3-Lie bialgebras (Lb,Cd) and (Lb,Ce)

(1)

3 -Lie bialgebras

(

L

b

, C

d

)

and

(

L

b

, C

e

)

BAI Ruipu

College of Mathematics and Information Science, Hebei University, Baoding, 071002, China

email: bairuipu@hbu.edu.cn

LIN Lixin

College of Mathematics and Information Science, Hebei University, Baoding, 071002, China

email: llxhebeidaxue@163.com

Abstract

In this paper, we continue to study the structure of four dimensional 3-Lie bialgebras. We discuss the existence of 3-Lie bialgebras of types (Lb, Cd) and (Lb, Ce). It is proved that there do not exist 3-Lie bialge-bras of types (Lb1, Ce), (Lb2, Ce) and (Lb2, Cd). There exists only one class of 3-Lie bialgebras of type (Lb, Cd), that is (Lb1, Cd,∆1) ( Theorem 3.4 ).

2010 Mathematics Subject Classification: 17B05 17D30 Keywords: 3-Lie algebra, 3-Lie coalgebra, 3-Lie bialgebra.

1 Introduction

(2)

which is denoted by (Lb, Cd) and (Lb, Ce), respectively. And we suppose that 3-Lie algebras and 3-Lie coalgebras are over a field F of characteristic zero, and omit the zero multiplication of basis vectors in 3-Lie algebras and 3-Lie coalgebras.

2 Preliminaries

A 3-Lie algebra [1] is a vector space L endowed with a linear multiplication

µ:L∧3

→L satisfying that, for all x, y, z, u, v_∈L,

µ(u, v, µ(x, y, z)) =µ(x, y, µ(u, v, z))+µ(y, z, µ(u, v, x))+µ(z, x, µ(u, v, y)).

For defining 3-Lie coalgebras, we need to define following linear mapps

ωi :L⊗L⊗L⊗L⊗L→L⊗L⊗L⊗L⊗L, 1≤i≤3, by

ω1(x1⊗x2⊗x3⊗x4⊗x5) =x3⊗x4⊗x1⊗x2⊗x5,

ω2(x1⊗x2⊗x3⊗x4⊗x5) =x4⊗x5⊗x1⊗x2⊗x3,

ω3(x1⊗x2⊗x3⊗x4⊗x5) =x5⊗x3⊗x1⊗x2⊗x4.

A 3-Lie coalgebra (L,∆) [5] is a vector space L with a linear map ∆ :

L_→L_⊗L_⊗L satisfying

Im(∆) ⊂L_∧L_∧L, and (1−ω₁₋ω₂₋ω₃)(1⊗1⊗∆)∆ = 0,

where 1 : L⊗₅

→L⊗₅

is identity.

Let (L1,∆1) and (L2,∆2) be 3-Lie coalgebras. If there is a linear

isomor-phism ϕ : L₁ _→ L₂ satisfying (ϕ_⊗ϕ_⊗ϕ)(∆1(e)) = ∆2(ϕ(e)), for all e ∈ L1,

then (L1,∆1) is isomorphic to (L2,∆2), and ϕ is called a 3-Lie coalgebra iso-morphism, where (ϕ_⊗ϕ_⊗ϕ)P

i (

ai⊗bi⊗ci) =

P

i

ϕ(ai)⊗ϕ(bi)⊗ϕ(ci).

A 3-Lie bialgebra [5] is a triple (L, µ,∆) such that

(1) (L, µ) is a 3-Lie algebra with the multiplication µ:L_∧L_∧L_→L,

(2) (L,∆) is a 3-Lie coalgebra with ∆ : L_→L_∧L_∧L,

(3) ∆ and µ satisfy the following identities, forx, y, u, v, w_∈L, ∆µ(x, y, z) =ad(3)_µ (x, y)∆(z) +ad_µ(3)(y, z)∆(x) +ad(3)_µ (z, x)∆(y),

where ad(3)_µ (x, y), ad(3)_µ (z, x), ad_µ(3)(y, z) : L_⊗L_⊗L _→ L_⊗L_⊗L are linear maps defined by (similar for ad(3)_µ (z, x) and ad(3)_µ (y, z))

ad(3)_µ (x, y)(u_⊗v_⊗w) = (adµ(x, y)_⊗1_⊗1)(u_⊗v _⊗w)

