Varieties of Nilpotent Complex Leibniz Algebras of Dimension Less than Five

Varieties of nilpotent complex Leibniz algebras of dimension

less than five

S.Albeverio1,B.A.Omirov2, I.S.Rakhimov3

Abstract

The aim of this work is to describe the irreducible components of the nilpotent complex Leibniz

algebras varieties of dimension less than 5. We construct degenerations between one-parametric

families of nilpotent Leibniz algebras and study rigidity of these families.

1 Institut fuur Angewandte Mathematik, Universitat Bonn, Wegelerstr.6, D-53115 Bonn (Ger-

many).

2 Institute of Mathematics, Uzbekistan Academy of Sciences, F.Hodjaev str.29, 700143, Tashkent

(Uzbekistan), e-mails: [email protected], [email protected]

3 National University of Uzbekistan, VUZ gorodok NUU, 700174, Tashkent (Uzbekistan), e-mails:

[email protected], [email protected]

AMS classification numbers: 14D06, 14L30, 14R20, 16D70, 17A32

Key words: variety of Leibniz algebras, nilpotence, irreducible component, degeneration, rigid

algebra.

0. Introduction

Let V be a vector space of dimension n over the field of complex numbers C. An n-dimensional

Leibniz algebra L may be considered as an element λ of the affine variety Hom(V ⊗V, V ) via the bilinear

mapping λ : L⊗ L→ L defining the Leibniz bracket on L. The set of Leibniz algebra structures is an

algebraic subset Leibn(C) of the variety Hom(V ⊗ V, V ) and the linear reductive group GLn(C) acts

on Leibn(C) by (g ∗ λ)(x, y) = g(λ(g−1(x), g−1(y))). The orbits under this action are the isomorphic

classes of algebras. For given two algebras λ and µ we will say that λ degenerates to µ, if µ lies in the

Zariski closure of the orbit λ. We denote this by λ → µ. It is easy to see that any nilpotent Leibniz

algebra degenerates to the abelian algebra.

In the present paper we are going to investigate the subvariety LNn(C) of Leibn(C) consisting of

nilpotent Leibniz algebras. This subvariety is invariant relatively to the above mentioned action of

GLn(C).

1

There are algebras the orbits of which are open in LNn(C). These algebras are called rigid. The

orbits of the rigid algebras give irreducible components of the variety LNn(C). Hence to describe the

variety LNn(C) it is enough to describe all rigid nilpotent Leibniz algebras and rigid family of the

nilpotent Leibniz algebras. By Neotherian consideration they are finite number.

We describe the variety LNn(C) when n ≤ 4.

1. Preliminaries

Definition 1. An algebra L over a field F is called a Leibniz algebra if it satisfies the following

Leibniz identity:

[x, [y, z]] = [[x, y], z]− [[x, z], y],

where [·, ·] denotes the multiplication in L.

Let L be a Leibniz algebra. We put:

L1 = L, Lk+1 = [Lk, L], k ∈ N.

Definition 2. A Leibniz algebra L is called nilpotent if there exists an integer s ∈ N, such that

L1 ⊃ L2 ⊃ ... ⊃ Ls = {0}. The smallest integer s for which Ls = 0 is called the nilindex of L.

Let V be an n-dimensional vector space over the field of complex numbers C. The bilinear maps

V ×V → V form an n3-dimensional affine space B(V) over C. We will consider the Zariski topology on

this space. We recall that the set is called irreducible if it can not be represent as a union of two non

trivial closed sets, otherwise it is called reducible. The maximal irreducible closed subset of a variety

is called an irreducible component.

We consider the set of all n-dimensional nilpotent Leibniz algebras LNn(C). The linear reductive

group GLn(C) acts on LNn(C) in the following way:

(g ∗ µ)(x, y) := g(µ(g−1(x), g−1(y))), where g ∈ GLn(C), µ ∈ LNn(C).

The orbits relatively to this action consist of all algebras which are isomorphic to each other.

The set LNn(C) can be included into the above mentioned n3-dimensional affine space B(V) in the

following way: let {e1, e2, . . . , en} be a basis of the Leibniz algebra λ. Then the table of its multiplication

is represented by a point (γkij) of this affine space as follows:

λ(ei, ej) =

n∑k=1

γkijek.

2

Thus, the algebra λ correspondents to the point (γkij) of B(V). They are called structural constants

of λ.

The Leibniz identity and nilpotence give polynomial relations among γkij . Therefore, LNn(C) is a

closed subset of B(V).

Definition 3. A Leibniz algebra λ is said to degenerate to another Leibniz algebra µ, if µ is

represented by a structure which lies in the Zariski closure of the GLn(C)-orbit of a structure which

represents λ. In this case the entire orbit Orb(µ) lies in the closure of Orb(λ). We denote this by λ→ µ.

