Nilpotent N-Lie Algebras

Abstract

Williams, Michael Peretzian. Nilpotent N-Lie Algebras. (Dissertation Director: Ernie Stitzinger).

Nilpotent N-Lie Algebras

by

Michael Peretzian Williams

Bachelor of Science, University of South Carolina, 1998 Master of Science, University of South Carolina, 2001

Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the

Department of Mathematics North Carolina State University

2004

Dr. Ernie Stitzinger Advisory Committee

Dr. Ronald Fulp Committee Member

Dr. Kwangil Koh Committee Member

Biography

Acknowledgements

First, I thank God for making me, making such a marvelous, intricate world and for the mathematical laws that govern it which I will never tire studying. Thank you to Jesus the Christ who hung the stars and also hung on a cross for my treachery. By whose death I may live, love and be who I was meant to be.

intelligent people who know that Jesus is the Christ. Thank you to ”H” for showing me my first glimpse of algebra, how to prove things, and for dubbing me ”Skew” and later just ”S”. Thank you to Dr. Roberts for teaching me complex variables and being a point of humorous hope against the drudgery of graduate analysis. Thank you to Ralph for teaching linear algebra for the third time, that there is no canonicle way to spell cononacle, and for introducing me to the sub-blob. A special thank you to Dr. Filaseta who either directly or indirectly or indirectly taught me everything I know about number theory, particularly thank you for teaching about Bernoulli numbers and polynomials and most importantly for being my Master’s Thesis advisor. Thank you to Dr. Luh for teaching me linear algebra for the fourth time. Thank you to Dr. Fulp for teaching me differential geometry, many theoremitos and for being on my committee. Thank you to Dr. Misra for teaching me about Lie algebras and for being on my committee. A huge thank you to Dr. Stitzinger for teaching me algebra and Galois theory, and especially for agreeing to be my thesis advisor. With out his enthusiasm, insight and great patience, this work would not have been possible.

I would like to thank my friends: Thank you to the Gowens sisters (I’m not sure if it was Pam, Cindy, Donna, or Kim), for teaching me to count past 19. Thank you to Sam for encouragement in one of the most difficult times of my life, for your humor, and for being a brother in Christ. Thank you to Joe also for being an encour-agement through two bouts with analysis. Be it Vlad’s ”And that’s it,” or Pencho’s, ”Uncountable... ah, ah,ah.” Thank you to James; you’re terrible at math but you’ve been about the best friend I’ve had. Thank you to Garrick; you introduced me to Neil and have been with me through many a tough time dulling pain with rock’n’roll. Fortunately for me this current math project is better than those we did together. Thank you to Danny for helping me to laugh through many hardships with non-linear driving. Again you’ve disappeared, I hope you come back soon.

Contents

Introduction 1

Section I: Examples of Nilpotent N-Lie Algebras 7

Section II: Conditions Equivalent To Nilpotentcy 13

Section III: Theorems That Fail To Generalize To N-Lie Algebras 17

Section IV: Some Automorphism and Derivation Theorems 20

Section V: Theorems Of Hall and Chao 29

Section VI: Uncountably Many Non-Isomorphic Real N-Lie Algebras 33

Section VII: Frattini Theory 41

Introduction

In 1960 Kurosh introduced the Ω-algebra, an algebra equipped with an n-ary,

n-linear product. He also discussed skew-symmetric Ω-algebras and noted that the class of Lie algebras is contained in them. In 1985 Fillipov introducedn-Lie algebras, a skew-symmetric Ω-alebgra with the identity

[[x₁, x₂, . . . , x_n]y₂, . . . , y_n] = n

X

i=1

[x₁, . . . , x_i−1[x_i, y₂, . . . , y_n], x_i+1, . . . , x_n]

which is called the Jacobian identity for n-Lie algebras. Note that if n = 2 this is precisely the familiar Jacobian identity for Lie algebras. Indeed if n = 2 we in fact have a Lie algebra.

In his work, Fillipov [FV] gives two main examples of n-Lie algebras.

N-cross product.

LetA have basis {x₁, . . . , xn+1}with product [x₁, x₂, . . . ,xˆ_i. . . , x_n+1] = (−1)n+1+ix_i. We shall examine this n-Lie algebras to gain insight into Frattini theory for n-Lie algebras in section VII.

Jacobian n-Lie algebra.

Let A = F[x₁, . . . , x_n]. If f₁, . . . , f_n ∈ A then [f₁, . . . , f_n] is the determinant of the Jacobian matrix of the partial derivatives of f₁, . . . , f_n.

To begin a discussion on n-Lie algebras we need some algebraic structures and linear transformations to work with. Some of the first topics discussed in Lie alge-bras and algealge-bras in general are subalgealge-bras and ideals, homomorphisms, derivations and representations. We now define these structures and transformations for n-Lie algebras. Throughout we let A be an n-Lie algebra.

Subalgebra

IfSis a subspace ofAthat is closed under then-product, then we callS asubalgebra.

Ideal

If I is a subalgebra of A such that [I, A, . . . , A] ⊂ I we call I an ideal of A and we denote it I CA.

Derivations

If δ is a linear transformation such that ([a₁, . . . , a_n])δ = P_i=1n [a₁, . . . ,(a_i)δ, . . . , a_n] for all a_j ∈A then δ is a derivation of A.

The n-Lie algebra analogue of the adjoint representation are the right multiplica-tions defined as follows:

Right Multiplications

R_(y) = [ , y₂, . . . , y_n] where (y) will always denote the sety₂, . . . , y_n of n−1 vectors right justified in the n-bracket. Example: xR_(y) = [x, y₂, . . . , y_n].

R(A)

Associative Algebra of A

A∗ is generated by the associative, binary, matrix multiplication onR(A)

Representations

Ifρ is a linear transformation such that ρ:A×A×. . .×A=An−1 −→g`(k,F) and

ρ_aρ_b−ρ_bρ_a = [ρ_a, ρ_b] = n−1

X

i=1

(a₁, . . . , a_iR_b, . . . , a_n−1)ρ

ρ([a₁, . . . , a_n], b₂, . . . , b_n) = n

X

i=1

(−1)i+1(a_i, b₂, . . . , b_n)ρ(a₁, . . . ,aˆ_i, . . . , a_n)ρ

where ρ_a=ρ(a₁, . . . , a_n) and ρ_b =ρ(b₁, . . . , b_n) then ρ is a representation of A.

