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Deformations and Extensions of 3-Lie Algebras

Tao Zhang

Abstract

We give a characterization of the cohomology and deformation theory of 3-Liealgebras. The notion of Nijenhuis operators for 3-Lie algebras is introduced to de-scribe trivial deformations. The theory of extending structures and unified productfor 3-Lie algebras are developed. It is proved that the extending structures can beclassified by using some non-abelian cohomology and deformation map theory.

1 Introduction

The concept of n-Lie algebra dates back to Nambu’s work [22] to generalize the classicalHamiltonian mechanics. He derived the following equation

h1, h2, x, y, z = h1, h2, x, y, z+ x, h1, h2, y, z+ x, y, h1, h2, z,

where h1, h2 are Hamiltonian and ·, ·, · is a ternary product.In 1985, Filippov [12] introduced the concept of n-Lie algebra. An n-Lie algebra (also

called Filippov algebra, Nambu algebra, Lie n-algebra, and so on) is a vector space g withan n-ary totally skew-symmetric linear map (n-bracket) from

∧ng to g: (x1, · · · , xn) 7→

[x1, · · · , xn] satisfying the n-Jacobi identity

[x1, · · · , xn−1, [y1, y2, · · · , yn]] =n∑

i=1

[y1, · · · , [x1, · · · , xn−1, yi], · · · yn] (1)

for all xi, yi ∈ g. When the n-ary linear map is not skew-symmetric, it is called n-Leibnizalgebra or Leibniz n-algebra.

The study of n-Lie algebra is closely related to many fields in mathematics and math-ematical physics. For example, L. Takhtajan in [25] developed the foundations of thetheory of Nambu-Poisson manifolds. More recently, 3-Lie algebras is applied to the studyof gauge symmetry and supersymmetry of multiple coincident M2-branes.

On the other hand, the algebraic theory of n-Lie algebras have been studied by manyauthors, see [8, 9, 11, 14, 16, 26]. Let’s now review briefly the story of cohomology theoryfor n-Lie algebra studied in [26, 9, 8, 7, 23].

0Keyword: 3-Lie algebras, cohomology, deformations, Nijenhuis operators, extensions, unified product0Mathematics Subject Classification (2010): 17A30, 17B99, 17B56.

1

In [26], L. Takhtajan give the cohomology theory for n-Lie algebra with trivial rep-resentation, and then he tried to give deformation cohomology for n-Lie algebra withadjoint representation, but he didn’t succeed. Next, it was noted by P. Gautheron in [9]that there is a natural Leibniz algebra structures on the the vector space L := ⊗n−1g. Atlast, it was Yu. Daletskii and L. Takhtajan himself [8] who revisited the Nambu algebrastructures and found the relationship between the deformation cohomology of Nambu al-gebra g and deformation cohomology of Leibniz algebra L. The main result of their workis that they introduced a homotopy operator between C•(g, g) (the cochain complex ofNambu algebra g) and Γ•(L,L) (the cochain complex of Leibniz algebra L)

C•(g, g) : Hom(g⊗2n−1, g)δ

//

∆

Hom(g⊗2n+1, g)

∆

Γ•(L,L) : Hom(g⊗2n, g⊗2)d

// Hom(g⊗2n+2, g⊗2)

and found ∆ δ = d ∆. Then they derived the deformation cohomology for Nambualgebra g from the deformation cohomology for Leibniz algebra L since the second one isknown in [19]. For anther way to interpret the deformation cohomology of n-Lie algebrasby using Nijenhuis-Richardson bracket, see [23].

Some years later, the authors in [7] claimed that ”Let g be a Leibniz n-algebra and Vis a representation of g. Then Hom(g, V ) is a representation of Leibniz algebra L”. Thenthey suggest that one can define the cochain complex C•(g, V ) to be C•(L,Hom(g, V )),and the cohomology of a Leibniz n-algebra g with coefficient in a representation V to bethe cohomology of the Leibniz algebra L with coefficients in the representation Hom(g, V ).

In the first part of this paper, we spell out the cohomology theory more precisely for3-Lie algebras. We use the same idea as in [7]. The difference is that in this case we findthat V is a representation of 3-Lie algebra g if and only if Hom(g, V ) is a representationof Leibniz algebra L. Furthermore, we study infinitesimal deformations of 3-Lie algebras.We also introduce the notion of Nijenhuis operators for 3-Lie algebras, which is analogyto the case of ordinary Lie algebras in [10, 15] and of associative algebras in [6]. It isproved that this kind of operator gives trivial deformation. We also prove that this kindof cohomology theory can be used to classify abelian extensions of 3-Lie algebras.

The second part of this paper is devoted to study extending structures for 3-Lie al-gebras. We follow closely to the theory of unified product which was well developed byA. L. Agore and G. Militaru in [1, 2, 3]. Let g be a 3-Lie algebra and E a vector spacecontaining g as a subspace. We will describe and classify all 3-Lie algebras structures on Esuch that g is a subalgebra of E. We show that associated to any extending structures ofg by a complement space V , there is a unified product on the direct sum space E ∼= g⊕V .At the last section, we give two special cases of the unified product, namely the crossedproduct and bicrossed product of 3-Lie algebras which are related to the study of thenon-abelian extension problem and classifying complements problem respectively. It isproved that they can be classified by using some non-abelian cohomology and deformationmap theory.

Throughout this paper, all vector spaces are assumed to be over an algebraically closedfield of characteristic not equal to 2 and 3. Let V and W be two vector spaces. The space

2

of linear maps from V to W is denoted by Hom(V,W ).

2 3-Lie algebras and cohomology

A 3-Lie algebra consists of a vector space g together with a bracket [·, ·, ·] :∧3

g → g suchthat the fundamental identity

[x1, x2, [y1, y2, y3]] = [[x1, x2, y1], y2, y3] + [y1, [x1, x2, y2], y3] + [y1, y2, [x1, x2, y3]] (2)

holds for all xi, yi ∈ g.Since the bracket [·, ·, ·] is skew-symmetric, the above fundamental identity (2) can be

rewritten as

[x1, x2, [y1, y2, y3]] = [[x1, x2, y1], y2, y3] + [[x1, x2, y2], y3, y1] + [[x1, x2, y3], y1, y2] (3)

, [[x1, x2, y1], y2, y3] + c.p. (4)

where c.p. means cyclic permutations with respect to elements y1, y2 and y3.Denote by x = (x1, x2) and ad(x)yi = [x1, x2, yi], then the above equality can be

rewritten in the form

ad(x)[y1, y2, y3] = [ad(x)y1, y2, y3] + [y1, ad(x)y2, y3] + [y1, y2, ad(x)y3]. (5)

Denoted by L :=∧2

g, which is called fundamental set. The elements x = (x1, x2) ∈∧2g are called fundamental object. Define an operation on fundamental object by

x y = ([x1, x2, y1], y2) + (y1, [x1, x2, y2]). (6)

In [8], the authors proved that L is a Leibniz algebra satisfying the following Leibniz rule

x (y z) = (x y) z + y (x z),

andad(x)ad(y)w − ad(y)ad(x)w = ad(x y)w,

for all x, y, z ∈ L, w ∈ g, i.e. ad : L → End(g) is a homomorphism of Leibniz algebras.Recall that for a Leibniz algebra L, a representation is a vector space V together with

two bilinear maps[·, ·]L : L × V → V and [·, ·]R : V × L → V

satisfying the following three axioms

• (LLM) [x y,m]L = [x, [y,m]L]L − [y, [x,m]L]L,

• (LML) [m, x y]R = [[m, x]R, y]R + [x, [m, y]R]L,

• (MLL) [m, x y]R = [x, [m, y]R]L − [[x,m]L, y]R.

By (LML) and (MLL) we also have

3

• (MMM) [[m, x]R, y]R + [[x,m]L, y]R = 0.

In fact, assume (LLM), one of (LML),(MLL),(MMM) can be derived from the other two.Given a 3-Lie algebra g and a vector spaces V , we define the maps

[·, ·]L : L ⊗ Hom(g, V ) → Hom(g, V ) and [·, ·]R : Hom(g, V )⊗ L → Hom(g, V )

by

[(x1, x2), φ]L(x3) = ρ(x1, x2)φ(x3)− φ([x1, x2, x3]), (7)

[φ, (x1, x2)]R(x3) = φ([x1, x2, x3])− ρ(x1, x2)φ(x3)− ρ(x2, x3)φ(x1)

−ρ(x3, x1)φ(x2), (8)

for all φ ∈ Hom(g, V ), xi ∈ g, where ρ is a map from L =∧

2g to End(V ).

Proposition 2.1. Let g be a 3-Lie algebra. Then Hom(g, V ) equipped with the abovetwo maps [·, ·]L and [·, ·]R is a representation of the Leibniz algebra L if and only if thefollowing two conditions are satisfied for any xi, yi ∈ g,

• (R1) [ρ(x1, x2), ρ(y1, y2)] = ρ((x1, x2) (y1, y2)),

• (R2) ρ(x1, [y1, y2, y3]) = ρ(y1, y2)ρ(x1, y3) + ρ(y2, y3)ρ(x1, y1) + ρ(y3, y1)ρ(x1, y2).

Proof. For x = (x1, x2), y = (y1, y2) ∈ L, y3 ∈ g, first we compute the equality

[x y, φ]L(y3) = [x, [y, φ]L]L(y3)− [y, [x, φ]L]L(y3).

By definition, the left hand side is equal to

[x y, φ]L(y3) = ρ(x y)φ(y3)− φ(ad(x y)y3),

and the right hand side is equal to

[x, [y, φ]L]L(y3)− [y, [x, φ]L]L(y3)

= ρ(x)[y, φ]L(y3)− [y, φ]L(ad(x)y3)− ρ(y)[x, φ]L(y3) + [x, φ]L(ad(y)y3)

= ρ(x)ρ(y)φ(y3)− ρ(x)φ(ad(y)y3)− ρ(y)φ(ad(x)y3) + φ(ad(y)ad(x)y3)

−ρ(y)ρ(x)φ(y3) + ρ(y)φ(ad(x)y3) + ρ(x)φ(ad(y)y3)− φ(ad(x)ad(y)y3)

= ρ(x)ρ(y)φ(y3) + φ(ad(y)ad(x)y3)− ρ(y)ρ(x)φ(y3)− φ(ad(x)ad(y)y3)

= [ρ(x), ρ(y)]φ(y3)− φ([ad(x), ad(y)]y3).

Since ad : L → End(g) is a homomorphism of Leibniz algebras, thus (LLM) is valid for[·, ·]L if and only if (R1) is valid for ρ.

Next we compute the equality

[[φ, x]R, y]R(y3) + [[x, φ]L, y]R(y3) = 0.

By (7) and (8) we have

[(x1, x2), φ]L(w) + [φ, (x1, x2)]R(w) = ρ(x1, w)φ(x2)− ρ(x2, w)φ(x1),

4

thus[(x1, x2), φ]L + [φ, (x1, x2)]R = ρ(x1, ·)φ(x2)− ρ(x2, ·)φ(x1),

where we denote ρ(x1, ·)φ(x2) : g → V by ρ(x1, ·)φ(x2)(w) = ρ(x1, w)φ(x2).Now replace xi by yi and φ by −(ρ(x1, ·)φ(x2) + ρ(x2, ·)φ(x1)) in (8), then we have

[[φ, x]R + [x, φ]L, y]R(y3)

= (ρ(x1, ·)φ(x2)− ρ(x2, ·)φ(x1))([y1, y2, y3])

−ρ(y1, y2)(ρ(x1, ·)φ(x2) + ρ(x2, ·)φ(x1))(y3)

−ρ(y3, y1)(ρ(x1, ·)φ(x2) + ρ(x2, ·)φ(x1))(y2)

−ρ(y2, y3)(ρ(x1, ·)φ(x2) + ρ(x2, ·)φ(x1))(y1)

= ρ(x1, [y1, y2, y3])φ(x2)− ρ(x2, [y1, y2, y3])φ(x1)

−ρ(y1, y2)ρ(x1, y3)φ(x2) + ρ(y1, y2)ρ(x2, y3)φ(x1))

−ρ(y3, y1)ρ(x1, y2)φ(x2) + ρ(y1, y3)ρ(x2, y2)φ(x1))

−ρ(y2, y3)ρ(x1, y1)φ(x2) + ρ(y2, y3)ρ(x2, y1)φ(x1)).

Thus (MMM) is valid for [·, ·]L and [·, ·]R if and only if (R2) is valid for ρ.At last, we compute the equality

[φ, x y]R(y3) = [x, [φ, y]R]L(y3)− [[x, φ]L, y]R(y3).

By definition, the left hand side is equal to

[φ, x y]R(y3) = [φ, ([x1, x2, y1], y2)]R(y3) + [φ, (y1, [x1, x2, y2])]R(y3)

= φ([[x1, x2, y1], y2, y3])− ρ([x1, x2, y1], y2)φ(y3)

−ρ(y3, [x1, x2, y1])φ(y2)− ρ(y2, y3)φ([x1, x2, y1])

+φ([y1, [x1, x2, y2], y3])− ρ(y1, [x1, x2, y2])φ(y3)

−ρ(y3, y1)φ([x1, x2, y2])− ρ([x1, x2, y2], y3)φ(y1),

and the right hand side is

[x, [φ, y]R]L(y3) = ρ(x1, x2)[φ, (y1, y2)]R(y3)− [φ, (y1, y2)]R([x1, x2, y3])

= ρ(x1, x2)φ([y1, y2, y3])− ρ(y1, y2)φ(y3)

−ρ(y3, y1)φ(y2)− ρ(y2, y3)φ(y1)

−φ([y1, y2, [x1, x2, y3]])− ρ(y1, y2)φ([x1, x2, y3])

−ρ([x1, x2, y3], y1)φ(y2)− ρ(y2, [x1, x2, y3])φ(y1),

[[x, φ]L, y]R(y3) = [x, φ]L([y1, y2, y3])− ρ(y1, y2)[x, φ]L(y3)

−ρ(y3, y1)[x, φ]L(y2)− ρ(y2, y3)[x, φ]L(y1)

= ρ(x1, x2)φ([y1, y2, y3])− φ([x1, x2, [y1, y2, y3]])

−ρ(y1, y2)ρ(x1, x2)φ(y3)− φ([x1, x2, y3])

−ρ(y3, y1)ρ(x1, x2)φ(y2)− φ([x1, x2, y2])

5

−ρ(y2, y3)ρ(x1, x2)φ(y1)− φ([x1, x2, y1]).

