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SOLVABLE 3-LIE ALGEBRAS WITH A MAXIMAL HYPO-NILPOTENT IDEAL N

RUI-PU BAI, CAI-HONG SHEN, AND YAO-ZHONG ZHANG

Abstract. This paper obtains all solvable 3-Lie algebras with them-dimensional filiform 3-Lie algebraN (m5) as a maximal hypo-nilpotent ideal, and proves that them-dimensional filiform 3-Lie algebra N can’t be as the nilradical of solvable non-nilpotent 3-Lie algebras. By means of one dimensional extension of Lie algebras to the 3-Lie algebras, we get some classes of solvable Lie algebras directly.

Key words. 3-Lie algebra, hypo-nilpotent ideal, filiform n-Lie algebra.

AMS subject classifications.17B05, 17D99.

1. Introduction. The concept ofn-Lie algebras appeared in two different con- texts [1, 2]. In [1], Nambu introducedn-ary multilinear operations in his description of simultaneous classical dynamics ofnparticles, and extended the Poisson bracket to the n-ary multilinear bracket. In [2], Filippov formulated a theory ofn-Lie algebras based on his proposed (2n−1)-fold Jacobi type identity and gave a classification forn-Lie algebras of lower (≤n+ 1) dimensions. The connection between the Nambu mechan- ics and the Filippov’s theory ofn-Lie algebras was established in 1994 by Takhtajan [3]. Recently n-Lie algebras have found important applications in string and mem- brane theories. For instance, in [4, 5] Bagger and Lambert proposed a supersymmetric field theory model for multiple M2-branes based on the metric 3-Lie algebras. More application ofn-Lie algebras can be found in e.g., [6, 7, 8, 9, 10, 11, 12, 13].

In recent years, the structure ofn-Lie algebras has been widely studied. Kasymov [14] developed the structure and representation theory of n-Lie algebras. Ling [15]

proved that there is a unique (n+ 1)-dimensional simplen-Lie algebra forn >2 over an algebraically closed field of characteristic zero. The first author of the current paper and her collaborators showed in [16] that there exist only [n2] + 1 classes of (n+ 1)-dimensional simplen-Lie algebras over a complete field of characteristic 2 and

Received by the editors on June 10, 2009. Accepted for publication on July 31, 2010. Handling Editors: Roger A. Horn and Fuzhen Zhang.

College of Mathematics and Computer Science, Key Lab in Machine Learning and Computational Intelligence, Hebei University, Baoding (071002), China (bairp1@yahoo.com.cn, sch0925@163.com).

Partially supported by NSF(10871192) of China, NSF(A2007000138) of Hebei Province, China.

School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Aus- tralia (yzz@maths.uq.edu.au). Supported by the Australian Research Council.

43

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gave a complete classification in [17] for six dimensional 4-Lie algebras. There are other results on structures and representations of n-Lie algebras.

The structure of n-Lie algebras is very different from that of Lie algebras, due to the n-ary multilinear operations involved. In particular, it turns out that the fundamental identity for an n-Lie algebra is much more restrictive than the Jocobi identity for a Lie algebra. One consequence is that higher, finite dimensional n-Lie algebras may be rare and are difficult to find. So it is very important to construct new examples ofn-Lie algebras.

Filiformn-Lie algebras, i.e., nilpotentn-Lie algebrasLsatisfying dimLi= dimL−

n−i, are important class of nilpotent n-Lie algebras. In [18] we introduced the concept of hypo-nilpotent ideals ofn-Lie algebras, and proved that anm-dimensionalsimplest filiform 3-Lie algebraN0can’t be a nilradical of solvable non-nilpotent 3-Lie algebras.

Bym-dimensionalsimplestfiliform 3-Lie algebra, we mean anm-dimensional filiform 3-Lie algebra with the following multiplication table in the basise1, e2, . . . , em,

[e1, e2, ej] =ej1,4≤j≤m.

(1.1)

Moreover, it was shown that there are only four classes of (m+ 1)-dimensional and one class of (m+2)-dimensional solvable non-nilpotent 3-Lie algebras withN0as their maximal hypo-nilpotent ideal.

In this paper we generalize the results of [18]. Namely we consider a more compli- catedm-dimensional filiform 3-Lie algebraN (m≥5) defined by the multiplication table (3.1) below (c.f. (1.1)). We obtain all solvable 3-Lie algebras with such anN as a maximal hypo-nilpotent ideal and prove thatN can’t be a nilradical of solvable non-nilpotent 3-Lie algebras.

