Chapter 1 Introduction

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dimension 5

by

c Iren Darijani M. Sc.

A thesis submitted to the School of Graduate Studies

in partial fulfillment of the requirements for the degree of

Master of Science.

Department of Mathematics and Statistics Memorial University of Newfoundland

March 5, 2015

ST. JOHN’S NEWFOUNDLAND

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Abstract

In this thesis, we will classifyp-nilpotent restricted Lie algebras of dimension 5. Any finite dimensionalp-nilpotent restricted Lie algebras is nilpotent by Engel’s theorem. Therefore, we use as our starting point, the classification of nilpotent Lie algebras of dimension 5 and classify the possible equivalence classes ofp-maps on these Lie algebras. We first explain the method that we used to classifyp-nilpotent restricted Lie algebras of dimension 5 which is the analogue of Skjelbred-Sund method for classifying nilpotent Lie algebras. Then, we will give a complete classification ofp-nilpotent restricted Lie algebras of dimension 5 over perfect fields of characteristicp>5.

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Acknowledgements

I believe that apart from personal efforts, the success of any project depends largely on the support and guidance of many others around us. I take this opportunity to express my gratitude to the people who have been instrumental in the completeness of this work.

First of all, I would like to express my deepest gratitude to my supervisor Dr. Hamid Usefi, as you have been a tremendous mentor for me during these last two years. Your support and encouragement have not only enabled me to accomplish this work, but also inspired me to pursue my goals. I specially thank you for your patience and being available at all times. I also wish to express my appreciation for your providing me with financial assistance throughout my Master’s program.

Sincere appreciations are also given to all the staff at the Department of Mathematics and Statistics. I also gratefully acknowledge financial support from the school of Graduate studies.

I can never express my gratitude enough to my dear family who have a special place in my heart. Your constant care and attention helped me to get through every step of my life.

Thank you for your unconditional love and your endless support. I love you.

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Canada. Finally, I would like to thank my friends in Canada for their moral support when it was needed most.

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Chapter 1 Introduction

Let L be a Lie algebra over a fieldF of positive characteristic p. Recall that Lis called restricted ifLaffords ap-map that satisfies the following conditions for all x, y ∈ Land λ∈F

1. (λx)^[p]=λ^px^[p]; 2. (adx)^p =ad(x^[p]);

3. (x+y)^[p] =x^[p]+y^[p]+Pp−1

j=1s_j(x, y),

wherejs_j(x, y)is the coefficient oft^j−1in(ad(tx+y))^p−1(x),tan indeterminate.

Recall that a restricted Lie algebraLis calledp-nilpotent, if there exists an integernsuch that x^[p]ⁿ = 0, for all x ∈ L. The purpose of this thesis is to classify all p-nilpotent restricted Lie algebras of dimension 5 over perfect fields of characteristicp > 5. It follows from the Engel Theorem that ifLis finite-dimensional andp-nilpotent thenLis nilpotent.

Our work builds upon the recent work of Schneider and Usefi [12] on the classification of p-nilpotent restricted Lie algebras of dimension up to 4 over perfect fields of characteristic p. Our method is different than what is used in [12] as we describe below. The analogous

5

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classification for small dimensional nilpotent Lie algebras has a long history. The classification of all nilpotent Lie algebras of dimension up to five over any field has been known for a long time. However, in dimension 6, the characterization depends on the underlying field. In 1958 Morozov [8] gave a classification of nilpotent Lie algebras of dimension 6 over a field of characteristic zero, see also [1, 7, 9, 11] for a classification over other fields.

These classifications, however, differ and it was not easy to compare them until recently that de Graaf [4] gave a complete classification over any field of characteristic other than 2. de Graaf’s approach can be verified computationally and was later revised and extended to characteristic 2 in [2]. The classification in dimensions more than 6 is still in progress, see for example [13, 10].

We now describe the method used in [12] and explain why this method is not applicable in dimension 5. Note that in order to define a p-map onL, it is enough to define it on a basis of L and then extend it to linear combinations using properties (1) and (3). Let ϕ₁, ϕ₂ :L→Lbe twop-maps onL. Then the restricted Lie algebras(L, ϕ₁)and(L, ϕ₂) are isomorphic if and only if there existsA∈Aut(L)such that

A(ϕ₁(x)) =ϕ₂(A(x)) holds for all x∈L.