Homomorphism

If π is a linear transformation such that [a₁, . . . , a_n]π= [a₁π, . . . , a_nπ] for all a_i ∈A, thenπ is a homomorphism ofA. If in addition,π is non-singular, we say thatπ is an isomorphism. Furthermore, if π is an isomorphism that takes A to A, we say thatπ

is an automorphism of A.

Simple

If A has no ideals other thanA and 0, we say that A is simple.

Lower Central Series

If A=A1, thenAk = [Ak−1, A, . . . , A].

Nilpotent

Derived Series

If A=A(1), then A(k) = [A(k−1), A(k−1), A, . . . , A].

Solvable

If A(s)= 0 for some s we say A issolvable.

The Center of A

Letz ∈A such that [z, A . . . , A] = 0. We call the collection of all such elements in A

the center of A and denote it Z(A).

The Centeralizer

LetC_A(I) = {x∈A|[x, I, A, . . . , A] = 0} and call it the centralizer of I in A.

The Normalizer

LetN(H) = {x∈A|[x, H, A, . . . , A]⊂H}and call it the normalizer of H.

The Frattini Subalgebra

φ(A) = TM where M is a maximal proper subalgebra of A.

In 1986, Kasymov [KS1-KS2] introduced the concept of nilpotent n-Lie algebras, proved an analogue of Engel’s Theorem and later proved an analogue of Jacobson’s refinement of Engel’s Theorem. They are as follows:

Engel’s Theorem

Jacobson’s Refinement of Engel’s Theorem

Let S be a spanning n-Lie set of A. If R is nilpotent for all R ∈ R(S) then A is nilpotent. Where an n-Lie set is a set of vectors that are closed under the n-bracket but not necessarily closed under addition.

Despite these achievements, the subject of nilpotentcy in n-Lie algebras has not been examined in great detail. We shall explore the concept of nilpotent n-Lie alge-bras by examining, and proving where possible, other classical nilpotent group theory and nilpotent Lie algebra results, in the n-Lie algebra setting.

We shall do so in the following ways:

I) Find examples of nilpotent n-Lie algebras by truncating Fillipov’s Jacobian n-Lie algebra.

II) Apply Kasymov’s theorems to establish results similar to those of Wielandt, in group theory and Barnes and Chao in Lie algebras. The theorem is as follows:

The following are equivalent:

1)A is a nilpotent Lie algebra (nilpotent group).

2) All maximal subalgebras (subgroups) of A are ideals (normal subgroups). 3) For H a subalgebra (subgroup) ofA, H is properly containedN(H).

Where N(H) is the normalizer ofH.

4) φ(A) = A2 (φ(A) ⊃ A2 for groups) where φ(A) is the Frattini subalgebra (sub-group) of A.

IV) Prove n-Lie analogues of J. Thompson’s and Jacobson’s work in automorphisms and derivations that imply nilpotentcy. In 1959, Thompson [TJ] proved if A admits an automorphism T such that Tp = I and T has no fixed points then A is nilpo-tent. Jacobson did this for Lie algebras. Jacobson [JN] proved in Lie algebras that if

Aadmits an non-singular derivation over a field of characteristic 0 thenAis nilpotent.

V) Prove analogues of Chao and P. Hall’s work on groups and Lie algebras respec-tively. In 1957, P. Hall [HP] proved the following for p-groups. If A/N2 and N are nilpotent then A is nilpotent. More precisely, if cl(N) = t and cl(A/N2) = m, then

cl(A)≤¡t+1₂ ¢m−¡₂t¢. In 1967, Chao [CC1] got this same result for Lie algebras.

VI) In 1962 Chao [CC2] proved that there are uncountably many non-isomorphic real Lie algebras of dimension 10 or greater. We will prove an analogue of this result.

Section I: Examples of Nilpotent N-Lie

Algebras

For Lie algebras the way to generate nilpotent examples is to look at strictly upper triangular matrices. Strictly upper triangular matrices form a nilpotent Lie algebra under the commutator bracket. Unfortunately matrices do not provide examples of

n-Lie algebras. We seek a method to generate examples that is to n-Lie algebras what strictly upper triangular matrices are for Lie algebras. To do this we examine the Jacobian n-Lie algebra, an example given by Fillipov.

LetJ =F[x₁, . . . , x_n]. Iff₁, . . . , f_n∈A then [f₁, . . . , f_n] is the determinant of the Jacobian matrix of the partial derivatives of f₁, . . . , f_n. We noted before that this is an infinite dimensional n-Lie algebra. We seek a nilpotent version of this example. To do this we will truncate the dimension making it finite.

Let deg(f) be the total degree of f for all f ∈ F[x₁, x₂, . . . , x_n]. Consider the monomials of degree 3 or more and letS be the subspace linearly generated by them. Now consider the monomials of degree r or more wherer ≥3 and let I_r be the sub-space linearly generated by them. Clearly I_r⊂S and we define J_r =S/I_r.

Theorem 1.1

J_r is a nilpotent n-Lie algebra. More precisely, cl(J_r) =

»

r−3 2n−3

¼

, where cl(J_r) = s

if s is the smallest integer such that J_rs+1 = 0.

will need to show nilpotentcy and calculate the class of J_r. To do this we will prove the following lemma.

Lemma 1.2

If f_j =Qn_i=1x_ipij ∈J are monomials for j = 1,2, . . . , n then

[f₁, f₂, . . . , f_n] =det([p_ij]) n

Y

i=1

xq_ii

whereq_i =p_i1+p_i2. . .+p_in−1.Furthermore,deg([f₁, . . . , f_n]) =

µ_P

n

j=1deg(fj)

¶

−n.

Proof of Lemma 1.2

Note that

∂f_j

∂x_i =pijx

−1 i fj

Letf_j =Qn_i=1xp_iij wherePn_i=1p_ij =deg(f_j). Using the definition of the determinant we see that

[f₁, f₂, . . . , f_n] =det

µ·

∂f_j ∂x_i

¸¶

= X

σ∈Sn

sgn(σ)

µ

∂f_σ(1) ∂x₁

∂f_σ(2) ∂x₂ . . .

