitary
$\lambda$-infBH-bialgebra and $(M,\rho,\psi_{M},\omega_{M})$ (resp.
$(M,\widetilde{\rho},\psi_{M},\omega_{M})$) be a left
$(C,\Delta,\psi_{C},\omega_{C})$-comodule,
$\alpha_{M},\beta_{M}:M\longrightarrow M$ be linear maps such that
$\beta_{M}\circ\psi_{M}=\psi_{M}\circ\beta_{M}$,
$\rho\circ\beta_{M}=(\beta_{A}\otimes\beta_{M})\circ\rho$ (resp.
$\widetilde{\rho}\circ\beta_{M}=(\beta_{A}\otimes\beta_{M})\circ\widetilde{\rho}$).
Then $(M,\gamma,\rho,\alpha_{M},\beta_{M},\psi_{M},\omega_{M})$ (resp.
$(M,\widetilde{\gamma},\widetilde{\rho},\alpha_{M},\beta_{M},\psi_{M},\omega_{M})$)
becomes a $\lambda$-infBH-Hopf module with an action $\gamma:C\otimes
M\longrightarrow M$ given by

$\displaystyle\gamma(c\otimes
m):=-\sigma(\omega_{C}(c),m_{-1})\beta_{M}\psi_{M}^{-1}(m_{0})$

(resp.

$\displaystyle\widetilde{\gamma}(c\otimes
m):=-\sigma(\omega_{C}(c),m_{-1})\beta_{M}\psi_{M}^{-1}(m_{0})-\lambda\varepsilon(c)\beta_{M}(m)$
$\displaystyle)$

for all $c\in C,m\in M$.

###### Proof.

We first prove that $(M,\gamma,\alpha_{M},\beta_{M})$ is a left
$(C,\mu,\alpha_{C},\beta_{C})$-module. For all $c,d\in C$ and $m\in M$, we
have

$\displaystyle\gamma(\mu_{\sigma}\otimes\beta_{M})(c\otimes d\otimes m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:01.06})}}{{=}}$
$\displaystyle\sigma(\omega_{C}^{2}(d),m_{-11})\sigma(\alpha_{C}\psi_{C}\omega_{C}(c),\beta_{C}(m_{-12}))\beta_{M}^{2}\psi_{M}^{-1}(m_{0})$
$\displaystyle=$ $\displaystyle\gamma(\alpha_{C}\otimes\gamma)(c\otimes
d\otimes m).$

Then in the rest we check the compatibility condition for $\lambda$-infBH-Hopf
module.

$\displaystyle\omega_{C}(c)m_{-1}\otimes\beta_{M}(m_{0})+\alpha_{C}(c_{1})\otimes
c_{2}\triangleright\psi_{M}(m)+\lambda\alpha_{C}\omega_{C}(c)\otimes\beta_{M}\psi_{M}(m)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:01.04})}}{{=}}$
$\displaystyle-\sigma(\omega_{C}^{2}(c),\omega_{C}(m_{-1}))\beta_{C}\psi_{C}^{-1}(m_{0-1})\otimes\beta_{M}\psi_{M}^{-1}(m_{00})$
$\displaystyle=$ $\displaystyle(c\triangleright m)_{-1}\otimes(c\triangleright
m)_{0},$

completing the proof. ∎

###  3.5. BiHom-pre-Lie coalgebras from $\lambda$-infBH-bialgebras

In this subsection, we provide two approaches to construct BiHom-pre-Lie
coalgebras from $\lambda$-infBH-bialgebras, one of which can also provide a
commutative diagram corresponding to the one in Sec. 3.3.

######  Definition 3.37.

([18, Definition 3.18]) A (left) BiHom-pre-Lie coalgebra is a 4-tuple
$(C,\Delta,\psi,\omega)$ where $C$ is a linear space and $\Delta:C\rightarrow
C\otimes C$ (write $\Delta(c)=c_{[1]}\otimes c_{[2]}$),
$\psi,\omega:C\rightarrow C$ are linear maps satisfying

$\displaystyle\psi\circ\omega=\omega\circ\psi,~{}\Delta\circ\psi=(\psi\otimes\psi)\circ\Delta,~{}\Delta\circ\omega=(\omega\otimes\omega)\circ\Delta,$
$\displaystyle\bar{\Delta}-\Phi_{(12)}\bar{\Delta}=0,$

where $\bar{\Delta}(c)=\omega\psi(c_{[1]})\otimes\omega(c_{[2][1]})\otimes
c_{[2][2]}-\psi(c_{[1][1]})\otimes\omega(c_{[1][2]})\otimes\psi(c_{[2]})$ and
$\Phi_{(12)}(a\otimes b\otimes c)=b\otimes a\otimes c$.

