Manin Triples and Bialgebras of Left-Alia Algebras Associated with Invariant Theory

Abstract

A left-Alia algebra is a vector space together with a bilinear map satisfying the symmetric Jacobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notions of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras.

1. Introduction and Main Statements

1.1. Introduction

Let G be a finite group and $\mathbb{K}$ an algebraic closed field of characteristic zero. Suppose that V is an n-dimensional faithful representation of G and $S=\mathbb{K}\left[V\right]=\mathbb{K}[x_1,\dots ,x_n]$ is the coordinate ring of V.

The goal of invariant theory is to study the structures of the ring of invariants

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

in which the group action is extended from the representation of G (see Section 2.1 for more details). In particular, Hilbert proved that $S^G$ is always a finite generated $\mathbb{K}$-algebra [1] and Chevalley [2], Shephard and Todd [3] proved that $S^G$ is a polynomial algebra if and only if G is generated by pseudo-reflections (see Section 2.2 for precise definition).

Twisted derivations [4] (also named $\sigma$-derivations) play an important role in the study of deformations of Lie algebras. Motivated by the above Chevalley’s Theorem, we apply pseudo-reflections to induce a class of twisted derivations on S (see Theorem 3 for more details). Based on twisted derivations on commutative associative algebras, we obtain a class of left-Alia (left anti-Lie-admissible) algebras [5], which appears in the study of a special class of algebras with a skew-symmetric identity of degree three. Furthermore, we construct Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras.

Throughout this paper, unless otherwise specified, all vector spaces are finite-dimensional over an algebraically closed field K of characteristic zero and all K-algebras are commutative and associative with the finite Krull dimension, although many results and notions remain valid in the infinite-dimensional case.

1.2. Left-Alia Algebras Associated with Invariant Theory

The notion of a left-Alia algebra was defined for the first time in the table in the Introduction of [5].

Definition 1

([5]). A left-Alia algebra (also named a 0-Alia algebra) is a vector space A together with a bilinear map $[·,·]:A\otimes A\to A$ satisfying the symmetric Jacobi identity:

$$\left[\right[x,y],z]+\left[\right[y,z],x]+\left[\right[z,x],y]=\left[\right[y,x],z]+\left[\right[z,y],x]+\left[\right[x,z],y],\forall x,y,z\in A.$$

(1)

There are some typical examples of left-Alia algebras. Firstly, when the bilinear map $[·,·]$ is skew-symmetric, $(A,[·,·\left]\right)$ is a Lie algebra. By contrast, any commutative algebra is a left-Alia algebra and, in particular, a mock-Lie algebra [6] (also known as a Jacobi–Jordan algebra in [7]) with a symmetric bilinear map that satisfies the Jacobi identity is a left-Alia algebra. Secondly, the notion of an anti-pre-Lie algebra [8] was recently studied as a left-Alia algebra with an additional condition. Anti-pre-Lie algebras are the underlying algebra structures of nondegenerate commutative 2-cocycles [9] on Lie algebras and are characterized as Lie-admissible algebras whose negative multiplication operators compose representations of commutator Lie algebras. Condition (1) of the identities of an anti-pre-Lie algebra is just to guarantee $(A,[·,·\left]\right)$ is a Lie-admissible algebra. Additionally, we also studied left-Alia algebras in terms of their relationships with Leibniz algebras [10] and Lie triple systems [11].

Let $(A,·)$ be a commutative associative algebra and $R:A\to A$ a linear map on A. For brevity, the operation · will be omitted. A linear map $D:A\to A$ is called a twisted derivation with respect to an R (also named a $\sigma$-derivation in [4]) if D satisfies the twisted Leibniz rule:

$$D\left(fg\right)=D\left(f\right)g+R\left(f\right)D\left(g\right),f,g\in A.$$

(2)

Non-trivial examples of twisted derivations can be constructed in invariant theory. In particular, each pseudo-reflection R on a vector space V induces a twisted derivation $D_R$ on the polynomial ring $\mathbb{K}\left[V\right]$ (see Section 2.2 for details).

Define

$${[f,g]}_R=D\left(f\right)g-R\left(f\right)D\left(g\right).$$

We then obtain a class of left-Alia algebras in Theorem A.

Theorem A (Theorems 3 and 4)

(a)

For each twisted derivation D on A, $(A,{[·,·]}_R)$ is a left-Alia algebra.

(b)

Each pseudo-reflection R on V induces a left-Alia algebra $(\mathbb{K}\left[V\right],{[·,·]}_R)$.

This applies when $R=I$, ${[f,g]}_R$ is skew-symmetric and $(A,{[·,·]}_R)$ is a Lie algebra of the Witt type [12]. Moreover, Theorem A also provides a class of left-Alia algebras on polynomial rings from invariant theory. As a corollary of Theorem A, we see that when $S^G$ is a polynomial algebra, each generator $g\in G$ corresponds to a left-Alia algebra $(S,{[·,·]}_{R_g})$. The collection of left-Alia algebras is also an interesting research object for further study.

In addition, if we define that $[f,h]=D_R\left(f\right)h-f{D}_R\left(h\right)$ on S, $(S,[·,·\left]\right)$, then it is not a left-Alia algebra in general. However, when $[·,·]$ is restricted to $V^{\ast }$, we obtain a finite-dimensional Lie algebra, which induces a linear Poisson structure on V, and figure out the entrance to the study of twisted relative Poisson structures on graded algebras. See [13] for reference.

1.3. Manin Triples and Bialgebras of Left-Alia Algebras

A bialgebra structure is a vector space equipped with both an algebra structure and a coalgebra structure satisfying certain compatible conditions. Some well-known examples of such structures include Lie bialgebras [14,15], which are closely related to Poisson–Lie groups and play an important role in the infinitesimalization of quantum groups, and antisymmetric infinitesimal bialgebras [16,17,18,19,20] as equivalent structures of double constructions of Frobenius algebras which are widely applied in the 2d topological field and string theory [21,22]. Recently, the notion of anti-pre-Lie bialgebras was studied in [23], which serves as a preliminary to supply a reasonable bialgebra theory for transposed Poisson algebras [24]. The notions of mock-Lie bialgebras [25] and Leibniz bialgebras [26,27] were also introduced with different motivations. These bialgebras have a common property in that they can be equivalently characterized by Manin triples which correspond to nondegenerate invariant bilinear forms on the algebra structures. In this paper, we follow such a procedure to study left-Alia bialgebras.

To develop the bialgebra theory of left-Alia algebras, we first define a representation of a left-Alia algebra to be a triple $(l,r,V)$, where V is a vector space and $l,r:A\to \mathrm{End}\left(V\right)$ are linear maps such that the following equation holds:

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

A representation $(\rho ,V)$ of a Lie algebra $(A,[·,·\left]\right)$ renders representations $(\rho ,-\rho ,V)$ and $(\rho ,2\rho ,V)$ of $(A,[·,·\left]\right)$ as left-Alia algebras.

Furthermore, we introduce the notion of a quadratic left-Alia algebra, defined as a left-Alia algebra $(A,[·,·\left]\right)$ equipped with a nondegenerate symmetric bilinear form $\mathcal{B}$ which is invariant in the sense that

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

A quadratic left-Alia algebra gives rise to the equivalence between the adjoint representation and the coadjoint representation.

Last, we introduce the notions of a matched pair (Definition 8) of left-Alia algebras, a Manin triple of left-Alia algebras (Definition 11) and a left-Alia bialgebra (Definition 13). Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras in Theorem B.

Theorem B (Theorems 5 and 6)

Let $(A,{[·,·]}_A)$ be a left-Alia algebra. Suppose that there is a left-Alia algebra structure $(A^{\ast },{[·,·]}_{A^{\ast }})$ on the dual space $A^{\ast }$, and $\delta :A\to A\otimes A$ is the linear dual of ${[·,·]}_{A^{\ast }}$. Then, the following conditions are equivalent:

(a)

There is a Manin triple of left-Alia algebras $\left((A\oplus A^{\ast },{[·,·]}_d,{\mathcal{B}}_d),A,A^{\ast }\right)$, where

$${\mathcal{B}}_d(x+a^{\ast },y+b^{\ast })=⟨x,b^{\ast }⟩+⟨a^{\ast },y⟩,\forall x,y\in A,a^{\ast },b^{\ast }\in A^{\ast }.$$

(b)

$(A,{[·,·]}_A,\delta )$ is a left-Alia bialgebra.

Theorem B naturally leads to the study of Yang–Baxter equations and relative Rota–Baxter operators for left-Alia algebras [28].