Hence,ϕ₁ andϕ₂ define isomorphic restricted Lie algebras if and only if there existsA ∈ Aut(L)such thatAϕ₁A⁻¹ =ϕ₂; that is, they are conjugate under the automorphism group ofL. In this case we say that thep-mapsϕ₁andϕ₂areequivalent. This defines a left action of Aut(L) on the set of p-maps and the isomorphism classes of restricted Lie algebras correspond to the Aut(L)-orbits under this action. The main task using this approach would be then to find the Aut(L)-orbits. This is exactly what the authors did in [12] to determine all p-nilpotent restricted Lie algebras of dimension up to 4 over perfect fields.

However, this task becomes computationally infeasible to carry out in dimension 5.

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The method we use to classifyp-nilpotent restricted Lie algebras of dimension 5 is the analogue of Skjelbred-Sund method for classifying nilpotent Lie algebras. We describe this method below.

Let L be a Lie algebra over F and M a vector space, a q-dimensional cochain of L with coefficients in M is a skew-symmetric, q-linear map on L taking values in M. We denote the space ofq-dimensional cochains of a Lie algebra Lwith coefficients in M by C_cl^q(L, M). The coboundary mapδ^q:C_cl^q(L, M)→C_cl^q+1(L, M)is defined by

(δ^qφ)(l1, . . . , lq+1) = X

1≤s<t≤q+1

(−1)^s+t−1φ([ls, lt], l1, . . . ,lbs, . . . ,blt, . . . , lq+1),

where the symbol lb_s indicates that this term is to be omitted. Now, let L be a restricted Lie algebra and M a vector space. We view M as a trivialL-module. Let φ be a skew- symmetric, 2-linear map onLtaking values inM andω : L → M a map. We sayω has

?-property with respect toφ,if for everyx, y ∈Landλ∈F, we have 1. ω(λx) =λ^pω(x)

2. ω(x+y) = ω(x) +ω(y) + X

xj=xory x1=x,x2=y

1

#xφ([x₁, x₂, . . . , xp−1], x_p), where #xis the number ofx.

We define the space of 2-dimensional cochains of a restricted Lie algebra L with coefficients in M as the subspace spanned by all such (φ, ω). We denote this vector space by C²(L, M). Let ψ : L → M a map. It can be verified thatψ˜ : L → M defined by ψ(x) =˜ ψ(x^[p])has the?-property with toδ¹ψ. We define

Z²(L, M) ={(φ, ω)∈C²(L, M)|δ²φ = 0, φ(x, y^[p]) = φ([x, y, . . . , y

| {z }

p−1

], y)},

B²(L, M) ={(φ, ω)∈C²(L, M)| ∃ψ ∈C_cl¹(L, M), s.t. δ¹ψ =φ,ψ˜=ω}.

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It can be seen that B²(L, M)⊆ Z²(L, M),so that, the quotientZ²(L, M)/B²(L, M) is well-defined. The quotientZ²(L, M)/B²(L, M)is called the second cohomology group of the restricted Lie algebraLwith coefficients inM and we denote it byH²(L, M).

Let[θ] = [(φ, ω)]∈ H²(L, M). We setL_θ = L⊕M as a vector space and define the Lie bracket andp-map onL_θ by:

[(x1 +m1),(x2+m2)] = φ(x1, x2) + [x1, x2], (x+m)^[p] =ω(x) +x^[p]. ThenL_θ with the given bracket andp-map is a restricted Lie algebra.

Now letK be ap-nilpotent restricted Lie algebra. Then its centerZ(K)is nonzero and there existsx∈Z(K)such thatx^[p]= 0. LetM be the one dimensional restricted ideal of K spanned byx, and setL= K/M. Letπ : K →Lbe the projection map. We have the exact sequence of restricted Lie algebras:

0→M →K →L→0.

Choose an injective linear mapσ :L→K such thatπσ = 1_L. Defineφ :L×L→M by φ(x₁, x₂) = [σ(x₁), σ(x₂)]−σ([x₁, x₂])andω : L → M byω(x) = σ(x)^[p]−σ(x^[p]). It turns out that[θ] = [(φ, ω)]∈H²(L, M)andK ∼=L_θ. Therefore, anyp-nilpotent restricted Lie algebra K of dimension n can be constructed as a central extensions of ap-nilpotent restricted Lie algebras of dimensionn−1.

