3-Lie bialgebras and 3-Lie classical Yang-Baxter equations in low dimensions

TL;DR

This paper explores low-dimensional examples of 3-Lie bialgebras, providing explicit formulas for solutions to the 3-Lie classical Yang-Baxter equation and classifying related structures in dimensions 3 and 4.

Contribution

It introduces practical formulas for skew-symmetric solutions of the 3-Lie CYBE and classifies 3-Lie bialgebras and Manin triples in low dimensions.

Findings

Explicit solutions to 3-Lie CYBE in dimensions 3 and 4

Classification of 3-Lie bialgebras in low dimensions

Construction of 8-dimensional pseudo-metric 3-Lie algebras

Abstract

In this paper, we give some low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang-Baxter equation and Manin triples for 3-Lie algebras. We give an explicit and practical formula to compute the skew-symmetric solutions of the 3-Lie classical Yang-Baxter equation (CYBE). As an illustration, we obtain all skew-symmetric solutions of the 3-Lie CYBE in complex 3-Lie algebras of dimension 3 and 4 and then the induced local cocycle 3-Lie bialgebras. On the other hand, we classify the double construction 3-Lie bialgebras for complex 3-Lie algebras in dimensions 3 and 4 and then give the corresponding 8-dimensional pseudo-metric 3-Lie algebras.

Equations

$$[x_1,x_2,[x_3,x_4,x_5]]=[[x_1,x_2,x_3],x_4,x_5]+[x_3,[x_1,x_2,x_4],x_5]+[x_3,x_4,[x_1,x_2,x_5]],$$

$${\text{ad}}_{x_1,x_2}:A\to A,{\text{ad}}_{x_1,x_2}x=[x_1,x_2,x]$$

$${\text{ad}}_{x_1,x_2}[x_3,x_4,x_5]=[{\text{ad}}_{x_1,x_2}{x}_3,x_4,x_5]+[x_3,{\text{ad}}_{x_1,x_2}{x}_4,x_5]+[x_3,x_4,{\text{ad}}_{x_1,x_2}{x}_5].$$

$$(i)\rho (x_1,x_2)\rho (x_3,x_4)-\rho (x_3,x_4)\rho (x_1,x_2)=\rho ([x_1,x_2,x_3],x_4)-\rho ([x_1,x_2,x_4],x_3);$$

$$(ii)\rho ([x_1,x_2,x_3],x_4)=\rho (x_1,x_2)\rho (x_3,x_4)+\rho (x_2,x_3)\rho (x_1,x_4)+\rho (x_3,x_1)\rho (x_2,x_4).$$

$$⟨{\rho }^{\ast }(x_1,x_2)\alpha ,v⟩=-⟨\alpha ,\rho (x_1,x_2)v⟩,\forall \alpha \in V^{\ast },x_1,x_2\in A,v\in V.$$

$$[e_1,e_2,e_3]=e_1.$$

$$(1)[e_1,e_2,e_3]=e_4,[e_1,e_2,e_4]=e_3,[e_1,e_3,e_4]=e_2,[e_2,e_3,e_4]=e_1;$$

$$f([x_1,x_2,x_3])=\rho (x_1,x_2)f(x_3)+\rho (x_2,x_3)f(x_1)+\rho (x_3,x_1)f(x_2),\forall x_1,x_2,x_3\in A.$$

$$(1){\Delta }_1\text{is a 1-cocycle associated to the representation }(A\otimes A\otimes A,\text{ad}\otimes \mathrm{id}\otimes \mathrm{id});$$

$$(2){\Delta }_2\text{is a 1-cocycle associated to the representation }(A\otimes A\otimes A,\mathrm{id}\otimes \text{ad}\otimes \mathrm{id});$$

$$(3){\Delta }_3\text{is a 1-cocycle associated to the representation }(A\otimes A\otimes A,\mathrm{id}\otimes \mathrm{id}\otimes \text{ad}).$$

$$r_{pq}:=\sum_{i}z_{i1}\otimes\cdots\otimes z_{in},\quad\text{ where }z_{ij}=\left\{\begin{array}[]{ll}x_{i},&j=p,\\ y_{i},&j=q,\\ 1,&i\neq p,q,\end{array}\right.$$

$$[[r,r,r]]$$

$$[[r,r,r]]=0$$

$$\left\{\begin{array}[]{ccc}\Delta_{1}(x)&:=&\sum_{i,j}[x,x_{i},x_{j}]\otimes y_{j}\otimes y_{i};\\ \Delta_{2}(x)&:=&\sum_{i,j}y_{i}\otimes[x,x_{i},x_{j}]\otimes y_{j};\\ \Delta_{3}(x)&:=&\sum_{i,j}y_{j}\otimes y_{i}\otimes[x,x_{i},x_{j}],\end{array}\right.$$

$$[[r,r,r]]=0.$$

$$\Delta ([x,y,z])=(\mathrm{id}\otimes \mathrm{id}\otimes {\text{ad}}_{y,z})\Delta (x)+(\mathrm{id}\otimes \mathrm{id}\otimes {\text{ad}}_{z,x})\Delta (y)+(\mathrm{id}\otimes \mathrm{id}\otimes {\text{ad}}_{x,y})\Delta (z),$$

$$\Delta ([x,y,z])=(\mathrm{id}\otimes \mathrm{id}\otimes {\text{ad}}_{y,z})\Delta (x)+(\mathrm{id}\otimes {\text{ad}}_{y,z}\otimes \mathrm{id})\Delta (x)+({\text{ad}}_{y,z}\otimes \mathrm{id}\otimes \mathrm{id})\Delta (x),$$

$$([x_1,x_2,x_3],x_4{)}_A+([x_1,x_2,x_4],x_3{)}_A=0,\forall x_1,x_2,x_3,x_4\in A.$$

$$(x+\xi ,y+\eta {)}_{+}=⟨x,\eta ⟩+⟨\xi ,y⟩,\forall x,y\in A,\xi ,\eta \in A^{\ast }.$$

$$[x+\xi ,y+\eta ,z+\gamma {]}_{A\oplus A^{\ast }}$$

$$r=\sum _{i,j}{a}^{ij}{e}_i\otimes e_j=\sum _i{e}_i\otimes (\sum _j{a}^{ij}{e}_j)\in A\otimes A.$$

$$[r_{12},r_{13},r_{14}]$$

$$\sum _{i,j,k}{a}^{ip}{a}^{jq}{a}^{kr}[e_i,e_j,e_k]\otimes e_p\otimes e_q\otimes e_r$$

