††*Corresponding Author: Shizhuo Yu. Email<EMAIL_ADDRESS>
# Yang-Baxter equations and relative Rota-Baxter operators for left-Alia
algebras associated to invariant theory

Chuangchuang Kang  School of Mathematical Sciences
Zhejiang Normal University
Jinhua 321004
China<EMAIL_ADDRESS>,  Guilai Liu  Chern Institute of Mathematics & LPMC
Nankai University
Tianjin 300071
China<EMAIL_ADDRESS>and  Shizhuo Yu  School of Mathematical
Sciences & LPMC
Nankai University
Tianjin 300317
China<EMAIL_ADDRESS>
###### Abstract.

Left-Alia algebras are a class of algebras with symmetric Jacobi identities.
They contain several typical types of algebras as subclasses, and are closely
related to the invariant theory. In this paper, we study the construction
theory of left-Alia bialgebras. We introduce the notion of the left-Alia Yang-
Baxter equation. We show that an antisymmetric solution of the left-Alia Yang-
Baxter equation gives rise to a left-Alia bialgebra that we call triangular.
The notions of relative Rota-Baxter operators of left-Alia algebras and pre-
left-Alia algebras are introduced to provide antisymmetric solutions of the
left-Alia Yang-Baxter equation.

###### Key words and phrases:

Left-Alia algebras; Reflection groups; Yang-Baxter equations; Relative Rota-
Baxter operators

###### 2020 Mathematics Subject Classification:

17A30, 17A36, 17B38, 17B60, 17B62, 16W22

###### Contents

  1. 1 Introduction
    1. 1.1 Yang-Baxter equations and relative Rota-Baxter operators
    2. 1.2 Motivation
    3. 1.3 Outline of the paper
  2. 2 Left-Alia algebras associated to invariant theory
    1. 2.1 Definition of left Alia algebras
    2. 2.2 Examples of left-Alia algebras associated to invariant theory
    3. 2.3 Representation of left-Alia algebras
  3. 3 Left-Alia bialgebras
  4. 4 Triangular left-Alia bialgebras and the left-Alia Yang-Baxter equation
  5. 5 Relative Rota-Baxter operators of left-Alia algebras
  6. 6 Pre-left-Alia algebras

##  1\. Introduction

A left-Alia algebra [11] is a pair $(A,[\cdot,\cdot])$ consisting of a vector
space $A$ and a bilinear map $[\cdot,\cdot]:A\otimes A\rightarrow A$
satisfying the symmetric Jacobi identity

$[[x,y],z]+[[y,z],x]+[[z,x],y]=[[y,x],z]+[[z,y],x]+[[x,z],y],\;\forall
x,y,z\in A.$ (1)

There are various typical examples of left-Alia algebras, including Lie
algebars, anti-pre-Lie algebras [16] and mock-Lie algebras [27]. Moreover, a
class of non-Lie algebra examples can be realized from twisted derivations in
invariant theory.

###  1.1. Yang-Baxter equations and relative Rota-Baxter operators

The classical Yang-Baxter equation (CYBE) was initially presented in tensor
form as

$[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0,$

where $r\in\mathfrak{g}\otimes\mathfrak{g}$ and $(\mathfrak{g},[-,-])$ is a
Lie algebra (see [7] for more details). The CYBE serves as the “semi-classical
limit” of the quantum Yang-Baxter equation and is closely related to classical
integrable systems and quantum groups . Moreover, antisymmetric solutions of
the CYBE give rise to (triangular) Lie bialgebras [7, 10].

The notion of Rota-Baxter operators on associative algebras was originally
investigated by Baxter in his study of fluctuation theory in probability [6]
and further developed by Rota [20]. The study of Rota-Baxter operators has
experienced a great expansion in recent years due to their broad connections
with combinatorics, quantum field theory, number theory and operads (see [12,
21] and the references therein). In particular, Rota-Baxter operators on Lie
algebras arose independently as the operator forms of the CYBE. Explicitly, it
was shown in [22] that an antisymmetric solution of the CYBE on a quadratic
Lie algebra is equivalent to a linear map
$P:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying

$[P(x),P(y)]=P\big{(}[P(x),y]+[x,P(y)]\big{)},\;\forall x,y\in\mathfrak{g}.$
(2)

The equation (2) is just what we call nowadays a Rota-Baxter operator (of
weight $0$) on a Lie algebra. The operator form of the CYBE was later
restudied in [14]. Identifying an $r\in\mathfrak{g}\otimes\mathfrak{g}$ as a
linear map $r^{\sharp}:\mathfrak{g}^{*}\rightarrow\mathfrak{g}$, Kupershmidt
showed that when $r$ is antisymmetric, $r$ is a solution of the CYBE if and
only if $r^{\sharp}$ satisfies

$[r^{\sharp}(a^{*}),r^{\sharp}(b^{*})]=r^{\sharp}\Big{(}\mathrm{ad}^{*}\big{(}r^{\sharp}(a^{*})\big{)}b^{*}-\mathrm{ad}^{*}\big{(}r^{\sharp}(b^{*})\big{)}a^{*}\Big{)},\;\forall
a^{*},b^{*}\in\mathfrak{g}^{*}.$ (3)

Unifying (2) and (3), the notion of a relative Rota-Baxter operator
(originally named as an $\mathcal{O}$-operator [14]) of a Lie algebra
$(\mathfrak{g},[-,-])$ associated to an arbitrary representation $(\rho,V)$
was also introduced as a natural generalization, which takes the form

$[T(u),T(v)]=T\Big{(}\rho\big{(}T(u)\big{)}v-\rho\big{(}T(v)\big{)}u\Big{)},\;\forall
u,v\in V.$ (4)

Relative Rota-Baxter operators have inspired subsequent studies on
deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
[23], relative Rota-Baxter operators on Hopf algebras [15] and bimodules over
relative Rota-Baxter algebras [9].

###  1.2. Motivation

A bialgebra structure is a vector space equipped with an algebra structure and
a coalgebra structure satisfying some compatible conditions. In addition to
the aforementioned Lie bialgebras, some known examples are antisymmetric
infinitesimal bialgebras [1, 24], Leibniz bialgebras [26], anti-pre-Lie
bialgebras [17], mock-Lie bialgebras [3] and so on. These bialgebras have a
common property, that is, they have equivalent characterizations in terms of
Manin triples which correspond to nondegenerate (symmetric or antisymmetric)
bilinear forms on the algebra structures satisfying certain invariant
conditions. It is natural to consider the bialgebra theory for left-Alia
algebras via the Manin triple approach.

In the previous article [13], we obtain a class of left-Alia algebras based on
twisted derivations of commutative associative algebras. The notions of Manin
triples of left-Alia algebras (with respect to the symmetric invariant
bilinear form) and left-Alia bialgebras are also studied. Moreover, their
equivalence is figured out via specific matched pairs of left-Alia algebras.

In this paper, we aim to solve the problem on how to construct a left-Alia
bialgebra. As an analogue of the studies on Lie bialgebras [7, 10], we first
introduce the notion of a left-Alia Yang-Baxter equation. Then we show that an
antisymmetric solution of the left-Alia Yang-Baxter equation gives rise to a
left-Alia bialgebra that we call triangular. Finally, we study solutions of
the left-Alia Yang-Baxter equation in terms of the operator forms, namely the
relative Rota-Baxter operators of left-Alia algebras.

###  1.3. Outline of the paper

This paper is organized as follows. In Section 2, we recall the notion of
left-Alia algebras and their representations, including a class of non-trivial
examples in invariant theory. In Section 3, we recall the notions of Manin
triples of left-Alia algebras and left-Alia bialgebras. In Section 4, we
introduce the concept of the left-Alia Yang-Baxter equation. We show that
antisymmetric solutions of the left-Alia Yang-Baxter equation give rise to
triangular left-Alia bialgebras (see Proposition 4.2). In Section 5, we
introduce relative Rota-Baxter operators of left-Alia algebras as the operator
forms of solutions of the left-Alia Yang-Baxter equation. We show that in the
antisymmetric condition, a solution of the left-Alia Yang-Baxter equation is
one-to-one correspondence to a relative Rota-Baxter operator with respect to
the coadjoint representation (Theorem 5.6). In Section 6, we introduce the
notion of a pre-left-Alia algebra $(A,\succ,\prec)$, which is the underlying
algebraic structure of a relative Rota-Baxter operator with respect to a
representation $(l,r,V)$ of a left-Alia algebra. We prove that there is a
natural antisymmetric solution of the left-Alia Yang-Baxter equation in the
left-Alia algebra
$A\ltimes_{\mathcal{L}^{*}_{\succ},\mathcal{L}^{*}_{\succ}-\mathcal{R}^{*}_{\prec}}A^{*}$
(Theorem 6.6).

Throughout this paper, unless otherwise specified, all vector spaces are
finite-dimensional over an algebraically closed field $\mathbb{K}$ of
characteristic zero, although many notions and results remain valid in the
infinite-dimensional case.

##  2\. Left-Alia algebras associated to invariant theory

###  2.1. Definition of left Alia algebras

######  Definition 2.1.

[11] A left-Alia algebra is a vector space $A$ together with a bilinear map
$[\cdot,\cdot]:A\times A\rightarrow A$ satisfying the symmetric Jacobi
property (1).

######  Remark 2.2.

A left-Alia algebra $(A,[\cdot,\cdot])$ is a Lie algebra if and only if the
bilinear map $[\cdot,\cdot]$ is antisymmetric.

######  Definition 2.3.

Let $(A,\cdot)$ be a commutative associative algebra. A linear map
$D:A\rightarrow A$ is called a twisted derivation respect to an
$R:A\rightarrow A$ if $D$ satisfies the twisted Lebniz rule:

$D(f\cdot g)=D(f)\cdot g+R(f)\cdot D(g),\ f,g\in A.$ (5)

The following theorem in [13] provide a class of examples of left-Alia
algebras.

######  Theorem 2.4.

[13] Let $(A,\cdot)$ be a commutative associative algebra and $D$ be a twisted
derivation. Define the bilinear map $[\cdot,\cdot]_{R}:A\times A\rightarrow A$
by

$[x,y]_{R}:=x\cdot D(y)-R(y)\cdot D(x),\ x,y\in A.$ (6)

Then $(A,[\cdot,\cdot]_{R})$ is a left-Alia algebra.

A more general construction of left-Alia algebras is given in [11].

######  Theorem 2.5.

[11] Let $(A,\cdot)$ be a commutative associative algebra with linear maps
$f,g:A\rightarrow A$. There is a left-Alia algebra $(A,[\cdot,\cdot])$ given
by

$[x,y]=x\cdot f(y)+g(x\cdot y),\;\forall x,y\in A.$ (7)

which is called a special left-Alia algebra with respect to $(A,\cdot,f,g)$.