######  Theorem 3.38.

Let $(C,\mu,\Delta,\alpha,\beta,\psi,\omega)$ be a $\lambda$-infBH-bialgebra
such that $\alpha,\beta,\psi,\omega$ are invertible. Then
$(C,\Delta_{\star},\psi,\omega)$ is a BiHom-pre-Lie coalgebra, where

$\displaystyle\Delta_{\star}:C\longrightarrow C\otimes C,\
\Delta_{\star}(c)=\psi^{-1}(c_{12})\otimes\alpha^{-1}\psi\omega^{-2}(c_{11})\beta^{-1}(c_{2}).$

###### Proof.

For all $c,d\in C$, we calculate

$\displaystyle(\omega\otimes{\rm id})\Delta_{\star}(cd)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:12.2})(\ref{eq:12.4})}}{{=}}$
$\displaystyle\beta\psi^{-1}\omega(d_{12})\otimes(\alpha^{-1}\psi(c)\alpha^{-1}\psi\omega^{-2}(d_{11}))d_{2}+\psi^{-1}\omega^{2}(c_{2})\omega(d_{1})\otimes\psi\omega^{-1}(c_{1})d_{2}$
$\displaystyle+\lambda\beta\omega(d_{1})\otimes\psi(c)d_{2}+\alpha\psi^{-1}\omega(c_{12})\otimes\psi\omega^{-2}(c_{11})(\beta^{-1}(c_{2})\beta^{-1}\psi(d))$
$\displaystyle+\lambda\alpha\psi^{-1}\omega^{2}(c_{2})\otimes\psi\omega^{-1}(c_{1})\psi(d),$

and

$\displaystyle({\rm id}\otimes\omega\otimes{\rm
id})(\omega\psi\otimes\Delta_{\star})\Delta_{\star}(c)$ $\displaystyle=$
$\displaystyle\omega(c_{12})\otimes\psi^{-1}\omega(c_{212})\otimes(\alpha^{-2}\psi^{2}\omega^{-2}(c_{11})\alpha^{-1}\beta^{-1}\psi\omega^{-2}(c_{211}))\beta^{-1}(c_{22})$
$\displaystyle+\omega(c_{12})\otimes\alpha^{-1}(c_{112})\beta^{-1}\omega(c_{21})\otimes\alpha^{-1}\psi^{2}\omega^{-3}(c_{111})\beta^{-1}(c_{22})$
$\displaystyle+\lambda\omega(c_{12})\otimes\omega(c_{21})\otimes\alpha^{-1}\psi^{2}\omega^{-2}(c_{11})\beta^{-1}(c_{22})$
$\displaystyle+\omega(c_{12})\otimes\omega^{-1}(c_{1112})\otimes\alpha^{-1}\psi^{2}\omega^{-4}(c_{1111})(\alpha^{-1}\beta^{-1}\psi\omega^{-2}(c_{112})\beta^{-2}\psi(c_{2}))$
$\displaystyle+\lambda\omega(c_{12})\otimes
c_{112}\otimes\alpha^{-1}\psi^{2}\omega^{-3}(c_{111})\beta^{-1}\psi(c_{2}).$

Morever,

$\displaystyle(\psi\otimes\omega\otimes{\rm
id})(\Delta_{\star}\otimes\psi)\Delta_{\star}(c)\stackrel{{\scriptstyle(\ref{eq:1.9})}}{{=}}\omega(c_{12})\otimes\alpha^{-1}(c_{112})\beta^{-1}\omega(c_{21})\otimes\alpha^{-1}\psi^{2}\omega^{-3}(c_{111})\beta^{-1}(c_{22}).$

Hence,

$\displaystyle\bar{\Delta}_{\star}(c)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:1.3})(\ref{eq:1.9})}}{{=}}$
$\displaystyle\omega(c_{12})\otimes\omega^{-1}(c_{1112})\otimes\alpha^{-1}\psi^{2}\omega^{-4}(c_{1111})(\alpha^{-1}\beta^{-1}\psi\omega^{-2}(c_{112})\beta^{-2}\psi(c_{2}))$
$\displaystyle+\omega^{-1}(c_{1112})\otimes\omega(c_{12})\otimes\alpha^{-1}\psi^{2}\omega^{-4}(c_{1111})(\alpha^{-1}\beta^{-1}\psi\omega^{-2}(c_{112})\beta^{-2}\psi(c_{2}))$
$\displaystyle+\lambda\omega(c_{12})\otimes
c_{112}\otimes\alpha^{-1}\psi^{2}\omega^{-3}(c_{111})\beta^{-1}\psi(c_{2})+\lambda
c_{112}\otimes\omega(c_{12})\otimes\alpha^{-1}\psi^{2}\omega^{-3}(c_{111})\beta^{-1}\psi(c_{2}).$

Thus we have $\bar{\Delta}_{\star}-\Phi_{(12)}\bar{\Delta}_{\star}=0$. The
proof is completed. ∎

The following way does not need the condition that the structure maps
$\alpha,\beta,\psi,\omega$ are invertible.