$$=\sum _{i<j<k}\sum _{\sigma \in S_3}{a}^{\sigma (i)p}{a}^{\sigma (j)q}{a}^{\sigma (k)r}[e_{\sigma (i)},e_{\sigma (j)},e_{\sigma (k)}]\otimes e_p\otimes e_q\otimes e_r$$

$$\displaystyle=\sum_{i<j<k}\big{(}\sum_{\sigma\in S_{3}}\mathrm{sgn}(\sigma)a^{\sigma(i)p}a^{\sigma(j)q}a^{\sigma(k)r}\big{)}[e_{i},e_{j},e_{k}]\otimes e_{p}\otimes e_{q}\otimes e_{r}.$$

$$[r_{12},r_{13},r_{14}]=\sum _{p,q,r}\sum _{i<j<k}(\sum _{\sigma \in S_3}\mathrm{sgn}(\sigma )a^{\sigma (i)p}{a}^{\sigma (j)q}{a}^{\sigma (k)r})[e_i,e_j,e_k]\otimes e_p\otimes e_q\otimes e_r.$$

$$[r_{12},r_{23},r_{24}]$$

$$[r_{13},r_{23},r_{34}]$$

$$[r_{14},r_{24},r_{34}]$$

$$M_{pqr}^{ijk}(1)$$

$$M_{pqr}^{ijk}(2)$$

$$M_{pqr}^{ijk}(3)$$

$$M_{pqr}^{ijk}(4)$$

Full text

1. 1 Introduction

2. 2 3-Lie algebras and 3-Lie bialgebras

3. 2.1 3-Lie algebras

4. 2.2 Local cocycle 3-Lie bialgebras and the 3-Lie classical Yang-Baxter equation

5. 2.3 Double construction 3-Lie bialgebras

6. 3 Skew-symmetric solutions of the 3-Lie CYBE and local cocycle 3-Lie bialgebras

7. 3.1 Notational simplification of the 3-Lie CYBE

8. 3.2 Skew-symmetric solutions of the 3-Lie CYBE

9. 3.3 Skew-symmetric solutions of the 3-Lie CYBE in the complex 3-Lie algebras in dimensions 3 and 4

10. 3.4 The induced local cocycle 3-Lie bialgebras

11. 4 Double construction 3-Lie bialgebras and Manin triples

12. 4.1 The double construction 3-Lie bialgebras for complex 3-Lie algebras in dimensions 3 and 4

13. 4.2 Pseudo-metric 3-Lie algebras in dimension 8

1. Introduction

Lie algebras have been generalized to higher arities as $n$-Lie algebras ([8, 12, 13]), which have connections with several fields of mathematics and physics. For example, the algebraic structure of $n$-Lie algebras correspond to Nambu mechanics ([1, 9, 14, 16]). As a special case of $n$-Lie algebras, 3-Lie algebras play an important role in string theory ([5, 7, 10, 11, 15]). As an instance, the structure of 3-Lie algebras applied in the study of the supersymmetry and gauge symmetry transformations of the world-volume theory of multiple coincident M2-branes. In particular, the metric 3-Lie algebras, or more generally, the 3-Lie algebras with invariant symmetric bilinear forms even attract more attentions in physics. In fact, the invariant inner product of a 3-Lie algebra is very useful in order to obtain the correct equations of motion for the Begger-Lambert theory from a Lagrangian that is invariant under certain symmetries. In order to find some new Bagger-Lambert Lagrangians, it is an approach by concerning 3-Lie algebras with metrics having signature $(p,q)$, or with a degenerate invariant symmetric bilinear form. Therefore, it is worthwhile to find new 3-Lie algebras with invariant symmetric bilinear forms.

On the other hand, it is interesting to consider the bialgebra structures of 3-Lie algebras. Giving a 3-Lie algebra $(A,[\cdot,\cdot,\cdot])$, a coalgebra $(A,\Delta)$ such that $(A^{\ast},\Delta^{\ast})$ is also a 3-Lie algebra, the most important part for a bialgebra theory is the compatibility conditions. As pointed out in [2], it is quite common for an algebraic system to have multiple bialgebra structures that differ only by their compatibility conditions. A good compatibility condition is prescribed on one hand by a strong motivation and potential applications, and on the other hand by a rich structure theory and effective constructions. Motivated by the well-known Lie bialgebra theory, the following compatibility conditions are applied in the construction of the bialgebra theory for 3-Lie algebras:

1. (a)

the comultiplication $\Delta$ satisfies certain “derivation” condition; 2. (b)

the comultiplication $\Delta$ is a 1-cocycle on $A$; 3. (c)

there is a Manin triple $(A\oplus A^{\ast},~{}A,~{}A^{\ast})$.

The above three conditions are equivalent in Lie algebras, but the equivalences are lost when they are extended in the context of 3-Lie algebras. Hence these conditions lead to the following three approaches respectively.

1. (a)

Based on Condition (a), there is an approach of bialgebra theory for 3-Lie algebras introduced in [3]. Unfortunately, it is a formal generalization in certain sense and neither a coboundary theory nor the structure on the double space $A\oplus A^{*}$ is known. 2. (b)

Motivated by Condition (b) with certain adjustments on the so-called “1-cocycles”, there is a bialgebra theory for 3-Lie algebras which is called “local cocycle 3-Lie algebra” in [2]. There is a coboundary theory which leads to the introduction of an analogue of the classical Yang-Baxter equation (CYBE), namely, 3-Lie CYBE. That is, from a skew-symmetric solution $r\in A\otimes A$ of 3-Lie CYBE in a 3-Lie algebra $A$, a local cocycle 3-Lie bialgebra is obtained. 3. (c)

Based on Condition (c) with the introduction of an analogue of Manin triple for 3-Lie algebras, a bialgebra theory for 3-Lie bialgebras which is called “double construction 3-Lie algebra” was given in [2]. Such a construction naturally provides a pseudo-metric 3-Lie algebra structure over the double space $A\oplus A^{\ast}$ with signature $(n,n)$, where $n=\dim A$, for the aforementioned study of Bagger-Lambert Lagrangians.

In this paper, we continue the study of local cocycle and the double construction 3-Lie bialgebras. The main purpose is to provide examples of such 3-Lie bialgebras systematically. For local cocycle 3-Lie bialgebras, we determine the examples from all skew-symmetric solutions of the 3-Lie CYBE. For double construction 3-Lie bialgebras, we obtain a complete classification. We give an explicit and practical formula to compute the skew-symmetric solutions of 3-Lie CYBE and then as an illustration, we give all skew-symmetric solutions of 3-Lie CYBE in the complex 3-Lie algebras in dimension 3 and 4 whose classification is already known (cf. [4]). Hence the induced local cocycle 3-Lie bialgebras are obtained. Besides, we classify the double construction 3-Lie bialgebras for the complex 3-Lie algebras in dimensions 3 and 4. As a byproduct, for the non-trivial cases, certain 8-dimensional pseudo-metric 3-Lie algebras are obtained explicitly. These examples can be regarded as a guide for a further development.