The organization for the rest of this paper is as follows. Section 2 introduces some basic notions. Section 3 describes the structure of solvable 3-Lie algebras with the maximal hypo-nilpotent ideal N. Section 4 studies the solvable 3-Lie algebras with nilradicalN. Section 5 gives an application of one dimensional extension of Lie algebras.

Throughout this paper we consider 3-Lie algebras over a fieldF of characteristic zero.

2. Fundamental notions. First we introduce some notions of n-Lie algebras (see [2, 14, 18]). A vector space A over a field F is an n-Lie algebra if there is an n-ary multilinear operation [,· · ·, ] satisfying the following identities

[x1,· · ·, xn] = (−1)τ(σ)[xσ(1),· · ·, xσ(n)], (2.1)

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and

[[x1,· · ·, xn], y2,· · ·, yn] =

n

X

i=1

[x1,· · ·,[xi, y2,· · ·, yn],· · ·, xn], (2.2)

where σ runs over the symmetric group Sn and the numberτ(σ) is equal to 0 or 1 depending on the parity of the permutationσ.

A derivation of ann-Lie algebraAis a linear map D:A→A, such that for any elementsx1, . . . ,xn ofA

D([x1,· · ·, xn]) =

n

X

i=1

[x1,· · ·, D(xi),· · ·, xn].

The set of all derivations of A is a subalgebra of Lie algebra gl(A). This subal- gebra is called the derivation algebra of A, and is denoted by DerA. The map ad(x1,· · ·, xn1) : A → A defined by ad(x1,· · ·, xn1)(xn) = [x1,· · ·, xn] for x1, . . . ,xn ∈Ais called a left multiplication. It follows from (2.2) that ad(x1,· · ·, xn1) is a derivation. The set of all finite linear combinations of left multiplications is an ideal of DerA and is denoted by ad(A). Every element in ad(A) is by definition an inner derivation, and for all x1, . . . , xn1, y1, . . . , yn1ofA,

[ad(x1,· · ·, xn1),ad(y1,· · ·, yn1)]

(2.3)

= ad(x1,· · ·, xn1)ad(y1,· · ·, yn1)−ad(y1,· · ·, yn1)ad(x1,· · ·, xn1)

=

n1

X

i=1

ad(y1,· · ·,[x1,· · ·, xn1, yi],· · ·, yn1).

Let A1, A2, . . . , An be subalgebras ofn-Lie algebra A and let [A1, A2, · · ·, An] denote the subspace ofAgenerated by all vectors [x1, x2,· · ·, xn], wherexi ∈Ai for i= 1,2, . . . , n. The subalgebra [A, A,· · ·, A] is called the derived algebra ofA, and is denoted byA1. IfA1= 0, thenA is called an abeliann-Lie algebra.

An ideal of an n-Lie algebra A is a subspace I such that [I, A,· · ·, A] ⊆ I. If A1 6= 0 andA has no ideals except for 0 and itself, thenA is by definition a simple n-Lie algebra.

An ideal I of an n-Lie algebraA is called a solvable ideal, if I(r) = 0 for some r≥0, whereI(0)=IandI(s) is defined by induction,

I(s+1)= [I(s), I(s), A,· · ·, A]

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fors≥0. WhenA=I, Ais a solvablen-Lie algebra.

An idealI of ann-Lie algebra A is called a nilpotent ideal, ifI satisfiesIr = 0 for somer≥0, whereI0=I and Ir is defined by induction, Ir+1= [Ir, I, A,· · ·, A]

forr≥0. IfI=A,Ais called a nilpotentn-Lie algebra.

The sum of two nilpotent ideals ofA is nilpotent, and the largest nilpotent ideal ofAis called the nilradical ofA, and is denoted byN R(A).

Denote by A an associative algebra generated by all operators ad(x), where x= (x1, · · ·, xn1)∈A(n1). IfI is an ideal ofA, denote byI,K(I) and ad(I, A) respectively the subalgebra ofA, the ideal ofA and the subalgebra of ad(A) gener- ated by the operators of the form ad(c, x1,· · ·, xn2), c∈I, xi∈A, i= 1, . . . , n−2.

It follows at once from (2.3) thatK(I) =I·A=A·I,and ad(I, A) is an ideal of ad(A).

Lemma 2.1. [14] An idealI of ann-Lie algebraAis a nilpotent ideal if and only if K(I)is a nilpotent ideal of the associative algebraA.