Thus (MLL) is valid for [·, ·]L if and only if (R1) and (R2) hold.

Definition 2.2. Let g be a 3-Lie algebra and V be a vector space. Then (V, ρ) is calleda (left) representation of g if and only if the conditions (R1) and (R2) in the aboveProposition 2.1 are satisfied.

If we denote by (x1, x2) ⊲ u.= ρ(x1, x2)(u), then the conditions (R1) and (R2) can be

rewritten as

• (R1’) [(x1, x2), (y1, y2)] ⊲ u =((x1, x2) (y1, y2)

)⊲ u,

• (R2’) (x1, [y1, y2, y3])⊲u = (y1, y2)⊲((x1, y3)⊲u

)+(y2, y3)⊲

((x1, y1)⊲u

)+(y3, y1)⊲(

(x1, y2) ⊲ u).

For example, given a 3-Lie algebra g, there is a natural adjoint representation on itself.The corresponding representation ad(x1, x2) is given by

ad(x1, x2)(x3) = [x1, x2, x3].

Now we define the cochain complex for a 3-Lie algebra g with coefficients in V by

Cn(g, V ) := Hom(∧

2n+1g, V)⊆ Hom

((∧2ng

)⊗ g, V

)∼= Hom (Ln,Hom(g, V ))

anddn−1 : C

n−1(g, V ) → Cn(g, V )

where

dn−1ω(x1, x2, · · · , xn, w)

= dn−1ω(x1, x2, · · · , xn)(w)

=n−1∑

i=1

(−1)k+1[xk, ω(x1, · · · , xi, · · · , xn)]L(w) + (−1)n[ω(x1, · · · , xn−1), xn]R(w)

+∑

1≤k<j≤n

(−1)kω(x1, · · · , xj−1, xi xj , xj+1, · · · , xn)(w),

for all xi ∈ L =∧

2g, w ∈ g. In other words, we define the cohomology of a 3-Lie algebrag with coefficients in V to be the cohomology of Leibniz algebra L with coefficients inHom(g, V ). For more details of cohomology of Leibniz algebras, see [19].

Theorem 2.3. Let g be a 3-Lie algebra and (V, ρ) be a representation of g. Then thereexists a cochain complex

C(g, V ) =

⊕n≥0C

n(g, V ), d, where the coboundary operator

is given by

dn−1ω(x1, x2, · · · , x2n+1)

6

= (−1)n+1ρ(x2n+1, x2n−1)ω(x1, x2, · · · , x2n−2, x2n)

+(−1)n+1ρ(x2n, x2n+1)ω(x1, x2, · · · , x2n−1)

+n∑

k=1

(−1)k+1ρ(x2k−1, x2k)ω(x1, x2, · · · , x2k−1, x2k, · · · , x2n+1)

+n∑

k=1

2n+1∑

j=2k+1

(−1)kω(x1, x2, · · · , x2k−1, x2k, · · · , [x2k−1, x2k, xj ], · · · , x2n+1) (9)

such that d d = 0.

Proof. Put xk = (x2k−1, x2k), w = x2n+1 and [·, ·]L, [·, ·]R as in (7) and (8), then we get acoboundary operator dn−1 : C

n−1(g, V ) → Cn(g, V ) as the following:

dn−1ω(x1, x2, · · · , x2n+1) := dn−1ω(x1, x2, · · · , xn)(w)

=n−1∑

k=1

(−1)k+1[x2k−1, x2k, ω(x1, x2, · · · , x2k−1, x2k, · · · , x2n)]L(x2n+1)

+(−1)n[ω(x1, x2, · · · , x2n−2), x2n−1, x2n]R(x2n+1)

+n∑

k=1

2n∑

j=2k+1

(−1)kω(x1, · · · , x2k−1, x2k, · · · , [x2k−1, x2k, xj], · · · , x2n)(x2n+1)

=

n−1∑

k=1

(−1)k+1ρ(x2k−1, x2k)ω(x1, x2, · · · , x2k−1, x2k, · · · , x2n+1)

−ω(x1, x2, · · · , x2k−1, x2k, · · · , [x2k−1, x2k, x2n+1])

+(−1)nω(x1, x2, · · · , [x2n−1, x2n, x2n+1])− ρ(x2n−1, x2n)ω(x1, x2, · · · , x2n+1)

−ρ(x2n+1, x2n−1)ω(x1, x2, · · · , x2n−2, x2n)− ρ(x2n, x2n+1)ω(x1, x2, · · · , x2n−1)

+

n∑

k=1

2n∑

j=2k+1

(−1)kω(x1, x2, · · · , x2k−1, x2k, · · · , [x2k−1, x2k, xj ], · · · , x2n+1)

= (−1)n+1ρ(x2n+1, x2n−1)ω(x1, x2, · · · , x2n−2, x2n)

+(−1)n+1ρ(x2n, x2n+1)ω(x1, x2, · · · , x2n−1)

+

n∑

k=1

(−1)k+1ρ(x2k−1, x2k)ω(x1, x2, · · · , x2k−1, x2k, · · · , x2n+1)

+

n∑

k=1

2n+1∑

j=2k+1

(−1)kω(x1, x2, · · · , x2k−1, x2k, · · · , [x2k−1, x2k, xj ], · · · , x2n+1).

Definition 2.4. The quotient space H•(g, V ) = Z•(g, V )/B•(g, V ), where Z•(g, V ) =ω ∈ Cn(g, V )|dω = 0 is the space of cocycles and B•(g, V ) = ω = dν|ν ∈ Cn−1(g, V )is the space of coboundaries, is called the cohomology group of a 3-Lie algebra g withcoefficients in V .

7

According to the above definition, a 0-cochain is a map ν ∈ Hom(g, V ), a 1-cochain isa map ω ∈ Hom (

∧3g, V ), and the coboundary operator is give by

d0ν(x1, w) = d0ν(x

1)(w) = −[ν, x1]R(w), (10)

d1ω(x1, x2, w) = [x1, ω(x2)]L(w) + [ω(x1), x2]R(w)− ω(x1 x2)(w). (11)

Put x1 = (x1, x2) ∈ L, w = x3 ∈ g in the equality (10), then by (8) we have

d0ν(x1, x2, x3)

= ρ(x1, x2)ν(x3) + ρ(x1, x3)ν(x2) + ρ(x2, x3)ν(x1)− ν([x1, x2, x3]). (12)

Definition 2.5. Let g be a 3-Lie algebra and (V, ρ) be a representation of g. Then a mapν ∈ Hom(g, V ) is called 0-cocycle if and only if ∀ x1, x2, x3 ∈ g,

ρ(x1, x2)ν(x3) + ρ(x1, x3)ν(x2) + ρ(x2, x3)ν(x1)− ν([x1, x2, x3]) = 0, (13)

and a map ω ∈ Hom (∧

3g, V ) is called a 1-coboudary if there exists a map ν ∈ Hom(g, V )such that ω = d0ν.

Put x1 = (x1, x2) ∈ L, x2 = (y1, y2) ∈ L, w = y3 ∈ g in the equality (11), then we have

d1ω(x1, x2, y1, y2, y3) = [x1, x2, ω(y1, y2)]L(y3) + [ω(x1, x2), y1, y2]R(y3)

−ω((x1, x2) (y1, y2))(y3),

where

[x1, x2, ω(y1, y2)]L(y3) = ρ(x1, x2)ω(y1, y2)(y3)− ω(y1, y2)([x1, x2, y3])

= ρ(x1, x2)ω(y1, y2, y3)− ω(y1, y2, [x1, x2, y3]),

[ω(x1, x2), y1, y2]R(y3) = ω(x1, x2)([y1, y2, y3])− ρ(y1, y2)ω(x1, x2)(y3)

−ρ(y1, y3)ω(x1, x2)(y2)− ρ(y2, y3)ω(x1, x2)(y1)

= ω(x1, x2, [y1, y2, y3])− ρ(y1, y2)ω(x1, x2, y3)

−ρ(y1, y3)ω(x1, x2, y2)− ρ(y2, y3)ω(x1, x2, y1),

ω((x1, x2) (y1, y2))(y3) = ω(([x1, x2, y1], y2) + (y1, [x1, x2, y2]))(y3)

= ω(([x1, x2, y1], y2, y3) + ω(y1, [x1, x2, y2], y3).

Definition 2.6. Let g be a 3-Lie algebra and (V, ρ) be a representation of g. Then a mapω ∈ Hom (

∧3g, V ) is called 1-cocycle if and only if ∀ x1, x2, y1, y2, y3 ∈ g,

ω(x1, x2, [y1, y2, y3]) + ρ(x1, x2)ω(y1, y2, y3)

= ω([x1, x2, y1], y2, y3) + ω([x1, x2, y2], y3, y1) + ω(y1, y2, [x1, x2, y3])

+ρ(y2, y3)ω(x1, x2, y1) + ρ(y3, y1)ω(x1, x2, y2) + ρ(y1, y2)ω(x1, x2, y3). (14)

8

3 Infinitesimal Deformations

Let g be a 3-Lie algebra and ω :∧

3g → g be a linear map. Consider a λ-parametrizedfamily of linear maps:

[x1, x2, x3]λ , [x1, x2, x3] + λω(x1, x2, x3).

If [·, ·, ·]λ endow g with a 3-Lie algebra structure which is denoted by gλ, then we saythat ω generates a λ-parameter infinitesimal deformation of the 3-Lie algebra g.

Theorem 3.1. ω generates a λ-parameter infinitesimal deformation of the 3-Lie algebrag is equivalent to (i) ω itself defines a 3-Lie algebra structure on g and (ii) ω is a 1-cocycleof g with the coefficients in the adjoint representation.

Proof. For the equality

[x1, x2, [y1, y2, y3]λ]λ

= [[x1, x2, y1]λ, y2, y3]λ + [y1, [x1, x2, y2]λ, y3]λ + [y1, y2, [x1, x2, y3]λ]λ,

the left hand side is equal to

[x1, x2, [y1, y2, y3] + λω(y1, y2, y3)]λ

= [x1, x2, [y1, y2, y3]] + λω(x1, x2, [y1, y2, y3])

+[x1, x2, λω(y1, y2, y3)] + λω(x1, x2, λω(y1, y2, y3))

= [x1, x2, [y1, y2, y3]] + λω(x1, x2, [y1, y2, y3]) + [x1, x2, ω(y1, y2, y3)]

+λ2ω(x1, x2, ω(y1, y2, y3)),

and the right hand side is equal to

[[x1, x2, y1] + λω(x1, x2, y1), y2, y3]λ + [y1, [x1, x2, y2] + λω(x1, x2, y2), y3]λ

+[y1, y2, [x1, x2, y3] + λω(x1, x2, y3)]λ

= [[x1, x2, y1], y2, y3] + [y1, [x1, x2, y2], y3] + [y1, y2, [x1, x2, y3]]

+λω([x1, x2, y1], y2, y3) + [ω(x1, x2, y1), y2, y3]

+ω(y1, [x1, x2, y2], y3) + [y1, ω(x1, x2, y2), y3]

+ω(y1, y2, [x1, x2, y3]) + [y1, y2, ω(x1, x2, y3)]

+λ2ω(ω(x1, x2, y1), y2, y3) + ω(y1, ω(x1, x2, y2), y3) + ω(y1, y2, ω(x1, x2, y3)).

Thus we have

ω(x1, x2, [y1, y2, y3]) + [x1, x2, ω(y1, y2, y3)]

= ω([x1, x2, y1], y2, y3) + ω(y1, [x1, x2, y2], y3) + ω(y1, y2, [x1, x2, y3])

+[ω(x1, x2, y1), y2, y3] + [y1, ω(x1, x2, y2), y3] + [y1, y2, ω(x1, x2, y3)], (15)

ω(x1, x2, ω(y1, y2, y3))

= ω(ω(x1, x2, y1), y2, y3) + ω(y1, ω(x1, x2, y2), y3) + ω(y1, y2, ω(x1, x2, y3)). (16)

Therefore ω defines a 3-Lie algebra structure on g and ω is a 1-cocycle of g with thecoefficients in the adjoint representation.

9

4 Nijenhuis operators

In this section, we introduce the notion of Nijenhuis operators for 3-Lie algebras, whichis analogy to the case of ordinary Lie algebras in [10, 15] and of associative algebras in[6]. This kind of operator gives trivial deformation. We study propositions of Nijenhuisoperators in details.

A deformation is said to be trivial if there exists a linear map N : g → g such that forTλ = id + λN : gλ → g there holds

Tλ([x1, x2, x3]λ) = [Tλx1, Tλx2, Tλx3]. (17)

By definition we have

Tλ([x1, x2, x3]λ) = [x1, x2, x3] + λω(x1, x2, x3) + λN([x1, x2, x3] + λω(x1, x2, x3))

= [x1, x2, x3] + λ(ω(x1, x2, x3) +N [x1, x2, x3]) + λ2Nω(x1, x2, x3),

and

[Tλx1, Tλx2, Tλx3] = [x1 + λNx1, x2 + λNx2, x3 + λNx3]

= [x1, x2, x3] + λ([Nx1, x2, x3] + [x1, Nx2, x3] + [x1, x2, Nx3])

+λ2([Nx1, Nx2, x3] + [Nx1, x2, Nx3] + [x1, Nx2, Nx3])

+λ3[Nx1, Nx2, Nx3].