2. Pseudo-Reflections and Twisted Deviations in Invariant Theory

2.1. Preliminary on Invariant Theory

Let G be a finite group and $\mathbb{K}$ an algebraic closed field of characteristic zero. Suppose that $(\rho ,V)$ is an n-dimensional faithful representation of G and its dual representation is denoted by $(\rho ,V^{\ast })$. Let $S=\mathbb{K}\left[V\right]=\mathbb{K}[x_1,\dots ,x_n]$ be the coordinate ring of V. Define a G-action on S as

$$g·{\displaystyle \sum _{i_1,\dots ,i_n}}{k}_{i_1,\dots ,i_n}{x}_1^{i_1}\dots x_n^{i_n}:={\displaystyle \sum _{i_1,\dots ,i_n}}{k}_{i_1,\dots ,i_n}{\left(\rho \left(g\right)x_1\right)}^{i_1}\dots {\left(\rho \left(g\right)x_n\right)}^{i_n},\forall g\in G.$$

(3)

Define the ring of invariants as

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

Theorem 1

([1,29]). (a) $S^G$ is a finitely generated $\mathbb{K}$-algebra.

(b) S is a finitely generated $S^G$-module.

Definition 2

([29]). A linear automorphism $R\in Aut\left(V\right)$ is called a pseudo-reflection if $R^m=I$ for some $m\in {\mathbb{N}}^{\ast }$ and $Im(I-R)$ is one-dimensional.

In invariant theory, the following theorem gives the equivalent condition that $S^G$ is a polynomial algebra:

Theorem 2

([2,3]). $S^G$ is a polynomial algebra if and only if $G\cong \rho \left(G\right)$ is generated by pseudo-reflections.

Then, we figure out the relation between a pseudo-reflection on V and a twisted deviation on $S=\mathbb{K}\left[V\right]$.

Lemma 1.

Let R be a pseudo-reflection on V. Then, R induces a pseudo-reflection on $V^{\ast }$ (also denoted by R).

Proof. 

Let $\{e_1,\dots ,e_n\}$ be a basis of V such that $W=\mathrm{Span}\{e_1,\dots ,e_{n-1}\}$ is fixed by R. By $R^m={\left(\begin{array}{ccccc}1& & & & a_1\\ & 1& & & a_2\\ & & \ddots & & ⋮\\ & & & 1& a_{n-1}\\ & & & & a_n\end{array}\right)}^m=I$, we see that R is given by the diagonal matrix $diag(1,\dots ,1,\omega )$, where $\omega \ne 1$ is an m-th primitive root over $\mathbb{K}$. Denote $\{x_1,\dots ,x_n\}$, the dual basis of $V^{\ast }$, such that $x_i\left(e_j\right)={\delta }_{ij}$. Thus, the induced automorphism on $V^{\ast }$, defined by $R\left(x_i\right)\left(e_j\right):=x_i\left(R^{-1}\left(e_j\right)\right)$, satisfies that $R\left(x_i\right)=x_i$, $1\le i\le n-1$ and $R\left(x_n\right)=(1/\omega )x_n$. Therefore, R is a pseudo-reflection on $V^{\ast }$. □

2.2. Pseudo-Reflections Induced by Twisted Deviations

Let $(A,·)$ be a commutative associative algebra and $R:A\to A$ a linear map on A. Recall from [4] the definition of a twisted derivation (also named a $\sigma$-derivation).

Definition 3.

A linear map $D:A\to A$ is called a twisted derivation with respect to R if D satisfies the twisted Leibniz rule:

$$D\left(fg\right)=D\left(f\right)g+R\left(f\right)D\left(g\right),f,g\in A.$$

(4)

Remark 1.

When $R=I$, D is a derivation on A.

Recall from Section 2.1 that for a fixed non-zero $v_R\in Im(I-R)\subset V$, there exists a ${\Delta }_R\in V^{\ast }$ such that

$$(I-R)v={\Delta }_R\left(v\right)v_R,\forall v\in V.$$

(5)

By Lemma 1, for a fixed non-zero $l_R\in Im(I-R)\subset V^{\ast }$, there also exists a ${\Delta }_R\in V$ such that

$$(I-R)x={\Delta }_R\left(x\right)l_R,\forall x\in V^{\ast }.$$

(6)

Also, denote $R:S\to S$ as an extension of $R\in \mathrm{Aut}\left(V\right)$ satisfying

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

Theorem 3.

For each $f\in S$, there exists a twisted derivation $D_R:S\to S$ with respect to R such that

$$R\left(f\right)=f-D_R\left(f\right)l_R.$$

(7)

Proof. 

First, we prove that $R\left(f\right)$ can be uniquely written as $R\left(f\right)=f-D_R\left(f\right)l_R$ for some $D_R:S\to S$. It follows from (6) that, for $1\le a_i\le n$,

$$\begin{array}{ccc}\hfill R(x_{a_1}\dots x_{a_k})& =& \left(R{x}_{a_1}\right)\dots \left(R{x}_{a_k}\right)\hfill \\ & =& (x_{a_1}-{\Delta }_R\left(x_{a_1}\right)l_R)\dots (x_{a_k}-{\Delta }_R\left(x_{a_k}\right)l_R)\hfill \end{array}$$

which can be expressed as

$$x_{a_1}\dots x_{a_k}-D_R(x_{a_1}\dots x_{a_k})l_R,$$

where $D_R$ maps the monomial to a polynomial in S. As a consequence, $R\left(f\right)$ can be written as

$$R\left(f\right)=f-D_R\left(f\right)l_R,$$

(8)

where $D_R:S\to S$ is a linear map. Then, we prove that $D_R$ is a twisted derivation on S with respect to R. On the one hand,

$$R\left(fh\right)=fh-D_R\left(fh\right)l_R.$$

On the other hand,

$$\begin{array}{ccc}\hfill R\left(f\right)R\left(h\right)=R\left(f\right)(h-D_R\left(h\right)l_R)& =& (f-D_R\left(f\right)l_R)h-R\left(f\right)D_R\left(h\right)l_R\hfill \\ & =& fh-(D_R\left(f\right)h+R\left(f\right)D_R\left(h\right))l_R.\hfill \end{array}$$

Therefore, $R\left(fh\right)=D_R\left(f\right)h+R\left(f\right)D_R\left(h\right)$. □

Remark 2.

When restricting $D_R$ to $V^{\ast }$, $D_R={\Delta }_R$ on $V^{\ast }$. When restricting $(I-D_R)$ to $V^{\ast }$, $(I-D_R)$ is a pseudo-reflection on $V^{\ast }$.

3. Left-Alia Algebras and Their Representations

3.1. Left-Alia Algebras and Twisted Derivations

Definition 4

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

$$\left[\right[x,y],z]+\left[\right[y,z],x]+\left[\right[z,x],y]=\left[\right[y,x],z]+\left[\right[z,y],x]+\left[\right[x,z],y],\forall x,y,z\in A.$$

Remark 3.

A left-Alia algebra $(A,[·,·\left]\right)$ is a Lie algebra if and only if the bilinear map $[·,·]$ is skew-symmetric. On the other hand, any commutative algebra $(A,[·,·\left]\right)$ in the sense that $[·,·]$ is symmetric is a left-Alia algebra.

We can obtain a class of left-Alia algebras from twisted derivations.

Lemma 2.

Let $D:A\to A$ be a twisted derivation of the commutative associative algebra $(A,·)$. Then, D satisfies

$$xD\left(y\right)-D\left(x\right)y=R\left(x\right)D\left(y\right)-D\left(x\right)R\left(y\right),\forall x,y\in A.$$

(9)

Proof. 

By the commutative property of $(A,·)$ and (4), we have

$$\begin{array}{c}\hfill D\left(xy\right)-D\left(yx\right)=D\left(x\right)y+R\left(x\right)D\left(y\right)-D\left(y\right)x-R\left(y\right)D\left(x\right)=0.\end{array}$$

Therefore, (9) holds. □

Theorem 4.

Let $(A,·)$ be a commutative associative algebra and D be a twisted derivation. For all $x,y\in A$, define the bilinear map ${[·,·]}_R:A\times A\to A$ by

$${[x,y]}_R:={[x,y]}_D=xD\left(y\right)-R\left(y\right)D\left(x\right).$$

(10)

Then, $(A,{[·,·]}_R)$ is a left-Alia algebra.

Proof. 