∂f_σ(n) ∂x_n

¶

= X

σ∈Sn

sgn(σ)

µ

p_1σ(1)x−₁1f_σ(1)p_2σ(2)x₂−1f_σ(2). . . p_nσ(n)x−_n1f_σ(n)

¶

= X

σ∈Sn

sgn(σ)

µ

p_1σ(1)p_2σ(2). . . p_nσ(n)x₁−1x−₂1. . . x−_n1

n

Y

i,j=1

xp_iiσ(j)

¶

= X

σ∈Sn

sgn(σ)

µ

p_1σ(1)p_2σ(2). . . p_nσ(n)

n

Y

i=1

xp_iiσ(1)+piσ(2)...+piσ(n)−1

Since x_ipi1+pi2...+pin−1 is invariant under S_n, that is to say

xpi1+pi2...+pin−1

i =xipiσ(1)+piσ(2)...+piσ(n)−1, we observe,

X

σ∈Sn

sgn(σ)

µ

p_1σ(1)p_2σ(2). . . p_nσ(n)

n

Y

i=1

xp_iiσ(1)+piσ(2)...+piσ(n)−1

¶

=X

σ∈Sn

sgn(σ)

µ

p_1σ(1)p_2σ(2). . . p_nσ(n)

n

Y

i=1

xp_ii1+pi2...+pin−1

¶

= n

Y

i=1

xpi1+pi2...+pin−1

i

X

σ∈Sn

sgn(σ)

µ

p_1σ(1)p_2σ(2). . . p_nσ(n)

¶

=det([p_ij]) n

Y

i=1

xqi

i .

where q_i =p_i1+. . .+p_in−1.

This proves the first part of the lemma. Also,

deg([f₁, . . . , f_n]) = n X i=1 q_i = n X i=1

(p_i1+. . .+p_in−1)

=

µ_Xn

i=1 n X j=1 p_ij ¶ −n =

µ_Xn

j=1 n X i=1 p_ij ¶ −n =

µ_Xn

j=1

deg(f_j)

¶

−n.

This proves the second part of the lemma.

It is clear that the monomials f_j =Qn_i=1x_ipij form a basis for J. If g₁, . . . , g_n∈S

[g₁, . . . , g_n] = [ n

X

j1=1

c_j1f_1j1, . . . ,

n

X

j1=1

c_jnf_njn]

= n

X

j1=1

c_j1. . .

n

X

jn=1

c_jn[f_1j1, . . . , f_njn]

What is important to note is that we can apply second part of lemma 1.2 to any monomial term in the polynomial above and find its total degree.

For any fixed j₁, . . . , j_n

deg([f_1j1, . . . , f_njn]) =

µ_Xn

r=1

deg(f_rjr)

¶

−n ≥

µ_Xn

j=1 3

¶

−n= 2n ≥2·2 = 4.

This proves that [g₁, . . . , g_n] ∈ S and that S is a subalgebra of J. Similarly if

g₁, . . . ,gˆ_i, . . . , g_n∈S and g_i ∈I_r then for any fixed j₁, . . . , j_n

deg([f₁, . . . , f_n])≥µX j6=i

deg(f_j)

¶

+deg(f_i)−n

=3(n−1) +r−n= 2n+r−3≥2·2 +r−3 = r+ 1.

This proves that [g₁, . . . , g_n]∈I_r and that I_r CS. We see that S/I_R=J_r is ann-Lie algebra. Now to show nilpotentcy we utilize the next lemma.

Lemma 1.3

If f ∈J_rs then deg(f)≥2(s−1)n−3(s−2).

Proof Lemma 1.3

g₁ ∈J_rs and g₂, . . . , g_n∈J_r. Using lemma 1.2 we obtain

deg([g₁, g₂, . . . , g_n])

≥2(s−1)n−3(s−2) + (3(n−1)−n)

=2sn−2n−3s+ 6 + 3n−3−n = 2sn−3(s−1).

This proves the lemma.

Proof of Theorem 1.1

By lemma 1.3, it is clear that J_rs+1 ⊂ J_rs and that J_rk+1 = 0 for some k and hence J_r is nilpotent of class k. To obtain the class of J_r we set J_rs+1 = 0 for s

minimal. By the lemma 1.3, if f ∈ J_rs+1 then deg(f) ≥ 2sn− 3s + 3 ≥ r and

r−3

2n−3 ≤ s. But

r−3

2n−3 need not be an integer. We set k =

»

r−3 2n−3

¼

(where d r−3

2n−3e denotes the smallest integer greater than 2nr−−33). Then if f ∈ Jrk+1 we obtain

deg(f)≥k(2n−3) + 3≥(2n−3) r−3

2n−3+ 3 =r. Hence,cl(Jr) =

»

r−3 2n−3

¼

.

The following is an example. We take J₅ with n= 2. A basis for J₅ is {xy2, x2y, x3, y3, x3y, xy3, x2y2, x4, y4} and the multiplication table is

[ , ] xy2 x2y x3 y3 x3y xy3 x2y2 x4 y4

xy2 0xy3 -3x2y2 -5x3y 3y4 -5x3y2 1xy4 -2x2y3 7x4y 4y5 x2y 3x2y2 0x3y1 -1x4 6x14y3 -1x4y1 5x2y3 2x3y2 -4x5 8xy4 x3 5x3y1 1x4 0x5 9x2y2 3x5 9x3y2 6x4y 0x6y 12x2y3

y3 -3y4 -6xy3 -9x2y2 0y5 -9x2y3 -3y5 -6xy4 -12x2y3 0y6 x3y 5x3y2 1x4y -3x5 9x2y3 0x5y 8x3y3 4x4y2 -4x6 12x2y4

xy3 -1xy4 -5x2y3 -9x3y2 3y5 -8x3y3 0xy5 -4x2y4 -12x4y2 4y6 x2y2 2x2y3 -2x3y2 -6x4y 6xy4 -4x4y2 4x2y4 0x3y3 -8x5y 8xy5 x4 7x4y 4x5 0x6 12x2y3 4x6 12x4y2 8x5y 0x7 16x3y

We note that everything in the multiplication is of degree 4 or more. Every entry other than the upper left hand 4×4 is inI₅ and is hence 0.