###  2.2. Examples of left-Alia algebras associated to invariant theory

Let $G$ be a finite group and $\mathbb{K}$ an algebraic closed field of
characteristic zero. Suppose that $(\rho,V)$ is a $n$-dimensional faithful
representation of $G$ and its dual representation is denoted by
$(\rho,V^{*})$. Let $S=\mathbb{K}[V]=\mathbb{K}[x_{1},\ldots,x_{n}]$ be the
coordinate ring of $V$. Define a $G$-action on $S$ by

$g\cdot\sum_{i_{1},\ldots,i_{n}}k_{i_{1},\ldots,i_{n}}x_{1}^{i_{1}}\ldots
x_{n}^{i_{n}}:=\sum_{i_{1},\ldots,i_{n}}k_{i_{1},\ldots,i_{n}}(\rho(g)x_{1})^{i_{1}}\ldots(\rho(g)x_{n})^{i_{n}}.$
(8)

Define the ring of invariants by

$S^{G}=\\{f\in S:a\cdot f=f,\ \forall a\in G\\}.$

######  Definition 2.6.

[4] A linear automorphism $R\in GL(V)$ is called a pseudo-reflection if
$R^{m}=I$ for some $m\in\mathbb{N}^{*}$ and ${\rm Im}(I-R)$ is 1-dimensional.

In invariant theory, Shephard, Todd and Chevalley figure out the following
equivalent condition that $S^{G}$ is a polynomial algebra.

######  Theorem 2.7.

[8, 25] $S^{G}$ is a polynomial algebra if and only if $G\cong\rho(G)$ is
generated by pseudo-reflections.

For a pseudo-reflection $R$ on $V$, it naturally induces a pseudo-reflection
on $V^{*}$ (also denoted by $R$). For a fixed non-zero $l_{R}\in{\rm
Im}(I-R)\subset V^{*}$, there exists a $\Delta_{R}\in V$ such that

$(I-R)x=\Delta_{R}(x)l_{R},\ \ \forall x\in V^{*}.$ (9)

Denote also $R:S\rightarrow S$ to be an extension of $R\in GL(V)$ satisfying

$R(k_{1}f+k_{2}h)=k_{1}R(f)+k_{2}R(h)\ {\rm and}\ R(fh)=R(f)R(h).$

######  Theorem 2.8.

[4] For each $f\in S$, there exists a twisted derivation $D_{R}:S\rightarrow
S$ with respect to $R$ such that

$R(f)=f-D_{R}(f)l_{R}.$ (10)

Motivated by Theorem 2.8 and Theorem 2.4, we know that each pseodo-reflection
$R$ on $V$ naturally induces a left-Alia algebra
$(\mathbb{K}[V],[\cdot,\cdot]_{R})$. The following is a concrete example.

######  Example 2.9.

Let $R$ be a reflection defined by
$R(x_{1})=x_{2},R(x_{2})=x_{1},R(x_{3})=x_{3}$ on three-dimensional vector
space $V^{*}$ with a basis $\\{x_{1},x_{2},x_{3}\\}$. On the coordinate ring
$S=\mathbb{K}[x_{1},x_{2},x_{3}]$ of $V$, $R$ can be also denoted an extension
of $R$ satisfying $R(fg)=R(f)R(g)$ and $R(k_{1}f+k_{2}h)=k_{1}R(f)+k_{2}R(h)$.
Let $D$ be the twisted derivation on $S$ induced by the reflection $R$. It
follows from Theorem 2.8 that $R(f)=f-D(f)(x_{1}-x_{2})$. Take two polynomials
in $S$, denoted by
$f=\underset{i}{\sum}k_{i}f_{i},g=\underset{j}{\sum}h_{j}g_{j},$ where
$f_{i}=x_{1}^{n_{i,1}}x_{2}^{n_{i,2}}x_{3}^{n_{i,3}},g_{j}=x_{1}^{m_{j,1}}x_{2}^{m_{j,2}}x_{3}^{m_{j,3}}$
are basis of $S$ and $k_{i},h_{j}\in\mathbb{K}$. We have

$\displaystyle D(x_{1})=$ $\displaystyle 1,D(x_{2})=-1,D(x_{3})=0,$
$\displaystyle D(x_{1}^{n_{1}})=$ $\displaystyle
x_{1}^{n_{1}-1}+x_{1}^{n_{1}-2}x_{2}+\cdots+x_{2}^{n_{1}-1},$ $\displaystyle
D(x_{2}^{n_{2}})=$ $\displaystyle-x_{1}^{n_{2}-1}-x_{1}^{n_{2}-1}x_{2}-\cdots-
x_{2}^{n_{2}-1},$ $\displaystyle D(x_{3}^{n_{3}})=$ $\displaystyle 0,$

and

$\displaystyle
D(\underset{i}{\sum}k_{i}x_{1}^{n_{i,1}}x_{2}^{n_{i,2}}x_{3}^{n_{i,3}})=$
$\displaystyle\underset{i}{\sum}k_{i}(D(x_{1}^{n_{i,1}}x_{2}^{n_{i,2}})x_{3}^{n_{i,3}}+R(x_{1}^{n_{i,1}}x_{2}^{n_{i,2}})D(x_{3}^{n_{i,3}}))$
$\displaystyle=$
$\displaystyle\underset{i}{\sum}k_{i}(x_{1}^{n_{i,1}}D(x_{2}^{n_{i,2}})+R(x_{2}^{n_{i,2}})D(x_{1}^{n_{i,1}}))x_{3}^{n_{i,3}}$
$\displaystyle=$
$\displaystyle\underset{i}{\sum}k_{i}(x_{1}^{n_{i,1}+n_{i,2}-1}+x_{1}^{n_{i,1}+n_{i,2}-2}x_{2}+\cdots+x_{1}^{n_{i,2}}x_{2}^{n_{i,1}-1}$
$\displaystyle-x_{1}^{n_{i,1}+n_{i,2}-1}-\cdots-
x_{1}^{n_{1}}x_{2}^{n_{2}-1})x_{3}^{n_{i,3}}.$

Moreover, the bilinear map

$\displaystyle[f,g]_{R}=$
$\displaystyle\underset{i,j}{\sum}k_{i}h_{j}[f_{i},g_{j}]_{R}$
$\displaystyle=$
$\displaystyle\underset{i,j}{\sum}k_{i}h_{j}(f_{i}D(g_{j})-R(g_{j})D(f_{i}))$
$\displaystyle=$
$\displaystyle\underset{i,j}{\sum}k_{i}h_{j}(x_{1}^{n_{i,1}+m_{j,1}+m_{j,2}-1}x_{2}^{n_{i,2}}+\cdots+x_{1}^{n_{i,1}+m_{j,2}}x_{2}^{n_{i,2}+m_{j,1}-1}$
$\displaystyle-x_{1}^{n_{i,1}+m_{j,1}+m_{j,2}-1}x_{2}^{n_{i,2}}-\cdots-
x_{1}^{m_{j,1}+n_{i,1}}x_{2}^{n_{i,2}+m_{j,2}-1}$ $\displaystyle-
x_{1}^{m_{j,2}+n_{i,1}+n_{i,2}-1}x_{2}^{m_{j,1}}-\cdots-
x_{1}^{n_{i,2}+m_{j,2}}x_{2}^{n_{i,1}+m_{j,1}-1}$
$\displaystyle+x_{1}^{m_{j,2}+n_{i,1}+n_{i,2}-1}x_{2}^{m_{j,1}}+\cdots+x_{1}^{n_{i,1}+m_{j,2}}x_{2}^{n_{i,2}+m_{j,1}-1})x_{3}^{m_{j,3}+n_{i,3}}$

makes $(S,[\cdot,\cdot]_{R})$ into a left-Alia algebra.

###  2.3. Representation of left-Alia algebras

In this subsection, we recall the representation theory [13] of left-Alia
algebras.

######  Definition 2.10.

A representation of a left-Alia algebra $(A,[\cdot,\cdot])$ is a triple
$(l,r,V)$, where $V$ is a vector space and $l,r:A\rightarrow\mathrm{End}(V)$
are linear maps such that the following equation holds:

$l([x,y])-l([y,x])=r(x)r(y)v-r(y)r(x)+r(y)l(x)v-r(x)l(y),\;\forall x,y\in
A,v\in V.$ (11)

######  Proposition 2.11.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra, $V$ be a vector space and
$l,r:A\rightarrow\mathrm{End}(V)$ be linear maps. Then $(l,r,V)$ is a
representation of $(A,[\cdot,\cdot])$ if and only if there is a left-Alia
algebra on the direct sum $A\oplus V$ of vector spaces given by

$[x+u,y+v]_{d}=[x,y]+l(x)v+r(y)u,\;\forall x,y\in A,u,v\in V.$ (12)

In this case, we denote $(A\oplus V,[\cdot,\cdot]_{d})=A\ltimes_{l,r}V$.

######  Example 2.12.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra. Set linear maps
$\mathcal{L}_{[\cdot,\cdot]},\mathcal{R}_{[\cdot,\cdot]}:A\rightarrow\mathrm{End}(A)$
by

$\mathcal{L}_{[\cdot,\cdot]}(x)y=[x,y]=\mathcal{R}_{[\cdot,\cdot]}(y)x,\;\forall
x,y\in A.$

Then, $(\mathcal{L}_{[\cdot,\cdot]},\mathcal{R}_{[\cdot,\cdot]},A)$ is a
representation of $(A,[\cdot,\cdot])$, which is called an adjoint
representation.

Let $A$ and $V$ be vector spaces. For a linear map
$l:A\rightarrow\mathrm{End}(V)$, define a linear map
$l^{*}:A\rightarrow\mathrm{End}(V^{*})$ by

$\langle l^{*}(x)u^{*},v\rangle=-\langle u^{*},l(x)v\rangle,\;\forall x\in
A,u^{*}\in V^{*},v\in V.$

######  Proposition 2.13.

[13] Let $(l,r,V)$ be a representation of a left-Alia algebra
$(A,[\cdot,\cdot])$. Then, $(l^{*},l^{*}-r^{*},V^{*})$ is also a
representation of $(A,[\cdot,\cdot])$. In particular,
$(\mathcal{L}^{*}_{[\cdot,\cdot]},\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]},A^{*})$
is a representation of $(A,[\cdot,\cdot])$, which is called the coadjoint
representation.

##  3\. Left-Alia bialgebras

In this section, we recall some basic results on left-Alia bialgebras [13].

######  Definition 3.1.

A quadratic left-Alia algebra is a triple $(A,[\cdot,\cdot],\mathcal{B})$,
where $(A,[\cdot,\cdot])$ is a left-Alia algebra and $\mathcal{B}$ is a
nondegenerate symmetric bilinear form on $A$ which is invariant in the sense
that

$\mathcal{B}([x,y],z)=\mathcal{B}(x,[z,y]-[y,z]),\;\forall x,y,z\in A.$ (13)

There is a typical class of quadratic left-Alia algebras realized from Theorem
2.5.