The paper is organized as follows. In Section 2, we give some elementary facts on 3-Lie algebras, the local cocycle 3-Lie bialgebras, the 3-Lie CYBE and the double construction 3-Lie bialgebras. In Section 3, we find all skew-symmetric solutions of 3-Lie CYBE in the complex 3-Lie algebras in dimension 3 and 4 and the induced local cocycle 3-Lie bialgebras are given. In Section 4, we classify the double construction 3-Lie bialgebras for the complex 3-Lie algebras in dimensions 3 and 4 and hence give certain corresponding 8-dimensional pseudo-metric 3-Lie algebras.

2. 3-Lie algebras and 3-Lie bialgebras

In this section we recall notions and results on 3-Lie algebras and 3-Lie bialgebras which will be needed later in the paper. We follow [2] to which we refer the reader for further details.

2.1. 3-Lie algebras

Definition 2.1**.**

([8])* *A 3-Lie algebra is a vector space $A$ with a skew-symmetric linear map (3-Lie bracket) $[\cdot,\cdot,\cdot]:\otimes^{3}A\rightarrow A$ such that the following Fundamental Identity holds:

[TABLE]

for any $x_{1},\cdots,x_{5}\in A$.**

The fundamental identity could be rewritten with the operator

[TABLE]

in the form as

[TABLE]

Definition 2.2**.**

([6, 12])* Let $V$ be a vector space. A representation of a 3-Lie algebra $A$ on $V$ is a skew-symmetric linear map $\rho:\otimes^{2}A\rightarrow gl(V)$ such that for any $x_{1},x_{2},x_{3},x_{4}\in A$,*

[TABLE]

Let $(V,\rho)$ be a representation of a $3$-Lie algebra $A$. Define $\rho^{\ast}:\otimes^{2}A\longrightarrow\mathfrak{gl}(V^{\ast})$ by

[TABLE]

Proposition 2.3**.**

With the above notations, $(V^{\ast},\rho^{\ast})$ is a representation of $A$, called the dual representation.

Example 2.4**.**

Let $A$ be a 3-Lie algebra. The linear map ${\rm ad}:\otimes^{2}A\rightarrow\mathfrak{g}\mathfrak{l}(A)$ with $x_{1},x_{2}\rightarrow{\rm ad}_{x_{1},x_{2}}$ for any $x_{1},x_{2}\in A$ defines a representation $(A,{\rm ad})$ which is called the adjoint representation of $A$, where ${\rm ad}_{x_{1},x_{2}}$ is given by Eq. (2.2). The dual representation $(A^{*},{\rm ad}^{*})$ of the adjoint representation $(A,{\rm ad})$ of a 3-Lie algebra $A$ is called the coadjoint representation.

The classification of complex 3-Lie algebras in dimension 3 and 4 has been known (cf. [4]).

Proposition 2.5**.**

There is a unique non-trivial 3-dimensional complex 3-Lie algebra. It has a basis $\{e_{1},e_{2},e_{3}\}$ with respect to which the non-zero product is given by

[TABLE]

Proposition 2.6**.**

Let $A$ be a non-trivial 4-dimensional complex 3-Lie algebra. Then $A$ has a basis $\{e_{1},e_{2},e_{3},e_{4}\}$ with respect to which the non-zero product of the 3-Lie algebra is given by one of the following.

[TABLE]

2.2. Local cocycle 3-Lie bialgebras and the 3-Lie classical Yang-Baxter equation

Most of the facts in this subsection and next subsection can be found in [2].

Definition 2.7**.**

Let $A$ be a 3-Lie algebra and $(V,\rho)$ be a representation of $A$. A linear map $f:A\rightarrow V$ is called a 1-cocycle of $A$ associated to $(V,\rho)$ if it satisfies**

[TABLE]

Definition 2.8**.**

A local cocycle 3-Lie bialgebra is a pair $(A,\Delta)$, where $A$ is a 3-Lie algebra, and $\Delta=\Delta_{1}+\Delta_{2}+\Delta_{3}:A\rightarrow A\otimes A\otimes A$ is a linear map, such that $\Delta^{\ast}:A^{\ast}\otimes A^{\ast}\otimes A^{\ast}\rightarrow A^{\ast}$ defines a 3-Lie algebra structure on $A^{\ast}$, and the following conditions are satisfied:

[TABLE]

In order to define the 3-Lie classical Yang-Baxter equation, we first give some necessary notations. Let $A$ be a 3-Lie algebra and $r=\sum_{i}x_{i}\otimes y_{i}\in A\otimes A$. For any $1\leq p\neq q\leq 4$, define an inclusion $\cdot_{pq}:\otimes^{2}A\longrightarrow\otimes^{4}A$ by

[TABLE]

where 1 is a symbol playing a similar role of the unit. Hence define $[[r,r,r]]\in\otimes^{4}A$ by

[TABLE]

Definition 2.9**.**

Let $A$ be a $3$-Lie algebra and $r\in A\otimes A$. The equation

[TABLE]

is called the $3$-Lie classical Yang-Baxter equation (3-Lie CYBE). **

Lemma 2.10**.**

Let $A$ be a 3-Lie algebra and $r=\sum_{i}x_{i}\otimes y_{i}\in A\otimes A$. Set

[TABLE]

where $x\in A$. Then

1. (1)

$\Delta_{1}$* is a $1$-cocycle associated to the representation $(A\otimes A\otimes A,{\rm ad}\otimes{\rm id}\otimes{\rm id})$;* 2. (2)

$\Delta_{2}$* is a $1$-cocycle associated to the representation $(A\otimes A\otimes A,{\rm id}\otimes{\rm ad}\otimes{\rm id})$;* 3. (3)

$\Delta_{3}$* is a $1$-cocycle associated to the representation $(A\otimes A\otimes A,{\rm id}\otimes{\rm id}\otimes{\rm ad})$.*

Moreover, $\Delta^{*}:A^{*}\otimes A^{*}\otimes A^{*}\rightarrow A^{*}$ defines a skew-symmetric operation, where $\Delta=\Delta_{1}+\Delta_{2}+\Delta_{3}$.