We obtain

[ , ] xy2 x2y x3 y3 x3y xy3 x2y2 x4 y4

xy2 0 -3x2y2 -5x3y 3y4 0 0 0 0 0

x2y 3x2y2 0 -1x4 6x14y3 0 0 0 0 0

x3 5x3y1 1x4 0 9x2y2 0 0 0 0 0

y3 -3y4 -6xy3 -9x2y2 0 0 0 0 0 0

x3y 0 0 0 0 0 0 0 0 0

xy3 0 0 0 0 0 0 0 0 0

x2y2 0 0 0 0 0 0 0 0 0

x4 0 0 0 0 0 0 0 0 0

y4 0 0 0 0 0 0 0 0 0

We see thatJ₅2 is spanned by{x3y, xy3, x2y2, x4, y4}so if f ∈J₅2 thendeg(f) = 4. This fits our formula asdeg(f)≥(2n−3)s+ 3 = (2·2−3)1 + 3 = 4. Also we see that

cl(J₅) = 2. This also fits with our formula as

»

r−3 2n−3

¼

=

»

5−3 2·2−3

¼

Section II: Conditions Equivalent To

Nilpotentcy

Kasymov’s [KS1] n-Lie algebra version of Engel’s Theorem is as follows:

Theorem 2.1

LetRbe the right multiplications on ann-Lie algebraAin a finite dimensional vector spaceV. Suppose that the operatorsR(a₂, . . . , a_n) are nilpotent for alla₂, . . . , a_n∈A, then R(A) is nilpotent.

The following is ann-Lie algebra analogue of famous theorems of Lie algebra and group theory. These results were done by Wielandt (1936-39) in group theory and by Barnes and Chao for Lie algebras in the 1960’s. It will follow from Kasymov’s work discussed above and Fitting’s Lemma.

Theorem 2.2

LetA be a finite dimensional n-Lie Algebra where the dimension of A is k. The following are equivalent:

1)A is nilpotent.

2) All maximal subalgebras of A are ideals.

3) For H a subalgebra ofA, H is properly containedN(H). Where N(H) is the set such that [N(H), H, A, . . . , A]⊂H

Proof Theorem 2.2

We will conduct the proof in the following order: A) 1 =⇒3 =⇒2 =⇒1

B) 3⇐⇒4.

For A and most of B we closely follow the group theory proofs in W.R. Scott’s group theory book [SW]. To show φ(A) = A2 we will follow closely Scott [SW] and Chao [CC3].

Proof of 1=⇒3

We assumeAis nilpotent andH is a proper subalgebra ofA. By Engel’s Theorem forn-Lie Algebras [KS1],A∗, the associative algebra generated byR(A), is nilpotent. If we let R₁R₂R₃. . . R_t−1R_t = 0 for t minimal for all R_i ∈ A∗, then there exist R_i’s ∈A∗andv 6= 0∈Asuch thatvR₁R₂R₃. . . R_t−1 6= 0. LetX =vR₁R₂R₃. . . R_t−1. Ob-serve X is a common eigen-vector for allR ∈A∗ indeed XR=vR₁R₂R₃. . . R_t−1R= 0.

We observe that R acts nilpotently on A/H for all R ∈R(H). By the comments above there exists v 6= 0 ∈ A/H such that vR = 0 for all R ∈ R(H). This means

vR ∈ H and hence v ∈ N(H) but since v 6= 0 ∈ A/H we have that v is not in H

hence H ⊂N(H).

Proof of 3=⇒2

We observe that if every proper subalgebra H is properly contained in N(H) then any maximal subalgebra M is properly contained in N(M). But since M is a subalgebra, N(M) is also a subalgebra properly containing M. The only way this can happen is if N(M) = A (otherwise we contradict the maximality of M). Since

Proof of 2=⇒1

Assume A is not nilpotent and all maximal subalgebras are ideals. By Theorem 2.1 there exists R ∈ R(A) such that R: A−→ A is a non-nilpotent derivation on A

definedxR +[x, a₂, . . . , a_n] wherea_i ∈A.

We observe that the a₂, . . . , a_n ∈ kerR as a_iR = [a_i, a₂, . . . , a_i, . . . , a_n] = 0 for all such a_i where 2≤i≤n. Now by Fitting’s Lemma we obtain

K₁ ⊂K₂ ⊂. . .⊂K_s=K_s+1 +K. I₁ ⊃I₂ ⊃. . .⊃I_s =I_s+1 +I.

Where K_i =kerRi and I_i =ImRi and A=I ⊕K. Recall R acts non-singularly on

I. Since R is not nilpotent, K 6=A.

Now let M be a proper maximal subalgebra ofA such thatK ⊂M. By assump-tion of 2),M CA. SincexR= [x, a₂, . . . , a_n] we see thata_iR= [a_i, a₂, . . . , a_n] = 0 for 1≤ i≤ n and consequently, a_i ∈K₁ ⊂K ⊂M. We observe that R : A−→ M and henceR:I −→I∩M. SinceR acts non-singularly onI, this implies thatI =I∩M. But

I+K =A6=M =A∩M = (I+K)∩M = (I∩M)+(K∩M) = (I∩M)+K =I+K.

This is obviously a contradiction. Hence, there exist no non-nilpotent R and A is nilpotent by Theorem 2.1.

Proof of 4=⇒3

We need only assume A2 ⊂ φ(A) and M is any maximal subalgebra of A. Then

A2 ⊂ M by definition of φ(A). We observe A2 = [A, A, . . . , A] ⊂ M which implies [M, A, . . . , A]⊂M. Hence M CA.

To show 3 =⇒4 we first show that dim(M) =k−1 for all maximal subalgebras

Assume dim(M) < k−1. By 3) we get that M ⊂ N(M) with proper containment and since 3 =⇒ 2 we see that M C A. Consequently A/M is defined and there exists 06=x∈A/M such that xR(M)∈M. This means thatx+M is a subalgebra whose dimension is greater than that ofM contradicting the maximality ofM. Hence

dim(M) = k−1.

Since M C A, we note A/M is a subalgebra and it follows from the statements above that dim(A/M) = 1. Furthermore A/M is Abelian soA2 ⊂M for all such M

and consequently A2 ⊂φ(A) by definition.

To show A2 ⊃ φ(A) we observe that because A is nilpotent, there exists a nonzero compliment to A2 in A. Let u₁, . . . , u_k be a basis for the compliment and let U_i = L_{`6=i</sub> < u<sub>`} > ⊕A2. Obviously the U_i’s are maximal subalgebras and

Section III: Theorems That Fail To Generalize

To N-Lie Algebras

The following are in contrast to theorems of Lie algebras that are a consequence of Jacobson’s refinement to Engel’s theorem. Interestingly, they do not hold forn-Lie algebras withn >2.