Let $x,y,z\in A$. By (10), we have

$$\begin{array}{cc}\hfill {[x,y]}_R-{[y,x]}_R& =xD\left(y\right)-R\left(y\right)D\left(x\right)-yD\left(x\right)+R\left(x\right)D\left(y\right)\hfill \\ \hfill & \stackrel{\left(9\right)}{=}2(xD\left(y\right)-yD\left(x\right)),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill D\left(xD\right(y)-yD(x\left)\right)& \stackrel{\left(4\right)}{=}D\left(x\right)D\left(y\right)-R\left(y\right)D^2\left(x\right)-D\left(y\right)D\left(x\right)+R\left(x\right)D^2\left(y\right)\hfill \\ \hfill & =R\left(x\right)D^2\left(y\right)-R\left(y\right)D^2\left(x\right).\hfill \end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill & {⥀}_{x,y,z}{[{[x,y]}_R-{[y,x]}_R,z]}_R\hfill \\ \hfill =& {⥀}_{x,y,z}2{[xD\left(y\right)-yD\left(x\right),z]}_R\hfill \\ \hfill \stackrel{\left(10\right)}{=}& {⥀}_{x,y,z}2\left(xD\left(y\right)D\left(z\right)-yD\left(x\right)D\left(z\right)-R\left(x\right)D^2\left(y\right)R\left(z\right)+R\left(y\right)D^2\left(x\right)R\left(z\right)\right)\hfill \\ \hfill =& 0.\hfill \end{array}$$

Therefore, the conclusion holds. □

Remark 4.

Theorem 4 can also be verified in the following way. Let $(A,·)$ be a commutative associative algebra with linear maps $f,g:A\to A$. By [5], there is a left-Alia algebra $(A,[·,·\left]\right)$ given by

$$[x,y]=x·f\left(y\right)+g(x·y),\forall x,y\in A,$$

(11)

which is called a special left-Alia algebra with respect to $(A,·,f,g)$. If D is a twisted derivation of $(A,·)$ with respect to R, then we see that $(A,{[·,·]}_R)$ satisfies (11) for

$$f=2D,g=-D.$$

Hence, $(A,{[·,·]}_R)$ is left-Alia.

3.2. Examples of Left-Alia Algebras

Example 1.

Let R be a reflection defined by $R\left(x_1\right)=x_2,R\left(x_2\right)=x_1,R\left(x_3\right)=x_3$ on three-dimensional vector space $V^{\ast }$ 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 $R\left(fg\right)=R\left(f\right)R\left(g\right)$ and $R(k_{1}f+k_{2}h)=k_{1}R\left(f\right)+k_{2}R\left(h\right)$. Let D be the twisted derivation on S induced by the reflection R. It follows from Theorem 3 that $R\left(f\right)=f-D\left(f\right)(x_1-x_2)$. Take two polynomials, $f={\displaystyle \sum _i}{k}_i{f}_i,g={\displaystyle \sum _j}{h}_j{g}_j,k_i,h_j\in \mathbb{K}$, in S, where $f_i,g_j$ are monomials, $f_i=x_1^{n_{i,1}}{x}_2^{n_{i,2}}{x}_3^{n_{i,3}},g_j=x_1^{m_{j,1}}{x}_2^{m_{j,2}}{x}_3^{m_{j,3}}$. We have

$$\begin{array}{cc}\hfill D\left(x_1\right)=& 1,D\left(x_2\right)=-1,D\left(x_3\right)=0.\hfill \\ \hfill D\left(x_1^{n_1}\right)=& x_1^{n_1-1}+x_1^{n_1-2}{x}_2+\dots +x_2^{n_1-1}.\hfill \\ \hfill D\left(x_2^{n_2}\right)=& -x_1^{n_2-1}-x_1^{n_2-1}{x}_2-\dots -x_2^{n_2-1}.\hfill \\ \hfill D\left(x_3^{n_3}\right)=& 0.\hfill \\ \hfill D({\displaystyle \sum _i}{k}_i{x}_1^{n_{i,1}}{x}_2^{n_{i,2}}{x}_3^{n_{i,3}})=& {\displaystyle \sum _i}{k}_i(D\left(x_1^{n_{i,1}}{x}_2^{n_{i,2}}\right)x_3^{n_{i,3}}+R\left(x_1^{n_{i,1}}{x}_2^{n_{i,2}}\right)D\left(x_3^{n_{i,3}}\right))\hfill \\ \hfill =& {\displaystyle \sum _i}{k}_i(x_1^{n_{i,1}}D\left(x_2^{n_{i,2}}\right)+R\left(x_2^{n_{i,2}}\right)D\left(x_1^{n_{i,1}}\right))x_3^{n_{i,3}}\hfill \\ \hfill =& {\displaystyle \sum _i}{k}_i(x_1^{n_{i,1}+n_{i,2}-1}+x_1^{n_{i,1}+n_{i,2}-2}{x}_2+\dots +x_1^{n_{i,2}}{x}_2^{n_{i,1}-1}\hfill \\ \hfill & -x_1^{n_{i,1}+n_{i,2}-1}-\dots -x_1^{n_1}{x}_2^{n_2-1})x_3^{n_{i,3}}.\hfill \end{array}$$

Let ${[·,·]}_R:S\times S\to S$ be the bilinear map defined in Theorem 4. Then,

$$\begin{array}{cc}\hfill {[f,g]}_R=& {\displaystyle \sum _{i,j}}{k}_i{h}_j{[f_i,g_j]}_R\hfill \\ \hfill =& {\displaystyle \sum _{i,j}}{k}_i{h}_j(f_{i}D\left(g_j\right)-R\left(g_j\right)D\left(f_i\right))\hfill \\ \hfill =& {\displaystyle \sum _{i,j}}{k}_i{h}_j(x_1^{n_{i,1}+m_{j,1}+m_{j,2}-1}{x}_2^{n_{i,2}}+\dots +x_1^{n_{i,1}+m_{j,2}}{x}_2^{n_{i,2}+m_{j,1}-1}\hfill \\ \hfill & -x_1^{n_{i,1}+m_{j,1}+m_{j,2}-1}{x}_2^{n_{i,2}}-\dots -x_1^{m_{j,1}+n_{i,1}}{x}_2^{n_{i,2}+m_{j,2}-1}\hfill \\ \hfill & -x_1^{m_{j,2}+n_{i,1}+n_{i,2}-1}{x}_2^{m_{j,1}}-\dots -x_1^{n_{i,2}+m_{j,2}}{x}_2^{n_{i,1}+m_{j,1}-1}\hfill \\ \hfill & +x_1^{m_{j,2}+n_{i,1}+n_{i,2}-1}{x}_2^{m_{j,1}}+\dots +x_1^{n_{i,1}+m_{j,2}}{x}_2^{n_{i,2}+m_{j,1}-1})x_3^{m_{j,3}+n_{i,3}}.\hfill \end{array}$$

Since $(S,·)$ is a commutative associative algebra, by Theorem 4 $(S,{[·,·]}_R)$ is a left-Alia algebra.

Proposition 1.

Let $(A,[·,·\left]\right)$ be an n-dimensional ($n\ge 2$) left-Alia algebra and $\left\{e_1,\cdots ,e_n\right\}$ be a basis of A. For all positive integers $1\le i,j,t\le n$ and structural constants $C_{ij}^t\in \mathbb{C}$, set

$$[e_i,e_j]={\displaystyle \sum _{t=1}^n}{C}_{ij}^t{e}_t.$$

(12)

Then, $(A,[·,·\left]\right)$ is a left-Alia algebra if and only if the structural constants $C_{ij}^t$ satisfy the following equation:

$${\displaystyle \sum _{k,m=1}^n}\left((C_{ij}^k-C_{ji}^k)C_{kl}^m+(C_{jl}^k-C_{lj}^k)C_{ki}^m+(C_{li}^k-C_{il}^k)C_{kj}^m\right)=0,\forall 1\le i,j,l\le n.$$

(13)

Proof. 

By (1), for all $e_i,e_j,e_l\in \left\{e_1,\cdots ,e_n\right\}$,

$$\begin{array}{c}\hfill [[e_i,e_j]-[e_j,e_i],e_l]+[[e_j,e_l]-[e_l,e_j],e_i]+[[e_l,e_i]-[e_i,e_l],e_j]=0.\end{array}$$

(14)

Set

$$[e_i,e_j]={\displaystyle \sum _{k=1}^n}{C}_{ij}^k{e}_k,[e_j,e_l]={\displaystyle \sum _{k=1}^n}{C}_{jl}^k{e}_l,[e_l·e_i]={\displaystyle \sum _{k=1}^n}{C}_{li}^k{e}_k,C_{ij}^k,C_{jl}^k,C_{li}^k\in \mathbb{C}.$$

Therefore, Equation (13) holds. □

As a direct consequence, we obtain the following:

Proposition 2.

Let A be a two-dimensional vector space over the complex field $\mathbb{C}$ with a basis $\{e_1,e_2\}$. Then, for any bilinear map $[·,·]$ on A, $(A,[·,·\left]\right)$ is a left-Alia algebra.