As is well-known, a skew-symmetric solution of the CYBE in a Lie algebra gives a Lie bialgebra. As its 3-Lie algebra analogue, we have

Theorem 2.11**.**

Let $A$ be a $3$-Lie algebra and let $r\in A\otimes A$ be a skew-symmetric solution of the 3-Lie CYBE:

[TABLE]

Define $\Delta:=\Delta_{1}+\Delta_{2}+\Delta_{3}:A\rightarrow A\otimes A\otimes A$, where $\Delta_{1},\Delta_{2},\Delta_{3}$ are induced by $r$ as in Eq. (2.6). Then $\Delta^{*}$ defines a $3$-Lie algebra structure on $A^{*}$. Furthermore, $(A,\Delta)$ is a local cocycle 3-Lie bialgebra.

2.3. Double construction 3-Lie bialgebras

We end this preparational section with recalling the notion of a double construction 3-Lie bialgebra and its related Minin triple.

Definition 2.12**.**

Let $A$ be a 3-Lie algebra and $\Delta:A\rightarrow A\otimes A\otimes A$ a linear map. Suppose that $\Delta^{\ast}:A^{\ast}\otimes A^{\ast}\otimes A^{\ast}\to A^{\ast}$ defines a 3-Lie algebra structure on $A^{\ast}$. If for all $x,y,z\in A$, $\Delta$ satisfies the following conditions,

[TABLE]

[TABLE]

then we call $(A,\Delta)$ a double construction 3-Lie bialgebra.**

Definition 2.13**.**

Let $A$ be a $3$-Lie algebra. A bilinear form $(\cdot,\cdot)_{A}$ on $A$ is called invariant if it satisfies

[TABLE]

A $3$-Lie algebra $A$ is called pseudo-metric if there is a nondegenerate symmetric invariant bilinear form on $A$. **

Definition 2.14**.**

A Manin triple of $3$-Lie algebras consists of a pseudo-metric $3$-Lie algebra $(\mathcal{A},(\cdot,\cdot)_{\mathcal{A}})$ and $3$-Lie algebras $A_{1},A_{2}$ such that

1. (1)

$A_{1},A_{2}$ are isotropic $3$-Lie subalgebras of $\mathcal{A}$; 2. (2)

$\mathcal{A}=A_{1}\oplus A_{2}$ as the direct sum of vector spaces; 3. (3)

For all $x_{1},y_{1}\in A_{1}$ and $x_{2},y_{2}\in A_{2}$, we have $\mathrm{pr}_{1}[x_{1},y_{1},x_{2}]=0$ and $\mathrm{pr}_{2}[x_{2},y_{2},x_{1}]=0$, where $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ are the projections from $A_{1}\oplus A_{2}$ to $A_{1}$ and $A_{2}$ respectively.

Let $(A,[\cdot,\cdot,\cdot])$ and $(A^{\ast},[\cdot,\cdot,\cdot]^{\ast})$ be 3-Lie algebras. On $A\oplus A^{\ast}$, there is a natural nondegenerate symmetric bilinear form $(\cdot,\cdot)_{+}$ given by

[TABLE]

There is also a bracket operation $[\cdot,\cdot,\cdot]_{A\oplus A^{\ast}}$ on $A\oplus A^{\ast}$ given by

[TABLE]

where $\mathrm{ad}^{\ast}$ and $\mathfrak{ad}^{\ast}$ are the coadjoint representations of $A$ and $A^{\ast}$ on $A^{\ast}$ and $A$ respectively. Note that the bracket operation $[\cdot,\cdot,\cdot]_{A\oplus A^{\ast}}$ is naturally invariant with respect to the symmetric bilinear form $(\cdot,\cdot)_{+}$, and satisfies Condition (3) in Definition 2.14. If $(A\oplus A^{\ast},[\cdot,\cdot,\cdot]_{A\oplus A^{\ast}})$ is a 3-Lie algebra, then obviously $A$ and $A^{\ast}$ are isotropic subalgebras. Consequently, $((A\oplus A^{\ast},(\cdot,\cdot)_{+}),A,A^{\ast})$ is a Manin triple, which is called the standard Manin triple of 3-Lie algebras.

Theorem 2.15**.**

Let $A$ be a $3$-Lie algebra and $\Delta:A\rightarrow A\otimes A\otimes A$ a linear map. Suppose that $\Delta^{*}:A^{*}\otimes A^{*}\otimes A^{*}\rightarrow A^{*}$ defines a $3$-Lie algebra structure on $A^{*}$. Then $(A,\Delta)$ is a double construction 3-Lie bialgebra if and only if $((A\oplus A^{*},(\cdot,\cdot)_{+}),A,A^{*})$ is a standard Manin triple, where the bilinear form $(\cdot,\cdot)_{+}$ and the $3$-Lie bracket $[\cdot,\cdot,\cdot]_{A\oplus A^{*}}$ are given by Eqs. (2.10) and (2.11) respectively.

3. Skew-symmetric solutions of the 3-Lie CYBE and local cocycle 3-Lie bialgebras

In this section, we obtain a computable formula of the 3-Lie CYBE and apply it to obtain all skew-symmetric solutions of the 3-Lie CYBE in the complex 3-Lie algebras in dimensions 3 and 4. We then obtain the local cocycle 3-Lie bialgebras induced from these solutions.

3.1. Notational simplification of the 3-Lie CYBE

Let $\{e_{1},\cdots,e_{n}\}$ be a basis of $A$. Set

[TABLE]

Then

[TABLE]

If any two of $\{i,j,k\}$ are equal, then $[e_{i},e_{j},e_{k}]=0$. So we can assume that $\{i,j,k\}$ are distinct in the sum.

Let $S_{3}$ denote the symmetric group of order 3. In the following, we let $\sigma\in S_{3}$ act on $\{i,j,k\}$ by permuting the three locations. So denoting $(i_{1},i_{2},i_{3}):=(i,j,k)$, we define $\{\sigma(i),\sigma(j),\sigma(k)\}=\{i_{\sigma(1)},i_{\sigma(2)},i_{\sigma(3)}\}$. This applies even if $i,j,k$ are not distinct. Then for fixed $\{p,q,r\}$, we have

[TABLE]

Therefore $[r_{12},r_{13},r_{14}]$ is rewritten as

[TABLE]

Similarly, we have

[TABLE]

For all $i<j<k,p,q,r$, set

[TABLE]

Thus, we have

[TABLE]

Moreover, it is obvious that $M_{pqr}^{ijk}(m)$ is invariant under the permutations on $\{i,j,k\}$, i.e.,

[TABLE]

Let $V$ be a vector space and let $\wedge$ denote the exterior product. For example,

[TABLE]

Let $\wedge^{k}(V)$ denote the $k$-th exterior power of $V$.