######  Theorem 3.39.

Let $(C,\mu,\Delta,\alpha,\beta,\psi,\omega)$ be a $\lambda$-infBH-bialgebra.
Then $(C,\Delta_{\star},\psi\omega^{2},\alpha\beta\psi^{2}\omega^{2})$ is a
BiHom-pre-Lie coalgebra, where

$\displaystyle\Delta_{\star}:C\longrightarrow C\otimes C,\
\Delta_{\star}(c)=\omega(c_{12})\otimes\beta\psi^{2}(c_{11})\alpha\psi\omega^{2}(c_{2}).$

###### Proof.

Similar to Theorem 3.38. ∎

##  4\. Further research

Let $(A,\mu,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$ be a
$\lambda$-infBH-bialgebra, $M$ a vector space, $\alpha_{M}$, $\beta_{M}$,
$\psi_{M}$, $\omega_{M}:\ M\longrightarrow M$ be four linear maps such that
any two of them commute. A $\lambda$-infBH-Hopf bimodule over
$(A,\mu,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$ is a 9-tuple
$(M,\gamma,\nu,\rho,\varphi,\alpha_{M},\beta_{M},\psi_{M},\omega_{M})$, where

$\displaystyle\gamma:A\otimes M\longrightarrow M,\ \nu:M\otimes
A\longrightarrow M,\ \rho:M\longrightarrow A\otimes M\ \hbox{and}\
\varphi:M\longrightarrow M\otimes A$

are linear maps satisfying the following conditions:

(1) $(M,\gamma,\rho,\alpha_{M},\beta_{M},\psi_{M},\omega_{M})$ is a left
$\lambda$-infBH-Hopf module over
$(A,\mu,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$.

(2) $(M,\nu,\varphi,\alpha_{M},\beta_{M},\psi_{M},\omega_{M})$ is a right
$\lambda$-infBH-Hopf module over
$(A,\mu,\Delta,\alpha_{A},\beta_{A},\psi_{A},\omega_{A})$.

(3) $(M,\gamma,\nu,\alpha_{M},\beta_{M})$ is a BiHom-bimodule over
$(A,\mu,\alpha_{A},\beta_{A})$.

(4) $(M,\rho,\varphi,\psi_{M},\omega_{M})$ is a BiHom-bicomodule over
$(A,\Delta,\psi_{A},\omega_{A})$.

(5) the following equations hold:

$\displaystyle(\gamma\otimes\beta_{A})(\omega_{A}\otimes\varphi)=\varphi\circ\gamma,$
(4.1) $\displaystyle(\alpha_{A}\otimes\nu)(\rho\otimes\psi_{A})=\rho\circ\nu.$
(4.2)

In the forthcoming paper [19], motivated by a class of $\lambda$-infinitesimal
BiHom-biproduct bialgebra, we discuss the notion above, which makes the
following diagram commutative.

$\textstyle{\text{Quasitriangular-
infBH}\atop\text{Bimodules}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{$\lambda$-Rota
Baxter}\atop\text{BiHom-
Bimodules}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{Dendriform}\atop\text{BiHom-
Bimodules}}$$\textstyle{\text{$\lambda$-infBH}\atop\text{Bimodules}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{BiHom-
Pre-
Lie}\atop\text{Bimodules}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

## Acknowledgment

Ma is supported by Natural Science Foundation of Henan Province
(No.212300410365).