Theorem 3.1

LetA be a finite dimensional n-Lie algebra. Then the following do not hold:

1) If A admits an automorphism T of prime order with no fixed points, then A is nilpotent.

2) IfAis over a field of characteristic 0 and admits a non-singular derivation D, then

A is nilpotent.

Proof of Theorem 3.1

We give counter examples. In both cases we examine ann-Lie algebra Awith dimen-sion n and multiplication [x₁, x₂, . . . , x_n] = x₂. Note that if R_x = [ , x₂, x₃, . . . , x_n], then ARm_x = x₂ 6= 0 for all m. Hence A is not nilpotent. We will show that this algebra admits both an automorphism and a derivation as described above.

Indeed we observe A admits the following automorphism. Let x₁T = νp−n+2x₁

is obviously a non-singular linear transformation and is an automorphism as

[x₁, x₂, . . . , x_n]T = [x₁T, x₂T, . . . , x_nT]

= [νp−n+2x₁, νx₂, . . . , νx_n] =νp−n+2νn−1[x₁, x₂, . . . , x_n]

=νp+1[x₁, x₂, . . . , x_n] =νx₂ =x₂T.

If n is odd A admits the following derivation. Let x₁D = x₁, x_iD = (−1)ix_i for alli≥2. D is obviously non-singular and derivation as

[x₁, x₂, . . . , x_n]D= n

X

i=1

[x₁, x₂, . . . , x_iD, . . . , x_n]

= [x₁D, x₂, . . . , x_n] + n

X

i=2

[x₁, x₂, . . . ,(−1)ix_i, . . . , x_n]

= [x₁, x₂, . . . , x_n] + n

X

i=2

(−1)i[x₁, x₂, . . . , x_i, . . . , x_n]

=x₂+ n

X

i=2

(−1)ix₂

=x₂+ 0

If n is even, then n ≥ 4 and A admits the following derivation. Let x₁D = x₁,

x₂D = x₂,x₃D = −1/2x₃, x_iD = (−1)i+1/2x_i for all i ≥ 4. D is obviously non-singular and a derivation as

[x₁, x₂, . . . , x_n]D= n

X

i=1

[x₁, x₂, . . . , x_iD, . . . , x_n]

=[x₁, x₂, . . . , x_n] + [x₁, x₂, . . . , x_n]

+ [x₁, x₂,−1/2x₃, . . . , x_n] + n

X

i=4

[x₁, x₂, . . . ,(−1)i+1/2x_i, . . . , x_n]

=2[x₁, x₂, . . . , x_n]−1/2[x₁, x₂, x₃, . . . , x_n]

+ n

X

i=4

(−1)i+1/2[x₁, x₂, . . . , x_n]

=3/2x₂+ n

X

i=4

(−1)i+1/2x₂

=3/2x₂+−1/2x₂

=x₂ =x₂D.

Theorem 3.2

IfA=M₁+M₂, M₁ CA,M₂ CA and both M₁ and M₂ are nilpotent, thenAis not necessarily nilpotent if n >2.

Proof of Theorem 3.2

Let {x₁, . . . , x_n} span A and the only non-zero bracket is [x₁, . . . , x_n] =x_n. Also let

M₁ ={xˆ₁, x₂. . . , xn}and let M₂ ={x₁,xˆ₂, . . . , xn}. Clearly A=M₁+M₂, M₁ CA,

Section IV: Some Automorphism and Derivation

Theorems

We begin by proving some theorems about automorphisms and derivations of n -Lie algebras.

Theorem 4.1

LetD be a derivation ofA, letλ₁, λ₂, . . . , λ_r be the corresponding characteristic val-ues of D and let A_λ1, A_λ2, . . . , A_λr be the characteristic spaces of A with respect to

D. Then the n-bracket preserves the characteristic spaces, i.e.

[A_λi

1, Aλi2, . . . , Aλin] =Aλi1+λi2+,...,+λin

Here ifA_λi

1+λi2+,...,+λin is not a a characteristic space then we sayAλi1+λi2+,...,+λin = 0.

Proof of Theorem 4.1

(∗∗)[x₁, x₂, . . . , x_n]¡D−

n

X

i=1

λ_i¢= [x₁, x₂, . . . , x_i, . . . , x_n]D−

n

X

i=1

λ_i[x₁, x₂, . . . , x_n]

= n

X

i=1

[x₁, x₂, . . . , x_iD, . . . , x_n]

− n

X

i=1

[x₁, x₂, . . . , λ_ix_i, . . . , x_n]

= n

X

i=1

We claim that

[x₁, x₂, . . . , x_n]

µ D− n X i=1 λ_i ¶_k = X

p1+...+pn=k

[x₁(D−λ₁)p1_{, x}

2(D−λ2)p2, . . . , xn(D−λn)pn].

We induct on k. For k = 1 we obtain (∗∗). Now we assume the result is true for k

and show fork+ 1. We observe

[x₁, x₂, . . . , x_n]

µ D− n X i=1 λ_i ¶_k+1

=[x₁, x₂, . . . , x_n]

µ D− n X i=1 λ_i

¶_kµ

D− n X i=1 λ_i ¶ = X

p1+...+pn=k

[x₁(D−λ₁)p1_{, x}

2(D−λ2)p2, . . . , xn(D−λn)pn]

µ D− n X i=1 λ_i ¶ = X

p1+...+pn=k

n

X

i=1

[x₁(D−λ₁)p1_{, x}

2(D−λ2)p2, . . . , xi(D−λi)pi+1, . . . , xn(D−λn)pn]

= X

p1+...+pn=k+1

[x₁(D−λ₁)p1_{, x}

2(D−λ2)p2, . . . , xn(D−λn)pn].

Now let k_i for 1 ≤ i ≤ n, be minimal such that (D−λ_i)ki_{v = 0 for all v} ∈ _A

λi

and let t= (Pn_i=1(k_i −1)) + 1. Then if x_i ∈A_λi for 1≤i≤n, we obtain

[x₁, x₂, . . . , x_n]

µ D− n X i=1 λ_i ¶_t = X

p1+...+pn=t

[x₁(D−λ₁)p1_{, x}

2(D−λ2)p2, . . . , xn(D−λn)pn]

We claim this product is 0. If we assume to the contrary then

06= [x₁, x₂, . . . , x_n]

µ D− n X i=1 λ_i ¶_t = X

p1+...+pn=t

[x₁(D−λ₁)p1_{, x}

But this implies that there exist p₁, p₂, . . . , p_n such that

[x₁(D−λ₁)p1_{, x}

2(D−λ2)p2, . . . , xn(D−λn)pn]6= 0.