Thus we have

ω(x1, x2, x3) = [Nx1, x2, x3] + [x1, Nx2, x3] + [x1, x2, Nx3]−N [x1, x2, x3], (18)

Nω(x1, x2, x3) = [Nx1, Nx2, x3] + [Nx1, x2, Nx3] + [x1, Nx2, Nx3], (19)

0 = [Nx1, Nx2, Nx3]. (20)

From the cohomology theory discussed in section 2, (18) can be represented in termsof 1-coboundary as ω = d0N . Moreover, it follows from (18) and (19) that N must satisfythe following condition

N2[x1, x2, x3] = N [Nx1, x2, x3] +N [x1, Nx2, x3] +N [x1, x2, Nx3]

−([Nx1, Nx2, x3] + [Nx1, x2, Nx3] + [x1, Nx2, Nx3]). (21)

In the following, we denote by ω(x1, x2, x3) = [x1, x2, x3]N , then (21) is equivalent to

N [x1, x2, x3]N = [Nx1, Nx2, x3] + [Nx1, x2, Nx3] + [x1, Nx2, Nx3]. (22)

Definition 4.1. A linear operator N : g → g is called a Nijenhuis operator if and only if(21) and (20) hold. 1

1Recently, a more general Nijenhuis operator for n-Lie algebras is introduced in [18]. For the case of3-Lie algebras, it is defined as follows:

[N(x1), N(x2), N(x3)] = N([Nx1, Nx2, x3] + [Nx1, x2, Nx3] + [x1, Nx2, Nx3])

−N2([Nx1, x2, x3] + [x1, Nx2, x3] + [x1, x2, Nx3]−N [x1, x2, x3]).

10

We have seen that any trivial deformation produces a Nijenhuis operator. Conversely,any Nijenhuis operator gives a trivial deformation as the following theorem shows.

Theorem 4.2. Let N be a Nijenhuis operator for g. Then a deformation of g can beobtained by putting

ω(x1, x2, x3) = [Nx1, x2, x3] + [x1, Nx2, x3] + [x1, x2, Nx3]−N [x1, x2, x3].

Furthermore, this deformation is a trivial one.

Proof. We have known that ω = d0N and d1ω = d1d0N = 0, therefore ω is a 1-cocycleof g with the coefficients in the adjoint representation. Now we check the Jacobi identity(2) hold for ω. Denote by

J(x1, x2, y1, y2, y3) = [x1, x2, [y1, y2, y3]]− [[x1, x2, y1], y2, y3]

−[y1, [x1, x2, y2], y3]− [y1, y2, [x1, x2, y3]],

Jω(x1, x2, y1, y2, y3) = ω(x1, x2, ω(y1, y2, y3))− ω(ω(x1, x2, y1), y2, y3)

−ω(y1, ω(x1, x2, y2), y3)− ω(y1, y2, ω(x1, x2, y3)).

A direct computation shows that

Jω(x1, x2, y1, y2, y3)

= J(Nx1, Nx2, y1, y2, y3) +N2J(x1, x2, y1, y2, y3)

+[x1, x2, [Ny1, Ny2, y3] + [Ny1, y2, Ny3] + [y1, Ny2, Ny3]−Nω(y1, y2, y3)]

−([y1, y2, [Nx1, Nx2, y3] + [Nx1, x2, Ny3] + [x1, Nx2, Ny3]−Nω(x1, x2, y3)])

−([y2, y3, [Nx1, Nx2, y1] + [Nx1, x2, Ny1] + [x1, Nx2, Ny1]−Nω(x1, x2, y1)])

−([y3, y1, [Nx1, Nx2, y2] + [Nx1, x2, Ny2] + [x1, Nx2, Ny2]−Nω(x1, x2, y2)]).

Therefore we have Jω = 0 by the 3-Lie rule of g and Nijenhuis operator condition.Note that in the proof of above Theorem we have not used condition (19). But this

condition is important to us since only in this case the k’s power of a Nijenhuis operatoris also a Nijenhuis operator.

Lemma 4.3. Let N be a Nijenhuis operator. Then for any k > 0, we have

[x1, x2, x3]Nk+1 = ([x1, x2, x3]Nk)N . (23)

Proof. First we check that (23) is valid for k = 1.

([x1, x2, x3]N)N = [Nx1, x2, x3]N + [x1, Nx2, x3]N + [x1, x2, Nx3]N −N [x1, x2, x3]N

= [N2x1, x2, x3] + [Nx1, Nx2, x3] + [Nx1, x2, Nx3]−N [Nx1, x2, x3]

+[Nx1, Nx2, x3] + [x1, N2x2, x3] + [x1, Nx2, Nx3]−N [x1, Nx2, x3]

+[Nx1, x2, Nx3] + [x1, Nx2, Nx3] + [x1, x2, N2x3]−N [x1, x2, Nx3]

−N [Nx1, x2, x3]−N [x1, Nx2, x3]−N [x1, x2, Nx3] +N2[x1, x2, x3]

= [N2x1, x2, x3] + [x1, N2x2, x3] + [x1, x2, N

2x3]−N2[x1, x2, x3]

11

= [x1, x2, x3]N2,

where we have used (21) in the third equality.Second, assume that

[x1, x2, x3]Nk = ([x1, x2, x3]Nk−1)N . (24)

We compute

([x1, x2, x3]Nk)N

= [Nkx1, x2, x3]N + [x1, Nkx2, x3]N + [x1, x2, N

kx3]N −Nk[x1, x2, x3]N

= [Nk+1x1, x2, x3] + [Nkx1, Nx2, x3] + [Nkx1, x2, Nx3]−N [Nkx1, x2, x3]

+[Nx1, Nkx2, x3] + [x1, N

k+1x2, x3] + [x1, Nkx2, Nx3]−N [x1, N

kx2, x3]

+[Nx1, x2, Nkx3] + [x1, Nx2, N

kx3] + [x1, x2, Nk+1x3]−N [x1, x2, N

kx3]

−Nk[x1, x2, x3]N .

By (20) and (22) we have

[Nkx1, Nx2, x3] + [Nkx1, x2, Nx3]

= [Nkx1, Nx2, x3] + [Nkx1, x2, Nx3] + [Nk−1x1, Nx2, Nx3]

= [NNk−1x1, Nx2, x3] + [NNk−1x1, x2, Nx3] + [Nk−1x1, Nx2, Nx3]

= N [Nk−1x1, x2, x3]N

Similarily we have

[Nx1, Nkx2, x3] + [x,Nkx2, Nx3] = N [x1, N

k−1x2, x3]N ,

and[Nx1, x2, N

kx3] + [x,Nx2, Nkx3] = N [x1, x2, N

k−1x3]N .

These three items together with the last item in ([x1, x2, x3]Nk)N is equal to

N [Nk−1x1, x2, x3]N +N [x1, Nk−1x2, x3]N +N [x1, x2, N

k−1x3]N −Nk[x1, x2, x3]N

= N(([x1, x2, x3]Nk−1)N)

= N([x1, x2, x3]Nk) by assumption (24).

Thus ([x1, x2, x3]Nk)N is equal to

= [Nk+1x1, x2, x3] + [x1, Nk+1x2, x3] + [x1, x2, N

k+1x3]

−N [Nkx1, x2, x3]−N [x1, Nkx2, x3]−N [x, x2, N

kx3]

+N([x1, x2, x3]Nk)

= [Nk+1x1, x2, x3] + [x1, Nk+1x2, x3] + [x1, x2, N

k+1x3]−Nk+1[x1, x2, x3]

= [x1, x2, x3]Nk+1.

and by introduction the Lemma holds.By the above Lemma, we have

([x1, x2, x3]Nk)Nr = (([x1, x2, x3]Nk)N )Nr−1 = ([x1, x2, x3]Nk+1)Nr−1

= ([x1, x2, x3]Nk+2)Nr−2 = · · · = [x1, x2, x3]Nk+r .

12

Lemma 4.4. Let N be a Nijenhuis operator, then for any k, r > 0, we have

[x1, x2, x3]Nk+r = ([x1, x2, x3]Nk)Nr . (25)

Proposition 4.5. Let N be a Nijenhuis operator. Then for any k > 0, Nk is also aNijenhuis operator.

Proof. We prove by introduction. The Proposition is valid for k = 1. Assume

Nk[x1, x2, x3]Nk = [Nkx1, Nkx2, x3] + [Nkx1, x2, N

kx3] + [x1, Nkx2, N

kx3],

then we have

Nk+1[x1, x2, x3]Nk+1

= NkN(([x1, x2, x3]Nk)N)

= Nk([Nx1, Nx2, x3]Nk + [Nx1, x2, Nx3]Nk + [x1, Nx2, Nx3]Nk)

= [Nk+1x1, Nk+1x2, x3] + [Nk+1x1, Nx2, N

kx3] + [Nx1, Nk+1x2, N

kx3]

+[Nk+1x1, Nkx2, Nx3] + [Nk+1x1, x2, N

k+1x3] + [Nx1, Nkx2, N

k+1x3]

+[Nkx1, Nk+1x2, Nx3] + [Nkx1, Nx2, N

k+1x3] + [x1, Nk+1x2, N

k+1x3]

= [Nk+1x1, Nk+1x2, x3] + [Nk+1x1, x2, N

k+1x3] + [x1,k+1 x2, N

k+1x3]

+[Nk+1x1, Nx2, Nkx3] + [Nx1, N

k+1x2, Nkx3] + [Nk+1x1, N

kx2, Nx3]

+[Nx1, Nkx2, N

k+1x3] + [Nkx1, Nk+1x2, Nx3] + [Nkx1, Nx2, N

k+1x3].

The items in the last two line are zero by (20), thus the Proposition is valid for k+1.Two Nijenhuis operator N1 and N2 are said to be compatible if N1 + N2 is also a

Nijenhuis operator.

Proposition 4.6. Let N1 and N2 be two Nijenhuis operators. Then they are compatibleif and only if

N1[x1, x2, x3]N2+N2[x1, x2, x3]N1

= [N2x1, N1x2, x3] + [N2x1, x2, N1x3] + [x1, N2x2, N1x3]

+[N1x1, N2x2, x3] + [N1x1, x2, N2x3] + [x1, N1x2, N2x3], (26)

and

[N1x1, N1x2, N2x3] + [N1x1, N2x2, N2x3]

+[N2x1, N1x2, N1x3] + [N2x1, N2x2, N1x3] = 0, (27)

Proof. By definition, N1 +N2 is a Nijenhuis operator if and only if

(N1 +N2)([x1, x2, x3]N1+N2)

= [(N1 +N2)x1, (N1 +N2)x2, x3] + [(N1 +N2)x1, x2, (N1 +N2)x3]

+[x1, (N1 +N2)x2, (N1 +N2)x3],

and

[(N1 +N2)x1, (N1 +N2)x2, (N1 +N2)x3] = 0.

Now it is easy to see the above condition is equivalent to (26) and (27).

13

Lemma 4.7. Let N be a Nijenhuis operator. For any j, k > 0, we have

N j [x1, x2, x3]Nk +Nk[x1, x2, x3]Nj

= [Nkx1, Njx2, x3] + [Nkx1, x2, N

jx3] + [x1, Nkx2, N

jx3]

+[N jx1, Nkx2, x3] + [N jx1, x2, N

kx3] + [x1, Njx2, N

kx3]. (28)

Proof. If j > k, then by Proposition 4.5 and Lemma 4.4 we have

N j [x1, x2, x3]Nk +Nk[x1, x2, x3]Nj

= N j−k(Nk[x1, x2, x3]Nk) +Nk(([x1, x2, x3]Nj−k)Nk)

= N j−k([Nkx1, Nkx2, x3] + [Nkx1, x2, N

kx3] + [x1, Nkx2, N

kx3])

+[Nkx1, Nkx2, x3]Nj−k + [Nkx1, x2, N

kx3]Nj−k + [x1, Nkx2, N

kx3]Nj−k

= N j−k([Nkx1, Nkx2, x3] + [Nkx1, x2, N

kx3] + [x1, Nkx2, N

kx3])

+[N jx1, Nkx2, x3] + [Nkx1, N

jx2, x3] + [Nkx1, Nkx2, N

j−kx3]−N j−k[Nkx1, Nkx2, x3]

+[N jx1, x2, Nkx3] + [Nkx1, N

j−kx2, Nkx3] + [Nkx1, x2, N

jx3]−N j−k[Nkx1, x2, Nkx3]

+[N j−kx1, Nkx2, N

kx3] + [x1, Njx2, N

kx3] + [x1, Nkx2, N

jx3]−N j−k[x1, Nkx2, N

kx3]

= [Nkx1, Njx2, x3] + [Nkx1, x2, N

jx3] + [x1, Nkx2, N

jx3]

+[N jx1, Nkx2, x3] + [N jx1, x2, N

kx3] + [x1, Njx2, N

kx3].

The case of j < k can be proved similarly and the case of j = k is by Proposition 4.5.Let N1 = N j and N2 = Nk, then condition (27) in Proposition 4.6 is satisfied by (20)

and condition (26) is satisfied by Lemma 4.7, thus we get

Theorem 4.8. Let N be a Nijenhuis operator. Then N j and Nk are compatible for anyj, k > 0.