######  Proposition 3.2.

Let $(A,\cdot)$ be a commutative associative algebra and $f:A\rightarrow A$ be
a linear map. Let $\mathcal{B}$ be a nondegenerate symmetric invariant
bilinear form on $(A,\cdot)$ and $\hat{f}:A\rightarrow A$ be the adjoint map
of $f$ with respect to $\mathcal{B}$, given by

$\mathcal{B}\big{(}\hat{f}(x),y\big{)}=\mathcal{B}\big{(}x,f(y)\big{)},\;\forall
x,y\in A.$

Then there is a quadratic left-Alia algebra $(A,[\cdot,\cdot],\mathcal{B})$,
where $(A,[\cdot,\cdot])$ is the special left-Alia algebra with respect to
$(A,\cdot,f,-\hat{f})$, that is,

$[x,y]=x\cdot f(y)-\hat{f}(x\cdot y).$ (14)

######  Definition 3.3.

Let $(A,[\cdot,\cdot]_{A})$ and $(A^{*},[\cdot,\cdot]_{A^{*}})$ be left-Alia
algebras. Assume that there is a left-Alia algebra structure $(A\oplus
A^{*},[\cdot,\cdot]_{d})$ on $A\oplus A^{*}$ which contains
$(A,[\cdot,\cdot]_{A})$ and $(A^{*},[\cdot,\cdot]_{A^{*}})$ as left-Alia
subalgebras. Suppose that the natural nondegenerate symmetric bilinear form
$\mathcal{B}_{d}$ given by

$\mathcal{B}_{d}(x+a^{*},y+b^{*})=\langle x,b^{*}\rangle+\langle
a^{*},y\rangle,\;\forall x,y\in A,a^{*},b^{*}\in A^{*}.$ (15)

is invariant on $(A\oplus A^{*},[\cdot,\cdot]_{d})$, that is, $(A\oplus
A^{*},[\cdot,\cdot]_{d},\mathcal{B}_{d})$ is a quadratic left-Alia algebra.
Then we say $\big{(}(A\oplus
A^{*},[\cdot,\cdot]_{d},\mathcal{B}_{d}),A,A^{*}\big{)}$ is a Manin triple of
left-Alia algebras (with respect to the symmetric invariant bilinear form).

######  Definition 3.4.

A left-Alia coalgebra is a pair $(A,\delta)$, such that $A$ is a vector space
and $\delta:A\rightarrow A\otimes A$ is a co-multiplication satisfying

$(\mathrm{id}^{\otimes
3}+\xi+\xi^{2})(\tau\otimes\mathrm{id}-\mathrm{id}^{\otimes
3})(\delta\otimes\mathrm{id})\delta=0,$ (16)

where $\tau(x\otimes y)=y\otimes x$ and $\xi(x\otimes y\otimes z)=y\otimes
z\otimes x$ for all $x,y,z\in A$.

######  Proposition 3.5.

Let $A$ be a vector space and $\delta:A\rightarrow A\otimes A$ be a co-
multiplication. Let $[\cdot,\cdot]_{A^{*}}:A^{*}\otimes A^{*}\rightarrow
A^{*}$ be the linear dual of $\delta$, that is,

$\langle[a^{*},b^{*}]_{A^{*}},x\rangle=\langle\delta^{*}(a^{*}\otimes
b^{*}),x\rangle=\langle a^{*}\otimes b^{*},\delta(x)\rangle,\;\;\forall
a^{*},b^{*}\in A^{*},x\in A.$ (17)

Then $(A,\delta)$ is a left-Alia coalgebra if and only if
$(A^{*},[\cdot,\cdot]_{A^{*}})$ is a left-Alia algebra.

######  Definition 3.6.

A left-Alia bialgebra is a triple $(A,[\cdot,\cdot],\delta)$, such that
$(A,[\cdot,\cdot])$ is a left-Alia algebra, $(A,\delta)$ is a left-Alia
coalgebra and the following equation holds:

$(\tau-\mathrm{id}^{2})\big{(}\delta([x,y]-[y,x])+(\mathcal{R}_{[\cdot,\cdot]}(x)\otimes\mathrm{id})\delta(y)-(\mathcal{R}_{[\cdot,\cdot]}(y)\otimes\mathrm{id})\delta(x)\big{)}=0,\;\forall
x,y\in A.$ (18)

The following theorem figures out the one-one correspondence between left-Alia
bialgebras and Manin triples of left-Alia algebras.

######  Theorem 3.7.

Let $(A,[\cdot,\cdot]_{A})$ be a left-Alia algebra. Suppose that there is a
left-Alia algebra structure $(A^{*},[\cdot,\cdot]_{A^{*}})$ on the dual space
$A^{*}$, and $\delta:A\rightarrow A\otimes A$ is the linear dual of
$[\cdot,\cdot]_{A^{*}}$. Then $(A,[\cdot,\cdot]_{A},\delta)$ is a left-Alia
bialgebra if and only if there is a Manin triple of left-Alia algebras
$\big{(}(A\oplus A^{*},[\cdot,\cdot]_{d},\mathcal{B}_{d}),A,A^{*}\big{)}$. In
this case, the multiplication $[-,-]_{d}$ on $A\oplus A^{*}$ is given by

$\displaystyle[x+a^{*},y+b^{*}]_{d}$ $\displaystyle=$
$\displaystyle[x,y]_{A}+\mathcal{L}^{*}_{[\cdot,\cdot]_{A^{*}}}(a^{*})y+(\mathcal{L}^{*}_{[\cdot,\cdot]_{A^{*}}}-\mathcal{R}^{*}_{[\cdot,\cdot]_{A^{*}}})(b^{*})x$
$\displaystyle+[a^{*},b^{*}]_{A^{*}}+\mathcal{L}^{*}_{[\cdot,\cdot]_{A}}(x)b^{*}+(\mathcal{L}^{*}_{[\cdot,\cdot]_{A}}-\mathcal{R}^{*}_{[\cdot,\cdot]_{A}})(y)a^{*},$

for all $x,y\in A,a^{*},b^{*}\in A^{*}$.

##  4\. Triangular left-Alia bialgebras and the left-Alia Yang-Baxter equation

In this section, we introduce the notion of the left-Alia Yang-Baxter equation
in a left-Alia algebra. An antisymmetric solution of the left-Alia Yang-Baxter
equation gives rise to a left-Alia bialgebra which we call triangular.

######  Definition 4.1.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra and
$r=\sum\limits_{i}u_{i}\otimes v_{i}\in A\otimes A$. Set

$\mathrm{Al}(r)=\sum_{i,j}[u_{i},u_{j}]\otimes v_{i}\otimes
v_{j}+u_{i}\otimes([u_{j},v_{i}]-[v_{i},u_{j}])\otimes v_{j}-u_{i}\otimes
u_{j}\otimes[v_{j},v_{i}]\in A\otimes A\otimes A.$ (19)

We say $r$ is a solution of the left-Alia Yang-Baxter equation if
$\mathrm{Al}(r)=0$.

######  Proposition 4.2.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra and
$r=\sum\limits_{i}u_{i}\otimes v_{i}\in A\otimes A$. Let
$h:A\rightarrow\mathrm{End}(A\otimes A)$ be a linear map given by

$h(x)=(\mathcal{R}_{[\cdot,\cdot]}-\mathcal{L}_{[\cdot,\cdot]})(x)\otimes\mathrm{id}-\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(x),\;\forall
x\in A.$ (20)

Set a co-multiplication $\delta_{r}:A\rightarrow A\otimes A$ by

$\delta_{r}(x)=h(x)r=\big{(}(\mathcal{R}_{[\cdot,\cdot]}-\mathcal{L}_{[\cdot,\cdot]})(x)\otimes\mathrm{id}-\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}r,\;\forall
x\in A.$ (21)

  1. (a)

For all $x\in A$, (16) holds if and only if

$\displaystyle(\mathrm{id}+\xi+\xi^{2})\bigg{(}\big{(}\mathrm{id}\otimes\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}(\mathrm{id}\otimes\tau)\mathrm{Al}(r)+\sum_{j}h([u_{j},x]-[x,u_{j}])\big{(}r+\tau(r)\big{)}\otimes
v_{j}$ $\displaystyle+\tau h(u_{j})\big{(}r+\tau(r)\big{)}\otimes
v_{j}+v_{j}\otimes\big{(}\mathcal{R}_{[\cdot,\cdot]}(u_{j})\otimes\mathrm{id}\big{)}h(x)\big{(}r+\tau(r)\big{)}$
$\displaystyle+\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(v_{j})\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}\Big{(}u_{j}\otimes\big{(}r+\tau(r)\big{)}\Big{)}$
$\displaystyle+\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(u_{j})\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}\Big{(}v_{j}\otimes\big{(}r+\tau(r)\big{)}\Big{)}\Bigg{)}=0.\
\ \ \ \ \ $ (22)

  2. (b)

For all $x,y\in A$, (18) holds if and only if

$\tau
h([x,y]-[y,x])\big{(}r+\tau(r)\big{)}-\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(y)\big{)}\tau
h(x)\big{(}r+\tau(r)\big{)}+\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}\tau
h(y)\big{(}r+\tau(r)\big{)}=0.$ (23)

###### Proof.

(a) For all $x\in A$, we have

$\displaystyle(\delta_{r}\otimes\mathrm{id})\delta_{r}(x)=(\delta_{r}\otimes\mathrm{id})(\sum_{i}[u_{i},x]\otimes
v_{i}-[x,u_{i}]\otimes v_{i}-u_{i}\otimes[v_{i},x])$
$\displaystyle=\sum_{i,j}[u_{j},[u_{i},x]]\otimes v_{j}\otimes
v_{i}-[[u_{i},x],u_{j}]\otimes v_{j}\otimes
v_{i}-u_{j}\otimes[v_{j},[u_{i},x]]\otimes v_{i}$ $\displaystyle\ \
-[u_{j},[x,u_{i}]]\otimes v_{j}\otimes v_{i}+[[x,u_{i}],u_{j}]\otimes
v_{j}\otimes v_{i}+u_{j}\otimes[v_{j},[x,u_{i}]]\otimes v_{i}$ $\displaystyle\
\ -[u_{j},u_{i}]\otimes v_{j}\otimes[v_{i},x]+[u_{i},u_{j}]\otimes
v_{j}\otimes[v_{i},x]+u_{j}\otimes[v_{j},u_{i}]\otimes[v_{i},x].$

Then

$\displaystyle(\tau\otimes\mathrm{id}-\mathrm{id}^{\otimes
3})(\delta_{r}\otimes\mathrm{id})\delta_{r}(x)$
$\displaystyle=\sum_{i,j}v_{j}\otimes[u_{j},[u_{i},x]]\otimes
v_{i}-v_{j}\otimes[[u_{i},x],u_{j}]\otimes v_{i}-[v_{j},[u_{i},x]]\otimes
u_{j}\otimes v_{i}$ $\displaystyle\ \ -v_{j}\otimes[u_{j},[x,u_{i}]]\otimes
v_{i}+v_{j}\otimes[[x\,u_{i}],u_{j}]\otimes v_{i}+[v_{j},[x,u_{i}]]\otimes
u_{j}\otimes v_{i}$ $\displaystyle\ \
-v_{j}\otimes[u_{j},u_{i}]\otimes[v_{i},x]+v_{j}\otimes[u_{i},u_{j}]\otimes[v_{i},x]+[v_{j},u_{i}]\otimes
u_{j}\otimes[v_{i},x]$ $\displaystyle\ \ -[u_{j},[u_{i},x]]\otimes
v_{j}\otimes v_{i}+[[u_{i},x],u_{j}]\otimes v_{j}\otimes
v_{i}+u_{j}\otimes[v_{j},[u_{i},x]]\otimes v_{i}$ $\displaystyle\ \
+[u_{j},[x,u_{i}]]\otimes v_{j}\otimes v_{i}-[[x,u_{i}],u_{j}]\otimes
v_{j}\otimes v_{i}-u_{j}\otimes[v_{j},[x,u_{i}]]\otimes v_{i}$ $\displaystyle\
\ +[u_{j},u_{i}]\otimes v_{j}\otimes[v_{i},x]-[u_{i},u_{j}]\otimes
v_{j}\otimes[v_{i},x]-u_{j}\otimes[v_{j},u_{i}]\otimes[v_{i},x].$