This means that p_j ≤ k_i for alli and hence t=Pn_i=1p_i ≤Pn_i=1(k_i −1) =t−1 But this is a contradiction and thus proves the claim. Now since

[x₁, x₂, . . . , x_n]

µ

D−Pn_i=1λ_i

¶t

= 0 we see that [x₁, x₂, . . . , x_n] ∈ A_λ1+...+λn. This proves the lemma.

Theorem 4.2

IfT is ann-Lie algebra automorphism andA_λi are the characteristic spaces ofAwith respect to T, then the n-bracket preserves the characteristic spaces. That is

[A_λi

1, Aλi2, . . . , Aλin] =Aλi1λi2...λin.

Proof of Theorem 4.2

First we claim that

[x₁, x₂, . . . , x_n](T −λ₁λ₂. . . λ_n)

= n

X

i=1

[λ₁x₁, λ₂x₂, . . . , x_i(T −λ_i), x_i+1T, x_i+2T, . . . , x_nT].

Observe n

X

i=1

[λ₁x₁, λ₂x₂, . . . , x_i(T −λ_i), x_i+1T, x_i+2T, . . . , x_nT]

= n

X

i=1

[λ₁x₁, λ₂x₂, . . . , x_iT, x_i+1T, . . . , x_nT]

− n

X

i=1

[λ₁x₁, λ₂x₂, . . . , λ_ix_i, x_i+1T, x_i+2T, . . . , x_nT]

= n

X

i=1

[λ₁x₁, λ₂x₂, . . . , x_iT, x_i+1T, . . . , x_nT]

−

µ_Xn−1

i=1

[λ₁x₁, λ₂x₂, . . . , λ_ix_i, x_i+1T, x_i+2T, . . . , x_nT] + [λ₁x₁, λ₂x₂, . . . , λ_nx_n]

And we see

= n

X

i=1

[λ₁x₁, λ₂x₂, . . . , x_iT, x_i+1T, . . . , x_nT]

−

µ_Xn

k=2

[λ₁x₁, λ₂x₂, . . . , x_kT, x_k+1T, . . . , x_nT]−[λ₁x₁, λ₂x₂, . . . , λ_nx_n]

¶

=[x₁T, x₂T, . . . , x_nT]

+

µ_Xn

i=2

[λ₁x₁, λ₂x₂, . . . , x_iT, . . . , x_nT]− n

X

k=2

[λ₁x₁, λ₂x₂, . . . , x_kT, . . . , x_nT]

¶

−[λ₁x₁, λ₂x₂, . . . , λ_nx_n]

=[x₁T, x₂T, . . . , x_nT]−[λ₁x₁, λ₂x₂, . . . , λ_nx_n] =[x₁, x₂, . . . , x_n](T −λ₁λ₂. . . λ_n).

We claim that for x_i ∈A_λi

[x₁, x₂, . . . , x_n](T −λ₁λ₂. . . λ_n)k= X p1+...+pn=k

[y₁, y₂, . . . , y_n]

where y_i =λ_i(k−p1−...−pi)x_iT(p1+...+pi−1)_(T −_λ

i)pi and pi ∈ Z+ for all i. We induct on

k. If k = 1, then

[x₁, x₂, . . . , x_n](T −λ₁λ₂. . . λ_n)

= n

X

i=1

[λ₁x₁, λ₂x₂, . . . ,(T −λ_i)x_i, T x_i+1, . . . , T x_n]

= X

p1+...+pn=1

[λ1₁−p1x₁Tp1_{(T −λ}

1)p1, . . . , λ1i−1xnT1(T −λi)0]

= X

p1+...+pn=1

Now we assume the result is true for k and show it is true fork+ 1.

[x₁, x₂, . . . , x_n](T −λ₁λ₂. . . λ_n)k+1

= [x₁, x₂, . . . , x_n](T −λ₁λ₂. . . λ_n)(T −λ₁λ₂. . . λ_n)k

= X

p1+...+pn=k

[y₁, y₂, . . . , y_n](T −λ₁λ₂. . . λ_n)

= X

p1+...+pn=k

n

X

i=1

[λ₁y₁, . . . , y_i(T −λ_i), . . . , y_nT]

= X

p1+...+pn=k+1

[y₁, y₂. . . , y_n].

Letk_i for 1≤i≤n, be minimal such that A_λi(T −λ_i)ki _{= 0 and let}

t= (Pn_i=1k_i−1) + 1. We claim [x₁, x₂, . . . , x_n](T −λ₁λ₂. . . λ_n)t= 0. Assume to the contrary that it is non-zero. Then

[x₁, x₂, . . . , x_n](T −λ₁λ₂. . . λ_n)t= X p1+...+pn=k+1

[y₁, y₂. . . , y_n]6= 0.

This implies that there exist p₁, . . . , p_n such that the [y₁, y₂, . . . , y_n] 6= 0. Hence the

y_i’s all are non-zero. This means p_i ≤ k_i −1 for all i. We obtain t = Pn_i=1p_i ≤

P_n

i=1(ki−1) =t−1 which is impossible hence [x1, x2, . . . , xn](T −λ1λ2. . . λn)t = 0. As a result [A_λ1A_λ2. . . A_λn]∈A_λ1λ2...λn proving the lemma.

Theorem 4.3

Let D be a nilpotent derivation of A, an n-Lie algebra over a field F. Then expD=

P_m

i=0 D

i

i! is an automorphism of A under the following field considerations: either

char(F) = 0 or, if char(F) =p6= 0 andDk = 0 for some minimalk, then k < p.

Proof of Theorem 4.3

(∗)

·

x₁, x₂, . . . , x_n

¸

Dk k! =

1

k!

X

i1+...+in=k

µ

k i₁, i₂, . . . , i_n

¶

[x₁Di1_{, x}

2Di2, . . . , xnDin]

= X

i1+...+in=k

·

x₁Di1

i₁! ,

x₂Di2

i₂! , . . . ,

x_nDin

i_n!

¸

.

Hence

[expD(x₁), expD(x₂), . . . , expD(x_n)] =

·_Xm

i1=1

x₁Di1

i₁! , m

X

i2=1

x₂Di2

i₂! , . . . , m

X

in=1

x_nDin

i_n!