It is easy to see that if N is a Nijenhuis operator, then cN is also a Nijenhuis operator,where c is any constant. Now by Proposition 4.6 and Lemma 4.7 we have

Theorem 4.9. Let N be a Nijenhuis operator. Then all linear combinations of Nk arecompatible.

Corollary 4.10. Let N be a Nijenhuis operator. Then for any polynomial P (X) =∑n

i=1 ciXi, the operator P (N) is also a Nijenhuis operator.

5 Abelian extensions of 3-Lie algebras

In this section, we study abelian extensions of 3-Lie algebras. We show that associatedto any abelian extension, there is a representation and a 1-cocycle. Furthermore, abelianextensions can be classified by the first cohomology group.

Definition 5.1. Let (g, [·, ·, ·]g), (V, [·, ·, ·]V ), (g, [·, ·, ·]g) be 3-Lie algebras and i : V →g, p : g → g be homomorphisms. The following sequence of 3-Lie algebras is a short exactsequence if Im(i) = Ker(p), Ker(i) = 0 and Im(p) = g.

0 // Vi

// gp

// g // 0 (29)

14

In this case, we call g an extension of g by V , and denote it by Eg. It is called an abelianextension if V is abelian ideal of g, i.e. [u, v, ·]g = [u, ·, v]g = [·, u, v]g = 0, ∀u, v ∈ V .

A section σ : g → g of p : g → g consists of linear maps σ : g → g such that pσ = idg.

Definition 5.2. Two extensions of 3-Lie algebras Eg : 0 → Vi→ g

p→ g → 0 and

Eg : 0 → Vj→ g

q→ g → 0 are equivalent, if there exists a 3-Lie algebra homomorphism

F : g → g such that the following diagram commutes

0 // V

idV

i// g

F

p// g

idg

// 0

0 // Vj

// gq

// g // 0

(30)

The set of equivalent classes of extensions of g by V is denoted by Ext(g, V ).

Let g be an abelian extension of g by V , and σ : g → g be a section. Denote by

σ(x) = σ(x1, x2) = (σx1, σx2),

and define ρ : ∧2g → End(V ) by

ρ(x)(u) = ρ(x1, x2)(u) , [σ(x1), σ(x2), u]g = ad(σ(x))u, (31)

for all x = (x1, x2) ∈∧2

g, u ∈ V .

Lemma 5.3. With the above notations, ρ is a representation of g on V and does notdepend on the choice of the section σ. Moreover, equivalent abelian extensions give thesame representation of g on V .

Proof. First, we show that ρ is independent of the choice of σ. In fact, if we chooseanother section σ′ : g → g, then

p(σ(xi)− σ′(xi)) = xi − xi = 0 =⇒ σ(xi)− σ′(xi) ∈ V =⇒ σ′(xi) = σ′(xi) + u

for some u ∈ V .Since we have [·, u, v]g = 0 for all u, v ∈ V , which implies that

[σ′(x1), σ′(x2), w]g = [σ(x1) + u, σ(x2) + v, w]g

= [σ(x1), σ(x2) + v, w]g + [u, σ(x2) + v, w]g

= [σ(x1), σ(x2), w]g + [σ(x1), v, w]g

= [σ(x1), σ(x2), w]g,

thus ρ is independent on the choice of σ.Second, we show that ρ is a representation of g on V .By the equality

[σx1, σx2, [σy1, σy2, u]g]g

15

= [[σx1, σx2, σy1]g, σy2, u]g + [σy1, [σx1, σx2, σy2]g, u]g + [σy1, σy2, [σx1, σx2, u]g]g,

we have

ρ(x1, x2)ρ(y1, y2)u = ρ((x1, x2) (y1, y2))u+ ρ(y1, y2)ρ(x1, x2)u,

where we use the fact

σ([x1, x2, y1]g)− [σx1, σx2, σy1]g ∈ V ∼= Ker(p),

and that V is abelian ideal of g,

[σ([x1, x2, y1]g)− [σx1, σx2, σy1]g, σy2, u]g = 0,

thus we get the condition (R1).Similarily, by the equality

[u, σx1, [σy1, σy2, σy3]g]g

= [[u, σx1, σy1]g, σy2, σy3]g + [σy1, [u, σx1, σy2]g, σy3]g + [σy1, σy2, [u, σx1, σy3]g]g,

we have

ρ(x1, [y1, y2, y3])u = ρ(y2, y3)ρ(x1, y1)u+ ρ(y3, y1)ρ(x1, y2)u+ ρ(y1, y2)ρ(x1, y3)u.

thus we get the condition (R2). Therefore we see that ρ is a representation of g on V .At last, suppose that Eg and Eg are equivalent abelian extensions, and F : g → g is the

3-Lie algebra homomorphism satisfying F i = j, q F = p. Choose linear sections σ andσ′ of p and q, we get qFσ(xi) = pσ(xi) = xi = qσ′(xi), then Fσ(xi)−σ′(xi) ∈ Ker(q) ∼= V .Thus, we have

[σ(x1), σ(x2), u]g = [Fσ(x1), Fσ(x2), u]g = [σ′(x1), σ′(x2), u]g.

Therefore, equivalent abelian extensions give the same ρ. The proof is finished.

Let σ : g → g be a section of the abelian extension. Define the following map:

ω(x1, x2, x3) , [σ(x1), σ(x2), σ(x3)]g − σ([x1, x2, x3]g), (32)

for all x1, x2, x3 ∈ g.

Lemma 5.4. Let 0 → V→g→g → 0 be an abelian extension of g by V . Then ω definedby (32) is a 1-cocycle of g with coefficients in V , where the representation ρ is given by(31).

Proof. By the equality

[σx1, σx2, [σy1, σy2, σy3]g]g

= [[σx1, σx2, σy1]g, σy2, σy3]g + [σy1, [σx1, σx2, σy2]g, σy3]g

+[σy1, σy2, [σx1, σx2, σy3]g]g,

16

we get that the left hand side is equal to

[σx1, σx2, [σy1, σy2, σy3]g]g

= [σx1, σx2, ω(y1, y2, y3) + σ([y1, y2, y3]g)]g

= ρ(x1, x2)ω(y1, y2, y3) + [σx1, σx2, σ([y1, y2, y3]g)]g

= ρ(x1, x2)ω(y1, y2, y3) + ω(x1, x2, [y1, y2, y3]) + σ([x1, x2, [y1, y2, y3]]).

Similarily, the right hand side is equal to

ρ(y2, y3)ω(x1, x2, y1) + ω([[x1, x2, y1], y2, y3]) + σ([[x1, x2, y1], y2, y3])

+ρ(y3, y1)ω(x1, x2, y2) + ω([y1, [x1, x2, y2], y3]) + σ([y1, [x1, x2, y2], y3])

+ρ(y1, y2)ω(x1, x2, y3) + ω([y1, y2, [x1, x2, y3]]) + σ([y1, y2, [x1, x2, y3]]).

Thus we have

ω(x1, x2, [y1, y2, y3]) + ρ(x1, x2)ω(y1, y2, y3)

= ω([x1, x2, y1], y2, y3) + ω([x1, x2, y2], y3, y1) + ω(y1, y2, [x1, x2, y3])

+ρ(y2, y3)ω(x1, x2, y1) + ρ(y3, y1)ω(x1, x2, y2) + ρ(y1, y2)ω(x1, x2, y3).

This is exactly the 1-cocycle condition in Definition 2.6.Now we can transfer the 3-Lie algebra structure on g to the 3-Lie algebra structure

on g ⊕ V using the 1-cocycle given above. The proof of Lemma 5.5 below can be foundin [5, 17] so we omit the details.

Lemma 5.5. Let g be a 3-Lie algebra, (V, ρ) be an g-module and ω :∧3

g → V is a1-cocycle. Then g⊕ V is a 3-Lie algebra under the following bracket:

[x1 + u1, x2 + u2, x3 + u3]ω

= [x1, x2, x3] + ω(x1, x2, x3) + ρ(x1, x2)(u3) + ρ(x2, x3)(u1) + ρ(x3, x1)(u2),

where x1, x2, x3 ∈ g and u1, u2, u3 ∈ V . This 3-Lie algebra is denoted by g⊕ω V .

Lemma 5.6. Two abelian extensions of 3-Lie algebras 0 → V→g ⊕ω V→g → 0 and0 → V→g⊕ω′ V→g → 0 are equivalent if and only if ω and ω′ are in the same cohomologyclass.

Proof. Let F : g⊕ω V → g⊕ω′ V be the corresponding homomorphism, then

F [x1, x2, x3]ω = [F (x1), F (x2), F (x3)]ω′ . (33)

Since F is an equivalence of extensions, there exist ν : g → V such that

F (xi + u) = xi + ν(xi) + u, i = 1, 2, 3.

The left hand side of (33) is equal to

F1([x1, x2, x3] + ω(x1, x2, x3))

17

= [x1, x2, x3] + ω(x1, x2, x3) + ν([x1, x2, x3]),

and the right hand side of (33) is equal to

[x1 + ν(x1), x2 + ν(x2), x3 + ν(x3)]ω′

= [x1, x2, x3] + ω′(x1, x2, x3)

+ρ(x1, x2)ν(x3) + ρ(x1, x3)ν(x2) + ρ(x2, x3)ν(x1).

Thus we have

(ω − ω′)(x1, x2, x3) = ρ(x1, x2)ν(x3) + ρ(x1, x3)ν(x2) + ρ(x2, x3)ν(x1)

−ν([x1, x2, x3]), (34)

that is ω − ω′ = dν. Therefore ω and ω′ are in the same cohomology class.

Theorem 5.7. Let g be a 3-Lie algebra and (V, ρ) a representation of g. Then there is aone-to-one correspondence between equivalence classes Ext(g, V ) of abelian extensions ofthe 3-Lie algebra g by V and the first cohomology group H1(g, V ).

6 Extending structures for 3-Lie algebras

In this section, we study extending structures for 3-Lie algebras. Let g be a 3-Lie algebraand E a vector space containing g as a subspace. We are going to describe and classify all3-Lie algebras structures on E such that g is a subalgebra of E. We show that associatedto any extending structures of g by a complement space V , there is a unified product onthe direct sum space E ∼= g⊕ V .

Definition 6.1. Let (g, [·, ·, ·]), (E, [·, ·, ·]) be 3-Lie algebras and i : g → E be an inclusionmap. The following short exact sequence (as vector spaces)

0 // gi

// Ep

// V // 0 (35)

is called an extending structures of g by V if E containing g as a subalgebra.

Definition 6.2. Let g be a 3-Lie algebra and V a vector space. An extending datum ofg by V is a system Ω(g, V ) consisting four maps

⊲ : V × V × g → g, ⊳ : V × g× g → V,

: V × g× g → g, : V × V × g → V,

and two totally skew-symmetric maps

ω : V × V × V → g, ·, ·, · : V × V × V → V.

Let Ω(g, V ) = (⊲, ⊳,,, ω, ·, ·, ·) be an extending datum. Denote by gV the directsum vector space under the following bracket:

[x1 + u1, x2 + u2, x3 + u3]

18

= [x1, x2, x3] + (u1, u2) ⊲ x3 + (u2, u3) ⊲ x1 + (u3, u1) ⊲ x2

+u3 (x1, x2) + u1 (x2, x3) + u2 (x3, x1) + ω(u1, u2, u3)

+u1, u2, u3+ u1 ⊳ (x2, x3) + u2 ⊳ (x3, x1) + u3 ⊳ (x1, x2)

+(u2, u3) x1 + (u3, u1) x2 + (u1, u2) x3, (36)

where xi ∈ g and ui ∈ V . Then gV is called the unified product of g and V if it is 3-Liealgebra with the above bracket.

We will prove that there is a one-to-one correspondence between an extending struc-tures of g by V and the unified product gV .

Theorem 6.3. Let g be a 3-Lie algebra, V a vector space and Ω(g, V ) an extending datum.Then gV is a unified product if and only if the following compatibility conditions hold:

Case I:(1)

v3 ⊳ [(x1, x2), (y1, y2)] = (v3 ⊳ (x1, x2)) ⊳ (y1, y2)− (v3 ⊳ (y1, y2)) ⊳ (x1, x2), (37)

v3 [(x1, x2), (y1, y2)]

= [x1, x2, v3 (y1, y2)] + (v3 ⊳ (y1, y2)) (x1, x2)

−[y1, y2, v3 (x1, x2)]− (v3 ⊳ (x1, x2)) (y1, y2), (38)

(2)

u1 ⊳ (x2, [y1, y2, y3]) = (u1 ⊳ (x2, y1)) ⊳ (y2, y3) + (u1 ⊳ (x2, y2)) ⊳ (y3, y1)

+(u1 ⊳ (x2, y3)) ⊳ (y1, y2), (39)

u1 (x2, [y1, y2, y3])

= [u1 (x2, y1), y2, y3] + (u1 ⊳ (x2, y1)) (y2, y3)

+[u1 (x2, y2), y3, y1] + (u1 ⊳ (x2, y2)) (y3, y1)

+[u1 (x2, y3), y1, y2] + (u1 ⊳ (x2, y3)) (y1, y2), (40)

Cases II:(1)

[x1, x2, (v1, v2) ⊲ y3] + ((v1, v2) y3) (x1, x2)

= v2 (v1 (x1, x2), y3

)+(v1 ⊳ (x1, x2), v2

)⊲ y3

+v1 (v2 (x1, x2), y3

)+(v1, v2 ⊳ (x1, x2)

)⊲ y3

+(v1, v2) ⊲ [x1, x2, y3], (41)

((v1, v2) y3) ⊳ (x1, x2)

= v2 ⊳(v1 (x1, x2), y3

)+(v1 ⊳ (x1, x2), v2

) y3

19

+v1 ⊳(v2 (x1, x2), y3

)+(v1, v2 ⊳ (x1, x2)