An ideal I of an n-Lie algebraA may not be a nilpotent ideal although it is a nilpotent subalgebra. This property is different from that of Lie algebras. In the following, we concern such types of ideals ofn-Lie algebras.

Definition 2.2. Let A be an n-Lie algebra and I be an ideal of A. If I is a nilpotent subalgebra but is not a nilpotent ideal, thenIis called a hypo-nilpotent ideal of A. IfI is not properly contained in any hypo-nilpotent ideals, then I is called a maximal hypo-nilpotent ideal of A.

From (2.2), a hypo-nilpotent ideal ofAis a proper ideal, and the nilradicalN R(A) is properly contained in every maximal hypo-nilpotent ideals. But the sum of two hypo-nilpotent ideals ofA may not be hypo-nilpotent.

In the following, any brackets of basis vectors not listed in the multiplication table ofn-Lie algebras are assumed to be zero.

3. 3-Lie algebras with maximal hypo-nilpotent idealN. In the following we suppose thatN is anm-dimensional filiform 3-Lie algebra with the multiplication table

[e1, e2, ej] =ej1, 4≤j≤m, [e1, ej, em] =ej2, 5≤j≤m−1, (3.1)

wheree1, . . . , emis a basis of N.

Lemma 3.1. Let N be an m-dimensional 3-Lie algebra with a basis e1, . . . , em

satisfying (3.1). Then the inner derivation algebra ad(N) has a basis ad(e1, e2),

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ad(e1, ej), ad(e2, ej), j = 4,5, . . . , m. And with respect to the basis e1,· · ·, em, ad(ek, el)is represented by the following matrix form

ad(e1, e2) =

m

X

j=4

Ejj1,ad(e1, em) =

m1

X

j=5

Ejj2+E2m1,

ad(e1, ei) =E2i1+Emi2,ad(e2, ei) =E1i1 for 5≤i≤m−1, ad(e1, e4) =E23,ad(e2, e4) =E13,ad(e2, em) =E1m1, whereEij is the (m×m)matrix unit.

Proof. The result follows from a direct computation.

LetAbe an (m+ 1)-dimensional 3-Lie algebra with the idealN, andx, e1, . . . , em

be a basis ofA. Then the multiplication table ofAin the basisx, e1, . . . , emis given by





[e1, e2, ej] =ej1, 4≤j ≤m, [e1, ej, em] =ej2, 5≤j ≤m−1, [x, ei, ej] =

m

P

k=1

akijek, 1≤i, j≤m, (3.2)

whereakij ∈F, akij=−akji,1≤i, j≤m.Therefore, the following (m(m21)×m) matrix M determines the structure ofA

M =

a112 a212 a312 a412 · am121 am12 a113 a213 a313 a413 · am131 am13 a114 a214 a314 a414 · am141 am14 a115 a215 a315 a415 · am151 am15

... ... ... ... · ... ...

a11m2 a21m2 a31m2 a41m2 · am1m12 am1m2 a11m1 a21m1 a31m1 a41m1 · am1m11 am1m1 a11m a21m a31m a41m · am1m1 am1m

a123 a223 a323 a423 · am231 am23 a124 a224 a324 a424 · am241 am24 a125 a225 a325 a425 · am251 am25

... ... ... ... · ... ...

a12m2 a22m2 a32m2 a42m2 · am2m12 am2m2 a12m1 a22m1 a32m1 a42m1 · am2m11 am2m1 a12m a22m a32m a42m · am2m1 am2m

a134 a234 a334 a434 · am341 am34

... ... ... ... · ... ...

a1m1m a2m1m a3m1m a4m1m · amm11m amm1m

 (3.3) .

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The matrixM is called the structure matrix ofAwith respect to the basisx,e1, . . . , em.

By the above notations we have the following result.

Theorem 3.2. Let A be an (m+ 1)-dimensional 3-Lie algebra with a maximal hypo-nilpotent ideal N. Then A is solvable, and up to isomorphism the following is the only possibility for the structural matrixM ofA:

M =

0 1 0 0 0 · · · 0 0 0 0

0 0 m−1 0 0 · · · 0 0 0 0

0 0 0 m−2 0 · · · 0 0 0 0

0 0 0 0 m−3 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 4 0 0

0 0 0 0 0 · · · 0 0 3 0

0 0 0 0 0 · · · 0 0 0 2

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 0 0 0

 (3.4) .