Next, we give some example of three-dimensional left-Alia algebras.

Example 2.

Let A be a three-dimensional vector space over the complex field $\mathbb{C}$ with a basis $\{e_1,e_2,e_3\}$. Define a bilinear map $[·,·]:A\times A\to A$ by

$$\begin{array}{cc}\hfill & [e_1,e_2]=e_1,[e_1,e_3]=e_1,[e_2,e_1]=e_2,[e_3,e_1]=e_3,\hfill \\ \hfill & [e_1,e_1]=[e_2,e_2]=[e_3,e_3]=[e_2,e_3]=[e_3,e_2]=e_1+e_2+e_3.\hfill \end{array}$$

Then, $(A,[·,·\left]\right)$ is a three-dimensional left-Alia algebra.

Remark 5.

A right-Leibniz algebra [10] is a vector space A together with a bilinear operation $[·,·]:A\otimes A\to A$ satisfying

$$\left[\right[x,y],z]=\left[\right[x,z],y]+[x,[y,z\left]\right],\forall x,y,z\in A.$$

Then, we have

$$[x,[y,z\left]\right]+[y,[z,x\left]\right]+[z,[x,y\left]\right]=\left[\right[x,y]-[y,x],z]+\left[\right[y,z]-[z,y],x]+\left[\right[z,x]-[x,z],y].$$

Therefore, if a right-Leibniz algebra satisfies

$$[x,[y,z\left]\right]+[y,[z,x\left]\right]+[z,[x,y\left]\right]=0,$$

then $(A,[·,·\left]\right)$ is a left-Alia algebra.

3.3. From Left-Alia Algebras to Anti-Pre-Lie Algebras

Definition 5

([8]). Let A be a vector space with a bilinear map $·:A\times A\to A$. $(A,·)$ is called an anti-pre-Lie algebra if the following equations are satisfied:

$$\begin{array}{cc}\hfill & x·(y·z)-y·(x·z)=[y,x]·z,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & [x,y]·z+[y,z]·x+[z,x]·y=0,\hfill \end{array}$$

(16)

where

$$[x,y]=x·y-y·x,$$

(17)

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

Remark 6.

Let $(A,[·,·\left]\right)$ be a left-Alia algebra. If $[·,·]:A\times A\to A$ satisfies

$$[x,[y,z\left]\right]-[y,[x,z\left]\right]=\left[\right[y,x],z]-\left[\right[x,y],z],\forall x,y,z\in A,$$

(18)

then $(A,[·,·\left]\right)$ is an anti-pre-Lie algebra.

3.4. From Left-Alia Algebras to Lie Triple Systems

Lie triple systems originated from Cartan’s studies on the Riemannian geometry of totally geodesic submanifolds [11], which can be constructed using twisted derivations and left-Alia algebras.

Definition 6

([30]). A Lie triple system is a vector space A together with a trilinear operation $[·,·,·]:A\times A\times A\to A$ such that the following three equations are satisfied, for all $x,y,z,a,b$ in A:

$$\begin{array}{rr}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}[x,x,y]& =0,\end{array}$$

(19)

$$\begin{array}{rr}[x,y,z]+[y,z,x]+[z,x,y]& =0\end{array}$$

(20)

and

$$[a,b,[x,y,z\left]\right]=\left[\right[a,b,x],y,z]+[x,[a,b,y],z]+[x,y,[a,b,z\left]\right].$$

(21)

Proposition 3.

Let $(A,·)$ be a commutative associative algebra and D be a twisted derivation. Define the bilinear map ${[·,·]}_R:A\times A\to A$ by (10). And define the trilinear map ${[·,·,·]}_R:A\times A\times A\to A$ by

$${[x,y,z]}_R:=\frac{1}{2}{[{[x,y]}_R-{[y,x]}_R,z]}_R,\forall x,y,z\in A.$$

If ${[·,·,·]}_R$ satisfies (21), then $(A,{[·,·,·]}_R)$ is a Lie triple system.

Proof. 

For all $x,y\in A$, it is obvious that ${[x,x,y]}_R=0$. By the proof of Theorem 4, Equation (20) holds. If, in addition, ${[·,·,·]}_R$ satisfies (21), then $(A,{[·,·,·]}_R)$ is a Lie triple system. □

Remark 7.

Let $(A,[·,·\left]\right)$ be a left-Alia algebra. For all $x,y,z\in A$, set a trilinear map $[·,·,·]:A\times A\times A\to A$ by $[x,y,z]=\left[\right[x,y]-[y,x],z]$. If $[·,·,·]$ satisfies (21), then $(A,[·,·,·\left]\right)$ is a Lie triple system.

3.5. Representations and Matched Pairs of Left-Alia Algebras

Definition 7.

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

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

(22)

Two representations, $(l,r,V)$ and $(l^{\prime },r^{\prime },V^{\prime })$, of a left-Alia algebra $(A,[·,·\left]\right)$ are called equivalent if there is a linear isomorphism $\varphi :V\to V^{\prime }$ such that

$$\varphi \left(l\left(x\right)v\right)=l^{\prime }\left(x\right)\varphi \left(v\right),\varphi \left(r\left(x\right)v\right)=r^{\prime }\left(x\right)\varphi \left(v\right),\forall x\in A,v\in V.$$

(23)

Example 3.

Let $(\rho ,V)$ be a representation of a Lie algebra $(\mathfrak{g},[·,·\left]\right)$, that is, $\rho :\mathfrak{g}\to \mathrm{End}\left(V\right)$ is a linear map such that

$$\rho \left(\right[x,y\left]\right)v=\rho \left(x\right)\rho \left(y\right)v-\rho \left(y\right)\rho \left(x\right)v,\forall x,y\in \mathfrak{g},v\in V.$$

Then, both $(\rho ,-\rho ,V)$ and $(\rho ,2\rho ,V)$ satisfy (22) and, hence, are representations of $(\mathfrak{g},[·,·\left]\right)$ as a left-Alia algebra.

Proposition 4.

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

$${[x+u,y+v]}_d=[x,y]+l\left(x\right)v+r\left(y\right)u,\forall x,y\in A,u,v\in V.$$

(24)

In this case, we denote $(A\oplus V,{[·,·]}_d)=A{⋉}_{l,r}V$.

Proof. 

This is the special case of matched pairs of left-Alia algebras where $B=V$ is equipped with the zero multiplication in Proposition 6. □

For a vector space A with a bilinear map $[·,·]:A\times A\to A$, we set linear maps ${\mathcal{L}}_{[·,·]},{\mathcal{R}}_{[·,·]}:A\to \mathrm{End}\left(A\right)$ using

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

Example 4.

Let $(A,[·,·\left]\right)$ be a left-Alia algebra. Then, $({\mathcal{L}}_{[·,·]},{\mathcal{R}}_{[·,·]},A)$ is a representation of $(A,[·,·\left]\right)$, which is called an adjoint representation. In particular, for a Lie algebra $(\mathfrak{g},[·,·\left]\right)$ with the adjoint representation $\mathrm{ad}:\mathfrak{g}\to \mathrm{End}\left(\mathfrak{g}\right)$ given by $\mathrm{ad}\left(x\right)y=[x,y],\forall x,y\in \mathfrak{g}$,

$$({\mathcal{L}}_{[·,·]},{\mathcal{R}}_{[·,·]},\mathfrak{g})=(\mathrm{ad},-\mathrm{ad},\mathfrak{g})$$

is a representation of $(\mathfrak{g},[·,·\left]\right)$ as a left-Alia algebra.

Let A and V be vector spaces. For a linear map $l:A\to \mathrm{End}\left(V\right)$, we set a linear map $l^{\ast }:A\to \mathrm{End}\left(V^{\ast }\right)$ using

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

Proposition 5.

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

Proof. 

Let $x,y\in A,u^{\ast }\in V^{\ast },v\in V$. Then, we have

$$\begin{array}{ccc}& & ⟨(l^{\ast }[x,y]-l^{\ast }[y,x]+(l^{\ast }-r^{\ast })\left(x\right)l^{\ast }\left(y\right)-(l^{\ast }-r^{\ast })\left(x\right)(l^{\ast }-r^{\ast })\left(y\right)\hfill \\ & & +(l^{\ast }-r^{\ast })\left(y\right)(l^{\ast }-r^{\ast })\left(x\right)-(l^{\ast }-r^{\ast })\left(y\right)l^{\ast }\left(x\right))u^{\ast },v⟩\hfill \\ & & =⟨(l^{\ast }[x,y]-l^{\ast }[y,x]+(l^{\ast }-r^{\ast })\left(x\right)r^{\ast }\left(y\right)-(l^{\ast }-r^{\ast })\left(y\right)r^{\ast }\left(x\right))u^{\ast },v⟩\hfill \\ & & =⟨u^{\ast },\left(l[y,x]-l[x,y]+r\left(y\right)(l-r)\left(x\right)-r\left(x\right)(l-r)\left(y\right)\right)v⟩\hfill \\ & & \stackrel{\left(22\right)}{=}0.\hfill \end{array}$$

Hence, the conclusion follows. □

Example 5.