Now, the group of automorphisms Aut(L) acts on H²(L, M) in a natural way. Let A ∈ Aut(L) and [θ] = [(φ, ω)] ∈ H²(L, M). We define [Aθ] = [(Aφ, Aω)], where Aφ(x, y) = φ(A(x), A(y))andAω(x) =ω(A(x)). Let[θ₁],[θ₂] ∈H²(L, M). It turns out that[θ₁]and[θ₂]are in the same Aut(L)-orbit if and only if there exists an isomorphism f :L_θ₁ →L_θ₂ such thatf(M) =M. Therefore, we use the action ofAut(L)to reduce the number of isomorphic restricted Lie algebras.

There are nine nilpotent Lie algebras of dimension 5 that we denote them byL5,i, for 1 6i 6 9. Letibe in the range1 6 i6 9and setK =L5,i. As we mentioned before, a

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p-map is determined by its action on anF-basisx1, . . . , x5 ofK. Since thep-maps arep- nilpotent, there exists a central elementx∈K such thatx^[p] = 0. Then we letL=K/hxi and use the methods above to find all 1-dimensional central extensions ofLthat lead toK. That is we choose those [θ] = [(φ, ω)] ∈ H²(L,hxi)such that L_θ is isomorphic as a Lie algebra toK. Then we list all possiblep-maps that are obtained via different choices ofθ andx. We still need to detect and remove the isomorphic algebras from this list. Finally, we shall prove that the remaining algebras in the list are pairwise non-isomorphic. Please note that a copy of this thesis exists in Arxiv.

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Chapter 2 Preliminaries

2.1 Restricted Lie algebras

We first recall some definitions and notations that are mostly adopted from [15].

Definition 2.1.1 A restricted Lie algebra of characteristic p > 0 is a Lie algebra L of characteristicptogether with a mapL→L, denoted byx7→x^[p], that satisfies

• (λx)^[p]=λ^px^[p],

• (x+y)^[p] =x^[p]+y^[p]+Pp−1

j=1s_j(x, y)

wherejs_j(x, y)is the coefficient oft^j−1 in(ad(tx+y))^p−1(x),tan indeterminate,

• [x, y^[p]] = [x, y, . . . , y

| {z }

p

],

for allx, y ∈Land allλ∈F.

The mapx7→x^[p]is referred to as thep-operator. Note that the second property is equivalent to

(x+y)^[p]=x^[p]+y^[p]+ X

xj=xory x1=x,x2=y

1

#x[x₁, x₂, . . . , x_p],

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where#xdenotes the number ofx’s among thexj. Note that long commutators are left-tapped, that is

[x₁, . . . , x_k, x_k+1] = [[x₁, . . . , x_k], x_k+1].

If L is a Lie algebra over F and M is vector space, a q-dimensional cochain of L with coefficients in M is a skew-symmetric, q-linear map on L taking values in M. We denote the space ofq-dimensional cochains of a Lie algebra Lwith coefficients in M by C_cl^q(L, M). So, we have

C_cl^q(L, M) = Hom_F(Λ^qL, M).

The coboundary mapδ^q :C_cl^q(L, M)→C_cl^q+1(L, M)is defined by (δ^qφ)(l₁, . . . , l_q+1) = X

1≤s<t≤q+1

(−1)^s+t−1φ([l_s, l_t], l₁, . . . ,lb_s, . . . ,bl_t, . . . , l_q+1),

where the symbollb_sindicates that this term is to be omitted.

Definition 2.1.2 LetL be a restricted Lie algebra over F andM a vector space. Ifφ ∈ C_cl²(L, M)andω :L→M a function, we sayωhas the?-property with respect toφif for everyx, y ∈Landλ∈F, we have

1. ω(λx) =λ^pω(x)

2. ω(x+y) = ω(x) + ω(y) + X

xj=xory x1=x,x2=y

1

#xφ([x₁, x₂, . . . , xp−1], x_p), where#xis the number ofx.

Now, we define the space of 2-dimensional cochains of a restricted Lie algberaLwith coefficients inM as the subspace spanned by all(φ, ω)such thatφis skew-symmetric and ω :L→M has the?-property with respect toφ. We denote this vector space byC²(L, M).