¸

= m

X

i1,...,in=0

·

x₁Di1

i₁! ,

x₂Di2

i₂! , . . . ,

x_nDin

i_n!

¸ = nm X k=1 X

i1+...+in=k

·

x₁Di1

i₁! ,

x₂Di2

i₂! , . . . ,

x_nDin

i_n!

¸

= nm

X

k=1

[x₁, x₂, . . . , x_n]Dk k!

=X

k=1

m[x1, x2, . . . , xn]D

k

Hence exp(D) is an n-Lie algebra homomorphism. Furthermore

(exp(D))(exp(−D)) =(exp(D))

µ_Xm

i=0

(−D)i

i! ¶ = m X i,j=0

Dj(−D)i

j!i!

= 2m X k=0 1 k! k X i=0 µ k i ¶

Di(−D)k−i

= 2m

X

k=0 1

k!(D−D) k

=I

Similarly, (exp(−D))(exp(D) = I. Hence exp(−D) = (exp(D))−1 and exp(D) is an automorphism. This proves lemma 4.3.

We now look at some applications of theorems 4.1 and 4.2. After we introduce the next theorem and corollary we will be able to find conditions for which the statements of theorem 3.1 will hold. As we have mentioned, Kasymov proved a generalization of an analogue of Jacobson’s refinement of Engel’s Theorem [KS2].

Theorem 4.4

LetAbe a finite dimensionaln-Lie algebra over a fieldF,SandT be subsets such that [S, T, T, . . . , T]⊂T and [S, S, T, T . . . , T]⊂Sand the operatorsR(x, t₂, t₃, . . . , t_n) be nilpotent for arbitraryx∈Sandt0_is ∈T. Then the associative algebraS∗is nilpotent.

From this we obtain the extremely useful n-Lie algebra analogue of Jacobson’s refinement of Engel’s theorem.

Corollary 4.5

under addition.

Proof of Corollary 4.5

Let S = T as in theorem 4.4 and span A, an n-Lie algebra. Then [S, S, . . . , S] ⊂ S

and is hence an n-Lie set. By the theorem, S∗ is nilpotent and as a result, A is nilpotent.

Now we introduce conditions that make the statements of theorem 3.1 hold.

Theorem 4.6

LetA be a finite dimensional n-Lie algebra. The following hold:

1) If A admits an automorphism T of prime order with characteristic spaces

A_λ1, A_λ2, . . . , A_λr and the product of anyn−1 of theλ0_is is not equal to 1 then A is nilpotent.

2) If A is over a field of characteristic 0, admits a non-singular derivation D with characteristic spaces A_λ1, A_λ2, . . . , A_λr, and the sum of any n−1 of the λ0_is is non-zero, then A is nilpotent.

Proof of Theorem 4.6

For 1 let R_λ = [ , x₂, x₃, . . . , x_n] for any x_i ∈ A_λi and any y ∈A_λ1. By theorem 4.2, [A_α, A_β, . . . , A_ω] = A_αβ...ω. As a result, yR_λk ∈ A_{λ1(λ2...λn)}k. Let λ₂. . . λ_n = ε and

λ₁ =η. Note that ε 6= 1 because the product of any n−1 of the λ0_is is not equal to 1. Since ε, η ∈F_p there exists i such thatεiη= 1 but this means

This is true for any choice y0s and x0_is. By theorem 4.2 the union of characteristic spaces of an automorphism form an n-Lie set. Applying corollary 4.5 we see that A

is nilpotent.

Section V: Theorems Of Hall and Chao

We now turn to work done by Hall and later Chao. In 1958 Hall proved the following in group theory. Let M C G be a nilpotent group and M2 its derived group. If G/M2 is a nilpotent group, then G is itself nilpotent. Furthermore, if

cl(M) =t and cl(G/M2) = m, then cl(G)≤¡t+1₂ ¢m−¡₂t¢.

Then in 1968, Chao proved the Lie algebra analogue of this theorem. Let L

be a Lie algebra and N C A. If N and L/N2 are nilpotent Lie algebras then L

is nilpotent. Furthermore Chao proved that if cl(L/N2) = m and cl(N) = t then

cl(L)≤¡t+1₂ ¢m−¡₂t¢ [C1]. We derive an n-Lie algebra analogue of these results.

Theorem 5.1

Let A be an n-Lie algebra with N C A. If Nt+1 = 0 and (A/N2)m+1 = 0, then

cl(A) ≤ tm+ 1₂t(t−1)(m−1)(n−1). Where cl(A) is the smallest natural number such that Acl(A)+1 = 0.

Before we start the proof we make some definitions and prove a lemma. Let

NiA= [Ni, A, . . . , A] and NiAj = [NiAj, A, . . . , A] forj >1.

Lemma 5.2

Let A/N2 be nilpotent. If Am+1 ⊂ N2 for some m, minimal then NrAu ⊂ Nr+1 for

r >0 whereu= (r−1)(n−1)(m−1) +m.

Proof of Lemma 5.2

and the base case is satisfied. We now assume the result holds for r and consider

r+ 1.

Lets=r(n−1)(m−1) +mand u= (r−1)(n−1)(m−1) +m. By the Leibnitz rule for n-Lie algebras we obtain

Nr+1As=[Nr, N, . . . , N]As

= X

s1+...+sn=s

[NrAs1_{, N A}s2_{, N A}s3_{, . . . , N A}sn_]

Suppose s₁ ≥u. Then by the induction hypothesis, NrAs1 ⊂_Nr+1 _and

X

s1+...+sn=s

[NrAs1_{, N A}s2_{, N A}s3_{, . . . , N A}sn_]⊂ X

s1+...+sn=s

[Nr+1, N, N, . . . , N]

⊂Nr+2.

Suppose s₁ < u. We claim there exists s_k ≥m. Assume to the contrary that s_j < m

for all j. We obtain

s= (s₁) + (s₂ +. . .+s_n)

< u+ (n−1)(m−1)

= (r−1)(n−1)(m−1) +m+ (n−1)(m−1)

=r(n−1)(m−1) +m =s.

But this is impossible. Hence there exists s_k ≥m for some k.