Let $(\mathfrak{g},[·,·\left]\right)$ be a Lie algebra. Then, the coadjoint representation of $(\mathfrak{g},[·,·\left]\right)$ as a left-Alia algebra is

$$({\mathrm{ad}}^{\ast },{\mathrm{ad}}^{\ast }-(-{\mathrm{ad}}^{\ast }),{\mathfrak{g}}^{\ast })=({\mathrm{ad}}^{\ast },2{\mathrm{ad}}^{\ast },{\mathfrak{g}}^{\ast }).$$

Hence, there is a left-Alia algebra structure $\mathfrak{g}{⋉}_{{\mathrm{ad}}^{\ast },2{\mathrm{ad}}^{\ast }}{\mathfrak{g}}^{\ast }$ on the direct sum $\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }$ of vector spaces.

Remark 8.

In [5], there is also the notion of a right-Alia algebra, defined as a vector space A together with a bilinear map ${[·,·]}^{\prime }:A\times A\to A$ satisfying

$${[x,{[y,z]}^{\prime }]}^{\prime }+{[y,{[z,x]}^{\prime }]}^{\prime }+{[z,{[x,y]}^{\prime }]}^{\prime }={[x,{[z,y]}^{\prime }]}^{\prime }+{[y,{[x,z]}^{\prime }]}^{\prime }+{[z,{[y,x]}^{\prime }]}^{\prime },\forall x,y,z\in A.$$

(25)

It is clear that $(A,{[·,·]}^{\prime })$ is a right-Alia algebra if and only if the opposite algebra $(A,[·,·\left]\right)$ of $(A,{[·,·]}^{\prime })$, given by $[x,y]={[y,x]}^{\prime }$, is a left-Alia algebra. Thus, our study on left-Alia algebras can straightforwardly generalize a parallel study on right-Alia algebras. Consequently, if $(l,r,V)$ is a representation of a right-Alia algebra $(A,{[·,·]}^{\prime })$, then $(r^{\ast }-l^{\ast },r^{\ast },V^{\ast })$ is also a representation of $(A,{[·,·]}^{\prime })$. Recall [23] that if $(l,r,V)$ is a representation of an anti-pre-Lie algebra $(A,{[·,·]}_{\mathrm{anti}})$, then $(r^{\ast }-l^{\ast },r^{\ast },V^{\ast })$ is also a representation of $(A,{[·,·]}_{\mathrm{anti}})$. Moreover, admissible Novikov algebras [8] are a subclass of anti-pre-Lie algebras. If $(l,r,V)$ is a representation of an admissible Novikov algebra $(A,{[·,·]}_{admissibleNovikov})$, then $(r^{\ast }-l^{\ast },r^{\ast },V^{\ast })$ is also a representation of the admissible Novikov algebra $(A,{[·,·]}_{admissibleNovikov})$. Therefore, we have the following algebras which preserve the form $(r^{\ast }-l^{\ast },r^{\ast },V^{\ast })$ of representations on the dual spaces:

{right-Alia algebras} ⊃ {anti-pre-Lie algebras} ⊃ {admissible Novikov algebras}.

Now, we introduce the notion of matched pairs of left-Alia algebras.

Definition 8.

Let $(A,{[·,·]}_A)$ and $(B,{[·,·]}_B)$ be left-Alia algebras and $l_A,r_A:A\to \mathrm{End}\left(B\right)$ and $l_B,r_B:B\to \mathrm{End}\left(A\right)$ be linear maps. If there is a left-Alia algebra structure ${[·,·]}_{A\oplus B}$ on the direct sum $A\oplus B$ of vector spaces given by

$${[x+a,y+b]}_{A\oplus B}={[x,y]}_A+l_B\left(a\right)y+r_B\left(b\right)x+{[a,b]}_B+l_A\left(x\right)b+r_A\left(y\right)a,\forall x,y\in A,a,b\in B,$$

then we say $\left((A,{[·,·]}_A),(B,{[·,·]}_B),l_A,r_A,l_B,r_B\right)$ is a matched pair of left-Alia algebras.

Proposition 6.

Let $(A,{[·,·]}_A)$ and $(B,{[·,·]}_B)$ be left-Alia algebras and $l_A,r_A:A\to \mathrm{End}\left(B\right)$ and $l_B,r_B:B\to \mathrm{End}\left(A\right)$ be linear maps. Then, $\left((A,{[·,·]}_A),(B,{[·,·]}_B),l_A,r_A,l_B,r_B\right)$ is a matched pair of left-Alia algebras if and only if the triple $(l_A,r_A,B)$ is a representation of $(A,{[·,·]}_A)$, the triple $(l_B,r_B,A)$ is a representation of $(B,{[·,·]}_B)$ and the following equations hold:

$$\begin{array}{ll}{r}_B\left(a\right)({[x,y]}_A-{[y,x]}_A)& =(l_B-r_B)\left(a\right){[y,x]}_A+(r_B-l_B)\left(a\right){[x,y]}_A\\ & +l_B\left((r_A-l_A)\left(y\right)a\right)x+l_B\left((l_A-r_A)\left(x\right)a\right)y,\end{array}$$

(26)

$$\begin{array}{ll}{r}_A\left(x\right)({[a,b]}_B-{[b,a]}_B)& =(l_A-r_A)\left(x\right){[b,a]}_B+(r_A-l_A)\left(x\right){[a,b]}_B\\ & +l_A\left((r_B-l_B)\left(b\right)x\right)a+l_A\left((l_B-r_B)\left(a\right)x\right)b,\end{array}$$

(27)

for all $x,y\in A,a,b\in B$.

Proof. 

The proof follows from a straightforward computation. □

3.6. Quadratic Left-Alia Algebras

Definition 9.

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

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

(28)

Remark 9.

Since $\mathcal{B}$ is symmetric, it follows from Definition 9 that

$$\mathcal{B}\left(\right[x,y],z)+\mathcal{B}(y,[x,z\left]\right)=0,\forall x,y,z\in A.$$

(29)

Lemma 3.

Let $(A,[·,·],\mathcal{B})$ be a quadratic left-Alia algebra. Then, $({\mathcal{L}}_{[·,·]},{\mathcal{R}}_{[·,·]},A)$ and $({\mathcal{L}}_{[·,·]}^{\ast },{\mathcal{L}}_{[·,·]}^{\ast }-{\mathcal{R}}_{[·,·]}^{\ast },A^{\ast })$ are equivalent as representations of $(A,[·,·\left]\right)$.

Proof. 

We set a linear isomorphism ${\mathcal{B}}^{♮}:A\to A^{\ast }$ using

$$⟨{\mathcal{B}}^{♮}\left(x\right),y⟩=\mathcal{B}(x,y).$$

(30)

Then, by (29) we have

$$⟨{\mathcal{B}}^{♮}\left({\mathcal{L}}_{[·,·]}\left(x\right)y\right),z⟩=\mathcal{B}([x,y],z)=-\mathcal{B}(y,[x,z])=-⟨{\mathcal{B}}^{♮}\left(y\right),[x,z]⟩=⟨{\mathcal{L}}_{[·,·]}^{\ast }\left(x\right){\mathcal{B}}^{♮}\left(y\right),z⟩,$$

that is, ${\mathcal{B}}^{♮}\left({\mathcal{L}}_{[·,·]}\left(x\right)y\right)={\mathcal{L}}_{[·,·]}^{\ast }\left(x\right){\mathcal{B}}^{♮}\left(y\right)$. Similarly, by (28), we have ${\mathcal{B}}^{♮}\left({\mathcal{R}}_{[·,·]}\left(x\right)y\right)=({\mathcal{L}}_{[·,·]}^{\ast }-{\mathcal{R}}_{[·,·]}^{\ast })\left(x\right){\mathcal{B}}^{♮}\left(y\right)$. Hence, the conclusion follows. □

Proposition 7.

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

$$\mathcal{B}\left(\widehat{f}\left(x\right),y\right)=\mathcal{B}\left(x,f\left(y\right)\right),\forall x,y\in A.$$

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

$$[x,y]=x·f\left(y\right)-\widehat{f}(x·y).$$

(31)

Proof. 