Moreover, the multiplication table ofA in a basisx, e1, . . . , em (m≥5) is as follows





[e1, e2, ej] =ej1 if 4≤j≤m, [e1, ej, em] =ej2 if 5≤j≤m−1, [x, e1, e2] =e2,

[x, e1, ek] = (m−k+ 2)ek if 3≤k≤m.

(3.5)

Proof. Since N is an ideal ofA and dimA =m+ 1, we haveA1 = [A, A, A] = [A, A, N]⊆N. Then the structural matrixM is of the form (3.3) with respect to a basisx, e1, . . . , em.

Firstly, imposing the Jacobi identities

[[x, e1, e2], e1, ej] = [[x, e1, ej], e1, e2] + [x, e1,[e2, e1, ej]] for 3≤j≤m−1,

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and

[[x, e1, e2], e1, em] = [[x, e1, em], e1, e2] + [x, e1,[e2, e1, em]], and using (3.2), we obtain

a11j=a21j= 0, aj+11j =aj+21j =· · ·=am1j= 0 for 3≤j≤m−2;

aj1j11=aj1j+a212,4≤j≤m−1;

am12=aj1j1−aj1j21, ak1j =ak1j11, k6=j−1, for 4≤k≤m,5≤j≤m−1, and

a212+am1m−am1m11= 0, a11m1=a21m1=am1m1= 0,

ai+112 +ai1m−ai1m11= 0,4≤i≤m−2, am1m1=am1m21 respectively.

Secondly, imposing the Jacobi identities on{[x, e1, e2], e2, ej} for 3≤j ≤m, we get

a12j=a22j= 0, aj+12j =aj+22j =· · ·=am2j= 0 for 3≤j≤m−1;

aj2j11=aj2j−a112,4≤j≤m;ak2j=ak2j11, for 4≤k < j ≤m.

From

[[x, e1, ei], e2, ej] = [[x, e2, ej], e1, ei] + [x,[e1, e2, ej], ei] = 0, for 3≤i, j≤m−1,

[[x, e1, e4], e1, em] = [[x, e1, em], e1, e4], and

[[x, e1, ei], e1, em] = [[x, e1, em], e1, ei]−[x, e1, ei2] for 5≤i≤m−1, we get

akij = 0,3≤i, j≤m−1,1≤k≤m, a21m= 0,

(8)

and

ai1i+am1m−ai1i22= 0, ak1i=ak1i22 fork6=i,5≤k≤m−1, a11i2=a21i2=am1i22=am1i12=am1i2= 0.

Then we haveam1m= 2a212, am12= 0.

Again from

[[x, e2, e4], e1, em] = [[x, e1, em], e2, e4], [[x, e1, em], e2, e4] = [[x, e2, e4], e1, em] + [x, e3, em],

[[x, e2, ei], e1, em] = [[x, e1, em], e2, ei]−[x, e2, ei2],5≤i≤m−1, [[x, e1, e4], e2, em] = [[x, e2, em], e1, e4],

and

[[x, e1, ei], e2, em] = [[x, e2, em], e1, ei], we get

a11m= 0, ak3m= 0,1≤k≤m, ak2i =ak2i22 for 5≤k≤m−1, a12i2=a22i2=am2i22=am2i12=am2i2= 0, a112= 0, a22m= 0 andam2m= 0.

By

[[x, e2, em], e1, ei] = [[x, e1, ei], e2, em]−[x, ei1, em] + [x, e2, ei2],5≤i≤m−1, [[x, e2, e4], e2, em] = [[x, e2, em], e2, e4],

and

[[x, e1, em], e2, em] = [[x, e2, em], e1, em] + [x, em1, em], we obtain

aji1m=aj2i2,1≤j≤m, a12m= 0, ai2m=aim21mfor 5≤i≤m−1, a1m1m=a2m1m=amm21m=amm11m=amm1m= 0.