3.2. Skew-symmetric solutions of the 3-Lie CYBE

We now give a simplified formula for the 3-Lie CYBE $[[r,r,r]]=0$ when $r=\sum\limits_{i,j}a^{ij}e_{i}\otimes e_{j}$ is skew-symmetric, i.e., $a^{ij}=-a^{ji}$.

Theorem 3.1**.**

Let $A$ be a 3-Lie algebra with a basis $\{e_{1},\cdots,e_{n}\}$. Let the ternary operation be given by

[TABLE]

Suppose that $r=\sum\limits_{i,j}a^{ij}e_{i}\otimes e_{j}\in A\otimes A$ is skew-symmetric. Then

[TABLE]

where

[TABLE]

The theorem has a direct consequence.

Corollary 3.2**.**

Let $A$ be a 3-Lie algebra. If $r\in\wedge^{2}(A)$, then $[[r,r,r]]\in\wedge^{4}(A)$.

Remark 3.3**.**

This corollary can be regarded as a generalization of the following result on Lie algebras in the context of 3-Lie algebras: for a Lie algebra $\mathfrak{g}$, if $r\in\wedge^{2}(\mathfrak{g})$, then

[TABLE]

We will prove Theorem 3.1 in several steps.

First by the skew-symmetry of $r$, Eqs. (3.2) – (3.3) become

[TABLE]

Lemma 3.4**.**

With the notations and conditions as above. Then

1. (a)

$D^{ijk}_{pqr}=\mathrm{sgn}(\tau)D^{\tau(i)\tau(j)\tau(k)}_{pqr}$* for any $\tau\in S_{3}$.* 2. (b)

$D^{ijk}_{pqr}=-D^{pqr}_{ijk}$. 3. (c)

$D^{ijk}_{pqr}=\mathrm{sgn}(\tau)D^{ijk}_{\tau(p)\tau(q)\tau(r)}$* for any $\tau\in S_{3}$.* 4. (d)

If $p,q,r$ are not distinct, then $M^{ijk}_{pqr}(m)=0$ for any $m=1,2,3,4$. 5. (e)

$D_{ijk}^{ijk}=0$.

Proof.

(a) First for any $\tau\in S_{3}$, since $S_{3}\tau=S_{3}$, we have

[TABLE]

(b) In fact, we have

[TABLE]

Hence $D^{ijk}_{pqr}=-D^{pqr}_{ijk}$.

(c) By (a) and (b), we have

[TABLE]

(d) It is a direct consequence due to (c) by taking $\sigma$ to be the transposition exchanging two of $p,q,r$ which are not distinct.

(e) This follows since by (b), $D^{ijk}_{ijk}=-D^{ijk}_{ijk}$ and hence must be zero.

Now the proof of the lemma is completed. ∎

By Lemma 3.4, we can assume that in Eqs. (3.6) – (3.9), both $\{i,j,k\}$ and $\{p,q,r\}$ consist of distinct elements. Then these equations can be simplified to

[TABLE]

Now we can give the proof of Theorem 3.1.

Proof of Theorem 3.1. By Eqs. (3.17) – (3.20), we have

[TABLE]

Next we need to show

[TABLE]

Define an operator $\phi_{pq}:A^{\otimes m}\rightarrow A^{\otimes m}$ by

[TABLE]

Obviously, $\phi_{pp}$ is the identity. Then by Eqs. (3.13) – (3.16), we have

[TABLE]

Moreover, by Lemma 3.4   (c), we have

[TABLE]

Therefore

[TABLE]

It remains to prove

[TABLE]

In fact, the right hand side gives

[TABLE]

Hence Theorem 3.1 holds. $\Box$

3.3. Skew-symmetric solutions of the 3-Lie CYBE in the complex 3-Lie algebras in dimensions 3 and

4

We first consider the dimension 3 case.

Theorem 3.5**.**

Let $A$ be a 3-dimensional 3-Lie algebra. Then for any $r\in\wedge^{2}(A)$, $[[r,r,r]]=0$. That is, any $r\in\wedge^{2}(A)$ is a solution of 3-Lie CYBE in a 3-dimensional 3-Lie algebra.

Proof.

By Corollary 3.2, $[[r,r,r]]\in\wedge^{4}(A)$. On the other hand, since $\mathrm{dim}~{}A=3$, any element in $\wedge^{4}(A)$ is zero. Therefore $[[r,r,r]]=0$. ∎

Next let $A$ be a 4-dimensional 3-Lie algebra with a basis $\{e_{1},\cdots,e_{4}\}$. Then we have

[TABLE]

for constants $T_{ijk}^{m}$.

Lemma 3.6**.**

With the notations and conditions as above, assume that $T_{ijk}^{l}\neq 0$ only if $i,j,k,l$ are distinct. Then for any skew-symmetric $r\in\wedge^{2}(A)$, $[[r,r,r]]=0$.

Proof.

First $e_{l}\wedge e_{p}\wedge e_{q}\wedge e_{r}\in\wedge^{4}(A)$ is not zero precisely when $l,p,q,r$ are distinct. By the assumption, the indices of a nonzero $T_{ijk}^{l}$ in Eq.  (3.12) are also distinct. But $i,j,k,p,q,r,l\in\{1,2,3,4\}$ and $i<j<k,~{}p<q<r$ in Eq. (3.12). Thus we have $i=p,~{}j=q,~{}k=r$. Then Eq. (3.12) gives

[TABLE]

By Lemma 3.4 (e), $[[r,r,r]]=0$. ∎

Theorem 3.7**.**

Let $A$ be a 4-dimensional 3-Lie algebra. If $A$ is one of the complex 3-Lie algebras of Cases (1), (3), (4),and (7) given in Proposition 2.6, then any skew-symmetric $r\in\wedge^{2}(A)$ satisfies $[[r,r,r]]=0$.

Proof.

The proof follows directly from Lemma 3.6. ∎

Theorem 3.8**.**

Let $A$ be a 4-dimensional 3-Lie algebra with a basis $\{e_{1},e_{2},e_{3},e_{4}\}$ in Case (2), Case (5) or Case (6) in Proposition 2.6. Let $r=\sum\limits_{i,j}a^{ij}e_{i}\otimes e_{j}\in A\otimes A$ be skew-symmetric.