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

$$\begin{array}{ccc}\hfill \mathcal{B}\left(\right[x,y],z)& =& \mathcal{B}\left(x·f\left(y\right)-\widehat{f}(x·y),z\right)\hfill \\ & =& \mathcal{B}\left(x,z·f\left(y\right)-y·f\left(z\right)\right)\hfill \\ & =& \mathcal{B}\left(x,z·f\left(y\right)-\widehat{f}(z·y)-\left(y·f\left(z\right)-\widehat{f}(y·z)\right)\right)\hfill \\ & =& \mathcal{B}(x,[z,y]-[y,z\left]\right).\hfill \end{array}$$

Hence, the conclusion follows. □

Example 6.

Let $(A,[·,·\left]\right)$ be a left-Alia algebra and $({\mathcal{L}}_{[·,·]},{\mathcal{R}}_{[·,·]},A)$ be the adjoint representation of $(A,[·,·\left]\right)$. By Propositions 4 and 5, there is a left-Alia algebra $A{⋉}_{{\mathcal{L}}_{[·,·]}^{\ast },{\mathcal{L}}_{[·,·]}^{\ast }-{\mathcal{R}}_{[·,·]}^{\ast }}{A}^{\ast }$ on $d=A\oplus A^{\ast }$, given by (24). There is a natural nondegenerate symmetric bilinear form ${\mathcal{B}}_d$ on $A\oplus A^{\ast }$, given by

$${\mathcal{B}}_d(x+a^{\ast },y+b^{\ast })=⟨x,b^{\ast }⟩+⟨a^{\ast },y⟩,\forall x,y\in A,a^{\ast },b^{\ast }\in A^{\ast }.$$

(32)

For all $x,y,z\in A,a^{\ast },b^{\ast },c^{\ast }\in A^{\ast }$, we have

$$\begin{array}{ccc}\hfill {\mathcal{B}}_d({[x+a^{\ast },y+b^{\ast }]}_d,z+c^{\ast })& =& {\mathcal{B}}_d\left([x,y]+{\mathcal{L}}_{[·,·]}^{\ast }\left(x\right)b^{\ast }+({\mathcal{L}}_{[·,·]}^{\ast }-{\mathcal{R}}_{[·,·]}^{\ast })\left(y\right)a^{\ast },z+c^{\ast }\right)\hfill \\ & =& ⟨[x,y],c^{\ast }⟩+⟨{\mathcal{L}}_{[·,·]}^{\ast }\left(x\right)b^{\ast }+({\mathcal{L}}_{[·,·]}^{\ast }-{\mathcal{R}}_{[·,·]}^{\ast })\left(y\right)a^{\ast },z⟩\hfill \\ & =& ⟨[x,y],c^{\ast }⟩-⟨[x,z],b^{\ast }⟩+⟨a^{\ast },[z,y]-[y,z]⟩,\hfill \\ \hfill {\mathcal{B}}_d\left(x+a^{\ast },{[z+c^{\ast },x+b^{\ast }]}_d\right)& =& ⟨[z,y],a^{\ast }⟩-⟨[z,x],b^{\ast }⟩+⟨c^{\ast },[y,x]-[x,y]⟩,\hfill \\ \hfill {\mathcal{B}}_d\left(x+a^{\ast },{[y+b^{\ast },z+c^{\ast }]}_d\right)& =& ⟨[y,z],a^{\ast }⟩-⟨[y,x],c^{\ast }⟩+⟨b^{\ast },[z,x]-[x,z]⟩.\hfill \end{array}$$

Hence, we have

$${\mathcal{B}}_d\left({[x+a^{\ast },y+b^{\ast }]}_d,z+c^{\ast }\right)={\mathcal{B}}_d\left(x+a^{\ast },{[z+c^{\ast },y+b^{\ast }]}_d-{[y+b^{\ast },z+c^{\ast }]}_d\right),$$

and, thus, $(A{⋉}_{{\mathcal{L}}_{[·,·]}^{\ast },{\mathcal{L}}_{[·,·]}^{\ast }-{\mathcal{R}}_{[·,·]}^{\ast }}{A}^{\ast },{\mathcal{B}}_d)$ is a quadratic left-Alia algebra.

Remark 10.

By Example 6, an arbitrary Lie algebra $(\mathfrak{g},[·,·\left]\right)$ renders a quadratic left-Alia algebra $(\mathfrak{g}{⋉}_{{\mathrm{ad}}^{\ast },2{\mathrm{ad}}^{\ast }}{\mathfrak{g}}^{\ast },{\mathcal{B}}_d)$, where $\mathrm{ad}:\mathfrak{g}\to \mathrm{End}\left(\mathfrak{g}\right)$ is the adjoint representation of $(\mathfrak{g},[·,·\left]\right)$.

We study the tensor forms of nondegenerate symmetric invariant bilinear forms on left-Alia algebras.

Definition 10.

Let $(A,[·,·\left]\right)$ be a left-Alia algebra and $h:A\to \mathrm{End}(A\otimes A)$ be a linear map given by

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

(33)

An element $r\in A\otimes A$ is called invariant on $(A,[·,·\left]\right)$ if $h\left(x\right)r=0$ for all $x\in A$.

Proposition 8.

Let $(A,[·,·\left]\right)$ be a left-Alia algebra. Suppose that $\mathcal{B}$ is a nondegenerate bilinear form on A and ${\mathcal{B}}^{♮}:A\to A^{\ast }$ is the corresponding map given by (30). Set $\tilde{\mathcal{B}}\in A\otimes A$ using

$$⟨\tilde{\mathcal{B}},a^{\ast }\otimes b^{\ast }⟩=⟨{\mathcal{B}}^{{♮}^{-1}}\left(a^{\ast }\right),b^{\ast }⟩,\forall a^{\ast },b^{\ast }\in A^{\ast }.$$

(34)

Then, $(A,[·,·],\mathcal{B})$ is a quadratic left-Alia algebra if and only if $\tilde{\mathcal{B}}$ is symmetric and invariant on $(A,[·,·\left]\right)$.

Proof. 

It is clear that $\mathcal{B}$ is symmetric if and only if $\tilde{\mathcal{B}}$ is symmetric. Let $x,y,z\in A$ and $a^{\ast }={\mathcal{B}}^{♮}\left(x\right),c^{\ast }={\mathcal{B}}^{♮}\left(z\right)$. Under the symmetric assumption, we have

$$\begin{array}{ccc}& & \mathcal{B}([x,y],z)=⟨[x,y],{\mathcal{B}}^{♮}\left(z\right)⟩=⟨[{\mathcal{B}}^{{♮}^{-1}}\left(a^{\ast }\right),y],c^{\ast }⟩\hfill \\ & & =-⟨{\mathcal{B}}^{{♮}^{-1}}\left(a^{\ast }\right),{\mathcal{R}}_{[·,·]}^{\ast }\left(y\right)c^{\ast }⟩=-⟨\tilde{\mathcal{B}},a^{\ast }\otimes {\mathcal{R}}_{[·,·]}^{\ast }\left(y\right)c^{\ast }⟩=⟨\left(\mathrm{id}\otimes {\mathcal{R}}_{[·,·]}\left(y\right)\right)\tilde{\mathcal{B}},a^{\ast }\otimes c^{\ast }⟩,\hfill \\ & & \mathcal{B}(x,[z,y]-[y,z])=⟨{\mathcal{B}}^{♮}\left(x\right),[z,y]-[y,z]⟩=⟨a^{\ast },[{\mathcal{B}}^{{♮}^{-1}}\left(c^{\ast }\right),y]-[y,{\mathcal{B}}^{{♮}^{-1}}\left(c^{\ast }\right)]⟩\hfill \\ & & =⟨({\mathcal{L}}_{[·,·]}^{\ast }-{\mathcal{R}}_{[·,·]}^{\ast })\left(y\right)a^{\ast },{\mathcal{B}}^{{♮}^{-1}}\left(c^{\ast }\right)⟩=⟨\tilde{\mathcal{B}},c^{\ast }\otimes ({\mathcal{L}}_{[·,·]}^{\ast }-{\mathcal{R}}_{[·,·]}^{\ast })\left(y\right)a^{\ast }⟩\hfill \\ & & =⟨\left(({\mathcal{R}}_{[·,·]}-{\mathcal{L}}_{[·,·]})\left(y\right)\otimes \mathrm{id}\right)\tilde{\mathcal{B}},a^{\ast }\otimes c^{\ast }⟩,\hfill \end{array}$$

that is, (28) holds if and only if $h\left(y\right)\tilde{\mathcal{B}}=0$ for all $y\in A$. Hence, the conclusion follows. □

4. Manin Triples of Left-Alia Algebras and Left-Alia Bialgebras

In this section, we introduce the notions of Manin triples of left-Alia algebras and left-Alia bialgebras. We show that they are equivalent structures via specific matched pairs of left-Alia algebras.