(9)

Therefore, we get ad(x, e1)|N

=

0 0 0 0 0 · 0 0 0 0

0 a212 a312 a412 a512 · am123 am122 am121 0

0 0 r1a212 0 0 · 0 0 0 0

0 0 am1m1 r2a212 0 · 0 0 0 0

0 0 am1m31 am1m1 r3a212 · 0 0 0 0 0 0 am1m41 am1m31 am1m1 · 0 0 0 0 ... ... ... ... ... ... ... ... ... ... 0 0 a41m1 a51m1 a61m1 · am1m1 rm4a212 0 0 0 0 a31m1 a41m1 a51m1 · am1m31 am1m1 rm3a212 0 0 0 a31m a41m a51m · am1m3 am1m2 am1m1 rm2a212

 ,

whererj=m−j for 1≤j≤m−2,ai1m11=ai+112 +ai1m for 4≤i≤m−2;

ad(x, e2)|N

=

0 −a212 −a312 −a412 −a512 · · · −am123 −am122 −am121 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 am2m1 0 0 · · · 0 0 0 0

0 0 am2m2 am2m1 0 · · · 0 0 0 0

0 0 am2m3 am2m2 am2m1 · · · 0 0 0 0

· · · ·

0 0 a52m a62m a72m · · · am2m1 0 0 0

0 0 a42m a52m a62m · · · am2m2 am2m1 0 0

0 0 a32m a42m a52m · · · am2m3 am2m2 am2m1 0

 ,

−ad(x, em)|N

=

0 0 a31m a41m a51m · · · am1m3 am1m2 am1m1 am1m 0 0 a32m a42m a52m · · · am2m3 am2m2 am2m1 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 am2m1 0 0 · · · 0 0 0 0

0 0 am2m2 am2m1 0 · · · 0 0 0 0

· · · ·

0 0 a62m a72m a82m · · · 0 0 0 0

0 0 a52m a62m a72m · · · am2m1 0 0 0

0 0 0 0 0 · · · 0 0 0 0

 .

If we replace xbyx−am2m1e1+am1m1e2−a312e4−a412e5− · · · −am121em, the above

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maps are reduced to ad(x, e1)|N

=

0 0 0 0 0 · 0 0 0 0

0 a212 0 0 0 · 0 0 0 0

0 0 r1a212 0 0 · 0 0 0 0

0 0 0 r2a212 0 · 0 0 0 0

0 0 bm1m2 0 r3a212 · 0 0 0 0

0 0 bm1m3 bm1m2 0 · 0 0 0 0

... ... ... ... ... ... ... ... ... ... 0 0 b51m b61m b71m · 0 rm4a212 0 0 0 0 b41m b51m b61m · bm1m2 0 rm3a212 0 0 0 b31m b41m b51m · bm1m3 bm1m2 0 rm2a212

 ,

whererj=m−j, for 1≤j≤m−2, ad(x, e2)|N

=

0 −a212 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 am2m2 0 0 · · · 0 0 0 0

0 0 am2m3 am2m2 0 · · · 0 0 0 0

· · · ·

0 0 a52m a62m a72m · · · 0 0 0 0

0 0 a42m a52m a62m · · · am2m2 0 0 0

0 0 a32m a42m a52m · · · am2m3 am2m2 0 0

 ,

−ad(x, em)|N

=

0 0 b31m b41m b51m · · · bm1m3 bm1m2 0 2a212 0 0 a32m a42m a52m · · · am2m3 am2m2 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 am2m2 0 0 · · · 0 0 0 0

· · · ·

0 0 a62m a72m a82m · · · 0 0 0 0

0 0 a52m a62m a72m · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

 .

Again by the Jacobi identities for vectors{[x, e1, e2], x, ei}for 3≤i≤m,we get

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a12a2m= 0 fori= 3,4, . . . , m−2.Sincea126= 0 (ifa12= 0, thenAis nilpotent), we getai2m= 0, fori= 3,4, . . . , m−2.

Therefore, ad(x, e1)|N

=

0 0 0 0 0 · 0 0 0 0

0 a212 0 0 0 · 0 0 0 0

0 0 r1a212 0 0 · 0 0 0 0

0 0 0 r2a212 0 · 0 0 0 0

0 0 bm1m2 0 r3a212 · 0 0 0 0

0 0 bm1m3 bm1m2 0 · 0 0 0 0

... ... ... ... ... ... ... ... ... ... 0 0 b51m b61m b71m · 0 rm4a212 0 0 0 0 b41m b51m b61m · bm1m2 0 rm3a212 0 0 0 b31m b41m b51m · bm1m3 bm1m2 0 rm2a212

 ,

whererj=m−j for 1≤j≤m−2, ad(x, e2)|N

=

0 −a212 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

 ,

−ad(x, em)|N

=

0 0 b31m b41m b51m · · · bm1m3 bm1m2 0 2a212

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

;

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that is

[x, e1, e2] =a212e2,[x, e1, e3] = (m−1)a212e3,[x, e1, e4] = (m−2)a212e4,

[x, e1, ek] =

k2

X

j=3

bm2mk+jej+ (m−k+ 2)a212ek, fork= 5,6, . . . , m, and other brackets of the basis vectors are equal to zero.