+(1_⊗adµ(x, y)_⊗1)(u_⊗v _⊗w) + (1_⊗1_⊗adµ(x, y))(u_⊗v_⊗w) =µ(x, y, u)_⊗v_⊗w+u_⊗µ(x, y, v)_⊗w+u_⊗v_⊗µ(x, y, w).

Two 3-Lie bialgebras (L1, µ1,∆1) and (L2, µ2,∆2) are called equivalentif

there exists a linear isomorphism f :L1 →L2 such that

(1) f : (L₁, µ₁)→(L₂, µ₂) is a 3-Lie algebra isomorphism,

(2) f : (L1,∆1)→(L2,∆2) is a 3-Lie coalgebra isomorphism, that is,

∆2(f(x)) = (f⊗f ⊗f)∆1(x) for all x∈L1.

Lemma 2.1[1] Let (L, µ) be a 4-dimensional3-Lie algebra with dimL1 ₆= 0,2, and e1, e2, e3, e4 be a basis of L. Then L is isomorphic to one and only one of the

(3)

Lb1. µ(e2, e3, e4) =e1. Lb2. µ(e1, e2, e3) =e1.

Ld. µd(e2, e3, e4) =e1, µd(e1, e3, e4) =e2, µd(e1, e2, e4) =e3.

Le. µe(e2, e3, e4) =e1, µe(e1, e3, e4) =e2, µe(e1, e2, e4) =e3, µe(e1, e2, e3) =e4.

Lemma 2.2 [5] Let L be a vector space over F, and µ: L_⊗L_⊗L_→ L be a

3-ary linear map. Then (L, µ) is a 3-Lie algebra if and only if (L∗

, µ∗

) is a 3-Lie coalgebra with µ∗

:L∗

→L∗

⊗L∗

, where µ∗

is the dual map ofµ.

3

3 -Lie bialgebras of types

(

L

b

, C

d

)

and

(

L

b

, C

e

)

First We give the classification of 3-Lie coalgebras of the types (L, Cd) and (L, Ce). Lemma 3.1 Let(L,∆) be a4-dimensional 3-Lie coalgebra withm-dimensional derived algebra (3 _≤m _≤4), and e1, e2, e3, e4 be a basis of L. Then L isomorphic to one and only one of the following

Cd.∆d(e1) =e2∧e3∧e4,∆d(e2) =e1∧e3∧e4,∆d(e3) =e1∧e2∧e4;

Ce.∆e(e1) =e2∧e3∧e4,∆e(e2) =e1∧e3∧e4,∆e(e3) =e1∧e2∧e4, ∆e(e4) =e1_∧e2_∧e3.

Proof The result follows from Lemma 2.1, Lemma 2.2 and a direct computation, we omit the computation process.

For convenience, in the following, for a 3-Lie bialgebra (L, µ,∆), if the 3-Lie algebra (L, µ) is the case (L, µbi) in Lemma 2.1 and the 3-Lie coalgebra (L,∆) is

the case (L,∆d) and (L,∆e) in Lemma 3.1, then the 3-Lie bialgebra (L, µbi,∆d) and

(L, µbi,∆e) are simply denoted by (Lbi, Cd) and (Lbi, Ce), which are calledthe 3-Lie

bialgebras of type (Lb, Cd), and (Lb, Ce), respectively.

For a given 3-Lie algebraL, in order to find all the 3-Lie bialgebra structures on

L, we should find all the 3-Lie coalgebra structures onLwhich are compatible with the 3-Lie algebra L. Although a permutation of a basis of Lgives isomorphic 3-Lie coalgebra, but it may lead to the non-equivalent 3-Lie bialgebra.