We have

$\displaystyle\sum_{i,j}-v_{j}\otimes[u_{j},u_{i}]\otimes[v_{i},x]+v_{j}\otimes[u_{i},u_{j}]\otimes[v_{i},x]+[v_{j},u_{i}]\otimes
u_{j}\otimes[v_{i},x]$ $\displaystyle\ \ +[u_{j},u_{i}]\otimes
v_{j}\otimes[v_{i},x]-[u_{i},u_{j}]\otimes
v_{j}\otimes[v_{i},x]-u_{j}\otimes[v_{j},u_{i}]\otimes[v_{i},x]$
$\displaystyle=-\sum_{j}\tau
h(u_{j})\big{(}r+\tau(r)\big{)}\otimes[v_{j},x]-\sum_{i,j}\big{(}[u_{i},u_{j}]\otimes
v_{j}\otimes[v_{i},x]+u_{i}\otimes[u_{j},v_{i}]\otimes[v_{j},x]\big{)},$

$\displaystyle\sum_{i,j}v_{j}\otimes[u_{j},[u_{i},x]]\otimes
v_{i}-v_{j}\otimes[[u_{i},x],u_{j}]\otimes v_{i}-[v_{j},[u_{i},x]]\otimes
u_{j}\otimes v_{i}$ $\displaystyle\ \ -[u_{j},[u_{i},x]]\otimes v_{j}\otimes
v_{i}+[[u_{i},x],u_{j}]\otimes v_{j}\otimes
v_{i}+u_{j}\otimes[v_{j},[u_{i},x]]\otimes v_{i}$
$\displaystyle=-\sum_{j}h([u_{j},x])\big{(}r+\tau(r)\big{)}\otimes
v_{j}-\sum_{i,j}\big{(}v_{j}\otimes[[u_{i},x],u_{j}]\otimes
v_{i}+[[u_{i},x],v_{j}]\otimes u_{j}\otimes v_{i}\big{)},$

$\displaystyle\sum_{i,j}-v_{j}\otimes[u_{j},[x,u_{i}]]\otimes
v_{i}+v_{j}\otimes[[x,u_{i}],u_{j}]\otimes v_{i}+[v_{j},[x,u_{i}]]\otimes
u_{j}\otimes v_{i}$ $\displaystyle\ \ +[u_{j},[x,u_{i}]]\otimes v_{j}\otimes
v_{i}-[[x,u_{i}],u_{j}]\otimes v_{j}\otimes
v_{i}-u_{j}\otimes[v_{j},[x,u_{i}]]\otimes v_{i}$
$\displaystyle=\sum_{j}h([x,u_{j}])\big{(}r+\tau(r)\big{)}\otimes
v_{j}+\sum_{i,j}v_{j}\otimes[[x,u_{i}],u_{j}]\otimes
v_{i}+[[x,u_{i}],v_{j}]\otimes u_{j}\otimes v_{i},$
$\displaystyle\sum_{i,j}v_{j}\otimes[[x,u_{i}],u_{j}]\otimes
v_{i}-v_{j}\otimes[[u_{i},x],u_{j}]\otimes v_{i}$
$\displaystyle=-\sum_{i,j}v_{j}\otimes\big{(}\mathcal{R}(u_{j})\otimes\mathrm{id}\big{)}h(x)\big{(}r+\tau(r)\big{)}+\sum_{i,j}-v_{j}\otimes[u_{i},u_{j}]\otimes[v_{i},x]$
$\displaystyle+v_{j}\otimes[[v_{i},x],u_{j}]\otimes
u_{i}-v_{j}\otimes[[x,v_{i}],u_{j}]\otimes
u_{i}-v_{j}\otimes[v_{i},u_{j}]\otimes[u_{i},x].$

By the action of $\xi$ we have

$\displaystyle(\mathrm{id}+\xi+\xi^{2})\big{(}v_{j}\otimes[[v_{i},x],u_{j}]\otimes
u_{i}-v_{j}\otimes[[x,v_{i}],u_{j}]\otimes u_{i}$
$\displaystyle-[[u_{i},x],v_{j}]\otimes u_{j}\otimes
v_{i}+[[x,u_{i}],v_{j}]\otimes u_{j}\otimes v_{i}\big{)}$
$\displaystyle=(\mathrm{id}+\xi+\xi^{2})\big{(}v_{j}\otimes[[v_{i},x],u_{j}]\otimes
u_{i}-v_{j}\otimes[[x,v_{i}],u_{j}]\otimes u_{i}$ $\displaystyle\ \
-v_{j}\otimes[[u_{j},x],v_{i}]\otimes
u_{i}+v_{j}\otimes[[x,u_{j}],v_{i}]\otimes u_{i}\big{)}$
$\displaystyle\overset{\eqref{eq:0-Alia}}{=}(\mathrm{id}+\xi+\xi^{2})\big{(}v_{j}\otimes[[v_{i},u_{j}],x]\otimes
u_{i}-v_{j}\otimes[[u_{j},v_{i}],x]\otimes u_{i}\big{)}.$

Hence we have

$\displaystyle(\mathrm{id}+\xi+\xi^{2})\big{(}\sum_{i,j}-v_{j}\otimes[u_{i},u_{j}]\otimes[v_{i},x]-v_{j}\otimes[v_{i},u_{j}]\otimes[u_{i},x]+u_{i}\otimes
v_{j}\otimes[[v_{i},u_{j}],x]$ $\displaystyle-u_{i}\otimes
v_{j}\otimes[[u_{j},v_{i}],x]-[u_{i},u_{j}]\otimes
v_{j}\otimes[v_{i},x]-u_{i}\otimes[u_{j},v_{i}]\otimes[v_{j},x]\big{)}$
$\displaystyle=(\mathrm{id}+\xi+\xi^{2})\Big{(}-\big{(}\mathrm{id}\otimes\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}(\mathrm{id}\otimes\tau)\mathrm{Al}(r)-\sum_{i,j}(v_{i}\otimes[u_{j},u_{i}]\otimes[v_{j},x]$
$\displaystyle\ \
+v_{i}\otimes[v_{j},u_{i}]\otimes[u_{j},x]+u_{i}\otimes[u_{j},v_{i}]\otimes[v_{j},x]+u_{i}\otimes[v_{j},v_{i}]\otimes[u_{j},x])\Big{)}$
$\displaystyle=-(\mathrm{id}+\xi+\xi^{2})\Bigg{(}\big{(}\mathrm{id}\otimes\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}(\mathrm{id}\otimes\tau)\mathrm{Al}(r)$
$\displaystyle\ \
+\sum_{j}\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(v_{j})\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}\Big{(}u_{j}\otimes\big{(}r+\tau(r)\big{)}\Big{)}$
$\displaystyle~{}~{}+\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(u_{j})\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}\Big{(}v_{j}\otimes\big{(}r+\tau(r)\big{)}\Big{)}\bigg{)}.$

In conclusion, we have

$\displaystyle(\mathrm{id}+\xi+\xi^{2})(\tau\otimes\mathrm{id}-\mathrm{id}^{\otimes
3})(\delta_{r}\otimes\mathrm{id})\delta_{r}(x)$
$\displaystyle=-(\mathrm{id}+\xi+\xi^{2})\bigg{(}\big{(}\mathrm{id}\otimes\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}(\mathrm{id}\otimes\tau)\mathrm{Al}(r)+\sum_{j}h([u_{j},x]-[x,u_{j}])\big{(}r+\tau(r)\big{)}\otimes
v_{j}$ $\displaystyle\ \ +\tau h(u_{j})\big{(}r+\tau(r)\big{)}\otimes
v_{j}+v_{j}\otimes\big{(}\mathcal{R}_{[\cdot,\cdot]}(u_{j})\otimes\mathrm{id}\big{)}h(x)\big{(}r+\tau(r)\big{)}$
$\displaystyle\ \
+\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(v_{j})\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}\Big{(}u_{j}\otimes\big{(}r+\tau(r)\big{)}\Big{)}+\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(u_{j})\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}\Big{(}v_{j}\otimes\big{(}r+\tau(r)\big{)}\Big{)}\Bigg{)}.$

Hence the conclusion follows.

(b) For all $x,y\in A$, we have

$\displaystyle\delta_{r}([x,y])$ $\displaystyle=$
$\displaystyle\sum_{i}[u_{i},[[x,y],v_{i}]]-[[x,y],u_{i}]\otimes
v_{i}-u_{i}\otimes[v_{i},[x,y]],$ $\displaystyle-\delta_{r}([y,x])$
$\displaystyle=$
$\displaystyle\sum_{i}-[[u_{i},[y,x]],v_{i}]+[[y,x],u_{i}]\otimes
v_{i}+u_{i}\otimes[v_{i},[y,x]],$
$\displaystyle\big{(}\mathcal{R}_{[\cdot,\cdot]}(x)\otimes\mathrm{id}\big{)}\delta_{r}(y)$
$\displaystyle=$ $\displaystyle\sum_{i}[[u_{i},y],x]\otimes
v_{i}-[[y,u_{i}],x]\otimes v_{i}-[u_{i},x]\otimes[v_{i},y],$
$\displaystyle-\big{(}\mathcal{R}_{[\cdot,\cdot]}(y)\otimes\mathrm{id}\big{)}\delta_{r}(x)$
$\displaystyle=$ $\displaystyle\sum_{i}-[[u_{i},x],y]\otimes
v_{i}+[[x,u_{i}],y]\otimes v_{i}+[u_{i},y]\otimes[v_{i},x].$

Hence

$\displaystyle(\tau-\mathrm{id}^{2})\big{(}\delta_{r}([x,y]-[y,x])+(\mathcal{R}_{[\cdot,\cdot]}(x)\otimes\mathrm{id})\delta_{r}(y)-(\mathcal{R}_{[\cdot,\cdot]}(y)\otimes\mathrm{id})\delta_{r}(x)\big{)}$
$\displaystyle\overset{\eqref{eq:0-Alia}}{=}\sum_{i}v_{i}\otimes[u_{i},[x,y]]-[v_{i},[x,y]]\otimes
u_{i}-v_{i}\otimes[u_{i},[y,x]]+[v_{i},[y,x]]\otimes u_{i}$ $\displaystyle\ \
-[v_{i},y]\otimes[u_{i},x]+[v_{i},x]\otimes[u_{i},y]-[u_{i},[x,y]]\otimes
v_{i}+u_{i}\otimes[v_{i},[x,y]]$ $\displaystyle\ \ +[u_{i},[y,x]]\otimes
v_{i}-u_{i}\otimes[v_{i},[y,x]]+[u_{i},x]\otimes[v_{i},y]-[u_{i},y]\otimes[v_{i},x].$