1. (a)

If $A$ is the complex 3-Lie algebra of Case (2), then $r$ satisfies $[[r,r,r]]=0$ if and only if

[TABLE] 2. (b)

If $A$ is the complex 3-Lie algebra of Case (5), then $r$ satisfies $[[r,r,r]]=0$ if and only if

[TABLE] 3. (c)

If $A$ is the complex 3-Lie algebra of Case (6), then $r$ satisfies $[[r,r,r]]=0$ if and only if

[TABLE]

Proof.

(a) For the 3-Lie algebra of Case (2), only $T^{1}_{123}\neq 0$. So

[TABLE]

Hence $[[r,r,r]]=0$ if and only if $D^{123}_{234}=a^{23}(a^{12}a^{34}-a^{13}a^{24}-a^{32}a^{14})=0$. (b)For the 3-Lie algebra of Case (5),only $T_{134}^{1}\neq 0$ and $T_{234}^{2}\neq 0$. So

[TABLE]

By Lemma 3.4 (b), $D^{134}_{234}-D^{234}_{134}=2D^{134}_{234}$. Hence $[[r,r,r]]=0$ if and only if $D^{123}_{234}=a^{34}(a^{12}a^{34}-a^{32}a^{14}+a^{42}a^{13})=0$.

(c) For the 3-Lie algebra of Case (6), only $T^{1}_{234}\neq 0$, $T^{2}_{234}\neq 0$ and $T^{2}_{134}\neq 0$. So

[TABLE]

Hence $[[r,r,r]]=0$ if and only if $D^{234}_{134}=a^{34}(a^{12}a^{43}+a^{32}a^{14}-a^{42}a^{13})=0$. ∎

3.4. The induced local cocycle 3-Lie bialgebras

We now provide the local cocycle 3-Lie bialgebras induced from skew-symmetric solutions of the 3-Lie CYBE.

Theorem 3.9**.**

Let $A$ be a 3-Lie algebra with a basis $\{e_{1},\cdots,e_{n}\}$. Let $r=\sum\limits_{i,j}a^{ij}e_{i}\otimes e_{j}\in A\otimes A$. Set $\Delta=\Delta_{1}+\Delta_{2}+\Delta_{3}:A\rightarrow A\otimes A\otimes A$, in which $\Delta_{1},\Delta_{2},\Delta_{3}$ are induced by $r$ as in Eq. (2.6). Then

[TABLE]

Proof.

By Eq. (2.6), for any $x\in A$, we have

[TABLE]

Note that $a^{ip}a^{jq}-a^{jp}a^{iq}=0$ when $p=q$. Then

[TABLE]

Due to Eq. (2.6) again, we have

[TABLE]

Therefore

[TABLE]

Hence the conclusion holds. ∎

Combining Theorem 2.11, Theorem 3.9 and the results in the previous subsection, we obtain the following conclusion on local cocycle 3-Lie bialgebras.

Proposition 3.10**.**

Let $A$ be a 3-Lie algebra with a basis $\{e_{1},\cdots,e_{n}\}$. For $r=\sum\limits_{i,j}a^{ij}e_{i}\otimes e_{j}\in A\otimes A$, denote

[TABLE]

Then every skew-symmetric solution of the 3-Lie CYBE in the complex 3-Lie algebras in dimension 3 and 4 gives a local cocycle 3-Lie bialgebra $(A,\Delta)$, where $\Delta$ is given by the following formula.

1. (1)

If $A$ is the 3-dimensional 3-Lie algebra in Proposition 2.5, then

[TABLE] 2. (2)

If $A$ is the 4-dimensional 3-Lie algebra of Case (1) in Proposition 2.6, then

[TABLE] 3. (3)

If $A$ is the 4-dimensional 3-Lie algebra of Case (2) in Proposition 2.6, then

[TABLE]

and the parameters satisfy an additional condition $a^{23}(a^{12}a^{34}-a^{13}a^{24}-a^{32}a^{14})=0$. 4. (4)

If $A$ is the 4-dimensional 3-Lie algebra of Case (3) in Proposition 2.6, then

[TABLE] 5. (5)

If $A$ is the 4-dimensional 3-Lie algebra of Case (4) in Proposition 2.6, then

[TABLE]

Here the parameters satisfy the condition $a^{34}(a^{12}a^{34}-a^{32}a^{14}+a^{42}a^{13})=0$. 6. (6)

If $A$ is the 4-dimensional 3-Lie algebra of Case (5) in Proposition 2.6, then

[TABLE] 7. (7)

If $A$ is the 4-dimensional 3-Lie algebra of Case (6) in Proposition 2.6, then

[TABLE]

and the parameters satisfy the condition $a^{34}(a^{12}a^{43}+a^{32}a^{14}-a^{42}a^{13})=0$. 8. (8)

If $A$ is the 4-dimensional 3-Lie algebra of Case (7) in Proposition 2.6, then

[TABLE]

4. Double construction 3-Lie bialgebras and Manin triples

In this section we classify double construction 3-Lie bialgebras for complex 3-Lie algebras in dimensions 3 and 4. We also give the corresponding Manin triples.

4.1. The double construction 3-Lie bialgebras for complex 3-Lie algebras in dimensions 3 and 4

Proposition 4.1**.**

Let $A$ be a 3-Lie algebra with a basis $\{e_{1},\cdots,e_{n}\}$. Let $\Delta:A\rightarrow A\otimes A\otimes A$ be a linear map. Set

[TABLE]

1. (1)

$\Delta$* satisfies Eq. (2.7) if and only if the following equation holds:*

[TABLE] 2. (2)

$\Delta$* satisfies Eq. (2.8) if and only if the following equation holds:*

[TABLE]

Proof.

It is obtained by a straightforward computation of Eqs. (2.7) and (2.8) followed by comparing the coefficients. ∎

With the conditions and notation as above, let $\{f_{1},f_{2},\cdots,f_{n}\}$ be the dual basis of $A^{\ast}$ and $\Delta^{*}:A^{*}\otimes A^{*}\otimes A^{*}\rightarrow A^{*}$ be the dual map. Then a direct computation shows that

[TABLE]

If in addition, $\Delta^{\ast}$ defines a 3-Lie algebra structure on $A^{\ast}$, then $\Delta$ is a skew-symmetric linear map, i.e., for any permutation $\tau$ on $\{p,q,r\}$,

[TABLE]

Lemma 4.2**.**

With the notations as above. If Eq. (4.4) holds, then $\Delta$ satisfies Eq. (2.7) if and only if Eq. (4.1) holds for any $p,q,r$ and $a<b<c$.

Proof.