Remark 1. Degeneration is transitive, that is if λ→ µ and µ→ ν then λ→ ν.

We will use a few facts from the algebraic groups theory. The first of them concerns constructive

subsets of algebraic varieties, the closures of which relatively to the Euclidean and the Zariski topologies

coincide. It is not hard to see that the GLn(C)-orbits are constructive sets. Therefore the usual

Euclidean topology on Cn3

leads to the same degenerations as does the Zariski topology, that is the

following condition will imply that λ→ µ :

∃gt ∈ GLn(C(t)) such that limt→0

gt ∗ λ = µ,

where C(t) is the field of fractions of the polynomial ring C[t].

The second fact concerns on closure of the GLn(C)-orbits, stating that the boundary of each orbit

consists of the union of orbits whose dimension is strictly less than the dimension of the given orbit.

It follows that each irreducible component of the variety, on which the algebraic group GLn(C) acts,

does not contain two orbits of maximal dimensions, that is the orbit of maximal dimension is unique.

It is obvious that its’ representatives are rigid.

Remark 2. It is easy to note that a rigid nilpotent algebra can not be obtained as degeneration

of any other nilpotent algebra.

The description of the variety of any class of algebras is a very difficult problem. Results in [1]-

[4], concerning applications of algebraic groups theory to the description of the variety of unitary

associative and Lie algebras gave a great momentum to the investigation of the classification problem

for other classes of algebras. Some investigations use for this problem nonstandard analysis methods

[5].

For a given Leibniz algebra λ we put:

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• R(λ) = {x ∈ λ|[λ, x] = 0} – the right annihilator of λ;

• L(λ) = {x ∈ λ|[x, λ] = 0} – the left annihilator of λ;

• Z(λ) = {x ∈ λ|[x, λ] = [λ, x] = 0} – the center of λ;

• Aut(λ) – the group of automorphisms of λ;

• λk = [λk−1, λ] – the k-th degree of λ;

• SA(λ) – the maximal abelian subalgebra of λ;

• Com(λ) – the maximal commutative subalgebra of λ;

• Lie(λ) – the maximal Lie subalgebra of λ.

Proposition 1. For any m, r ∈ N the following subsets of LNn(C) are closed relatively to the

Zariski topology:

1. {λ ∈ LNn(C) | dimλm ≤ r}

2. {λ ∈ LNn(C) | dimR(λ) ≥ m}

3. {λ ∈ LNn(C) | dimL(λ) ≥ m}

4. {λ ∈ LNn(C) | dimZ(λ) ≥ m}

5. {λ ∈ LNn(C) | dimAut(λ) > m}

6. {λ ∈ LNn(C) | dimSA(λ) ≥ m}

7. {λ ∈ LNn(C) | dimCom(λ) ≥ m}

8. {λ ∈ LNn(C) | dimLie(λ) ≥ m}

Proof. The proof of the parts 1-5 is similar to the one for Lie algebras [3]. As to the proof of parts

6-8 it suffices to realize that these statements are special cases of the following more general fact: let

B be a Borel subgroup of GLn(C) and λ, µ in LNn(C). If λ→ µ and λ lies in a B-stable closed subset

R ⊂ LNn(C) then µ must also lie in R. It is not hard to check that the subsets 6-8 are stable relatively

to the Borel subgroup consisting of lower triangular matrices. The proof is complete.

Corollary. An algebra λ does not degenerate to µ if one of the following conditions is valid:

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1. dimλm < dimµm for some µ,

2. dimR(λ) > dimR(µ),

3. dimL(λ) > dimL(µ),

4. dimZ(λ) > dimZ(µ),

5. dimAut(λ) ≥ dimAut(µ),

6. dimSA(λ) > dimSA(µ),

7. dimCom(λ) > dimCom(µ),

8. dimLie(λ) > dimLie(µ).

Below for the sake of convenience brackets [ , ] are omitted and we assume that the undefined

multiplications are zero.

Proposition 2. Let λ be a non Lie algebra in Leibn(C). Then λ → µ ⊕ Cn−2, where µ is two-

dimensional non abelian nilpotent Leibniz algebra.

Proof. Since λ is a non Lie Leibniz algebra then there exists x such that xx = y, where y 6= 0.

These two elements are linear independent. Indeed, if we have α1x+ α2y = 0 for these elements then

multiplying the both sides of this equality to x by the left we obtain that α1xx = α1y = 0, it follows

α1 = 0 and α2 = 0.

Thus, x and y can be included to the basis of λ : e1 = x, e2 = y, e3, ..., en. Then taking the following

family gt in GLn(C) : gt(e1) = t−1e1, gt(ei) = t−2ei (2 ≤ i ≤ n), we obtain that λ→ µ⊕Cn−2, where

µ is defined by the following table of multiplications: e1e1 = e2. But by J.-L. Loday the algebra µ is

unique non abelian two-dimensional nilpotent Leibniz algebra [6].