) y3

+(v1, v2) [x1, x2, y3], (42)

(2)

u1 (x2, v1 (y2, y3)

)+(v1, u1 ⊳ (x2, y3)

)⊲ y2

+[(u1, v1) ⊲ x2, y2, y3] + ((u1, v1) x2) (y2, y3)

= v1 (u1 (x2, y2), y3

)+(v1, u1 ⊳ (x2, y2)

)⊲ y3

+v1 (y2, u1 (x2, y3)

)+(u1, v1 ⊳ (y2, y3)

)⊲ x2, (43)

u1 ⊳(x2, v1 (y2, y3)

)+ ((u1, v1) x2) ⊳ (y2, y3)

+(v1, u1 ⊳ (x2, y3)

) y2

=(u1, v1 ⊳ (y2, y3)

) x2 + v1 ⊳

(u1 (x2, y2), y3

)

+(v1, u1 ⊳ (x2, y2)

) y3 + v1 ⊳

(y2, u1 (x2, y3)

), (44)

(3)

(u1, u2) ⊲ [y1, y2, y3]

= [(u1, u2) ⊲ y1, y2, y3] + ((u1, u2) y1) (y2, y3)

+[y1, (u1, u2) ⊲ y2, y3] + ((u1, u2) y2) (y3, y1)

+[y1, y2, (u1, u2) ⊲ y3] + ((u1, u2) y3) (y1, y2), (45)

(u1, u2) [y1, y2, y3]

= ((u1, u2) y1) ⊳ (y2, y3) + ((u1, u2) y2) ⊳ (y3, y1)

+((u1, u2) y3) ⊳ (y1, y2), (46)

Cases III:(1)

v1, v2, v3 (x1, x2) + [x1, x2, ω(v1, v2, v3)]

= (v2, v3) ⊲ (v1 (x1, x2)) + ω(v1 ⊳ (x1, x2), v2, v3)

+(v3, v1) ⊲ (v2 (x1, x2)) + ω(v2 ⊳ (x1, x2), v3, v1)

+(v1, v2) ⊲ (v3 (x1, x2)) + ω(v3 ⊳ (x1, x2), v1, v2), (47)

v1, v2, v3 ⊳ (x1, x2)

= (v2, v3) (v1 (x1, x2)) + v1 ⊳ (x1, x2), v2, v3

+(v3, v1) (v2 (x1, x2)) + v2 ⊳ (x1, x2), v3, v1

+(v1, v2) (v3 (x1, x2)) + v3 ⊳ (x1, x2), v1, v2, (48)

(2)

u2 ((v1, v2) ⊲ y3, x1

)+(u2, (v1, v2) y3

)⊲ x1

20

= v2 (y3, (u2, v1) ⊲ x1

)+((u2, v1) x1, v2

)⊲ y3

+v1 ((u2, v2) ⊲ x1, y3

)+(v1, (u2, v2) x1) ⊲ y3

+(v1, v2) ⊲(u2 (y3, x1)

)+ ω(v1, v2, u3 ⊳ (y3, x1)), (49)

+u2 ⊳((v1, v2) ⊲ y3, x1

)+(u2, (v1, v2) y3

) x1

= v2 ⊳(y3, (u2, v1) ⊲ x1

)+((u2, v1) x1, v2

) y3

+v1 ⊳((u2, v2) ⊲ x1, y3

)+(v1, (u2, v2) x1) y3

+(v1, v2)(u2 (y3, x1)

)+ v1, v2, u3 ⊳ (y3, x1), (50)

(3)

(u1, u2) ⊲(v1 (y2, y3)

)+ ω(u1, u2, v1 ⊳ (y2, y3))

= [ω(u1, u2, v1), y2, y3] + u1, u2, v1 (y2, y3)

+v1 ((u1, u2) ⊲ y2, y3

)+(v1, (u1, u2) y2

)⊲ y3

+v1 (y2, (u1, u2) ⊲ y3

)+((u1, u2) y3, v1

)⊲ y2, (51)

(u1, u2)(v1 (y2, y3)

)+ u1, u2, v1 ⊳ (y2, y3)

= u1, u2, v1 ⊳ (y2, y3)

+v1 ⊳((u1, u2) ⊲ y2, y3

)+(v1, (u1, u2) y2

) y3

+v1 ⊳(y2, (u1, u2) ⊲ y3

)+((u1, u2) y3, v1

) y2, (52)

Cases IV:(1)

(u1, u2), (v1, v2) ⊲ y3

= (u1, u2) ⊲ ((v1, v2) ⊲ y3)− (v1, v2) ⊲ ((u1, u2) ⊲ y3)

+ω(u1, u2, (v1, v2) y3)− ω(v1, v2, (u1, u2) y3)

+v2 (ω(u1, u2, v1), y3)− v1 (ω(u1, u2, v2), y3), (53)

(u1, u2), (v1, v2) y3

= (u1, u2) ((v1, v2) ⊲ y3) + [u1, u2, (v1, v2) y3]

−(v1, v2) ((u1, u2) ⊲ y3)− [v1, v2, (u1, u2) y3]

+v2 ⊳ (ω(u1, u2, v1), y3)− v1 ⊳ (ω(u1, u2, v2), y3), (54)

(2)

(u2, [v1, v2, v3]) ⊲ x1 + u2 (ω(v1, v2, v3), x1)

= (v2, v3) ⊲ ((u2, v1) ⊲ x1) + ω((u2, v1) x1, v2, v3)

+(v3, v1) ⊲ ((u2, v2) ⊲ x1) + ω((u2, v2) x1, v3, v1)

+(v1, v2) ⊲ ((u2, v3) ⊲ x1) + ω((u2, v3) x1, v1, v2), (55)

21

(u2, [v1, v2, v3]) x1 + u2 ⊳ (ω(v1, v2, v3), x1)

= (v2, v3) ⊲ ((u2, v1) x1) + (u2, v1) x1, v2, v3

+(v3, v1) ⊲ ((u2, v2) x1) + (u2, v2) x1, v3, v1

+(v1, v2) ⊲ ((u2, v3) x1) + (u2, v3) x1, v1, v2, (56)

Cases V:

u1, u2, v1, v2, v3 − u1, u2, v1, v2, v3

−v1, u1, u2, v2, v3 − v1, v2, u1, u2, v3

+(u1, u2) ω(v1, v2, v3)− (v2, v3) ω(u1, u2, v1)

−(v3, v1) ω(u1, u2, v2)− (v1, v2) ω(u1, u2, v3) = 0, (57)

ω(u1, u2, v1, v2, v3)− ω(u1, u2, v1v2, v3)

−ω(v2, u1, u2, v2, v3)− ω(v1, v2, u1, u2, v3)

+(u1, u2) ⊲ ω(v1, v2, v3)− (v2, v3) ⊲ ω(u1, u2, v1)

−(v3, v1) ⊲ ω(u1, u2, v2)− (v1, v2) ⊲ ω(u1, u2, v3) = 0. (58)

In the above conditions, (37) and (39) means that (V, ⊳) is a representation of g.

Proof. Assume gV is a unified product, then we have

[x1 + u1, x2 + u2, [y1 + v1, y2 + v2, y3 + v3]]

= [[x1 + u1, x2 + u2, y1 + v1], y2 + v2, y3 + v3]

+[y1 + v1, [x1 + u1, x2 + u2, y2 + v2], y3 + v3]

+[y1 + v1, y2 + v2, [x1 + u1, x2 + u2, y3 + v3]].

By direct computations of the fundamental identity for different combinations of elements,assuming some of xi and ui are equal to zero, we can check the above conditions musthold in the above five cases. For example, by letting u1, u2, y1, , y2, v3 to be zero, we havethe equality

[x1, x2, [v1, v2, y3]] = [[x1, x2, v1], v2, y3] + [v1, [x1, x2, v2], y3] + [v1, v2, [x1, x2, y3]].

By definition of unified product (36), we get

[x1, x2, [v1, v2, y3]] = [x1, x2, (v1, v2) ⊲ y3] + ((v1, v2) y3) (x1, x2)

+((v1, v2) y3) ⊳ (x1, x2),

[[x1, x2, v1], v2, y3] = [v1 (x1, x2), v2, y3] + [v1 ⊳ (x1, x2), v2, y3]

= v2 (v1 (x1, x2), y3) + v2 ⊳ (v1 (x1, x2), y3)

+(v1 ⊳ (x1, x2), v2) ⊲ y3 + (v1 ⊳ (x1, x2), v2) y3,

[v1, [x1, x2, v2], y3] = [v1, v2 (x1, x2), y3] + [v1, v2 ⊳ (x1, x2), y3]

= v1 (v2 (x1, x2), y3) + v1 ⊳ (v2 (x1, x2), y3)

22

+(v1, v2 ⊳ (x1, x2)) ⊲ y3 + (v1, v2 ⊳ (x1, x2)) y3,

[v1, v2, [x1, x2, y3]] = (v1, v2) ⊲ [x1, x2, y3] + (v1, v2) ([x1, x2, y3]).

Thus we obtain

[x1, x2, (v1, v2) ⊲ y3] + ((v1, v2) y3) (x1, x2)

= v2 (v1 (x1, x2), y3) + (v1 ⊳ (x1, x2), v2) ⊲ y3

+v1 (v2 (x1, x2), y3) + (v1, v2 ⊳ (x1, x2)) ⊲ y3

+(v1, v2) ⊲ ([x1, x2, y3]), (59)

and

((v1, v2) y3) ⊳ (x1, x2)

= v2 ⊳ (v1 (x1, x2), y3) + (v1 ⊳ (x1, x2), v2) y3

+v1 ⊳ (v2 (x1, x2), y3) + (v1, v2 ⊳ (x1, x2)) y3

+(v1, v2) ([x1, x2, y3]), (60)

which are conditions (41) and (42) in Case II:(1) respectively. The other conditions canobtained similarly.

Conversely, if the above conditions hold, it is straightforward to see that gV is a 3-Liealgebra.

Given an extending structure Ω(g, V ), it is obvious that g can be seen a 3-Lie subalge-bra of gV . Conversely, we will prove that any 3-Lie algebra structure on a vector spaceE containing g as a subalgebra is isomorphic to a unified product.

Theorem 6.4. Let (g, [·, ·, ·] and (E, [·, ·, ·]) be two 3-Lie algebras such that E containing gas a subalgebra of E. Then, there exists an extending structure Ω(g, V ) of g by a subspaceV of E and an isomorphism of 3-Lie algebras E ∼= gV which stabilizes g and co-stabilizesV .

Proof. Note that there is a natural linear map p : E → A such that p(x) = x for all x ∈ g.Set V = Ker(p) which is a complement of g in E. Then, we define the extending datumΩ(g, V ) of g by a subspace V of E as follows:

⊲ : V × V × g → g, (u1, u2) ⊲ x3 = p([u1, u2, x3]),

: V × g× g → g, u1 (x2, x3) = p([u1, x2, x3]),

ω : V × V × V → g, ω(u1, u2, u3) = p([u1, u2, u3]),

⊳ : V × g× g → V, u1 ⊳ (x2, x3) = [u1, x2, x3]− p([u1, x2, x3]),

: V × V × g → V, (u1, u2) x3 = [u1, u2, x3]− p([u1, u2, x3]),

·, ·, · : V × V × V → V, u1, u2, u3 = [u1, u2, u3]− p([u1, u2, u3]),

for any xi ∈ g, ui ∈ V . It is easy to see that ϕ : g × V → E defined as ϕ(x, u) = x + uis a linear isomorphism, whose inverse is as follows: ϕ−1(e) := (p(e), e − p(e)) for all

23

e ∈ E. Next, we should prove that Ω(g, V ) is a extending structure of g by V andϕ : AV → E is an isomorphism of 3-Lie algebras that stabilizes g and co-stabilizes V . Infact, if ϕ : g×V → E is an isomorphism of 3-Lie algebras, there exists a unique 3-bracketgiven by

[(x1, u1), (x2, u2), (x3, u3)] = ϕ−1[ϕ(x1, u1), ϕ(x2, u2), ϕ(x3, u3)] (61)

We are going to proof that the 3-bracket defined by (61) is just the one given in the aboveextending system Ω(g, V ). Indeed, for any xi ∈ g, ui ∈ V , we have

[(x1, u1), (x2, u2), (x3, u3)]

= ϕ−1[ϕ(x1, u1), ϕ(x2, u2), ϕ(x3, u3)]

= ϕ−1[x1 + u1, x2 + u2, x3 + u3]

= ϕ−1([x1, x2, x3] + [x1, x2, u3] + [x1, u2, u3]

+[u1, x2, x3] + [u1, u2, x3] + [u1, u2, u3])

=([x1, x2, x3] + p([u1, u2, x3])

+p([u1, x2, x3]) + c.p.+ p([u1, u2, u3]) + c.p.,

[u1, u2, u3] + [u1, u2, x3] + c.p. + [u1, x2, x3] + c.p.

−p([u1, u2, u3])− p([u1, x2, x3])− c.p.− p([u1, u2, x3]− c.p.)

=([x1, x2, x3] + (u1, u2) ⊲ x3 + (u2, u3) ⊲ x1 + (u3, u1) ⊲ x2

+u3 (x1, x2) + u1 (x2, x3) + u2 (x3, x1) + ω(u1, u2, u3),

u1, u2, u3+ u1 ⊳ (x2, x3) + u2 ⊳ (x3, x1) + u3 ⊳ (x1, x2)

+(u2, u3) x1 + (u3, u1) x2 + (u1, u2) x3

).

This bracket coincides with the bracket in (36) and thus ϕ : gV → E is an isomorphismof 3-Lie algebras. Moreover, the following diagram is commutative

g

id

i// gV

ϕ

q// V

id

gi

// Eπ

// V

where q : gV → V and π : E → V are the natural projections. The proof is finished.