Noticing that

$\displaystyle\sum_{i}v_{i}\otimes[u_{i},[x,y]]-[v_{i},[x,y]]\otimes
u_{i}-[u_{i},[x,y]]\otimes v_{i}+u_{i}\otimes[v_{i},[x,y]]$
$\displaystyle=\tau
h([x,y])\big{(}r+\tau(r)\big{)}+\sum_{i}v_{i}\otimes[[x,y],u_{i}]+u_{i}\otimes[[x,y],v_{i}],$
$\displaystyle\sum_{i}-v_{i}\otimes[u_{i},[y,x]]+[v_{i},[y,x]]\otimes
u_{i}+[u_{i},[y,x]]\otimes v_{i}-u_{i}\otimes[v_{i},[y,x]]$
$\displaystyle=-\Big{(}\tau
h([y,x])\big{(}r+\tau(r)\big{)}+\sum_{i}v_{i}\otimes[[y,x],u_{i}]+u_{i}\otimes[[y,x],v_{i}]\Big{)},$
$\displaystyle\sum_{i}v_{i}\otimes[[x,y],u_{i}]-v_{i}\otimes[[y,x],u_{i}]$
$\displaystyle\overset{\eqref{eq:0-Alia}}{=}\sum_{i}v_{i}\otimes[[x,u_{i}],y]-v_{i}\otimes[[u_{i},x],y]+v_{i}\otimes[[u_{i},y],x]-v_{i}\otimes[[y,u_{i}],x],$
$\displaystyle\sum_{i}u_{i}\otimes[[x,y],v_{i}]-u_{i}\otimes[[y,x],v_{i}]$
$\displaystyle\overset{\eqref{eq:0-Alia}}{=}\sum_{i}u_{i}\otimes[[x,v_{i}],y]-u_{i}\otimes[[v_{i},x],y]+u_{i}\otimes[[v_{i},y],x]-u_{i}\otimes[[y,v_{i}],x],$
$\displaystyle\sum_{i}[v_{i},x]\otimes[u_{i},y]+[u_{i},x]\otimes[v_{i},y]+v_{i}\otimes[[x,u_{i}],y]$
$\displaystyle-
v_{i}\otimes[[u_{i},x],y]+u_{i}\otimes[[x,v_{i}],y]-u_{i}\otimes[[v_{i},x],y]$
$\displaystyle=-\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(y)\big{)}\tau
h(x)\big{(}r+\tau(r)\big{)},$
$\displaystyle\sum_{i}-[v_{i},y]\otimes[u_{i},x]-[u_{i},y]\otimes[v_{i},x]-v_{i}\otimes[[y,u_{i}],x]$
$\displaystyle+v_{i}\otimes[[u_{i},y],x]-u_{i}\otimes[[y,v_{i}],x]+u_{i}\otimes[[v_{i},y],x]$
$\displaystyle=\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}\tau
h(y)\big{(}r+\tau(r)\big{)},$

we finally have

$\displaystyle(\tau-\mathrm{id}^{2})\big{(}\delta_{r}([x,y]-[y,x])+(\mathcal{R}_{[\cdot,\cdot]}(x)\otimes\mathrm{id})\delta_{r}(y)-(\mathcal{R}_{[\cdot,\cdot]}(y)\otimes\mathrm{id})\delta_{r}(x)\big{)}$
$\displaystyle=\tau
h([x,y]-[y,x])\big{(}r+\tau(r)\big{)}-\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(y)\big{)}\tau
h(x)\big{(}r+\tau(r)\big{)}$
$\displaystyle+\big{(}\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]}(x)\big{)}\tau
h(y)\big{(}r+\tau(r)\big{)}.$

Hence the conclusion follows. ∎

By Proposition 4.2, we have

######  Corollary 4.3.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra and $r\in A\otimes A$. Suppose
that $r$ is an antisymmetric solution of the left-Alia Yang-Baxter equation,
that is,

$r+\tau(r)=0,\;\mathrm{Al}(r)=0.$ (24)

Then $(A,[\cdot,\cdot],\delta_{r})$ is a left-Alia bialgebra, where
$\delta_{r}:A\rightarrow A\otimes A$ is given by (21).

######  Definition 4.4.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra. If $r\in A\otimes A$ is an
antisymmetric solution of the left-Alia Yang-Baxter equation, then we say the
resulting left-Alia bialgebra $(A,[\cdot,\cdot],\delta_{r})$ from Corollary
4.3 is triangular.

##  5\. Relative Rota-Baxter operators of left-Alia algebras

In this section, we study relative Rota-Baxter operators of left-Alia
algebras, which serve as operator forms of the left-Alia Yang-Baxter equation.

######  Definition 5.1.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra with a representation
$(l,r,V)$. A linear map $T:V\rightarrow A$ is called a relative Rota-Baxter
operator of $(A,[\cdot,\cdot])$ associated to $(l,r,V)$ if the following
equation holds:

$[T(u),T(v)]=T\big{(}l(Tu)v+r(Tv)u\big{)},\;\forall u,v\in V.$ (25)

For vector spaces $V$ and $A$, we can identify an element $r\in V\otimes A$ as
a linear map $r^{\sharp}:V^{*}\rightarrow A$ by

$\langle r^{\sharp}(u^{*}),a^{*}\rangle=\langle r,u^{*}\otimes
a^{*}\rangle,\;\forall u^{*}\in V^{*},a^{*}\in A^{*}.$ (26)

Now we study the operator form of $r$ in a triangular left-Alia bialgebra
$(A,[\cdot,\cdot],\delta_{r})$.

######  Proposition 5.2.

Let $(A,[\cdot,\cdot]_{A})$ be a left-Alia algebra and
$r=\sum\limits_{i}u_{i}\otimes v_{i}\in A\otimes A$ be antisymmetric. Let
$\delta_{r}:A\rightarrow A\otimes A$ be a co-multiplication given by (21) and
$[\cdot,\cdot]_{A^{*}}:A^{*}\otimes A^{*}\rightarrow A^{*}$ be the linear dual
of $\delta_{r}$. Then we have

$[a^{*},b^{*}]_{A^{*}}=\mathcal{L}^{*}_{[\cdot,\cdot]_{A}}\big{(}r^{\sharp}(a^{*})\big{)}b^{*}+(\mathcal{L}^{*}_{[\cdot,\cdot]_{A}}-\mathcal{R}^{*}_{[\cdot,\cdot]_{A}})\big{(}r^{\sharp}(b^{*})\big{)}a^{*},\;\forall
a^{*},b^{*}\in A^{*}.$ (27)

Moreover, we have

$\langle
r^{\sharp}([a^{*},b^{*}]_{A^{*}})-[r^{\sharp}(a^{*}),r^{\sharp}(b^{*})]_{A},c^{*}\rangle=\langle
a^{*}\otimes b^{*}\otimes
c^{*},(\tau\otimes\mathrm{id})\mathrm{Al}(r)\rangle.$ (28)

###### Proof.

Let $a^{*},b^{*},c^{*}\in A^{*},x\in A$. Then we have

$\displaystyle\langle[a^{*},b^{*}]_{A^{*}},x\rangle=\langle a^{*}\otimes
b^{*},\delta_{r}(x)\rangle$ $\displaystyle=\langle a^{*}\otimes
b^{*},\big{(}(\mathcal{R}_{[\cdot,\cdot]_{A}}-\mathcal{L}_{[\cdot,\cdot]_{A}})(x)\otimes\mathrm{id}-\mathrm{id}\otimes\mathcal{R}_{[\cdot,\cdot]_{A}}(x)\big{)}r\rangle$
$\displaystyle=\langle
r,(\mathcal{L}^{*}_{[\cdot,\cdot]_{A}}-\mathcal{R}^{*}_{[\cdot,\cdot]_{A}})(x)a^{*}\otimes
b^{*}+a^{*}\otimes\mathcal{R}^{*}_{[\cdot,\cdot]_{A}}(x)b^{*}\rangle$
$\displaystyle=\langle
r^{\sharp}(a^{*}),\mathcal{R}^{*}_{[\cdot,\cdot]_{A}}(x)b^{*}\rangle-\langle
r^{\sharp}(b^{*}),(\mathcal{L}^{*}_{[\cdot,\cdot]_{A}}-\mathcal{R}^{*}_{[\cdot,\cdot]_{A}})(x)a^{*}\rangle$
$\displaystyle=-\langle[r^{\sharp}(a^{*}),x]_{A},b^{*}\rangle+\langle[x,r^{\sharp}(b^{*})]_{A}-[r^{\sharp}(b^{*}),x]_{A},a^{*}\rangle$
$\displaystyle=\langle
x,\mathcal{L}^{*}_{[\cdot,\cdot]_{A}}\big{(}r^{\sharp}(a^{*})\big{)}b^{*}+(\mathcal{L}^{*}_{[\cdot,\cdot]_{A}}-\mathcal{R}^{*}_{[\cdot,\cdot]_{A}})\big{(}r^{\sharp}(b^{*})\big{)}a^{*}\rangle.$

Hence (27) holds. Moreover, we have

$\displaystyle\langle
r^{\sharp}([a^{*},b^{*}]_{A^{*}}),c^{*}\rangle=\sum_{i}\langle
u_{i},[a^{*},b^{*}]_{A^{*}}\rangle\langle
v_{i},c^{*}\rangle=\sum_{i}\langle\delta(u_{i}),a^{*}\otimes
b^{*}\rangle\langle v_{i},c^{*}\rangle$
$\displaystyle=\sum_{i,j}\langle[u_{j},u_{i}]_{A}\otimes
v_{j}-[u_{i},u_{j}]_{A}\otimes
v_{j}-u_{j}\otimes[v_{j},u_{i}]_{A},a^{*}\otimes b^{*}\rangle\langle
v_{i},c^{*}\rangle$ $\displaystyle=\sum_{i,j}\langle a^{*}\otimes b^{*}\otimes
c^{*},[u_{j},u_{i}]_{A}\otimes v_{j}\otimes v_{i}-[u_{i},u_{j}]_{A}\otimes
v_{j}\otimes v_{i}-u_{j}\otimes[v_{j},u_{i}]_{A}\otimes v_{i}\rangle.$

Similarly we have

$\displaystyle\langle
r^{\sharp}([a^{*}),r^{\sharp}(b^{*})]_{A},c^{*}\rangle=\sum_{i,j}\langle
a^{*}\otimes b^{*}\otimes c^{*},u_{i}\otimes
u_{j}\otimes[v_{i},v_{j}]_{A}\rangle.$

Thus

$\displaystyle\langle
r^{\sharp}([a^{*},b^{*}]_{A^{*}})-[r^{\sharp}(a^{*}),r^{\sharp}(b^{*})]_{A},c^{*}\rangle$
$\displaystyle=\sum_{i,j}\langle a^{*}\otimes b^{*}\otimes
c^{*},([u_{j},u_{i}]_{A}-[u_{i},u_{j}]_{A})\otimes v_{j}\otimes
v_{i}-u_{j}\otimes[v_{j},u_{i}]_{A}\otimes v_{i}-u_{i}\otimes
u_{j}\otimes[v_{i},v_{j}]_{A}\rangle$ $\displaystyle=\langle a^{*}\otimes
b^{*}\otimes
c^{*},(\tau\otimes\mathrm{id})\mathrm{Al}(r)+\sum_{j}h(u_{j})\big{(}r+\tau(r)\big{)}\otimes
v_{j}\rangle$ $\displaystyle=\langle a^{*}\otimes b^{*}\otimes
c^{*},(\tau\otimes\mathrm{id})\mathrm{Al}(r)\rangle.$

that is, (28) holds. ∎

######  Corollary 5.3.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra and
$r=\sum\limits_{i}u_{i}\otimes v_{i}\in A\otimes A$ be antisymmetric. Let
$\delta_{r}:A\rightarrow A\otimes A$ be a co-multiplication given by (21).
Then the following statements are equivalent:

  1. (a)

$r$ is a solution of the left-Alia Yang-Baxter equation in $(A,[\cdot,\cdot])$
such that $(A,[\cdot,\cdot],\delta_{r})$ is a triangular left-Alia bialgebra.