The proof is complete.

Definition 4. An n-dimensional Leibniz algebra L is said to be nulfiliform if dimLi = n − i + 1,

where 1 ≤ i ≤ n+ 1.

It should be noted that the nulfiliform Leibniz algebra is unique and it is rigid in LNn(C).

Definition 5. An n-dimensional Leibniz algebra L is said to be filiform if dimLi = n − i, where

2 ≤ i ≤ n.

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Proposition 3. Any n-dimensional nulfiliform and non Lie filiform (n ≥ 4) Leibniz algebra

degenerates to the algebra ν ⊕ Cn−3, where ν is three-dimensional non abelian nilpotent Leibniz

algebra with the following table of multiplications:

e2e2 = e1, e2e3 = e1.

Proof. First we consider of nulfiliform algebra case. By the lemma 1 of [7] a complex n-dimensional

nulfiliform Leibniz algebra can be represented on a basis {e1, e2, . . . , en} by the following table of

multiplications:

eie1 = ei+1 (1 ≤ i ≤ n− 1).

It is easy to check that the following family gt in GLn(C) :

gt(e1) = t−2e2, gt(e2) = t−4e1, gt(e3) = t−3e2 − t−3e3,

gt(e4) = t−5e1 + t−4e4, gt(ei) = t−1ei (5 ≤ i ≤ n)

gives us corresponding transformations.

Now, let us consider the filiform Leibniz algebras case. By the theorem 2 of [7] any complex

n-dimensional filiform non Lie Leibniz algebra is isomorphic to one of the following algebras:

e0e0 = e2, eie0 = ei+1 (1 ≤ i ≤ n− 2), e0e1 = α3e3 + α4e4 + · · ·+ αn−2en−2 + θen−1,

eie1 = α3ei+2 + α4ei+3 + · · ·+ αn−ien−1 (1 ≤ i ≤ n− 3), (with α3, α4, ..., αn−1 ∈ C and θ ∈ C)

and

e0e0 = e2, eie0 = ei+1 (2 ≤ i ≤ n− 2), e0e1 = β3e3 + β4e4 + · · ·+ βn−1en−1,

e1e1 = γen−1, eie1 = β3ei+2+β4ei+3+· · ·+βn−ien−1 (2 ≤ i ≤ n−3), (with β3, β4, ..., βn−1 ∈ C and γ ∈ C)

where {e0, e1, . . . , en−1} is a basis of L.

Checking up the following family gt of transformations from GLn(C) :

gt(e0) = t−1e2, gt(e1) = t−1e2 − t−1e3, gt(e2) = t−2e1,

gt(e3) = t−1e0, gt(ei) = t−1ei (4 ≤ i ≤ n− 1)

it is easy to conclude that the first class degenerates to ν ⊕ Cn−3.

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As for the second class, it can be degenerated to ν⊕Cn−3 by the following family of transformations

gt in GLn(C) :

gt(e0) = t−2e2, gt(e1) = t−3e0, gt(e2) = t−4e1 + t−2e2 − t−2e3,

gt(e3) = t−4e1, gt(ei) = tei (4 ≤ i ≤ n− 1).

The proof is complete.

2. Degenerations of nilpotent Leibniz algebras of dimension less than four

The classification up to isomorphisms of complex nilpotent Leibniz algebras in dimension 2 and 3

can be found in [6], [8]. In dimension two there are two non isomorphic nilpotent Leibniz algebras. One

of them is abelian, the other, λ, can be given by the following table e1e1 = e2. In this case, LN2(C) is

irreducible and coincides with Orb(λ).

In dimension three there are five non isomorphic each other algebras and one infinite family of

pairwise not isomorphic algebras. Below the volumes of invariants of these algebras are presented:

mult. table of λ dim λ2 dim λ3 dim λ4 dim R(λ) dim L(λ) dim Z(λ) dim Aut(λ)

λ1 abelian 0 0 0 3 3 3 9

λ2 e1e1 = e2 1 0 0 2 2 2 5

λ3 e2e3 = e1 1 0 0 1 1 1 6

e3e2 = −e1

e2e2 = e1

λ4 e3e3 = αe1 1 0 0 2(α = 0) 2(α = 0) 1 4

e2e3 = e1 (α ∈ C) 1(α 6= 0) 1(α 6= 0)

e2e2 = e1

λ5 e3e2 = e1 1 0 0 1 1 1 4

e2e3 = e1

λ6 e3e3 = e1 2 1 0 2 1 1 3

e1e3 = e2

Using this table and the corollary to the proposition 1 we write down all possibilities for degener-

7

ations of nilpotent three-dimensional Leibniz algebras:

λ4 (α = 0)→ λ1, λ2,

λ4 (α 6= 0)→ λ1, λ2, λ3,

λ5 → λ1, λ2,

λ6 → λ1, λ2, λ4(α = 0).