Definition 6.5. Let g be a 3-Lie algebra, E a vector space such that g is a subspace ofE and V a complement of g in E. For a linear map ϕ : E → E, the following diagram isconsidered:

g

id

i// E

ϕ

π// V

id

gi

// Eπ

// V

24

where π : E → V is the natural projection of E = g ⊕ V onto V and i : g → E is theinclusion map. We say that ϕ : E → E stabilizes g (resp. co-stabilizes V ) if the leftsquare (resp. the right square) of the above diagram is commutative.

Let [·, ·, ·] and [·, ·, ·]′ be two 3-Lie structures on E both containing g as a 3-Lie sub-algebra. If there exists a 3-Lie algebra isomorphism ϕ : (E, [·, ·, ·]) → (E, [·, ·, ·]′) whichstabilizes g, then (E, [·, ·, ·]) and (E, [·, ·, ·]′) are called equivalent, which is denoted by(E, [·, ·, ·]) ≡ (E, [·, ·, ·]′). If there exists a 3-Lie algebra isomorphism ϕ : (E, [·, ·, ·]) →(E, [·, ·, ·]′) which stabilizes g and co-stabilizes V , then (E, [·, ·, ·]) and (E, [·, ·, ·]′) arecalled cohomologous, which is denoted by (E, [·, ·, ·]) ≈ (E, [·, ·, ·]′).

Obviously, ≡ and ≈ are equivalence relations on the set of all 3-Lie algebra structureson E containing g as a 3-Lie subalgebra. Denote by Extd(E, g) (resp. Extd′(E, g)) the setof all equivalence classes via ≡ (resp. ≈). Thus, Extd(E, g) is the classifying object of theextending structures problem and Extd′(E, g) provides a classification of the extendingstructures problem from the point of the view of the extension problem. In addition, it iseasy to see that there exists a canonical projection Extd(E, g) ։ Extd′(E, g).

Next, by Theorem 6.4, for classifying all 3-Lie algebra structures on E containing g

as a subalgebra, we only need to classify all unified products gV associated to all 3-Liealgebra structures Ω(g, V ) for a given complement V of g in E.

Lemma 6.6. Let Ω(g, V ) = (⊲, ⊳,,, ω, ·, ·, ·) and Ω′(g, V ) = (⊲′, ⊳′,′,′, ω′, ·, ·, ·′)be two extending structures of g through V and gV, g′V the associated unified prod-ucts. Then there exists a bijection between the set of all homomorphisms of 3-Lie algebraψ : gV → g′V which stabilizes g and the set of pairs (r, ν), where r : V → g, ν : V → Vare two linear maps satisfying the following compatibility conditions for all xi ∈ g, ui ∈ V :

(M1)ν(u1 ⊳ (x2, x3)) = ν(u1) ⊳

′ (x2, x3),

(M2)

r(u1 ⊳ (x2, x3)) + u1 (x2, x3) = [x2, x3, r(u1)] + ν(u1)′ (x2, x3),

(M3)

ν((u1, u2) x3)

=(ν(u1), ν(u2)

)′ x3 − ν(u2) ⊳

′ (r(u1), x3) + ν(u1) ⊳′ (r(u2), x3),

(M4)

r((u1, u2) x3) + (u1, u2) ⊲ x3

= [r(u1), r(u2), x3] + (ν(u1), ν(u2)) ⊲′ x3

−ν(u2)′ (r(u1), x3) + ν(u1)

′ (r(u2), x3),

(M5)

ν(u1, u2, u3)

= ν(u1), ν(u2), ν(u3)′

25

+(ν(u1), ν(u2))′ r(u3) + c.p.

+ν(u1) ⊳′ (r(u2), r(u3)) + c.p.,

(M6)

r(u1, u2, u3) + ω(u1, u2, u3)

= [r(u1), r(u2), r(u3)] + ω′(ν(u1), ν(u2), ν(u3))

+(ν(u1), ν(u2)) ⊲′ r(u3) + c.p.

+ν(u1)′ (r(u2), r(u3)) + c.p.

where c.p. means cyclic permutations with respect to elements u1, u2 and u3.Under the above bijection the homomorphism of 3-Lie algebras ψ : gV → g′V corre-

sponding to (r, ν) is given by:

ψ(x, u) = (x+ r(u), ν(u)).

Moreover, ψ is an isomorphism if and only if ν : V → V is an isomorphism and ψco-stabilizes V if and only if ν = idV .

Proof. A linear map ψ : gV → g′V which stabilizes g is uniquely determined by twolinear maps r : V → g, ν : V → V such that ψ(x, u) = (x+ r(u), ν(u)), for all x ∈ g andu ∈ V . We rewrite the map ψ as an equivalent forms as

ψ(x+ u) = ψ(x) + ψ(u)

= x+ r(u) + ν(u).

Now we proof that ψ is a homomorphism if and only if (M1)–(M6) hold. By the equality

ψ([x1 + u1, x2 + u2, x3 + u3]) = [ψ(x1 + u1), ψ(x2 + u2), ψ(x3 + u3)]′,

we have the following cases.Case I:

ψ([u1, x2, x3]) = ψ(u1 (x2, x3)) + ψ(u1 ⊳ (x2, x3)),

= u1 (x2, x3) + r(u1 ⊳ (x2, x3)) + ν(u1 ⊳ (x2, x3)),

[ψ(u1), ψ(x2), ψ(x3)]′ = [r(u1), x2, x3] + [ν(u1), x2, x3]

′

= [r(u1), x2, x3] + ν(u1)′ (x2, x3) + (ν(u1)) ⊳

′ (x2, x3).

Thus we obtain (M1) and (M2).Case II:

ψ([u1, u2, x3]) = ψ((u1, u2) ⊲ x3) + ψ((u1, u2) x3),

= (u1, u2) ⊲ x3 + r((u1, u2) x3) + ν((u1, u2) x3),

[ψ(u1), ψ(u2), ψ(x3)]′ = [r(u1) + ν(u1), r(u2) + ν(u2), x3]

′

26

= [r(u1), r(u2), x3] + [ν(u1), ν(u2), x3]′

+[r(u1), ν(u2), x3]′ + [ν(u1), r(u2), x3]

′

= [r(u1), r(u2), x3]

+(ν(u1), ν(u2)) ⊲′ x3 + (ν(u1), ν(u2))

′ x3

−ν(u2)′ (r(u1), x3)− ν(u2) ⊳

′ (r(u1), x3)

+ν(u1)′ (r(u2), x3) + ν(u1) ⊳

′ (r(u2), x3).

Thus we obtain (M3) and (M4).Case III:

ψ([u1, u2, u3]) = ψ(u1, u2, u3) + ψ(ω(u1, u2, u3)),

= r(u1, u2, u3) + ν(u1, u2, u3) + ω(u1, u2, u3),

[ψ(u1), ψ(u2), ψ(u3)]′ = [r(u1) + ν(u1), r(u2) + ν(u2), r(u3) + ν(u3)]

′

= [r(u1), r(u2), r(u3)]

+ν(u2)′ (r(u3), r(u1)) + ν(u2) ⊳

′ (r(u3), r(u2))

+ν(u1)′ (r(u2), r(u3)) + ν(u1) ⊳

′ (r(u2), r(u3))

+ν(u3)′ (r(u1), r(u2)) + ν(u3) ⊳

′ (r(u1), r(u2))

+(ν(u3), ν(u1)) ⊲′ r(u2) + (ν(u3), ν(u2))

′ r(u2)

+(ν(u2), ν(u3)) ⊲′ r(u1) + (ν(u2), ν(u3))

′ r(u1)

+(ν(u1), ν(u2)) ⊲′ r(u3) + (ν(u1), ν(u2))

′ r(u3)

+ν(u1), ν(u2), ν(u3)′ + ω′(ν(u1), ν(u2), ν(u3)).

Thus we obtain (M5) and (M6).Assume that ν : V → V is bijective. Then ψ is an isomorphism of 3-Lie algebras

with the inverse given by ψ−1(x, u) =(x − r(ν−1(u)), ν−1(y)

), for all x ∈ g and u ∈ V .

Conversely, assume that ψ is bijective. It follows easily that ν is surjective. Now we provethat v is injective. Let u ∈ V such that ν(u) = 0. We have ψ(0, 0) = (0, 0) = (0, ν(u)) =ψ(−r(u), u), and hence we obtain u = 0, i.e. ν is a bijection. Finally, it is straightforwardto see that ψ co-stabilizes V if and only if ν = idV and the proof is now finished.

Definition 6.7. Two extending structures Ω(g, V ) = (⊲, ⊳,,, ω, ·, ·, ·) and Ω′(g, V ) =(⊲′, ⊳′,′,′, ω′, ·, ·, ·′) are called equivalent if there exists a pair (r, ν) of linear maps,where r : V → g and ν : V → V is an isomorphism satisfying the following conditions forall xi ∈ g, ui ∈ V :

u1 ⊳ (x2, x3) = ν−1(ν(u1) ⊳

′ (x2, x3)), (62)

u1 (x2, x3) = [x2, x3, r(u1)] + ν(u1)′ (x2, x3)− r ν−1

(ν(u1) ⊳

′ (x2, x3)), (63)

(u1, u2) x3

27

= ν−1((ν(u1), ν(u2))

′ x3 − ν(u2) ⊳′ (r(u1), x3) + ν(u1) ⊳

′ (r(u2), x3)), (64)

(u1, u2) ⊲ x3

= [r(u1), r(u2), x3] + (ν(u1), ν(u2)) ⊲′ x3

−ν(u2)′ (r(u1), x3) + ν(u1)

′ (r(u2), x3)

−r ν−1((ν(u1), ν(u2))

′ x3 − ν(u2) ⊳′ (r(u1), x3) + ν(u1) ⊳

′ (r(u2), x3)), (65)

u1, u2, u3

= ν−1(ν(u1), ν(u2), ν(u3)

′ + (ν(u1), ν(u2))′ r(u3) + c.p.

+ν(u1) ⊳′ (r(u2), r(u3)) + c.p.

), (66)

ω(u1, u2, u3)

= [r(u1), r(u2), r(u3)] + ω′(ν(u1), ν(u2), ν(u3))

+(ν(u1), ν(u2)) ⊲′ r(u3) + c.p.+ ν(u1)

′ (r(u2), r(u3)) + c.p.

−r ν−1(ν(u1), ν(u2), ν(u3)

′ + (ν(u1), ν(u2))′ r(u3) + c.p.

+ν(u1) ⊳′ (r(u2), r(u3)) + c.p.

), (67)

where c.p. means cyclic permutations with respect to elements u1, u2 and u3.

Definition 6.8. Two extending structures Ω(g, V ) = (⊲, ⊳,,, ω, ·, ·, ·) and Ω′(g, V ) =(⊲′, ⊳′,′,′, ω′, ·, ·, ·′) are called cohomologous if ⊳ = ⊳′,=′ and there exists a lin-ear maps, where r : V → g satisfying the following conditions for all xi ∈ g, ui ∈ V :

u1 (x2, x3) = u1 ′ (x2, x3) + [x2, x3, r(u1)]− r(u1 ⊳ (x2, x3)), (68)

u2 ⊳′ (r(u1), x3) = u1 ⊳

′ (r(u2), x3), (69)

(u1, u2) ⊲ x3 = (u1, u2) ⊲′ x3 + [r(u1), r(u2), x3]− r((u1, u2)

′ x3)

−u2 ′ (r(u1), x3) + u1

′ (r(u2), x3), (70)

u1, u2, u3 = u1, u2, u3′ + (u1, u2)

′ r(u3) + c.p.

+u1 ⊳′ (r(u2), r(u3)) + c.p., (71)

ω(u1, u2, u3) = ω′(u1, u2, u3) + [r(u1), r(u2), r(u3)]

+(u1, u2) ⊲′ r(u3) + c.p. + u1

′ (r(u2), r(u3)) + c.p.

−r(u1, u2, u3

′ + (u1, u2)′ r(u3) + c.p.

+u1 ⊳′ (r(u2), r(u3)) + c.p.

), (72)

where c.p. means cyclic permutations with respect to elements u1, u2 and u3.

28

We denote by T(g, V ) the set of extending structures Ω(g, V ) = (⊲, ⊳,,, ω, ·, ·, ·).

Theorem 6.9. Let g be a 3-Lie algebra, E a vector space that contains g as a subspaceand V a complement of g in E. Then, we get:(1)Denote EH2(V, g) := T(g, V )/ ≡. Then, the map

EH2(V, g) → Extd(E, g), Ω(g, V ) → gV (73)

is bijective, where Ω(g, V ) is the equivalence class of Ω(g, V ) under ≡.(2) Denote UH2(V, g) := T(g, V )/ ≈. Then, the map

UH2(V, g) → Extd′(E, g), Ω(g, V ) → gV (74)

is bijective, where Ω(g, V ) is the equivalence class of Ω(g, V ) under ≈.

7 Special cases of unified products

In this section, we show some special cases of unified products and extending structures.

7.1 Crossed products and extension problem

Now we give a first special case of the unified product, namely the crossed product of3-Lie algebras which is related to the study of the extension problem.

Let g and h be two given 3-Lie algebras. The extension problem asks for the clas-sification of all extensions of h by g, i.e. of all 3-Lie algebras E that fit into an exactsequence

0 // gi

// Eπ

// h // 0 . (75)

The classification is up to an isomorphism of 3-Lie algebras that stabilizes g and co-stabilizes h and we denote by EP(h, g) the isomorphism classes of all extensions of h byg up to this equivalence relation.

If g is abelian, then EP(h, g) ∼= H2(h, g), where H2(h, g) is the the second cohomologygroup in Section 5. We are going to study the non-abelian case.