At first, we claim that the following two equations are equivalent when $p,q,r$ are fixed.

[TABLE]

where $a_{2},b_{2},c_{2}$ are obtained by permuting $a_{1},b_{1},c_{1}$. In fact, without loss of generality, we assume $a_{1}=b_{2},b_{1}=a_{2},c_{1}=c_{2}$. Then

[TABLE]

Furthermore, if any two of $a,b,c$ are equal, then Eq. (4.1) holds automatically. In fact, assume $a=b$ without loss of generality. The left hand side of Eq. (4.1) is zero because $T^{i}_{aac}=0$, whereas the right hand side is also zero because $T_{aci}^{r}C_{a}^{pqi}=-T_{aci}^{q}C_{a}^{pir}$ and $T_{aai}^{r}=0$. Therefore the indices $a,b,c$ should be selected distinct and the sequence of $a,b,c$ makes no difference. Hence the lemma holds. ∎

Theorem 4.3**.**

Let $A$ be the 3-dimensional 3-Lie algebra given in Proposition 2.5. If a skew-symmetric linear map $\Delta:A\rightarrow A\otimes A\otimes A$ satisfies Eq. (2.7) or Eq. (4.1), then $\Delta=0$. Therefore there is no non-trivial double construction 3-Lie bialgebra for $A$.

Proof.

With the notations as in Proposition 4.1. Fix $p,q,r$. Then by lemma 4.2, we only need to consider the following three equations.

[TABLE]

Simplifying those equations, we have

[TABLE]

Since $p,q,r$ are chosen arbitrarily and $\Delta$ is skew-symmetric, this shows that $C_{i}^{abc}=0$, $\forall i,a,b,c=1,2,3$, i.e., $\Delta=0$. ∎

Lemma 4.4**.**

For any 3-Lie algebra in dimension 4, as displayed in Proposition 2.6, the skew-symmetric linear map $\Delta:A\rightarrow A\otimes A\otimes A$ satisfying Eq. (4.1) is given as follows (all the parameters are arbitrary constants).

1. (1)

If $A$ is the 4-dimensional 3-Lie algebra of Case (1) in Proposition 2.6, then

[TABLE] 2. (2)

If $A$ is the 4-dimensional 3-Lie algebra of Case (2) in Proposition 2.6, then

[TABLE] 3. (3)

If $A$ is the 4-dimensional 3-Lie algebra of Case (3) in Proposition 2.6, then

[TABLE] 4. (4)

If $A$ is the 4-dimensional 3-Lie algebra of Case (4) in Proposition 2.6, then

[TABLE] 5. (5)

If $A$ is the 4-dimensional 3-Lie algebra of Case (5) in Proposition 2.6, then

[TABLE] 6. (6)

If $A$ is the 4-dimensional 3-Lie algebra of Case (6) in Proposition 2.6, then

[TABLE] 7. (7)

If $A$ is the 4-dimensional 3-Lie algebra of Case (7) in Proposition 2.6, then

[TABLE]

Proof.

We give an explicit proof for the Case (1) as an example and we omit the proofs for the other cases since the proofs are similar. Fix $p,q,r$. Then by Lemma 4.2, we only need to consider Eq. (4.1) whose indices $(r,a,b,c)$ are given by the following quadruples.

[TABLE]

Let $A$ be the 3-Lie algebra of Case (1). For the quadruples $(i,1,2,3),1\leq i\leq 4$, we have

[TABLE]

By Eq. (4.13), we obtain

[TABLE]

By Eq. (4.14), we obtain

[TABLE]

Hence $C^{pq4}_{4}=0$, $\forall p,q=1,2,3,4$.

Similarly, for the other rows of equations, we show that

[TABLE]

That is, $C^{pqr}_{i}=0$ if any one of $p,q,r$ equals $i$. What remain unknown in $\{C^{abc}_{i}|a<b<c\}$ are $C^{123}_{4}$, $C^{124}_{3}$, $C^{134}_{2}$ and $C^{234}_{1}$. By Eq. (4.13) again, we obtain

[TABLE]

By Eq. (4.14), we obtain

[TABLE]

By Eq. (4.15), we obtain

[TABLE]

Therefore

[TABLE]

Furthermore, it is straightforward to check that Eq. (4.17) satisfies all the 16 equations in Eq. (4.12). Therefore $\Delta$ is determined explicitly in Eq. (4.5) by taking $C^{123}_{4}=k$. ∎

Lemma 4.5**.**

Let $A$ be a complex 3-Lie algebra with a basis $\{e_{1},e_{2},\cdots,e_{n}\}$. Let $p,q,r,s,t$ be fixed indexes, and $m_{1},m_{2},m_{3},m_{4}\in\mathbb{C}$. Assume that

[TABLE]

and set

[TABLE]

Then

[TABLE]

Proof.

Assume $m_{2}=m_{3}=m_{4}=0$. Then

[TABLE]

Adding Eqs. (4.19) – (4.21) together, we have

[TABLE]

Assume $m_{1}=m_{3}=m_{4}=0$. Then

[TABLE]

Assume $m_{1}=m_{2}=m_{4}=0$. Then

[TABLE]

Assume $m_{1}=m_{2}=m_{3}=0$. Then

[TABLE]

Adding Eqs. (4.25) – (4.27) together, we have

[TABLE]

Since $\Phi_{e_{s},e_{t}}$ is linear, Eqs. (4.22), (4.23), (4.24) and (4.28) together indicate that Eq. (4.18) holds. ∎

Theorem 4.6**.**

Let $A$ be one of the 4-dimensional 3-Lie algebras of Cases (2), (5) and (6) given in Proposition 2.6. Then any double construction 3-Lie bialgebra for $A$ is trivial.

Proof.

By Lemma 4.4, we need to show that for the mentioned cases, if in addition $\Delta$ satisfies Eq. (2.8), then $\Delta=0$.

Case (2): Substituting $x=e_{2},y=e_{2},z=e_{3}$ into Eq. (2.8), we get

[TABLE]

Hence $k=c_{1}=0$. Substituting $x=e_{3},y=e_{2},z=e_{3}$ into Eq. (2.8), we get

[TABLE]

Hence $c_{2}=0$. Therefore $\Delta=0$.

Case (5): Substituting $x=e_{3},y=e_{3},z=e_{4}$ into Eq. (2.8), we get

[TABLE]

Hence $k=c_{1}=0$. Substituting $x=e_{4},y=e_{3},z=e_{4}$ into Eq. (2.8), we get

[TABLE]

Hence $c_{2}=0$. Therefore $\Delta=0$.