  2. (b)

$r^{\sharp}:A^{*}\rightarrow A$ is a relative Rota-Baxter operator of
$(A,[\cdot,\cdot])$ associated to
$(\mathcal{L}^{*}_{[\cdot,\cdot]},\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]},A^{*})$,
that is,

$[r^{\sharp}(a^{*}),r^{\sharp}(b^{*})]=r^{\sharp}\Big{(}\mathcal{L}^{*}_{[\cdot,\cdot]}\big{(}r^{\sharp}(a^{*})\big{)}b^{*}+(\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]})\big{(}r^{\sharp}(b^{*})\big{)}a^{*}\Big{)},\;\forall
a^{*},b^{*}\in A^{*}.$ (29)

In this case,
$r^{\sharp}:(A^{*},[\cdot,\cdot]_{A^{*}})\rightarrow(A,[\cdot,\cdot])$ is a
homomorphism of left-Alia algebras, where $[\cdot,\cdot]_{A^{*}}:A^{*}\otimes
A^{*}\rightarrow A^{*}$ is the linear dual of $\delta_{r}$.

###### Proof.

It follows from Proposition 5.2. ∎

Now we consider the case when $r^{\sharp}$ is moreover an isomorphism.

######  Proposition 5.4.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra. Then there is an antisymmetric
$r\in A\otimes A$ such that $r^{\sharp}:A^{*}\rightarrow A$ is an invertible
relative Rota-Baxter operator associated to
$(\mathcal{L}^{*}_{[\cdot,\cdot]},\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]},A^{*})$
if and only if there is a nondegenerate antisymmetric bilinear form $\omega$
on $A$ such that the following equation holds:

$\omega(x,[y,z]-[z,y])=\omega([x,z],y)-\omega([x,y],z),\;\forall x,y,z\in A.$
(30)

###### Proof.

Suppose $r\in A\otimes A$ is antisymmetric and $r^{\sharp}$ is an invertible
relative Rota-Baxter operator of $(A,[\cdot,\cdot])$ associated to
$(\mathcal{L}^{*}_{[\cdot,\cdot]},\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]},A^{*})$.
Define a bilinear form $\omega$ on $A$ by

$\omega(x,y)=\langle(r^{\sharp})^{-1}x,y\rangle,\;\forall x,y\in A.$ (31)

Then $\omega$ is antisymmetric and nondegenerate. Let $a^{*},b^{*},c^{*}\in
A^{*}$ and $x=r^{\sharp}(a^{*}),y=r^{\sharp}(b^{*}),z=r^{\sharp}(c^{*})$. Then
we have

$\displaystyle\langle[r^{\sharp}(a^{*}),r^{\sharp}(b^{*})],c^{*}\rangle=\langle[r^{\sharp}(a^{*}),r^{\sharp}(b^{*})],(r^{\sharp})^{-1}z\rangle=\omega(z,[x,y]),$
$\displaystyle\langle
r^{\sharp}\big{(}\mathcal{L}^{*}_{[\cdot,\cdot]}\big{(}r^{\sharp}(a^{*})\big{)}b^{*}\big{)},c^{*}\rangle=\langle
r^{\sharp}\big{(}\mathcal{L}^{*}_{\circ}(x)b^{*}\big{)},(r^{\sharp})^{-1}z\rangle=-\langle\mathcal{L}^{*}_{[\cdot,\cdot]}(x)b^{*},z\rangle$
$\displaystyle=\langle
b^{*},[x,z]\rangle=\langle(r^{\sharp})^{-1}y,[x,z]\rangle=\omega(y,[x,z]),$
$\displaystyle\langle
r^{\sharp}\Big{(}(\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]})\big{(}r^{\sharp}(b^{*})\big{)}a^{*}\Big{)},c^{*}\rangle=\langle
r^{\sharp}\big{(}(\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]})(y)a^{*}\big{)},(r^{\sharp})^{-1}z\rangle$
$\displaystyle=-\langle(\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]})(y)a^{*},z\rangle$
$\displaystyle=\langle
a^{*},[y,z]-[z,y]\rangle=\langle(r^{\sharp})^{-1}x,[y,z]-[z,y]\rangle=\omega(x,[y,z]-[z,y]).$

Hence by (28) in Proposition 5.2, we have (30).

Conversely, suppose that $\omega$ is a nondegenerate antisymmetric bilinear
form on $(A,[\cdot,\cdot])$ satisfying (30). Then we can routinely check that
$r^{\sharp}:A^{*}\rightarrow A$ given by (31) is an invertible relative Rota-
Baxter operator of $(A,[\cdot,\cdot])$ associated to
$(\mathcal{L}^{*}_{[\cdot,\cdot]},\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]},A^{*})$.
∎

Recall that a Connes cocycle [1] on an associative algebra $(A,\cdot)$ is an
antisymmetric bilinear form $\omega$ such that

$\omega(x\cdot y,z)+\omega(y\cdot z,x)+\omega(z\cdot x,y)=0,\;\forall x,y,z\in
A.$ (32)

######  Proposition 5.5.

Suppose that $\omega$ is a nondegenerate Connes cocycle on a commutative
associative algebra $(A,\cdot)$. Let $f:A\rightarrow A$ be a linear map and
$\hat{f}:A\rightarrow A$ be the adjoint map of $f$ with respect to $\omega$,
that is,

$\omega\big{(}\hat{f}(x),y\big{)}=\omega\big{(}x,f(y)\big{)},\;\forall x,y\in
A.$

Then $\omega$ satisfies (30), where $(A,[\cdot,\cdot])$ is the special left-
Alia algebra given by (14).

###### Proof.

For all $x,y,z\in A$, we have

$\displaystyle\omega(x,[y,z]-[z,y])-\omega([x,z],y)+\omega([x,y],z)$
$\displaystyle=\omega\big{(}x,y\cdot f(z)-z\cdot
f(y)\big{)}-\omega\big{(}x\cdot f(z)-\hat{f}(x\cdot
z),y\big{)}+\omega\big{(}x\cdot f(y)-\hat{f}(x\cdot y),z\big{)}$
$\displaystyle=\omega\big{(}x,y\cdot f(z)-z\cdot
f(y)\big{)}-\omega\big{(}x\cdot f(z),y\big{)}+\omega\big{(}x\cdot
z,f(y)\big{)}+\omega\big{(}x\cdot f(y),z\big{)}-\omega(x\cdot y,f(z)\big{)}$
$\displaystyle\overset{\eqref{eq:cc}}{=}0.$

∎

######  Theorem 5.6.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra with a representation
$(l,r,V)$. Let $T:V\rightarrow A$ be a linear map and $T_{\sharp}\in
V^{*}\otimes A\subset(A\oplus V^{*})\otimes(A\oplus V^{*})$ be given by

$\langle T_{\sharp},u\otimes a^{*}\rangle=\langle T(u),a^{*}\rangle,\;\forall
u\in V,a^{*}\in A^{*}.$ (33)

Then $r=T_{\sharp}-\tau(T_{\sharp})$ is an antisymmetric solution of the left-
Alia Yang-Baxter equation in $A\ltimes_{l^{*},l^{*}-r^{*}}V^{*}$ if and only
if $T$ is a relative Rota-Baxter operator of $(A,[\cdot,\cdot])$ associated to
$(l,r,V)$.

###### Proof.

Let $\\{v_{1},\cdots,v_{n}\\}$ be a basis of $V$ and
$\\{v^{*}_{1},\cdots,v^{*}_{n}\\}$ be the dual basis. Then we have

$\displaystyle T_{\sharp}=\sum_{i=1}^{n}v^{*}_{i}\otimes Tv_{i}\in
V^{*}\otimes A\subset(A\oplus V^{*})\otimes(A\oplus V^{*}),$ $\displaystyle
r=T_{\sharp}-\tau(T_{\sharp})=\sum_{i=1}^{n}v^{*}_{i}\otimes
Tv_{i}-Tv_{i}\otimes v^{*}_{i}.$

Note that

$l^{*}(Tv_{i})v^{*}_{j}=-\sum_{k=1}^{n}\langle v^{*}_{j},l(Tv_{i})v_{k}\rangle
v^{*}_{k},\;r^{*}(Tv_{i})v^{*}_{j}=-\sum_{k=1}^{n}\langle
v^{*}_{j},r(Tv_{i})v_{k}\rangle v^{*}_{k}.$ (34)

Then we have

$\displaystyle\mathrm{Al}(r)$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{n}[Tv_{i},Tv_{j}]\otimes v^{*}_{i}\otimes
v^{*}_{j}-[v^{*}_{i},Tv_{j}]_{d}\otimes Tv_{i}\otimes
v^{*}_{j}-[Tv_{i},v^{*}_{j}]_{d}\otimes v^{*}_{i}\otimes Tv_{j}$
$\displaystyle+v^{*}_{i}\otimes\big{(}[v^{*}_{j},Tv_{i}]_{d}-[Tv_{i},v^{*}_{j}]_{d}\big{)}\otimes
Tv_{j}+v^{*}_{i}\otimes([Tv_{i},Tv_{j}]-[Tv_{j},Tv_{i}])\otimes v^{*}_{j}$
$\displaystyle+Tv_{i}\otimes\big{(}[Tv_{j},v^{*}_{i}]_{d}-[v^{*}_{i},Tv_{j}]_{d}\big{)}\otimes
v^{*}_{j}$ $\displaystyle-v^{*}_{i}\otimes
v^{*}_{j}\otimes[Tv_{j},Tv_{i}]+v^{*}_{i}\otimes
Tv_{j}\otimes[v^{*}_{j},Tv_{i}]_{d}+Tv_{i}\otimes
v^{*}_{j}\otimes[Tv_{j},v^{*}_{i}]_{d}$ $\displaystyle=$
$\displaystyle\sum_{i,j}[Tv_{i},Tv_{j}]\otimes v^{*}_{i}\otimes
v^{*}_{j}-(l^{*}-r^{*})(Tv_{j})v^{*}_{i}\otimes Tv_{i}\otimes
v^{*}_{j}-l^{*}(Tv_{i})v^{*}_{j}\otimes v^{*}_{i}\otimes Tv_{j}$
$\displaystyle-v^{*}_{i}\otimes r^{*}(Tv_{i})v^{*}_{j}\otimes
Tv_{j}+v^{*}_{i}\otimes([Tv_{i},Tv_{j}]-[Tv_{j},Tv_{i}])\otimes
v^{*}_{j}+Tv_{i}\otimes r^{*}(Tv_{j})v^{*}_{i}\otimes v^{*}_{j}$
$\displaystyle-v^{*}_{i}\otimes
v^{*}_{j}\otimes[Tv_{j},Tv_{i}]+v^{*}_{i}\otimes
Tv_{j}\otimes(l^{*}-r^{*})(Tv_{i})v^{*}_{j}+Tv_{i}\otimes v^{*}_{j}\otimes
l^{*}(Tv_{j})v^{*}_{i}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\Big{(}[Tv_{i},Tv_{j}]-T\big{(}l(Tv_{i})v_{j}+r(Tv_{j})v_{i}\big{)}\Big{)}\otimes
v^{*}_{i}\otimes v^{*}_{j}$
$\displaystyle+v^{*}_{i}\otimes([Tv_{i},Tv_{j}]-[Tv_{j},Tv_{i}]+T\big{(}(l-r)(Tv_{j})v_{i}+(r-l)(Tv_{i})v_{j}\big{)}\Big{)}\otimes
v^{*}_{j}$ $\displaystyle+v^{*}_{i}\otimes
v^{*}_{j}\otimes\Big{(}T\big{(}l(Tv_{j})v_{i}+r(Tv_{i})v_{j}\big{)}-[Tv_{j},Tv_{i}]\Big{)}.$

Hence the conclusion follows. ∎

##  6\. Pre-left-Alia algebras

In this section, we first introduce the notion of a pre-left-Alia algebra,
which is the underlying algebraic structure of a relative Rota-Baxter operator
with respect to a representation $(l,r,V)$ of a left-Alia algebra. Then we
prove that there is a natural antisymmetric solution of the left-Alia Yang-
Baxter equation in the left-Alia algebra
$A\ltimes_{\mathcal{L}^{*}_{\succ},\mathcal{L}^{*}_{\succ}-\mathcal{R}^{*}_{\prec}}A^{*}$.