For any l satisfying 3 ≤ l ≤ m−2, we take a series of linear transformations defined by

˜

ek =ek for 1≤k≤l+ 1 and ˜ek=ek− bm1ml+1

(l−1)a212ekl+1 forl+ 2≤k≤m.

Then the basis vectors ˜e1, . . . ,˜em satisfy (3.1). After replacingxby ax2

12, we get the structural matrixM ofAwith respect to the basis vectorsx,˜e1, . . . ,˜emas follows

M =

0 1 0 0 0 · · · 0 0 0 0

0 0 m−1 0 0 · · · 0 0 0 0

0 0 0 m−2 0 · · · 0 0 0 0

0 0 0 0 m−3 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 4 0 0

0 0 0 0 0 · · · 0 0 3 0

0 0 0 0 0 · · · 0 0 0 2

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 0 0 0

 .

Therefore, the multiplication table ofAis





[e1, e2, ej] =ej1 for 4≤j≤m, [e1, ej, em] =ej2 for 5≤j≤m−1, [x, e1, e2] =e2,

[x, e1, ek] = (m−k+ 2)ek for 3≤k≤m.

(13)

Let A be a solvable (m+k)-dimensional 3-Lie algebra with the maximal hypo-nilpotent idealN. Then we have k= 1.

Proof. Ifk≥2, letx1, . . . , xk, e1, . . . embe a basis ofA. Thanks to the solvability of A, we have [A, A, A] ⊆ N. By the discussions of the proof of Theorem 3.2, we might as well suppose

ad(x1, e1)|N = diag(0,1, m−1, m−2, . . . ,4,3,2),

ad(x1, e2)|N =

0 −1 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

 ,

ad(x1, em)|N =

0 0 0 0 0 · · · 0 0 0 −2

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

· · · ·

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

 ,

ad(x2, e1)|N

=

0 0 0 0 0 · 0 0 0 0

0 a212 0 0 0 · 0 0 0 0

0 0 r1a212 0 0 · 0 0 0 0

0 0 0 r2a212 0 · 0 0 0 0

0 0 bm1m2 0 r3a212 · 0 0 0 0

0 0 bm1m3 bm1m2 0 · 0 0 0 0

... ... ... ... ... ... ... ... ... ... 0 0 b51m b61m b71m · 0 rm4a212 0 0 0 0 b41m b51m b61m · bm1m2 0 rm3a212 0 0 0 b31m b41m b51m · bm1m3 bm1m2 0 rm2a212

 ,

(14)

whererj=m−j for 1≤j≤m−2

ad(x2, e2)|N =

0 −a212 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 am2m2 0 0 · · · 0 0 0 0

0 0 am2m3 am2m2 0 · · · 0 0 0 0

· · · ·

0 0 a52m a62m a72m · · · 0 0 0 0

0 0 a42m a52m a62m · · · am2m2 0 0 0

0 0 a32m a42m a52m · · · am2m3 am2m2 0 0

 ,

−ad(x2, em)|N =

0 0 b31m b41m b51m · · · bm1m3 bm1m2 0 2a212 0 0 a32m a42m a52m · · · am2m3 am2m2 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

0 0 am2m2 0 0 · · · 0 0 0 0

· · · ·

0 0 a62m a72m a82m · · · 0 0 0 0

0 0 a52m a62m a72m · · · 0 0 0 0

0 0 0 0 0 · · · 0 0 0 0

 .

Therefore, ad(x2, e1)|N −a212ad(x1, e1)|N, ad(x2, e2)|N −a212ad(x1, e2)|N and ad(x2, em)|N−a212ad(x1, em)|N

are nilpotent. It follows that I = F(x2−a212x1) +N is an (m+ 1)-dimensional hypo-nilpotent ideal ofA. This is a contradiction. Therefore, we havek= 1.

Corollary 3.4. There are no(m+k)-dimensional solvable3-Lie algebras with a maximal hypo-nilpotent ideal N whenk≥2.

4. 3-Lie algebras with nilradical N. In this section we study the solvable 3-Lie algebras with the nilradicalN.

Theorem 4.1. There are no solvable non-nilpotent3-Lie algebras with nilradical N.

Proof. First let A be an (m+k)-dimensional 3-Lie algebra with the nilpotent idealN, where 1≤k≤2. We will prove thatAis nilpotent.

参照

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