Definition 7.1. Let (g, [·, ·, ·]g) and (h, [·, ·, ·]h) be two 3-Lie algebras. Then (g, h) is calleda crossed system if there exits three maps

⊲ : h× h× g → g, : h× g× g → g, ω : h× h× h → g,

such that the direct sum space g⊕ h form a 3-Lie algebra under the following bracket

[x1 + u1, x2 + u2, x3 + u3]

= [x1, x2, x3] + ω(u1, u2, u3)

+(u1, u2) ⊲ x3 + (u2, u3) ⊲ x1 + (u3, u1) ⊲ x2

29

+u3 (x1, x2) + u1 (x2, x3) + u2 (x3, x1)

+[u1, u2, u3].

where xi ∈ g and ui ∈ h, the bracket in g and h is written as [x1, x2, x3] and [u1, u2, u3] forsimplicity. This 3-Lie algebra is called crossed product of g and h which will be denotedby g#ω

⊲,h.

Theorem 7.2. Let (g, [·, ·, ·]g) and (h, [·, ·, ·]h) be two 3-Lie algebras. Then (g, h) is acrossed system if and only if the following conditions hold:

v3 [(x1, x2), (y1, y2)]

= [x1, x2, v3 (y1, y2)]− [v3 (x1, x2), y1, y2], (76)

u1 (x2, [y1, y2, y3])

= [u1 (x2, y1), y2, y3] + [u1 (x2, y2), y3, y1]

+[u1 (x2, y3), y1, y2], (77)

[x1, x2, (v1, v2) ⊲ y3]

= v2 (v1 (x1, x2), y3

)+ v1

(v2 (x1, x2), y3

)

+(v1, v2) ⊲ [x1, x2, y3], (78)

u1 (x2, v1 (y2, y3)

)+ [(u1, v1) ⊲ x2, y2, y3]

= v1 (u1 (x2, y2), y3

)+ v1

(y2, u1 (x2, y3)

), (79)

(u1, u2) ⊲ [y1, y2, y3]

= [(u1, u2) ⊲ y1, y2, y3] + [y1, (u1, u2) ⊲ y2, y3]

+[y1, y2, (u1, u2) ⊲ y3], (80)

[v1, v2, v3] (x1, x2) + [x1, x2, ω(v1, v2, v3)]

= (v2, v3) ⊲ (v1 (x1, x2)) + (v3, v1) ⊲ (v2 (x1, x2))

+(v1, v2) ⊲ (v3 (x1, x2)), (81)

u2 ((v1, v2) ⊲ y3, x1

)

= v2 (y3, (u2, v1) ⊲ x1

)+ v1

((u2, v2) ⊲ x1, y3

)

+(v1, v2) ⊲(u2 (y3, x1)

), (82)

(u1, u2) ⊲(v1 (y2, y3)

)

= [ω(u1, u2, v1), y2, y3] + [u1, u2, v1] (y2, y3)

+v1 ((u1, u2) ⊲ y2, y3

)+ v1

(y2, (u1, u2) ⊲ y3

),

30

[(u1, u2), (v1, v2)] ⊲ y3

= (u1, u2) ⊲ ((v1, v2) ⊲ y3)− (v1, v2) ⊲ ((u1, u2) ⊲ y3)

+v2 (ω(u1, u2, v1), y3)− v1 (ω(u1, u2, v2), y3), (83)

(u2, [v1, v2, v3]) ⊲ x1 + u2 (ω(v1, v2, v3), x1)

= (v2, v3) ⊲ ((u2, v1) ⊲ x1) + (v3, v1) ⊲ ((u2, v2) ⊲ x1)

+(v1, v2) ⊲ ((u2, v3) ⊲ x1), (84)

ω(u1, u2, [v1, v2, v3])− ω([u1, u2, v1]v2, v3)

−ω(v2, [u1, u2, v2], v3)− ω(v1, v2, [u1, u2, v3])

+(u1, u2) ⊲ ω(v1, v2, v3)− (v2, v3) ⊲ ω(u1, u2, v1)

−(v3, v1) ⊲ ω(u1, u2, v2)− (v1, v2) ⊲ ω(u1, u2, v3) = 0. (85)

Now we explain how to classify the extension problem by using crossed product. Letg and h be two 3-Lie algebras and we denote by CS (h, g) the set of all triples (⊲, , ω)such that (g, h, ⊲, , ω) is a crossed system of 3-Lie algebras. First we remark that, if(g, h, ⊲, , ω) is a crossed system, then the crossed product g#ω

⊲, h is an extension of hby g via

0 // gi// g#ω

⊲, hπ

// h // 0 (86)

where i(x) = (x, 0) and π(x, u) = u are the canonical maps. Conversely, any extensionE of h by g is equivalent to a crossed product extension of the form (86). Thus, theclassification of all extensions of h by g reduces to the classification of all crossed productsg#ω

⊲, h associated to all crossed systems of 3-Lie algebras (g, h, ⊲,, ω).By lemma 6.6, in the special case of crossed systems, we obtain the following

Definition 7.3. Two crossed systems (⊲, , ω) and (⊲′, ′, ω′) of CS (h, g) are calledcohomologous and we denote this by (⊲, , ω) ≈ (⊲′, ′, ω′) if there exists a linear mapr : h → g satisfying the following conditions for all xi ∈ g, ui ∈ h:

u1 (x2, x3) = u1 ′ (x2, x3) + [x2, x3, r(u1)], (87)

(u1, u2) ⊲ x3 = (u1, u2) ⊲′ x3 + [r(u1), r(u2), x3], (88)

ω(u1, u2, u3) = ω′(u1, u2, u3) + [r(u1), r(u2), r(u3)]− r([u1, u2, u3])

+(u1, u2) ⊲′ r(u3) + c.p.+ ν(u1)

′ (r(u2), r(u3)) + c.p. (89)

Note that (⊲, , ω) ≈ (⊲′, ′, ω′) if and only if there exists ψ : g#ω⊲, h → g#f ′

⊲′,′ h

an isomorphism of 3-Lie algebras that stabilizes g and co-stabilizes h. Thus we obtain theclassifying result to the extension problem in the non-abelian case:

Proposition 7.4. Let g and h be two 3-Lie algebras. Then ≈ is an equivalence relationon the set CS (h, g) of all crossed systems and the map

NH2(h, g) := CS (h, g)/ ≈ −→ EP(h, g), (⊲, , ω) 7→ g#ω⊲, h

is a bijection, where (⊲, , ω) is the equivalence class of (⊲, , ω) under ≈.

31

Example 7.5. [24] Let g be the 3-dimensional 3-Lie algebra defined with respect to abasis x1, x2, x3 by the skew-symmetric bracket [x1, x2, x3] = x1, and let V be the same3-Lie algebra which we consider with respect to basis v1, v2, v3, that is [v1, v2, v3] = v1.Then we have a crossed system given by:

(ui, uj) ⊲ xk = 0,

u1 (x1, x2) = 0, u1 (x1, x3) = 0, u1 (x2, x3) = r1x1,

u2 (x1, x2) = 0, u2 (x1, x3) = 0, u2 (x2, x3) = r2x1,

u3 (x1, x2) = 0, u3 (x1, x3) = 0, u3 (x2, x3) = r3x1,

ω(u1, u2, u3) = −r1x1,

where ri are arbitrary parameters.

7.2 Matched pair of 3-Lie algebras

The concept of a matched pair of Lie algebras was introduced in quantum group theory [21,Theorem 4.1] and in Poisson Lie group thory [20, Theorem 3.9]. Now we shall introducethe concept of a matched pair of 3-Lie algebras.

Definition 7.6. Let (g, [·, ·, ·]g) and (h, [·, ·, ·]h) be two 3-Lie algebras. Then (g, h) is calleda matched pair if there exits four linear maps

⊲ : h× h× g → g, ⊳ : h× g× g → h,

: h× g× g → g, : h× h× g → h,

such that the direct sum space g⊕ h form a 3-Lie algebra under the following bracket:

[x1 + u1, x2 + u2, x3 + u3]

= [x1, x2, x3] + (u1, u2) ⊲ x3 + (u2, u3) ⊲ x1 + (u3, u1) ⊲ x2

+u3 (x1, x2) + u1 (x2, x3) + u2 (x3, x1)

+[u1, u2, u3] + u1 ⊳ (x2, x3) + u2 ⊳ (x3, x1) + u3 ⊳ (x1, x2)

+(u2, u3) x1 + (u3, u1) x2 + (u1, u2) x3,

where x1, x2, x3 ∈ g and u1, u2, u3 ∈ h. This 3-Lie algebra is called the bicrossed productof (g, [·, ·, ·]g) and (h, [·, ·, ·]h) . We will denoted it by g ⊲⊳ h.

Theorem 7.7. Let (g, [·, ·, ·]g) and (h, [·, ·, ·]h) be two 3-Lie algebras, Then (g, h) is amatched pair if and only if the following compatibility conditions hold:

v3 ⊳ [(x1, x2), (y1, y2)]

= (v3 ⊳ (x1, x2)) ⊳ (y1, y2)− (v3 ⊳ (y1, y2)) ⊳ (x1, x2), (90)

u1 ⊳ (x2, [y1, y2, y3])

= (u1 ⊳ (x2, y1)) ⊳ (y2, y3) + (u1 ⊳ (x2, y2)) ⊳ (y3, y1)

32

+(u1 ⊳ (x2, y3)) ⊳ (y1, y2), (91)

v3 [(x1, x2), (y1, y2)]

= [x1, x2, v3 (y1, y2)] + (v3 ⊳ (y1, y2)) (x1, x2)

−[v3 (x1, x2), y1, y2]− (v3 ⊳ (x1, x2)) (y1, y2), (92)

u1 (x2, [y1, y2, y3])

= [u1 (x2, y1), y2, y3] + (u1 ⊳ (x2, y1)) (y2, y3)

+[u1 (x2, y2), y3, y1] + (u1 ⊳ (x2, y2)) (y3, y1)

+[u1 (x2, y3), y1, y2] + (u1 ⊳ (x2, y3)) (y1, y2), (93)

[x1, x2, (v1, v2) ⊲ y3] + ((v1, v2) y3) (x1, x2)

= v2 (v1 (x1, x2), y3

)+(v1 ⊳ (x1, x2), v2

)⊲ y3

+v1 (v2 (x1, x2), y3

)+(v1, v2 ⊳ (x1, x2)

)⊲ y3

+(v1, v2) ⊲ [x1, x2, y3], (94)

((v1, v2) y3) ⊳ (x1, x2)

= v2 ⊳(v1 (x1, x2), y3

)+(v1 ⊳ (x1, x2), v2

) y3

+v1 ⊳(v2 (x1, x2), y3

)+(v1, v2 ⊳ (x1, x2)

) y3

+(v1, v2) [x1, x2, y3], (95)

u1 (x2, v1 (y2, y3)

)+(v1, u1 ⊳ (x2, y3)

)⊲ y2

+[(u1, v1) ⊲ x2, y2, y3] + ((u1, v1) x2) (y2, y3)

= v1 (u1 (x2, y2), y3

)+(v1, u1 ⊳ (x2, y2)

)⊲ y3

+v1 (y2, u1 (x2, y3)

)+(u1, v1 ⊳ (y2, y3)

)⊲ x2, (96)

u1 ⊳(x2, v1 (y2, y3)

)+ ((u1, v1) x2) ⊳ (y2, y3)

+(v1, u1 ⊳ (x2, y3)

) y2

=(u1, v1 ⊳ (y2, y3)

) x2 + v1 ⊳

(u1 (x2, y2), y3

)

+(v1, u1 ⊳ (x2, y2)

) y3 + v1 ⊳

(y2, u1 (x2, y3)

), (97)

(u1, u2) ⊲ [y1, y2, y3]

= [(u1, u2) ⊲ y1, y2, y3] + ((u1, u2) y1) (y2, y3)

+[y1, (u1, u2) ⊲ y2, y3] + ((u1, u2) y2) (y3, y1)

+[y1, y2, (u1, u2) ⊲ y3] + ((u1, u2) y3) (y1, y2), (98)

(u1, u2) [y1, y2, y3]

33

= ((u1, u2) y1) ⊳ (y2, y3) + ((u1, u2) y2) ⊳ (y3, y1)

+((u1, u2) y3) ⊳ (y1, y2), (99)

[v1, v2, v3] (x1, x2)

= (v2, v3) ⊲ (v1 (x1, x2)) + (v3, v1) ⊲ (v2 (x1, x2))

+(v1, v2) ⊲ (v3 (x1, x2)), (100)

[v1, v2, v3] ⊳ (x1, x2)

= (v2, v3) (v1 (x1, x2)) + [v1 ⊳ (x1, x2), v2, v3]

+(v3, v1) (v2 (x1, x2)) + [v2 ⊳ (x1, x2), v3, v1]

+(v1, v2) (v3 (x1, x2)) + [v3 ⊳ (x1, x2), v1, v2]. (101)

u2 ((v1, v2) ⊲ y3, x1

)+(u2, (v1, v2) y3

)⊲ x1

= v2 (y3, (u2, v1) ⊲ x1

)+((u2, v1) x1, v2

)⊲ y3

+v1 ((u2, v2) ⊲ x1, y3

)+(v1, (u2, v2) x1) ⊲ y3

+(v1, v2) ⊲(u2 (y3, x1)

), (102)

u2 ⊳((v1, v2) ⊲ y3, x1

)+(u2, (v1, v2) y3

) x1

= v2 ⊳(y3, (u2, v1) ⊲ x1

)+((u2, v1) x1, v2

) y3

+v1 ⊳((u2, v2) ⊲ x1, y3

)+(v1, (u2, v2) x1) y3

+(v1, v2)(u2 (y3, x1)