## References

  * [1]
  * [2] M. Aguiar, Infinitesimal Hopf algebras, in: New trends in Hopf algebra theory (La Falda,1999), 1–29, Contemp. Math. 267, Amer. Math. Soc., Providence, RI, 2000.
  * [3] M. Aguiar, On the associative analog of Lie bialgebras. J. Algebra 244 (2001), 492–532.
  * [4] M. Aguiar, Infinitesimal bialgebras, pre-Lie and dendriform algebras, Hopf algebras, 1-33, Lecture Notes in Pure and Appl. Math. 237, Dekker, New York, 2004.
  * [5] T. Brzeziński, Rota-Baxter systems, dendriform algebras and covariant bialgebras. J. Algebra 460(2016), 1–25.
  * [6] K. Ebrahimi-Fard, Rota-Baxter algebras and the Hopf algebra of renormalization, PhD. Thesis, Bonn University, 2006.
  * [7] L. Foissy, The infinitesimal Hopf algebra and the operads of planar forests, Int. Math. Res. Not. IMRN  3 (2010), 395–435.
  * [8] L. Foissy, The infinitesimal Hopf algebra and the poset of planar forests, J. Algebraic Combin. 30 (2009), 277–309.
  * [9] X. Gao and X. M. Wang, Infinitesimal unitary Hopf algebras and planar rooted forests. J. Algebraic Combin. 49 (2019), 437–460.
  * [10] G. Graziani, A. Makhlouf, C. Menini and F. Panaite, BiHom-Associative algebras, BiHom-Lie algebras and BiHom-Bialgebras. SIGMA 11 (2015),086,34 pages.
  * [11] J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations. J. Algebra 295 (2006), 340–361.
  * [12] S. A. Joni and G.-C. Rota, Coalgebras and Bialgebras in Combinatorics, Studies in Applied Mathematics  61(1979): 93-139. Reprinted in Gian-Carlo Rota on Combinatorics: Introductory papers and commentaries (Joseph P. S. Kung, Ed), Birkhäuser, Boston (1995).
  * [13] L. Liu, A. Makhlouf, C. Menini and F. Panaite, BiHom-pre-Lie algebras, BiHom-Leibniz algebras and Rota-Baxter operators on BiHom-Lie algebras. Georgian Math. J. 28 (2021), 581–594.
  * [14] L. Liu, A. Makhlouf, C. Menini and F. Panaite, Rota-Baxter operators on BiHom-associative algebras and related structures. Colloq. Math. 161 (2020), 263–294.
  * [15] L. Liu, A. Makhlouf, C. Menini and F. Panaite, BiHom-Novikov algebras and infinitesimal BiHom-bialgebras. J. Algebra 560 (2020), 1146-1172.
  * [16] J.-L. Loday and M. Ronco, On the structure of cofree Hopf algebras, J. Reine Angew. Math. 592 (2006), 123–155.
  * [17] T. Ma and J. Li, Nonhomogeneous associative Yang-Baxter equations. Bull. Math. Soc. SCI. Math. Roumanie (N.S.) 65(113)(1)(2022), 97–118.
  * [18] T. S. Ma, J. Li and T. Yang, Coquasitriangular infinitesimal BiHom-bialgebras and related structures. Comm. Algebra  49(6)(2021), 2423–2443.
  * [19] T. S. Ma and A. Makhlouf, Infinitesimal (BiHom-)bialgebras of any weight (II): Representations. Preprint (2023).
  * [20] T. S. Ma, A. Makhlouf and S. Silvestrov, Rota-Baxter cosystems and coquasitriangular mixed bialgebras. J. Algebra Appl. 20 (2021), 2150064.
  * [21] T. S. Ma and H. Y. Yang, Drinfeld double for infinitesimal BiHom-bialgebras. Adv. Appl. Clifford Algebr.  30 (2020), Paper No. 42, 22 pp.
  * [22] T. S. Ma, H. Y. Yang, L. Y. Zhang and H. H. Zheng, Quasitriangular covariant monoidal BiHom-bialgebras, associative monoidal BiHom-Yang-Baxter equations and Rota-Baxter paired monoidal BiHom-modules. Colloq. Math.  161 (2020), 189-221.
  * [23] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2) (2008), 51–64 .
  * [24] O. Ogievetsky and T. Popov, R-matrics in rime, Adv. Theor. Math. Phys, 14, (2010), 439–505.
  * [25] S. X. Wang and S. H. Wang, Drinfeld double for braided infinitesimal Hopf algebras, Comm. Algebra 42 (2014), 2195–2212.
  * [26] D. Yau, Infinitesimal Hom-bialgebras and Hom-lie bialgebras, arXiv: 1001.5000.
  * [27] Y. Zhang, D. Chen, X. Gao and Y. F. Luo, Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles, Pacific J. Math. 302 (2019), 741–766.
  * [28] Y. Zhang and X. Gao, Weighted infinitesimal bialgebras, arXiv:1810.10790v3 (2020).
  * [29] Y. Zhang, X. Gao and Y. F. Luo, Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras. J. Algebraic Combin. 53 (2021), 771–803.
  * [30] Y. Zhang, X. Gao and J. W. Zheng, Weighted infinitesimal unitary bialgebras on matrix algebras and weighted associative Yang-Baxter equations, arXiv:1811.00842.