4.1. Manin Triples of Left-Alia Algebras

Definition 11.

Let $(A,{[·,·]}_A)$ and $(A^{\ast },{[·,·]}_{A^{\ast }})$ be left-Alia algebras. Assume that there is a left-Alia algebra structure $(d=A\oplus A^{\ast },{[·,·]}_d)$ on $A\oplus A^{\ast }$ which contains $(A,{[·,·]}_A)$ and $(A^{\ast },{[·,·]}_{A^{\ast }})$ as left-Alia subalgebras. Suppose that the natural nondegenerate symmetric bilinear form ${\mathcal{B}}_d$, given by (32), is invariant on $(A\oplus A^{\ast },{[·,·]}_d)$, that is, $(A\oplus A^{\ast },{[·,·]}_d,{\mathcal{B}}_d)$ is a quadratic left-Alia algebra. Then, we say that $\left((A\oplus A^{\ast },{[·,·]}_d,{\mathcal{B}}_d),A,A^{\ast }\right)$ is a Manin triple of left-Alia algebras.

Recall [19] that a double construction of commutative Frobenius algebras $\left(\right(A\oplus A^{\ast },{·}_d$, ${\mathcal{B}}_d),A,A^{\ast })$ is a commutative associative algebra $(A\oplus A^{\ast },{·}_d)$ containing $(A,{·}_A)$ and $(A^{\ast },{·}_{A^{\ast }})$ as commutative associative subalgebras, such that the natural nondegenerate symmetric bilinear form ${\mathcal{B}}_d$ given by (32) is invariant on $(A\oplus A^{\ast },{·}_d)$. Now, we show that double constructions of commutative Frobenius algebras with linear maps naturally give rise to Manin triples of left-Alia algebras.

Corollary 1.

Let $\left((A\oplus A^{\ast },{·}_d,{\mathcal{B}}_d),A,A^{\ast }\right)$ be a double construction of commutative Frobenius algebras. Suppose that $P:A\to A$ and $Q^{\ast }:A^{\ast }\to A^{\ast }$ are linear maps. Then, there is a Manin triple of left-Alia algebras $\left((A\oplus A^{\ast },{[·,·]}_d,{\mathcal{B}}_d),A,A^{\ast }\right)$ given by

$$\begin{array}{ccc}& & {[x+a^{\ast },y+b^{\ast }]}_d=(x+a^{\ast }){·}_d\left(P\left(y\right)+Q^{\ast }\left(b^{\ast }\right)\right)-(Q+P^{\ast })\left((x+a^{\ast }){·}_d(y+b^{\ast })\right),\hfill \\ & & {[x,y]}_A=x{·}_{A}P\left(y\right)-Q\left(x{·}_{A}y\right),{[a^{\ast },b^{\ast }]}_{A^{\ast }}=a^{\ast }{·}_{A^{\ast }}{Q}^{\ast }\left(b^{\ast }\right)-P^{\ast }\left(a^{\ast }{·}_{A^{\ast }}{b}^{\ast }\right),\hfill \end{array}$$

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

Proof. 

The adjoint map of $P+Q^{\ast }$ with respect to ${\mathcal{B}}_d$ is $Q+P^{\ast }$. Hence, the conclusion follows from Proposition 7 by taking $f=P+Q^{\ast }$. □

Theorem 5.

Let $(A,{[·,·]}_A)$ and $(A^{\ast },{[·,·]}_{A^{\ast }})$ be left-Alia algebras. Then, there is a Manin triple of left-Alia algebras $\left((A\oplus A^{\ast },{[·,·]}_d,{\mathcal{B}}_d),A,A^{\ast }\right)$ if and only if

$$\left((A,{[·,·]}_A),(A^{\ast },{[·,·]}_{A^{\ast }}),{\mathcal{L}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }\right)$$

is a matched pair of left-Alia algebras.

Proof. 

Let $\left((A\oplus A^{\ast },{[·,·]}_d,{\mathcal{B}}_d),A,A^{\ast }\right)$ be a Manin triple of left-Alia algebras. For all $x,y\in A,a^{\ast },b^{\ast }\in A^{\ast }$, we have

$$\begin{array}{ccc}\hfill {\mathcal{B}}_d({[x,b^{\ast }]}_d,y)& \stackrel{\left(28\right)}{=}& -\mathcal{B}(b^{\ast },{[x,y]}_A)=-⟨b^{\ast },{[x,y]}_A⟩=⟨{\mathcal{L}}_{{[·,·]}_A}^{\ast }\left(x\right)b^{\ast },y⟩={\mathcal{B}}_d\left({\mathcal{L}}_{{[·,·]}_A}^{\ast }\left(x\right)b^{\ast },y\right),\hfill \\ \hfill {\mathcal{B}}_d({[x,b^{\ast }]}_d,a^{\ast })& \stackrel{\left(28\right)}{=}& {\mathcal{B}}_d(x,{[a^{\ast },b^{\ast }]}_{A^{\ast }}-{[b^{\ast },a^{\ast }]}_{A^{\ast }})=⟨x,{[a^{\ast },b^{\ast }]}_{A^{\ast }}-{[b^{\ast },a^{\ast }]}_{A^{\ast }}⟩\hfill \\ & =& ⟨({\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast })\left(b^{\ast }\right)x,a^{\ast }⟩={\mathcal{B}}_d\left(({\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast })\left(b^{\ast }\right)x,a^{\ast }\right).\hfill \end{array}$$

Thus,

$${\mathcal{B}}_d({[x,b^{\ast }]}_d,y+a^{\ast })={\mathcal{B}}_d\left(({\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast })\left(b^{\ast }\right)x+{\mathcal{L}}_{{[·,·]}_A}^{\ast }\left(x\right)b^{\ast },y+a^{\ast }\right)$$

and, by the nondegeneracy of ${\mathcal{B}}_d$, we have

$${[x,b^{\ast }]}_d=({\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast })\left(b^{\ast }\right)x+{\mathcal{L}}_{{[·,·]}_A}^{\ast }\left(x\right)b^{\ast }.$$

Similarly,

$${[y,a^{\ast }]}_d=({\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast })\left(y\right)a^{\ast }+{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }\left(a^{\ast }\right)y.$$

Therefore, we have

$$\begin{array}{cc}\hfill {[x+a^{\ast },y+b^{\ast }]}_d& ={[x,y]}_A+{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }\left(a^{\ast }\right)y+({\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast })\left(b^{\ast }\right)x\hfill \\ \hfill & +{[a^{\ast },b^{\ast }]}_{A^{\ast }}+{\mathcal{L}}_{{[·,·]}_A}^{\ast }\left(x\right)b^{\ast }+({\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast })\left(y\right)a^{\ast }.\hfill \end{array}$$

(35)

Hence, $\left((A,{[·,·]}_A),(A^{\ast },{[·,·]}_{A^{\ast }}),{\mathcal{L}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }\right)$ is a matched pair of left-Alia algebras.

Conversely, if $\left((A,{[·,·]}_A),(A^{\ast },{[·,·]}_{A^{\ast }}),{\mathcal{L}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }\right)$ is a matched pair of left-Alia algebras, then it is straightforward to check that ${\mathcal{B}}_d$ is invariant on the left-Alia algebra $(A\oplus A^{\ast },{[·,·]}_d)$ given by (35). □

4.2. Left-Alia Bialgebras

Definition 12.

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

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

(36)

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 9.

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

$$⟨{[a^{\ast },b^{\ast }]}_{A^{\ast }},x⟩=⟨{\delta }^{\ast }(a^{\ast }\otimes b^{\ast }),x⟩=⟨a^{\ast }\otimes b^{\ast },\delta \left(x\right)⟩,\forall a^{\ast },b^{\ast }\in A^{\ast },x\in A.$$

(37)

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

Proof. 