Theorem 3.2 There do not exist 3-Lie bialgebras of the types (Lb1, Ce) and (Lb2, Ce).

ProofBy Lemma 2.1 and 2.2, we need to verify that 3-Lie algebrasLb1 andLb2 are incompatible with the following six isomorphic 3-Lie coalgebras of the type Ce: (1).∆(e1) =e2∧e3∧e4,∆(e2) =e1∧e3∧e4,∆(e3) =e1∧e2∧e4,∆(e4) =e1∧e2∧e3;

(2).∆(e1) =e2∧e3∧e4,∆(e2) =e1∧e3∧e4,∆(e3) =e2∧e1∧e4,∆(e4) =e2∧e1∧e3;

(6).∆(e1) =e2∧e4∧e3,∆(e2) =e4∧e3∧e1,∆(e3) =e2∧e4∧e1,∆(e4) =e2∧e3∧e1.

Here we only check the case (1) is incompatible with the 3-Lie algebraLb1. Since ∆µb1(e2, e3, e4) = ∆(e1) =e2∧e3∧e4,but

ad(3)µb

1(e2, e3)∆(e4) +ad

(3)

µb

1(e3, e4)∆(e2) +ad

(3)

µb

1(e4, e2)∆(e3) = 0, we obtain that

∆µb1e2, e3, e4)6=ad

(3)

µb

1(e2, e3)∆(e4) +ad

(3)

µb

1(e3, e4)∆(e2) +ad

(3)

µb

(4)

Theorem 3.3There does not exist 3-Lie bialgebras of the type (Lb2, Cd). ProofBy Lemma 2.1 and 2.2, we need to verify that 3-Lie algebra Lb2 is incom-patible with following twenty-four isomorphic 3-Lie coalgebras of the type Cd:

(1).∆(e₁) =e₂_∧e₃_∧e₄,∆(e₂) =e₁_∧e₃_∧e₄,∆(e₃) =e₁_∧e₂_∧e₄; (2).∆(e1) =e2∧e3∧e4,∆(e2) =e3∧e1∧e4,∆(e3) =e2∧e1∧e4;

(3).∆(e1) =e2∧e3∧e4,∆(e2) =e1∧e3∧e4,∆(e3) =e2∧e1∧e4;

(4).∆(e1) =e3∧e2∧e4,∆(e2) =e3∧e1∧e4,∆(e3) =e1∧e2∧e4;

(5).∆(e1) =e3∧e2∧e4,∆(e2) =e3∧e1∧e4,∆(e3) =e2∧e1∧e4;

(6).∆(e1) =e3∧e2∧e4,∆(e2) =e1∧e3∧e4,∆(e3) =e1∧e2∧e4;

(7).∆(e₁) =e₂_∧e₄_∧e₃,∆(e₂) =e₁_∧e₄_∧e₃,∆(e₄) =e₂_∧e₁_∧e₃; (8).∆(e1) =e2∧e4∧e3,∆(e2) =e4∧e1∧e3,∆(e4) =e2∧e1∧e3;

(9).∆(e₁) =e₂_∧e₄_∧e₃,∆(e₂) =e₁_∧e₄_∧e₃,∆(e₄) =e₁_∧e₂_∧e₃; (10).∆(e1) =e4∧e2∧e3,∆(e2) =e4∧e1∧e3,∆(e4) =e2∧e1∧e3;

(11).∆(e1) =e4∧e2∧e3,∆(e2) =e1∧e4∧e3,∆(e4) =e1∧e2∧e3;

(12).∆(e1) =e4∧e2∧e3,∆(e2) =e4∧e1∧e3,∆(e4) =e1∧e2∧e3;

(13).∆(e1) =e3∧e4∧e2,∆(e3) =e4∧e1∧e2,∆(e4) =e3∧e1∧e2;

(14).∆(e1) =e3∧e4∧e2,∆(e3) =e1∧e4∧e2,∆(e4) =e1∧e3∧e2;