######  Definition 6.1.

A pre-left-Alia algebra is a triple $(A,\succ,\prec)$, where $A$ is a vector
space and $\succ,\prec:A\otimes A\rightarrow A$ are multiplications such that
the following equation holds:

$(x\succ y+x\prec y)\succ z-(y\succ x+y\prec x)\succ z+(y\succ z-z\prec
y)\prec x+(z\prec x-x\succ z)\prec y=0,$ (35)

for all $x,y,z\in A$.

Recall a Zinbiel algebra [18] is a vector space $A$ together with a
multiplication $\star:A\otimes A\rightarrow A$ such that the following
equation holds:

$x\star(y\star z)=(y\star x)\star z+(x\star y)\star z,\;\;\forall x,y,z\in A.$
(36)

From the operadic viewpoint, Zinbiel algebras are known as the successors of
commutative associative algebras, as well as the Koszul dual to Leibniz
algebras (see [19]). We shall show that there are pre-left-Alia algebras
constructed Zinbiel algebras with linear maps.

######  Proposition 6.2.

Let $(A,\star)$ be a Zinbiel algebra and $f,g:A\rightarrow A$ be linear maps.
Then there is a pre-left-Alia algebra $(A\succ,\prec)$ given by

$x\succ y=x\star f(y)+g(x\star y),\;x\prec y=f(y)\star x+g(y\star x),\;\forall
x,y\in A.$

###### Proof.

Let $x,y,z\in A$. We first observe that

$x\succ y-y\prec x=x\star f(y)-f(x)\star y.$

Then we have

$\displaystyle(x\succ y+x\prec y)\succ z-(y\succ x+y\prec x)\succ z+(y\succ
z-z\prec y)\prec x+(z\prec x-x\succ z)\prec y$ $\displaystyle=\big{(}x\star
f(y)-f(x)\star y\big{)}\succ z-\big{(}y\star f(x)-f(y)\star x\big{)}\succ z$
$\displaystyle\ \ +\big{(}y\star f(z)-f(y)\star z\big{)}\prec
x+\big{(}f(x)\star z-z\star f(x)\big{)}y$ $\displaystyle=\big{(}x\star
f(y)-f(x)\star y\big{)}\star z+g\Big{(}\big{(}x\star f(y)\big{)}\star
z-\big{(}f(x)\star y\big{)}\star z\Big{)}$ $\displaystyle\ \ -\big{(}y\star
f(x)-f(y)\star x\big{)}\star z-g\Big{(}\big{(}y\star f(x)\big{)}\star
z-\big{(}f(y)\star x\big{)}\star z\Big{)}$ $\displaystyle\ \
+f(x)\star\big{(}y\star f(z)-f(y)\star z\big{)}+g\Big{(}x\star\big{(}y\star
f(z)\big{)}-x\star\big{(}f(y)\star z\big{)}\Big{)}$ $\displaystyle\ \
+f(y)\star\big{(}f(x)\star z-f(z)\star x\big{)}+g\Big{(}y\star\big{(}f(x)\star
z\big{)}-y\star\big{(}x\star f(z)\big{)}\Big{)}$
$\displaystyle\overset{\eqref{eq:Zinb}}{=}0.$

Hence $(A\succ,\prec)$ is a pre-left-Alia algebra. ∎

On the other hand, recall a pre-Lie algebra [5, 2] is a vector space $A$
together with a multiplication $\star:A\otimes A\rightarrow A$ such that the
following equation holds:

$(x\star y)\star z-x\star(y\star z)=(y\star x)\star z-y\star(x\star
z),\;\forall x,y,z\in A.$ (37)

Pre-Lie algebras play an important role in diverse areas of mathematics and
mathematical physics. From the operadic viewpoint, pre-algebras are known as
the successors of Lie algebras.

######  Proposition 6.3.

Let $(A,\star)$ be a pre-Lie algebra. Then there is a pre-left-Alia algebra
$(A\succ,\prec)$ given by

$x\succ y=x\star y,\;x\prec y=-y\star x,\;\forall x,y\in A.$

###### Proof.

For all $x,y,z\in A$, we have

$\displaystyle(x\succ y+x\prec y)\succ z-(y\succ x+y\prec x)\succ z+(y\succ
z-z\prec y)\prec x+(z\prec x-x\succ z)\prec y$ $\displaystyle=2(x\star
y-y\star x)\star z-2x\star(y\star z)+2y\star(x\star z)$
$\displaystyle\overset{\eqref{eq:pre-Lie}}{=}0.$

Hence $(A\succ,\prec)$ is a pre-left-Alia algebra. ∎

######  Proposition 6.4.

Let $(A,\succ,\prec)$ be a pre-left-Alia algebra. Then the multiplication
$[\cdot,\cdot]:A\otimes A\rightarrow A$ given by

$[x,y]=x\succ y+x\prec y,\;\forall x,y\in A,$ (38)

defines a left-Alia algebra which is called the sub-adjacent left-Alia algebra
of $(A,\succ,\prec)$, and $(A,\succ,\prec)$ is called a compatible pre-left-
Alia algebra of $(A,[\cdot,\cdot])$. Moreover,
$(\mathcal{L}_{\succ},\mathcal{R}_{\prec},A)$ is a representation of
$(A,[\cdot,\cdot])$, and the identity map $\mathrm{id}:A\rightarrow A$ is a
relative Rota-Baxter operator of $(A,[\cdot,\cdot])$ with respect to
$(\mathcal{L}_{\succ},\mathcal{R}_{\prec},A)$.

###### Proof.

For all $x,y,z\in A$, we have

$\displaystyle[[x,y],z]+[[y\circ
z],x]+[[z,x],y]-[[y,x],z]-[[z,y],x]-[[x,z],y]$ $\displaystyle=(x\succ y+x\prec
y)\succ z+(x\succ y+x\prec y)\prec z+(y\succ z+y\prec z)\succ x+(y\succ
z+y\prec z)\prec x$ $\displaystyle\ \ +(z\succ x+z\prec x)\succ y+(z\succ
x+z\prec x)\prec y-(y\succ x+y\prec x)\succ z-(y\succ x+y\prec x)\prec z$
$\displaystyle\ \ -(z\succ y+z\prec y)\succ x-(z\succ y+z\prec y)\prec
x-(x\succ z+x\prec z)\succ y-(x\succ z+x\prec z)\prec y$
$\displaystyle\overset{\eqref{eq:pre}}{=}0.$

Hence $(A,[\cdot,\cdot])$ is a left-Alia algebra. The other results are
straightforward. ∎

######  Lemma 6.5.

Let $T:V\rightarrow A$ be a relative Rota-Baxter operator of a left-Alia
algebra with respect to a representation $(l,r,V)$. Then there is a pre-left-
Alia algebra $(V,\triangleright,\triangleleft)$ given by

$u\triangleright v=l(Tu)v,\;u\triangleleft v=r(Tv)u,\;\forall u,v\in V.$ (39)

###### Proof.

For all $u,v,w\in V$, we have

$\displaystyle(u\triangleright v+u\triangleleft v)\triangleright
w-(v\triangleright u+v\triangleleft u)\triangleright w+(v\triangleright
w-w\triangleleft v)\triangleleft u+(w\triangleleft u-u\triangleleft
w)\triangleleft v$
$\displaystyle=lT\big{(}l(Tu)v+r(Tv)u\big{)}w-lT\big{(}l(Tv)u+r(Tu)v\big{)}w$
$\displaystyle\ \ +r(Tu)r(Tv)w-r(Tu)r(Tv)w+r(Tv)r(Tu)w-r(Tv)l(Tu)w$
$\displaystyle\overset{\eqref{eq:oop}}{=}l([Tu,Tv])-l([Tv,Tu])w+r(Tu)l(Tv)w-r(Tu)r(Tv)w+r(Tv)r(Tu)w-r(Tv)l(Tu)w$
$\displaystyle\overset{\eqref{eq:rep 0-Alia}}{=}0.$

Hence the conclusion follows. ∎

######  Theorem 6.6.

Let $(A,[\cdot,\cdot])$ be a left-Alia algebra. Then there is a compatible
pre-left-Alia algebra $(A,\succ,\prec)$ of $(A,[\cdot,\cdot])$ if and only if
there exists an invertible relative Rota-Baxter operator $T:V\rightarrow A$
with respect to a representation $(l,r,V)$. Furthermore, the compatible pre-
left-Alia algebra is given by

$x\succ y=T\big{(}l(x)T^{-1}y\big{)},\;x\prec
y=T\big{(}r(y)T^{-1}x\big{)},\;\forall x,y\in A.$ (40)

###### Proof.

Let $T:V\rightarrow A$ be an invertible relative Rota-Baxter operator of
$(A,[\cdot,\cdot])$ with respect to $(l,r,V)$. Then by Lemma 6.5, there is a
pre-left-Alia algebra algebra on $V$ given by (39). Since $T$ is invertible,
there is a pre-left-Alia algebra structure $(A,\succ,\prec)$ on $A$ given by

$\displaystyle x\succ y=T(T^{-1}x\triangleright
T^{-1}y)\overset{\eqref{eq:42}}{=}T\big{(}l(x)T^{-1}y\big{)},$ $\displaystyle
x\prec y=T(T^{-1}x\triangleleft
T^{-1}y)\overset{\eqref{eq:42}}{=}T\big{(}r(y)T^{-1}x\big{)}.$

Moreover, we have

$\displaystyle x\succ y+x\prec
y=T\big{(}l(x)T^{-1}y+r(y)T^{-1}x\big{)}\overset{\eqref{eq:oop}}{=}[x,y],$

that is, $(A,\succ,\prec)$ is a compatible pre-left-Alia algebra of
$(A,[\cdot,\cdot])$.