Case (6): Substituting $x=e_{3},y=e_{3},z=e_{4}$ into Eq. (2.8), we get

[TABLE]

Hence $k=c_{1}=0$. Substituting $x=e_{4},y=e_{3},z=e_{4}$ into Eq. (2.8), we get

[TABLE]

Hence $c_{2}=0$. Therefore $\Delta=0$. ∎

Lemma 4.7**.**

With the notations as in Proposition 4.1. If Eq.  (4.4) holds, then $\Delta$ satisfies Eq.  (2.8) if and only if Eq. (4.2) holds for any $a$ and $b<c,~{}p<q<r$.

Proof.

It follows from a proof similar to the one for Lemma 4.2. ∎

Theorem 4.8**.**

Let $A$ be a 4-dimensional 3-Lie algebra with a basis $\{e_{1},\cdots,e_{4}\}$.

1. (1)

If $A$ is the 4-dimensional 3-Lie algebra of Case (1) given in Proposition 2.6, then $(A,\Delta)$ is a double construction 3-Lie bialgebra, where $\Delta$ is given by Eq. (4.5). 2. (2)

If $A$ is the 4-dimensional 3-Lie algebra of Case (3) given in Proposition 2.6, then $(A,\Delta)$ is a double construction 3-Lie bialgebra, where $\Delta$ is given by Eq. (4.7). 3. (3)

If $A$ is the 4-dimensional 3-Lie algebra of Case (4) given in Proposition 2.6, then $(A,\Delta)$ is a double construction 3-Lie bialgebra, where $\Delta$ is given by Eq. (4.8). 4. (4)

If $A$ is the 4-dimensional 3-Lie algebra of Case (7) given in Proposition 2.6, then $(A,\Delta)$ is a double construction 3-Lie bialgebra, where $\Delta$ is given by Eq. (4.11).

Proof.

In fact, for the 4-dimensional 3-Lie algebras of Cases (1), (3), (4) and (7), the corresponding $\Delta$ appearing in Lemma 4.4 satisfies Eq. (2.8), too. We give an explicit proof for the Case (1) as an example and we omit the proofs for the other cases since they are similar. For the 3-Lie algebra of Case (1), fix $a,b,c$. By Lemma 4.7, we only need to consider othe following four equations.

[TABLE]

For Eq. (4.29), the left hand side is

[TABLE]

whereas the right hand side is

[TABLE]

Therefore, Eq. (4.29) holds if and only if the following series of equations hold:

[TABLE]

It is straightforward to show that these equations hold for arbitrary $b,c=1,2,3,4$. Hence Eq. (4.29) holds. Similarly, Eqs. (4.30) – (4.32) also hold. Therefore Eq. (2.8) holds.

Moreover, it is straightforward to check (also see the remark after this proof) that, for every $\Delta$ appearing in the conclusion, the dual $\Delta^{*}$ defines a 3-Lie algebra on $A^{*}$. Hence the conclusion holds. ∎

Remark 4.9**.**

We give explicitly the 3-Lie algebra structure on the dual space $A^{*}$ obtained from $\Delta$ in the above double construction 3-Lie bialgebras as follows.

1. (1)

$A$ is the 4-dimensional 3-Lie algebra of Case (1) given in Proposition 2.6.

[TABLE] 2. (2)

$A$ is the 4-dimensional 3-Lie algebra of Case (3) given in Proposition 2.6.

[TABLE] 3. (3)

$A$ is the 4-dimensional 3-Lie algebra of Case (4) given in Proposition 2.6.

[TABLE] 4. (4)

$A$ is the 4-dimensional 3-Lie algebra of Case (7) given in Proposition 2.6.

[TABLE]

It is easy to show that $(A^{\ast},[\cdot,\cdot,\cdot]^{\ast})$ in Case (1) for $k\neq 0$ and in Case (7) for $c\neq 0$ are respectively isomorphic to the 3-Lie algebras of the Case (1) and Case (3) given in Proposition 2.6. For Cases (3) and (4) mentioned in the remark, $(A^{\ast},[\cdot,\cdot,\cdot]^{\ast})$ is still a 3-Lie algebra.**

4.2. Pseudo-metric 3-Lie algebras in dimension 8

By Theorem 2.15 and the results in the previous subsection, we can get the corresponding pseudo-metric 3-Lie algebras in dimension 8 (Manin triples of 3-Lie algebras) as follows.

Theorem 4.10**.**

Let $A$ be a 4-dimensional vector space with a basis $\{e_{1},\cdots,e_{4}\}$ and $\{e^{\ast}_{1},\cdots,e^{\ast}_{4}\}$ be the dual basis. On the vector space $A\oplus A^{*}$ define a bilinear form $(\cdot,\cdot)_{+}$ by Eq. (2.10), that is, with respect to the basis $\{e_{1},\cdots,e_{4},e^{\ast}_{1},\cdots,e^{\ast}_{4}\}$, it corresponds to the matrix $\begin{pmatrix}0&I_{n}\\ I_{n}&0\end{pmatrix}$. We can get the following families of 8-dimensional pseudo-metric 3-Lie algebras $(A\oplus A^{*},(\cdot,\cdot)_{+})$.

1. (1)

The non-zero product of 3-Lie algebra structure on $A\oplus A^{*}$ is given by

[TABLE]

where the last equation holds for $i<j<k$ and $m$ which is distinct from $i,j,k$. They correspond to the double construction bialgebras $(A,\Delta)$ given in Theorem 4.8, where $A$ is the 3-Lie algebra of Case (1) given in Proposition 2.6. 2. (2)

The non-zero product of 3-Lie algebra structure on $A\oplus A^{*}$ is given by

[TABLE]

They correspond to the double construction bialgebras $(A,\Delta)$ given in Theorem 4.8, where $A$ is the 3-Lie algebra of Case (3) given in Proposition 2.6. 3. (3)

The non-zero product of 3-Lie algebra structure on $A\oplus A^{*}$ is given by

[TABLE]

They correspond to the double construction bialgebras $(A,\Delta)$ given in Theorem 4.8, where $A$ is the 3-Lie algebra of Case (4) given in Proposition 2.6. 4. (4)

The non-zero product of 3-Lie algebra structure on $A\oplus A^{*}$ is given by

[TABLE]

They correspond to the double construction bialgebras $(A,\Delta)$ given in Theorem 4.8, where $A$ is the 3-Lie algebra of Case (7) given in Proposition 2.6.

The proof is by a straightforward computation.

Acknowledgements. This work was supported by the Natural Science Foundation of China (Grant Nos. 11371178, 11425104).