(15).∆(e1) =e3∧e4∧e2,∆(e3) =e1∧e4∧e2,∆(e4) =e3∧e1∧e2;

(16).∆(e1) =e4∧e3∧e2,∆(e3) =e4∧e1∧e2,∆(e4) =e1∧e3∧e2;

(17).∆(e₁) =e₄_∧e₃_∧e₂,∆(e₃) =e₄_∧e₁_∧e₂,∆(e₄) =e₃_∧e₁_∧e₂; (18).∆(e1) =e4∧e3∧e2,∆(e3) =e1∧e4∧e2,∆(e4) =e1∧e3∧e2;

(19).∆(e2) =e4∧e3∧e1,∆(e3) =e4∧e2∧e1,∆(e4) =e3∧e2∧e1;

(20).∆(e2) =e4∧e3∧e1,∆(e3) =e2∧e4∧e1,∆(e4) =e3∧e2∧e1;

(21).∆(e2) =e4∧e3∧e1,∆(e3) =e4∧e2∧e1,∆(e4) =e2∧e3∧e1;

(22).∆(e2) =e3∧e4∧e1,∆(e3) =e2∧e4∧e1,∆(e4) =e3∧e2∧e1;

(23).∆(e2) =e3∧e4∧e1,∆(e3) =e2∧e4∧e1,∆(e4) =e2∧e3∧e1;

(24).∆(e2) =e3∧e4∧e1,∆(e3) =e4∧e2∧e1,∆(e4) =e3∧e2∧e1.

The discussion is completely similar to Theorem 3.2. We omit the computing pro-cess.

Theorem 3.4The only non-equivalent 3-Lie bialgebras of the type (Lb1, Cd) is (Lb1, Cd,∆1) : ∆1(e2) =e1∧e4∧e3,∆1(e3) =e1∧e4∧e2,∆1(e4) =e1∧e3∧e2. Proof We need to discuss the compatibility of twenty-four isomorphic 3-Lie coalgebras of the type Ccd in Theorem 3.3 with the 3-Lie algebra Lb₁.

By a direct computation, only cases (19), (20), (21), (22), (23) and (24) are compatible with the 3-Lie algebra Lb1, respectively. Thanks to isomorphisms of 3-Lie bialgebras:

(19)_→(22) :f(e1) =−e1, f(e2) =e3, f(e3) =e2, f(e4) =e4;

(21)→(23),(23)→(24) :f(e1) =e1, f(e2) =e3, f(e3) =e4, f(e4) =e2;

(19)_→(20) :f(e1) =e1, f(e2) =−e2, f(e3) =√−1e3, f(e4) =√−1e4;

(19)→(21) :f(e1) =e1, f(e2) =−e2, f(e3) =−√−1e3, f(e4) =−√−1e4;

we get that the only non-equivalent 3-Lie bialgebras of the type (Lb1, Cd) is (Lb1, Cd,∆1).

Acknowledgements

(5)

References

[1] V. Filippov,n-Lie algebras,Sib. Mat. Zh., 26 (1985) 126-140.

[2] Y. Liu L. ChenY. Ma, Representations and module-extensions of 3-hom-Lie algebras.J. Geom. Phys.,98 (2015): 376-383.

[3] B. SunL. Chen, RotaCBaxter multiplicative 3-ary Hom-Nambu Lie algebras.J. Geom. Phys.,98 (2015): 400-413.

[4] 6.Y. Ma, L. Chen, On the cohomology and extensions of first-classn-Lie super-algebras.Commun. Algebra,42(10) (2014): 4578-4599.

[5] R. Bai, Y. Cheng, J. Li, W. Meng, 3-Lie bialgebras,Acta Math. Scientia, 2014, 34B(2):513-522.

[6] R. Bai, Y. Zhang, Classes of 3-Lie bialgebras (Lb, Cb), Mathematica Aeterna, 2016, 6(1): 25-29.