)+ [v1, v2, u3 ⊳ (y3, x1)]. (103)

(u1, u2) ⊲(v1 (y2, y3)

)

= [u1, u2, v1] (y2, y3)

+v1 ((u1, u2) ⊲ y2, y3

)+(v1, (u1, u2) y2

)⊲ y3

+v1 (y2, (u1, u2) ⊲ y3

)+((u1, u2) y3, v1

)⊲ y2, (104)

(u1, u2)(v1 (y2, y3)

)+ [u1, u2, v1 ⊳ (y2, y3)]

= [u1, u2, v1] ⊳ (y2, y3)

+v1 ⊳((u1, u2) ⊲ y2, y3

)+(v1, (u1, u2) y2

) y3

+v1 ⊳(y2, (u1, u2) ⊲ y3

)+((u1, u2) y3, v1

) y2, (105)

[(u1, u2), (v1, v2)] y3

= (u1, u2) ((v1, v2) ⊲ y3) + [u1, u2, (v1, v2) y3]

−(v1, v2) ((u1, u2) ⊲ y3)− [v1, v2, (u1, u2) y3], (106)

[(u1, u2), (v1, v2)] ⊲ y3

34

= (u1, u2) ⊲ ((v1, v2) ⊲ y3)− (v1, v2) ⊲ ((u1, u2) ⊲ y3), (107)

(u2, [v1, v2, v3]) ⊲ x1

= (v2, v3) ⊲ ((u2, v1) ⊲ x1) + (v3, v1) ⊲ ((u2, v2) ⊲ x1)

+(v1, v2) ⊲ ((u2, v3) ⊲ x1), (108)

(u2, [v1, v2, v3]) x1

= (v2, v3) ⊲ ((u2, v1) x1) + [(u2, v1) x1, v2, v3]

+(v3, v1) ⊲ ((u2, v2) x1) + [(u2, v2) x1, v3, v1]

+(v1, v2) ⊲ ((u2, v3) x1) + [(u2, v3) x1, v1, v2]. (109)

The equations (90)-(91) means that (h, ⊳) is a representation of g and the equations(107)-(108) means that (g, ⊲) is a representation of h.

The bicrossed product of two 3-Lie algebras is related to the so called factorizationproblem, which can be stated as follows: Let g and h be two given 3-Lie algebras. Describeand classify all 3-Lie algebras E that factorize through g and h, i.e. E contains g and h

as 3-Lie subalgebras such that E = g + h and g ∩ h = [0]. We use Theorem 6.4 to provethe following:

Proposition 7.8. A 3-Lie algebra E factorizes through g and h if and only if there existsa matched pair of 3-Lie algebras (g, h, ⊳, ⊲, , ) such that E ∼= g ⊲⊳ h.

Proof. We know that any bicrossed product g ⊲⊳ h factorizes through g ∼= g × [0] andh ∼= [0] × h. Conversely, assume that E factorizes through g and h. Let p : E → g bethe natural projection of E on g, i.e. p(x + u) := x, for all x ∈ g and u ∈ h. Now,we apply Theorem 6.4 for V := Ker(p) = h. Since V is a 3-Lie subalgebra of E, themap ω is the trivial map and the extending structure Ω(g, V ) =

(⊳, ⊲, , , [−, −]

)

constructed in the proof of Theorem 6.4 is precisely a matched pair of 3-Lie algebra.Thus the unified product gV is the bicrossed product g ⊲⊳ h. Explicitly, the matchedpair (g, h, ⊳, ⊲ , ) is given by:

⊲ : h× h× g → g, (u1, u2) ⊲ x3 = p([u1, u2, x3]),

: h× g× g → g, u1 (x2, x3) = p([u1, x2, x3]),

⊳ : h× g× g → h, u1 ⊳ (x2, x3) = [u1, x2, x3]− p([u1, x2, x3]),

: h× h× g → h, (u1, u2) x3 = [u1, u2, x3]− p([u1, u2, x3]),

for all ui ∈ h and xi ∈ g.

7.3 Classifying complements for 3-Lie algebras

This section is devoted to the classifying complements problem. Let g ⊆ E be a 3-Liesubalgebra of E. A 3-Lie subalgebra h of E is called a complement of g in E (or a g-complement of E) if E = g + h and g ∩ h = [0]. If h is a complement of g in E, then

35

we have E ∼= g ⊲⊳ h, where g ⊲⊳ h is the bicrossed product associated to the canonicalmatched pair of the factorization E = g+ h.

We denote by F(g, E) the (possibly empty) isomorphism classes of all g-complementsof E. The factorization index of g in E is defined by [E : g] := | F(g, E) |.

Definition 7.9. Let (g, h, ⊲, ⊳, , ) be a matched pair of 3-Lie algebras. A linearmap r : h → g is called a deformation map of the matched pair (g, h, ⊲, ⊳, , ) if thefollowing condition holds for any ui ∈ h:

r[u1, u2, u3]− [r(u1), r(u2), r(u3)]

= (u1, u2) ⊲ r(u3) + c.p.+ u1 (r(u2), r(u3)) + c.p.

−r((u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p.

). (110)

We denote by DM (h, g | (⊲, ⊳,, )) the set of all deformation maps of the matchedpair (g, h, ⊲, ⊳,,). The trivial map r(x) = 0, for all x ∈ h, is of course a deformationmap. The right hand side of (110) measures how far r : h → g is from being a 3-Lie algebramap. Using this concept of deformation map, we introduce the following deformation ofa 3-Lie algebra.

Theorem 7.10. Let g be a 3-Lie subalgebra of E, h a given g-complement of E andr : h → g a deformation map of the associated canonical matched pair (g, h, ⊲, ⊳,,).

(1) Let fr : h → E = g ⊲⊳ h be the linear map defined for any u ∈ h by:

fr(u) = (r(u), u).

Then h := Im(fr) is a g-complement of E.(2) Let hr := h, as a vector space, with the new bracket defined for any ui ∈ h by:

[u1, u2, u3]r := [u1, u2, u3] + (u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p. (111)

Then hr is a 3-Lie algebra which is called the r-deformation of h. Furthermore, hr ∼= h,as 3-Lie algebras.

Proof. (1) First we will prove that h = [(r(u), u

)| u ∈ h] is a 3-Lie subalgebra of

g ⊲⊳ h = E. Indeed, for all ui ∈ h we have:[(r(u1), u1), (r(u2), u2), (r(u3), u3)

]

=([r(u1), r(u2), r(u3)

]+(u1, u2) ⊲ r(u3) + c.p.+ u1 (r(u2), r(u3)) + c.p.,

[u1, u2, u3] + (u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p.)

(110)=

(r([u1, u2, u3] + (u1, u2) r(u3) + c.p. + u1 ⊳ (r(u2), r(u3)) + c.p.

),

[u1, u2, u3] + (u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p.).

Thus[(r(u1), u1), (r(u2), u2), (r(u3), u3)

]∈ h. Moreover, it is straightforward to see that

g ∩ h = [0] and (x, u) =(x − r(u), 0

)+(r(u), u

)∈ g + h for all x ∈ g, u ∈ h. Here, we

view g ∼= g× [0] as a subalgebra of g ⊲⊳ h. Therefore, h is a g-complement of E = g ⊲⊳ h.

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(2) We denote by fr : h → h the linear isomorphism induced by fr. We will prove that

fr is also a 3-Lie algebra map if we consider on h the bracket given by (111). In fact, forany ui ∈ h we have:

fr([u1, u2, u3]r

)

(111)= fr

([u1, u2, u3] + (u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p.

)

=(r([u1, u2, u3] + (u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p.

),

[u1, u2, u3] + (u1, u2) r(u3) + c.p. + u1 ⊳ (r(u2), r(u3)) + c.p.)

(110)=

([r(u1), r(u2), r(u3)

]+(u1, u2) ⊲ r(u3) + c.p.+ u1 (r(u2), r(u3)) + c.p.,

[u1, u2, u3] + (u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p.)

= [(r(u1), u1), (r(u2), u2), (r(u3), u3)] = [fr(u1), fr(u2), fr(u3)].

Therefore, hr is a 3-Lie algebra and the proof is finished.

The following is the converse of Theorem 7.10. It proves that all g-complements of Eare r-deformations of a given complement.

Theorem 7.11. Let g be a 3-Lie subalgebra of E, h a given g-complement of E withthe associated canonical matched pair of 3-Lie algebras (g, h, ⊲, ⊳, ,,). Then h is ag-complement of E if and only if there exists an isomorphism of 3-Lie algebras h ∼= hr,for some deformation map r : h → g of the matched pair (g, h, ⊲, ⊳,,).

Proof. Let h be an arbitrary g-complement of E. Since E = g ⊕ h = g ⊕ h we can findfour k-linear maps:

s : h → g, ν : h → h, t : h → g, : h → h

such that for all u ∈ h and v ∈ h we have:

u = s(u)⊕ ν(u), y = t(v)⊕(v). (112)

It is easy to see that ν : h → h is a linear isomorphism of vector spaces. We denote byν : h → g ⊲⊳ h the composition:

ν : hv

−→ hi→ E = g ⊲⊳ h.

Therefore, we have ν(u)(112)=

(−s(u), u

), for all u ∈ h. Then we shall prove that r := −s

is a deformation map and h ∼= hr. Indeed, h = Im(ν) = Im(ν) is a 3-Lie subalgebra ofE = g ⊲⊳ h and we have:

[(r(u1), u1), (r(u2), u2), (r(u3), u3)

]

=([r(u1), r(u2), r(u3)

]+(u1, u2) ⊲ r(u3) + c.p.+ u1 (r(u2), r(u3)) + c.p.,

37

[u1, u2, u3] + (u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p.)

= (r(v), v)

for some v ∈ h. Thus, we obtain:

r(v) =[r(u1), r(u2), r(u3)

]+(u1, u2) ⊲ r(u3) + c.p.+ u1 (r(u2), r(u3)) + c.p., (113)

v = [u1, u2, u3] + (u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p. (114)

By applying r to (114) it follows that r is a deformation map of the matched pair(g, h, ⊲, ⊳,,). Furthermore, (113) and (111) show that ν : hr → h is also a 3-Liealgebra map which finishes the proof.

In order to provide the classification of all complements we introduce the following:

Definition 7.12. Let (g, h, ⊲, ⊳,,) be a matched pair of 3-Lie algebras. Two defor-mation maps r, R : h → g are called equivalent and we denote this by r ∼ R if thereexists σ : h → h a k-linear automorphism of h such that for any x, y ∈ h:

σ[u1, u2, u3]− [σ(u1), σ(u2), σ(u3)]

= (σ(u1), σ(u2)) R(σ(u3)) + c.p.+ σ(u1) ⊳ (R(σ(u2)), R(σ(u3))) + c.p.

−σ((u1, u2) r(u3) + c.p.+ u1 ⊳ (r(u2), r(u3)) + c.p.

).

To conclude this section, the following result provides the answer to the classifyingcomplements problem for 3-Lie algebras:

Theorem 7.13. Let g be a 3-Lie subalgebra of E, h a g-complement of E and (g, h, ⊲, ⊳,,) the associated canonical matched pair. Then ∼ is an equivalence relation on the setDM (h, g | (⊲, ⊳,,)). If we denote by MH2(h, g) := DM (h, g | (⊲, ⊳,,))/ ∼, thenwe have

MH2(h, g) −→ F(g, E),

r 7→ hr

is a bijection between MH2(h, g) and the isomorphism classes of all g-complements of E.In particular, the factorization index of g in E is computed by the formula:

[E : g] = |MH2(h, g)|

Example 7.14. Let E be the 6-dimensional 3-Lie algebra defined with respect to a basisx1, x2, x3, u1, u2, u3 by the skew-symmetric bracket

[x1, x2, x3] = x1, [u1, u2, x2] = αx1 + u1,

[x1, u2, x2] = u1, [u1, x2, x3] = u1,

where α is an arbitrary parameter.

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Let g be the 3-Lie subalgebra of E with basis x1, x2, x3 and h be the abelian 3-Liealgebra of dimension 3 with basis u1, u2, u3 . Then h is a g-complement of E with theassociated matched pair (g, h) given as follows:

(u1, u2) ⊲ x2 = −αx1, (u1, u2) x2 = u1, u1 ⊳ (x2, x3) = u1, u2 ⊳ (x2, x1) = u1.

It is easy to see that the map r : h → g given by

r(u1) = 0, r(u2) = b1x1 + b2x2 + b3x3,

r(u3) = c1x1 + c3x3,

is a deformation map associated with the above matched pair of 3-Lie algebras, where bi, ciare arbitrary parameters. Furthermore, the r-deformation of h has the bracket given by

[u1, u2, u3]r = b2c3u1.

If b2c3 6= 0, then hr is not isomorphic to h as hr is not abelian. Since we have only twotypes of 3-Lie algebras of dimension 3, we obtain that [E : g] = 2.

It is natural to consider the special matched pair (g, g∗) where g∗, the dual spaceof g, is also a non-abelian 3-Lie algebra. This will lead to the general theory of ”3-Liebialgebras” and the 3-Lie algebra g ⊲⊳ g∗ is called the double construction of g. Anotherinteresting problem is to develop the theory of flag extending structures for 3-Lie algebrasas in [1, 3, 4]. Since these problems are both more difficult than the cases of Lie algebrasand Leibniz algebras, their solutions are left to future investigations.

Acknowledgements

Most of this work was done while the author was visiting Courant Research Centre, Georg-August Universitat Gottingen from June to September, 2013. He is grateful to ProfessorChenchang Zhu for invitation and hospitality.

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College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, P. R.China;E-mail address: zhangtao@htu.edu.cn.

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