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Evidently ifωandω⁰have the?-property with respect toφandφ⁰ respectively, thenω+ω⁰ has the ?-property with respect to φ+φ⁰, and hence C²(L, M) is a vector space over F by point wise addition in each coordinate. We have adopted these definitions from [5, 6], however, the definition of?-property in the whole generality given in [5, 6] is ambiguous.

Lemma 2.1.3 LetM be a vector space andψ :L → M a linear map. Then ψ˜: L →M defined byψ(x) =˜ ψ(x^[p])has the?-property with respect toδ¹ψ.

Proof. Clearly,ψ(λx) =˜ λ^pψ(x), for every˜ λ∈F. Sinceψ[x₁, . . . , x_p] = (δ¹ψ)([x₁, . . . , xp−1], x_p), we have

ψ(x˜ +y) = ψ((x+y)^[p]) = ψ(x^[p]) +ψ(y^[p]) +ψ( X

xj=xory x1=x,x2=y

1

#x[x₁, x₂, . . . , x_p])

=ψ(x^[p]) +ψ(y^[p]) + X

xj=xory x1=x,x2=y

1

#(x)(δ¹ψ)([x₁, . . . , xp−1], x_p),

for everyx, y ∈L.

Definition 2.1.4 We define

Z²(L, M) ={(φ, ω)∈C²(L, M)|δ²φ = 0, φ(x, y^[p]) = φ([x, y, . . . , y

| {z }

p−1

], y)},

B²(L, M) ={(φ, ω)∈C²(L, M)| ∃ψ ∈C_cl¹(L, M), s.t. δ¹ψ =φ,ψ˜=ω}.

Note that it is easy to verify thatZ²(L, M)andB²(L, M)are subspaces ofC²(L, M).

Theorem 2.1.5 B²(L, M)⊆Z²(L, M),so that, the quotient

H²(L, M) = Z²(L, M)/B²(L, M)

is well-defined.

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Proof. Let(δ¹ψ,ψ)˜ ∈B²(L, M). First, We claim thatδ²δ¹ψ = 0. Indeed, for allx, y, z ∈ L,we have

δ²δ¹ψ(x, y, z) =δ¹ψ([x, y], z) +δ¹ψ([y, z], x) +δ¹ψ([z, x], y)

=ψ([[x, y], z]]) +ψ([[y, z], x]]) +ψ([[z, x], y]])

=ψ([[x, y], z]] + [[y, z], x]] + [[z, x], y]]),

which is equal to zero by jacobi identity. Next, we claim that (δ¹ψ)(x, y^[p]) = (δ¹ψ)([x, y, ..., y

| {z }

p−1

], y),

for allx, y ∈L. Indeed, for allx, y ∈L, we have (δ¹ψ)(x, y^[p])−(δ¹ψ)([x, y, . . . , y

| {z }

p−1

], y) =ψ[x, y^[p]]−ψ([x, y, . . . , y

| {z }

p

])

=ψ([x, y^[p]]−[x, y, . . . , y

| {z }

p

])

=0.

The proof is complete.

We call H²(L, M) the second cohomology group of L with coefficients in M. Let θ = (φ, ω)∈Z²(L, M). Then we denote by[θ]the image ofθinH²(L, M).

Definition 2.1.6 A restricted Lie algebraM is called strongly abelian if,[M, M] = 0and M^[p]= 0.

2.2 Constructing p-nilpotent restricted Lie algebras

The analogue of the results of the remaining of this chapter for Lie algebras is known [14, 4]. However, we could not find a refrence for the corresponding results in the setting

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of restricted Lie algebras.

Let Lbe a restricted Lie algebra,M a vector space and θ = (φ, ω) ∈ Z²(L, M). We construct a restricted extension ofLbyM as follows.

Lemma 2.2.1 LetL_θ =L⊕M as a vector space and define the Lie bracket andp-map on L_θby:

[(x₁ +m₁),(x₂+m₂)] = [x₁, x₂] +φ(x₁, x₂), (x+m)^[p] =x^[p]+ω(x).

ThenL_θ with the given bracket andp-map is a resticted Lie algebra.

Proof. The bracket is clearly bilinear and skew symmetric and it is well known that the jacobi identity is equivalent toδ²φ = 0. We claim thatLis restricted with the givenp-map.