Conversely, if $(A,\succ,\prec)$ is a compatible pre-left-Alia algebra of
$(A,[\cdot,\cdot])$, then the identity map $\mathrm{id}:A\rightarrow A$ is an
invertible relative Rota-Baxter operator of the sub-adjacent left-Alia algebra
$(A,[\cdot,\cdot])$ with respect to
$(\mathcal{L}_{\succ},\mathcal{R}_{\prec},A)$. ∎

######  Corollary 6.7.

Let $\omega$ be a nondegenerate antisymmetric bilinear form on a left-Alia
algebra $(A,[\cdot,\cdot])$ satisfying (30). Then there is a compatible pre-
left-Alia algebra $(A,\succ,\prec)$ given by

$\omega(x\succ y,z)=-\omega(y,[x,z]),\;\omega(x\prec
y,z)=\omega(x,[z,y]-[y,z]),\;\forall x,y,z\in A.$ (41)

###### Proof.

By Proposition 5.4, $r^{\sharp}:A^{*}\rightarrow A$ given by (31) is an
invertible relative Rota-Baxter operator of $(A,[\cdot,\cdot])$ with respect
to
$(\mathcal{L}^{*}_{[\cdot,\cdot]},\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]},A^{*})$.
Then by Theorem 6.6, there is a compatible pre-left-Alia algebra
$(A,\succ,\prec)$ of $(A,[\cdot,\cdot])$ given by

$x\succ
y=r^{\sharp}\big{(}\mathcal{L}^{*}_{[\cdot,\cdot]}(x)(r^{\sharp})^{-1}y\big{)},\;x\prec
y=r^{\sharp}\big{(}(\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]})(y)(r^{\sharp})^{-1}x\big{)}.$
(42)

Moreover, we have

$\displaystyle\omega(x\succ y,z)$ $\displaystyle=$
$\displaystyle\omega\Big{(}r^{\sharp}\big{(}\mathcal{L}^{*}_{[\cdot,\cdot]}(x)(r^{\sharp})^{-1}y\big{)},z\Big{)}=\langle\mathcal{L}^{*}_{[\cdot,\cdot]}(x)(r^{\sharp})^{-1}y,z\rangle=-\langle(r^{\sharp})^{-1}y,[x,z]\rangle=-\omega(y,[x,z]),$
$\displaystyle\omega(x\prec y,z)$ $\displaystyle=$
$\displaystyle\omega\Big{(}r^{\sharp}\big{(}(\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]})(y)(r^{\sharp})^{-1}x\big{)},z\Big{)}=\langle(\mathcal{L}^{*}_{[\cdot,\cdot]}-\mathcal{R}^{*}_{[\cdot,\cdot]})(y)(r^{\sharp})^{-1}x,z\rangle$
$\displaystyle=$
$\displaystyle\langle(r^{\sharp})^{-1}x,[z,y]-[y,z]\rangle=\omega(x,[z,y]-[y,z]).$

Hence $(A,\succ,\prec)$ satisfies (41). ∎

######  Theorem 6.8.

Let $(A,\succ,\prec)$ be a pre-left-Alia algebra. Then

$r=\sum_{i=1}^{n}e^{*}_{i}\otimes e_{i}-e_{i}\otimes e^{*}_{i}$

is an antisymmetric solution of the left-Alia Yang-Baxter equation in the
left-Alia algebra
$A\ltimes_{\mathcal{L}^{*}_{\succ},\mathcal{L}^{*}_{\succ}-\mathcal{R}^{*}_{\prec}}A^{*}$,
where $\\{e_{1},\cdots,e_{n}\\}$ is a basis of $A$ and
$\\{e^{*}_{1},\cdots,e^{*}_{n}\\}$ is the dual basis.

###### Proof.

Since $(A,\succ,\prec)$ is a pre-left-Alia algebra, the identity map
$\mathrm{id}:A\rightarrow A$ is a relative Rota-Baxter operator of the sub-
adjacent left-Alia algebra with respect to
$(\mathcal{L}_{\succ},\mathcal{R}_{\prec},A)$. Then by Theorem 5.6,

$r=(\mathrm{id})_{\sharp}-\tau(\mathrm{id}_{\sharp})=\sum_{i=1}^{n}e^{*}_{i}\otimes
e_{i}-e_{i}\otimes e^{*}_{i}$

is an antisymmetric solution of the left-Alia Yang-Baxter equation in the
left-Alia algebra
$A\ltimes_{\mathcal{L}^{*}_{\succ},\mathcal{L}^{*}_{\succ}-\mathcal{R}^{*}_{\prec}}A^{*}$.
∎

######  Example 6.9.

Let $(A,\succ,\prec)$ be a 2-dimensional pre-left-Alia algebra with a basis
$\\{e_{1},e_{2}\\}$ and the non-zero multiplications $\succ,\prec$ given by

$\displaystyle e_{1}\succ e_{2}=2e_{1},\;e_{1}\prec e_{2}=-e_{1},\;e_{2}\succ
e_{1}=e_{1},\;e_{2}\prec e_{1}=-2e_{1},$ $\displaystyle e_{2}\succ
e_{2}=e_{1}+2e_{2},\;e_{2}\prec e_{2}=-e_{1}-2e_{2}.$

Then the non-zero multiplications of the sub-adjacent left-Alia algebra of
$(A,\succ,\prec)$ are

$[e_{1},e_{2}]=e_{1},\;[e_{2},e_{1}]=-e_{1}.$

If $\\{e_{1}^{*},e_{2}^{*}\\}$ is the dual basis, then

$\mathcal{L}^{*}_{\succ}(e_{1})(e_{1}^{*})=-2e_{2}^{*},\;\mathcal{L}^{*}_{\succ}(e_{2})(e_{1}^{*})=-e_{1}^{*}-e_{2}^{*},\;\mathcal{L}^{*}_{\succ}(e_{2})(e_{2}^{*})=-2e_{2}^{*}.$

The non-zero multiplications of the left-Alia algebra
$d=A\ltimes_{\mathcal{L}^{*}_{\succ},\mathcal{L}^{*}_{\succ}-\mathcal{R}^{*}_{\prec}}A^{*}$
are

$\displaystyle[e_{1},e_{2}]_{d}=e_{1},\;[e_{2},e_{1}]_{d}=-e_{1},\;[e_{1},e_{1}^{*}]_{d}=[e_{2},e_{2}^{*}]_{d}=-2e_{2}^{*},\;[e_{2},e_{1}^{*}]_{d}=-e_{1}^{*}-e_{2}^{*},$
$\displaystyle[e_{1}^{*},e_{1}]_{d}=-4e_{2}^{*},\;[e_{1}^{*},e_{2}]_{d}=-2e_{1}^{*}-2e_{2}^{*},\;[e_{2}^{*},e_{2}]_{d}=-4e_{2}^{*}.$

Furthermore, by Theorem 6.8,

$r=\sum_{i=1}^{2}e^{*}_{i}\otimes e_{i}-e_{i}\otimes e^{*}_{i},$

is an antisymmetric solution of the left-Alia Yang-Baxter equation in the
left-Alia algebra $d$. There exists a left-Alia bialgebra
$(d,[\cdot,\cdot]_{d},\delta)$ with a non-zero co-multiplication $\delta$ on
$d$, given by

$\displaystyle\delta(e_{1})=-4e_{2}^{*}\otimes e_{1}+2e_{2}^{*}\otimes
e_{1}+4e_{1}\otimes e_{2}^{*}+e_{2}^{*}\otimes e_{1}+2e_{1}\otimes e_{2}^{*},$
$\displaystyle\delta(e_{2})=-3e_{1}^{*}\otimes e_{1}-2e_{2}^{*}\otimes
e_{1}-4e_{1}\otimes e_{1}^{*}-2e_{1}\otimes e_{2}^{*}-2e_{2}^{*}\otimes
e_{2}-4e_{2}\otimes e_{2}^{*},$
$\displaystyle\delta(e_{1}^{*})=e_{1}^{*}\otimes e_{2}^{*}-e_{2}^{*}\otimes
e_{1}^{*}.$

Acknowledgements. The third author acknowledges support from the NSF China
(12101328) and NSF China (12371039).

## References

  * [1] C. Bai, Double constructions of Frobenius algebras, Connes cocycles and their duality. J. Noncommut. Geom. 4 (2010), 475-530.
  * [2] C. Bai, An introduction to pre-Lie algebras, in: Algebra and Applications 1: Nonssociative Algebras and Categories, Wiley Online Library (2021), 245-273.
  * [3] K. Benali, T. Chtioui, A. Hajjaji and S. Mabrouk, Bialgebras, the Yang-Baxter equation and Manin triples for mock-Lie algebras, Acta Comment. Univ. Tartu. Math. 27 (2023), 211-233.
  * [4] D. Beson, Polynomial Invariants of Finite Groups, Cambridge University Press (1993).
  * [5] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4 (2006), 323-357.
  * [6] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pac. J. Math. 10 (1960), 731 C742.
  * [7] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge (1994).
  * [8] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math.1955, 778-782.
  * [9] A. Das and S.K. Mishra, Bimodules over relative Rota-Baxter algebras and cohomologies, Algebr. Represent. Theory 26 (2023), no.5, 1823 C1848.
  * [10] V. Drinfeld, Halmiltonian structure on the Lie groups, Lie bialgebras and the geometric sense of the classical Yang-Baxter equations, Sov. Math. Dokl. 27 (1983), 68-71.
  * [11] A. Dzhumadil’daev, Algebras with skew-symmetric identity of degree 3, J. Math. Sci. 161 (2009), 11-30.
  * [12] L. Guo, An Introduction to Rota-Baxter Algebra, Surveys of Modern Mathematics, Somerville, MA: International Press; Beijing: Higher education press, 4 (2012).
  * [13] C. Kang, G. Liu, Z. Wang and S. Yu, Manin triples and bialgebras of Left-Alia algebras associated to invariant theory, arXiv:2403.05339v1.
  * [14] B. A. Kupershmidt, What a classical r-matrix really is, J. Nonlinear Math. Phys. 6 (1999), 448-488.
  * [15] Y. Li, Y. Sheng and R. Tang, Post-Hopf algebras, relative Rota-Baxter operators and solutions to the Yang-Baxter equation, J. Noncommut. Geom. 18 (2024), no.2, 605 C630.
  * [16] G. Liu and C. Bai, Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebras, J. Algebra 609 (2022), 337-379.
  * [17] G. Liu and C. Bai, A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras. Commun. Contemp. Math. (2023), https://doi.org/10.1142/S0219199723500505.
  * [18] J.-L. Loday, Cup product for Leibniz cohomology and dual Leibniz algebras, Math. Scand. 77, Univ. Louis Pasteur, Strasbourg, (1995), pp. 189-196.
  * [19] J.-L. Loday and B. Vallette, Algebraic Operads, Springer, Berlin (2012).
  * [20] G.C. Rota, Baxter algebras and combinatorial identities I, II,  Bull. Am. Math. Soc. 75 (1969), 330 C334.
  * [21] G.C. Rota, Baxter operators, an introduction, In: Gian-Carlo Rota on Combinatorics: Introductory Papers and Commentaries, Joseph P. S. Kung, Editor, Birkh auser, Boston, 1995.
  * [22] M. A. Semenov-Tian-Shansky, What is a classical R-matrix? Funct. Anal. Appl. 17 (1983), 259-272.
  * [23] Y. Sheng, A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras, Bull. Lond. Math. Soc. 54 (2022), no.6, 2045 C2065.
  * [24] Y. Sheng and Y. Wang, Quasi-triangular and factorizable antisymmetric infinitesimal bialgebras, J. Algebra 628 (2023), 415-433.
  * [25] G. Shephard and J. Todd, Finite unitary reflection groups, Canadian J. Math. 6(1954), 274-304.
  * [26] R. Tang and Y. Sheng, Leibniz bialgebras, relative Rota-Baxter operators and the classical Leibniz Yang-Baxter equation, J. Noncommut. Geom. 16 (2022), 1179-1211.
  * [27] P. Zusmanovich, Special and exceptional mock-Lie algebras, Linear Algebra Appl. 518 (2017), 79-96.