For all $x\in A,a^{\ast },b^{\ast },c^{\ast }\in A^{\ast }$, we have

$$\begin{array}{ccc}\hfill ⟨{[{[a^{\ast },b^{\ast }]}_{A^{\ast }},c^{\ast }]}_{A^{\ast }}-{[{[b^{\ast },a^{\ast }]}_{A^{\ast }},c^{\ast }]}_{A^{\ast }},x⟩& =& ⟨{\delta }^{\ast }({\delta }^{\ast }\otimes \mathrm{id})({\mathrm{id}}^{\otimes 3}-\tau \otimes \mathrm{id})a^{\ast }\otimes b^{\ast }\otimes c^{\ast },x⟩\hfill \\ & =& ⟨a^{\ast }\otimes b^{\ast }\otimes c^{\ast },({\mathrm{id}}^{\otimes 3}-\tau \otimes \mathrm{id})(\delta \otimes \mathrm{id})\delta \left(x\right)⟩,\hfill \\ \hfill ⟨{[{[b^{\ast },c^{\ast }]}_{A^{\ast }},a^{\ast }]}_{A^{\ast }}-{[{[c^{\ast },b^{\ast }]}_{A^{\ast }},a^{\ast }]}_{A^{\ast }},x⟩& =& ⟨b^{\ast }\otimes c^{\ast }\otimes a^{\ast },({\mathrm{id}}^{\otimes 3}-\tau \otimes \mathrm{id})(\delta \otimes \mathrm{id})\delta \left(x\right)⟩\hfill \\ & =& ⟨a^{\ast }\otimes b^{\ast }\otimes c^{\ast },{\xi }^2({\mathrm{id}}^{\otimes 3}-\tau \otimes \mathrm{id})(\delta \otimes \mathrm{id})\delta \left(x\right)⟩,\hfill \\ \hfill ⟨{[{[c^{\ast },a^{\ast }]}_{A^{\ast }},b^{\ast }]}_{A^{\ast }}-{[{[a^{\ast },c^{\ast }]}_{A^{\ast }},b^{\ast }]}_{A^{\ast }},x⟩& =& ⟨c^{\ast }\otimes a^{\ast }\otimes b^{\ast },({\mathrm{id}}^{\otimes 3}-\tau \otimes \mathrm{id})(\delta \otimes \mathrm{id})\delta \left(x\right)⟩\hfill \\ & =& ⟨a^{\ast }\otimes b^{\ast }\otimes c^{\ast },\xi ({\mathrm{id}}^{\otimes 3}-\tau \otimes \mathrm{id})(\delta \otimes \mathrm{id})\delta \left(x\right)⟩.\hfill \end{array}$$

Hence, (1) holds for $(A^{\ast },{[·,·]}_{A^{\ast }})$ if and only if (36) holds. □

Definition 13.

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

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

(38)

Theorem 6.

Let $(A,{[·,·]}_A)$ be a left-Alia algebra. Suppose that there is a left-Alia algebra structure $(A^{\ast },{[·,·]}_{A^{\ast }})$ on the dual space $A^{\ast }$, and $\delta :A\to A\otimes A$ is the linear dual of ${[·,·]}_{A^{\ast }}$. Then, $\left((A,{[·,·]}_A),(A^{\ast },{[·,·]}_{A^{\ast }}),{\mathcal{L}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }\right)$ is a matched pair of left-Alia algebras if and only if $(A,{[·,·]}_A,\delta )$ is a left-Alia bialgebra.

Proof. 

For all $x,y\in A,a^{\ast },b^{\ast }\in A^{\ast }$, we have

$$\begin{array}{ccc}\hfill ⟨({\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast })\left(a^{\ast }\right)({[x,y]}_A-{[y,x]}_A),b^{\ast }⟩& =& ⟨{[x,y]}_A-{[y,x]}_A,{[b^{\ast },a^{\ast }]}_{A^{\ast }}-{[a^{\ast },b^{\ast }]}_{A^{\ast }}⟩\hfill \\ & =& ⟨(\tau -{\mathrm{id}}^{\otimes 2})\delta ({[x,y]}_A-{[y,x]}_A),a^{\ast }\otimes b^{\ast }⟩,\hfill \\ \hfill ⟨{[{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }\left(a^{\ast }\right)y,x]}_A,b^{\ast }⟩& =& -⟨{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }\left(a^{\ast }\right)y,{\mathcal{R}}_{{[·,·]}_A}^{\ast }\left(x\right)b^{\ast }⟩\hfill \\ & =& ⟨y,{[{\mathcal{R}}_{{[·,·]}_A}^{\ast }\left(x\right)b^{\ast },a^{\ast }]}_{A^{\ast }}⟩\hfill \\ & =& -⟨\left({\mathcal{R}}_{{[·,·]}_A}\left(x\right)\otimes \mathrm{id}\right)\delta \left(y\right),b^{\ast }\otimes a^{\ast }⟩\hfill \\ & =& -⟨\tau \left({\mathcal{R}}_{{[·,·]}_A}\left(x\right)\otimes \mathrm{id}\right)\delta \left(y\right),a^{\ast }\otimes b^{\ast }⟩,\hfill \\ \hfill -⟨{[{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }\left(a^{\ast }\right)y,x]}_A,b^{\ast }⟩& =& ⟨\tau \left({\mathcal{R}}_{{[·,·]}_A}\left(y\right)\otimes \mathrm{id}\right)\delta \left(x\right),a^{\ast }\otimes b^{\ast }⟩,\hfill \\ \hfill -⟨{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }\left({\mathcal{R}}_{{[·,·]}_A}^{\ast }\left(y\right)a^{\ast }\right)x,b^{\ast }⟩& =& ⟨x,{[{\mathcal{R}}_{{[·,·]}_A}^{\ast }\left(y\right)a^{\ast },b^{\ast }]}_{A^{\ast }}⟩\hfill \\ & =& -⟨\left({\mathcal{R}}_{{[·,·]}_A}\left(y\right)\otimes \mathrm{id}\right)\delta \left(x\right),a^{\ast }\otimes b^{\ast }⟩,\hfill \\ \hfill ⟨{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }\left({\mathcal{R}}_{{[·,·]}_A}^{\ast }\left(x\right)a^{\ast })y\right),b^{\ast }⟩& =& ⟨\left({\mathcal{R}}_{{[·,·]}_A}\left(x\right)\otimes \mathrm{id}\right)\delta \left(y\right),a^{\ast }\otimes b^{\ast }⟩.\hfill \end{array}$$

Thus, (38) holds if and only if (26) holds for $l_A={\mathcal{L}}_{{[·,·]}_A}^{\ast },r_A={\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast },l_B={\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast },r_B={\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }$. Similarly, (38) holds if and only if (27) holds for $l_A={\mathcal{L}}_{{[·,·]}_A}^{\ast },r_A={\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast },l_B={\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast },r_B={\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }$. Hence, the conclusion follows. □

Summarizing Theorems 5 and 6, we have the following corollary:

Corollary 2.

Let $(A,{[·,·]}_A)$ be a left-Alia algebra. Suppose that there is a left-Alia algebra structure $(A^{\ast },{[·,·]}_{A^{\ast }})$ on the dual space $A^{\ast }$, and $\delta :A\to A\otimes A$ is the linear dual of ${[·,·]}_{A^{\ast }}$. Then, the following conditions are equivalent:

(a) There is a Manin triple of left-Alia algebras $\left((d=A\oplus A^{\ast },{[·,·]}_d,{\mathcal{B}}_d),A,A^{\ast }\right)$.

(b) $\left((A,{[·,·]}_A),(A^{\ast },{[·,·]}_{A^{\ast }}),{\mathcal{L}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast }\right)$ is a matched pair of left-Alia algebras.

(c) $(A,{[·,·]}_A,\delta )$ is a left-Alia bialgebra.

Example 7.

Let $(A,{[·,·]}_A)$ be the three-dimensional left-Alia algebra given in Example 2.

Then, there is a left-Alia bialgebra $(A,{[·,·]}_A,\delta )$ with a non-zero co-multiplication δ on A, given by

$$\delta \left(e_1\right)=e_1\otimes e_1.$$

(39)

Then, by Corollary 2, there is a Manin triple $\left((A\oplus A^{\ast },[·,·],{\mathcal{B}}_d),A,A^{\ast }\right)$. Here, the multiplication ${[·,·]}_{A^{\ast }}$ on $A^{\ast }$ is given through δ by (39), that is,

$${[e_1^{\ast },e_1^{\ast }]}_{A^{\ast }}=e_1^{\ast },$$

and the multiplication $[·,·]$ on $A\oplus A^{\ast }$ is given by (35). Moreover, $((A,{[·,·]}_A),(A^{\ast },{[·,·]}_{A^{\ast }}),{\mathcal{L}}_{{[·,·]}_A}^{\ast }$, ${\mathcal{L}}_{{[·,·]}_A}^{\ast }-{\mathcal{R}}_{{[·,·]}_A}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast },{\mathcal{L}}_{{[·,·]}_{A^{\ast }}}^{\ast }-{\mathcal{R}}_{{[·,·]}_{A^{\ast }}}^{\ast })$ is a matched pair of left-Alia algebras.