[NrAs1_{, N A}s2_{, N A}s3_{, . . . , N A}sk_{, . . . , N A}sn+1_]

= [Nr, N, N, . . . , N2, . . . , N]

= [N2, Nr, N, . . . , N]

= [[N, . . . , N], Nr, N, . . . , N]

= [[N, Nr. . . , N]N, . . . , N] + [N,[N, Nr. . . , N]N, . . . , N] +. . .+ [N, . . . , N,[N, Nr. . . , N]]

= [[N, Nr. . . , N]N, . . . , N]

= [Nr+1, N, . . . , N] =Nr+2.

This proves the lemma.

Proof of Theorem 5.1

Using lemma 5.2 we observe that

Am+1 ⊂N2

⊂N2Am+(n−1)(m−1) ⊂N3

⊂N3Am+2(n−1)(m−1) ⊂N4

⊂N4Am+3(n−1)(m−1) ⊂N5

.. .

Adding the exponents on the left hand side we see that Aw = 0 where w = tm+ 1

2t(t−1)(n−1)(m−1) + 1 proving the theorem.

Remark: If we set n= 2 we observe

cl(A)≤tm+1

2t(t−1)(2−1)(m−1) =tm+1

2(t

2_m−t2 _−tm+t)

= 1

2t(t+ 1)m− 1

2t(t−1) =

µ

t+ 1 2

¶

m−

µ

t

2

¶

.

Section VI: Uncountably Many Non-Isomorphic

Real N-Lie Algebras

Classifying nilpotent, real Lie algebras has been an often studied subject since Engel. In 1962, Chao [CC2] proved that there are uncountably many such Lie algebras of dimension 10 and greater that are non-isomorphic. We shall set about proving an

n-Lie algebra analogue of this theorem.

Theorem 6.1

There are uncountably many non-isomorphicn-Lie algebras of dimensionsdand nilpo-tent of length 2 when

1)n= 2 andd = 10. 2)n= 3 andd = 9. 3)n= 4 andd = 10. 4)n≥6 and d=n+ 4.

Definition 6.2

Let Fbe a subfield of R. An n-Lie algebra A over R is said to be an F-algebra if its structure constants with respect to some basis of A lie in F.

LetFbe a subfield ofRandC_ik_1,i2,...,inbe real numbers inFsuch thatσ(C_ik_1,i2,...,in) =

C_ik

σ(1),iσ(2),...,iσ(n) =sgnσ(Cik1,i2,...,in) for all σ ∈ Sn, the symmetric group. Let A be an

0. Note that this fits the anti-symmetric condition as,

σ([x_i1, x_i2, . . . , x_in]) = [x_iσ(1), x_iσ(2), . . . , x_iσ(n)]

=σ( n

X

k=1

C_ik_1,i2,...,iny_k)

= n

X

k=1

σ(C_ik_1,i2,...,in)y_k

= n

X

k=1

sgnσ(C_ik_1,i2,...,in)y_k

=sgnσ

µ_Xn

k=1

C_ik_1,i2,...,iny_k

¶

=sgnσ¡[x_i1, x_i2, . . . , x_in]¢.

Lemma 6.3

If the numbers C_ik_1,i2,...,in, for 1 ≤ i₁ < i₂ < . . . < i_n ≤ `, and 1 ≤ k ≤ m are algebraically independent over F and if ¡<sub>n</sub>`¢m > m2+`2, thenA is not an F-algebra.

Proof of Lemma 6.3

We want to show that A2 =< y₁, . . . , y_m >= Z(A). First we show that A2 =< y₁, . . . , y_m >. Since ¡_n`¢ > m we can pick m distinct sets of n integers between 1 and `. Each such set determines a set of vectors from x₁, . . . , x_` and we label these sets S_j, j = 1, . . . , m. Let z_j be the product of the elements in S_j where the indices are arranged in increasing order in the product. As a result z_j = Pm_k=1C_jky_k for

j = 1, . . . , m where C_jk =C_jk_1,j2,...,jn if x_j1, x_j2, . . . , x_jn ∈S_j. The polynomial det(x_ij) for i, j = 1, . . . , m has integer coefficients and thus lies in F[x₁₁, . . . , x_kj, . . . , x_mm]. If

det(C_jk) = 0, then theC_jk’s are not algebraically independent, which is a contradiction. Therefore (C_jk) is a non-singular matrix which generates < y₁, . . . , y_m > and hence

A2 =< y₁, . . . , y_m >.

Now we show thatZ(A) =< y₁, . . . , y_m >. Since the only non-zero products in A

Letz =P`_j=1a_jx_j+Pm_j=1b_ky_k ∈Z(A) and let R_π = [ , x_i2, x_i3, . . . , x_in] then,

0 =zR_π

=

µ_X`

j=1

a_jx_j

¶

R_π +

µ_Xm

k=1

b_ky_k

¶

R_π

= `

X

j=1

a_j(x_jR_π) + 0

= `

X

j=1 m

X

k=1

a_jC_jπk y_k

where C_jπk =C_jik_2i3...in.

By virtue of the linear independence of the y_k’s we obtain P`<sub>j=1</sub>a<sub>i</sub>C<sub>jπ</sub>k = 0. For each 1 ≤ t ≤ ` choose π_t = t₂, . . . , t_n such that t_r =6 t_s for r 6= s and t 6= t₂, . . . , t_n. Then P`<sub>j=1</sub>a<sub>i</sub>C<sub>jπ</sub>k<sub>t</sub> = 0. We observe that C<sub>tπ</sub>k<sub>t</sub> 6= 0 and C<sub>tπ</sub>k<sub>t</sub> is in the algebraically independent set. Repeating this process for each t, 1 ≤ t ≤ ` gives us a system of ` equations and ` unknowns. The coefficient matrix C has non-zero elements on the diagonal and hence are algebraically independent. Considering det(x_ij) as in the last paragraph, gives us a polynomial in `2 variables with coefficients +1. If C is singular then the elements of C satisfy det(x_ij). The non-zero elements of C satisfy a polynomial obtained from det(x_ij) by deleting terms if necessary, from any ele-ments of C that are 0. The resulting polynomial is non-zero because of the non-zero diagonal of C. This non-zero polynomial is satisfied by a set of algebraically inde-pendent elements. This is a contradiction and hence C is non-singular. As a result

a₁ =a₂ =. . .=a<sub>`</sub> = 0 andz =P`_j=1a_jx_j+Pm_j=1b_ky_k =Pm_j=1b_ky_k∈< y₁, . . . , y_m >. ThusZ(A) =< y₁, . . . , y_m >=A2.