Letx₁, . . . , x_k+1 ∈L,m₁, . . . m_k+1 ∈M. Note that by induction we have

[x₁+m₁, x₂+m₂, . . . , x_k+1+m_k+1] = [x₁, . . . , x_k+1] +φ([x₁, . . . , x_k], x_k+1). (2.1) Now, we have

[x₁+m₁,(x₂+m₂)^[p]] =[x₁+m₁, x^[p]₂ +ω(x)]

=[x₁, x^[p]₂ ] +φ(x₁, x^[p]₂ ).

On the other hand,

[x₁+m₁, x₂+m₂, . . . , x₂ +m₂

| {z }

p

] = [x₁, x₂, . . . , x₂

| {z }

p

] +φ([x₁, x₂, . . . , x₂

| {z }

p−1

], x₂),

by equation (2.1). We have

φ(x1, x^[p]₂ ) =φ([x1, x2, . . . , x2

| {z }

p−1

], x2),

also we have

[x₁, x^[p]₂ ] = [x₁, x₂, . . . , x₂

| {z }

p

].

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Therefore,

[x₁ +m₁,(x₂+m₂)^[p]] = [x₁+m₁, x₂+m₂, . . . , x₂ +m₂

| {z }

p

].

Next, we have

(λ(x+m))^[p]= (λx+λm)^[p]= (λx)^[p]+ω(λx) =λ^px^[p]+λ^pω(x)

=λ^p(x^[p]+ω(x))

=λ^p(x+m)^[p].

Finally, we have

((x₁+m₁) + (x₂+m₂))^[p]=((x₁+x₂) + (m₁+m₂))^[p]

=(x₁+x₂)^[p]+ω(x₁+x₂)

=x^[p]₁ +x^[p]₂ + X

xlj=x1 orx2 xl1=x1,xl2=x2

1

#x₁[x_l₁, x_l₂, . . . , x_l_p]

+ω(x1) +ω(x2) + X

1

#x₁φ([xl1, xl2, . . . , xlp−1], xlp).

On the other hand,

(x1+m1)^[p]+ (x2+m2)^[p]+ X

lj=1 or 2 l1=1,l2=2

1

#(x₁+m₁)[xl1 +ml1, xl2 +ml2, . . . , xlp +mlp]

=x^[p]₁ +ω(x₁) +x^[p]₂ +ω(x₂) + X

1

#x₁[x_l₁, x_l₂, . . . , x_l_p]

+ X

1

#x₁φ([x_l₁, x_l₂, . . . , x_l_p−1], x_l_p),

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by equation (2.1). Therefore,

((x₁+m₁) + (x₂+m₂))^[p]= (x₁+m₁)^[p]+ (x₂+m₂)^[p]

+ X

lj=1 or 2 l1=1,l2=2

1

#x₁ +m₁[x_l₁ +m_l₁, x_l₂ +m_l₂, . . . , x_l_p +m_l_p].

The proof is complete.

Now letK be a restricted Lie algebra, and suppose that its centerZ(K),is nonzero. Let 06=M ⊆Z(K)such thatM^[p]= 0, and setL=K/M. Letπ :K → Lbe the projection map. We have the exact sequence

0→M →K →L→0.

Choose an injective linear mapσ : L → K such that πσ = 1_L. Note that we can easily show thatπ([σ(x_i), σ(x_j)]−σ([x_i, x_j])) = 0,for everyx_i, x_j ∈Landπ(σ(x)^[p]−σ(x^[p])) = 0, for everyx∈L. Therefore,[σ(x_i), σ(x_j)]−σ([x_i, x_j]), σ(x)^[p]−σ(x^[p])∈M, for every xi, xj, x ∈L. Now, we defineφ :L×L→M byφ(xi, xj) = [σ(xi), σ(xj)]−σ([xi, xj]) andω :L→M byω(x) =σ(x)^[p]−σ(x^[p]). With these notations, we have:

Lemma 2.2.2 Letθ = (φ, ω). Thenθ ∈Z²(L, M)andK ∼=L_θ.

Proof. It is easy to see that φis a bilinear and skew-symmetric form onL. We claim that

Chapter 1 Introduction

Abstract

Acknowledgements

Contents

Chapter 1 Introduction

Chapter 2

Preliminaries