We note that by proposition 2 all non Lie Leibniz algebras degenerate to λ2.

As for λ3, it can be considered as an associative algebra and its nonrigidity follows from [9].

Moreover, the algebra λ6 degenerates to the algebra λ4 (α = 0) via the following family of matrices:

gt(e1) = e2 − e3, gt(e2) = t−1e1, gt(e3) = t−1e3, (t ∈ C, t 6= 0).

Thus, the possible degenerations are summarized in the following diagrams:

LN1(C) : consists of one point.

LN2(C) : λ2 −→ λ1,

LN3(C) : λ4(α 6= 0) λ5 λ6

↙ ↘ ↙ ↙

λ3 λ2 ←− λ4(α = 0)

↘ ↙

λ1

Summarizing, we have that LN2(C) has one irreducible component Orb(λ2) and dimLN2(C) = 2;

LN3(C) has four irreducible components:⋃α6=0

Orb(λ4), Orb(λ5), Orb(λ6) and dimLN3(C) = 6.

3. The description of the irreducible components of the nilpotent complex Leibniz

algebras varieties of dimension four.

In this section we are going to investigate the orbits of the Leibniz algebras in dimension four in order

to describe the irreducible components of LN4(C). For our purpose we need the list of classification

of all nilpotent Leibniz algebras dimensions less than five. The classification of nilpotent Lie algebras

in dimensions less than five can be found in many books on Lie algebras, see, for example [10]. As for

the Leibniz algebras case, the classification can be found in paragraph 1 of the present paper and in

[11]. There are 25 algebras in this list:

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µ1∼= λ1 ⊕ C : abelian

µ2∼= λ2 ⊕ C : e1e1 = e2

µ3∼= λ2 ⊕ λ2 : e1e1 = e2, e3e3 = e4

µ4∼= λ3 ⊕ C : e2e3 = e1, e3e2 = −e1

µ5∼= λ4 ⊕ C : e2e2 = e1, e3e3 = αe1, e2e3 = e1, α ∈ C

µ6∼= λ5 ⊕ C : e2e2 = e1, e3e2 = e1, e2e3 = e1

µ7∼= λ6 ⊕ C : e3e3 = e1, e1e3 = e2

µ8 : e1e4 = e3, e4e1 = −e3, e1e3 = e2, e3e1 = −e2

µ9 : e1e1 = e2, e2e1 = e3, e3e1 = e4

µ10 : e1e1 = e3, e2e1 = e3, e1e2 = e4, e3e1 = e4

µ11 : e1e1 = e3, e2e1 = e3, e3e1 = e4

µ12 : e1e1 = e3, e1e2 = αe4, e2e1 = e3, e2e2 = e4, e3e1 = e4, α ∈ C

µ13 : e1e1 = e3, e1e2 = e4, e3e1 = e4

µ14 : e1e1 = e3, e1e2 = αe4, e2e2 = e4, e3e1 = e4, α ∈ C

µ15 : e1e1 = e4, e1e2 = e3, e2e1 = −e3, e2e2 = −2e3 + e4

µ16 : e1e2 = e3, e2e1 = e4, e2e2 = −e3

µ17 : e1e1 = e3, e1e2 = e4, e2e1 = −αe3, e2e2 = −e4, α ∈ C

µ18 : e1e1 = e4, e1e2 = αe4, e2e1 = −αe4, e2e2 = e4, e3e3 = e4, α ∈ C

µ19 : e1e2 = e4, e1e3 = e4, e2e1 = −e4, e2e2 = e4, e3e1 = e4

µ20 : e1e1 = e4, e1e2 = e4, e2e1 = −e4, e3e3 = e4

µ21 : e1e2 = e3, e2e1 = e4

µ22 : e2e1 = e4, e2e2 = e3

µ23 : e1e2 = e4, e2e2 = e3, e2e1 = (1+α)(1−α)e4, α ∈ C\{1}

µ24 : e1e2 = e4, e2e1 = −e4, e3e3 = e4

µ25 : e1e2 = e3, e2e1 = −e3, e2e2 = e4

In order to decide which Leibniz algebra structures are degenerations or rigid ones, one has to

consider several isomorphism invariants which behave well under degeneration.

In the following table the volumes of invariants of these algebras are presented (in the table by dI

is denoted the dimension of I).

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N dµ2 dµ3 dR(µ) dL(µ) dZ(µ) dAut(µ) dSA(µ) dCom(µ) dLie(µ)

µ1 0 0 4 4 4 16 4 4 4

µ2 1 0 3 3 3 10 3 4 3

µ3 2 0 2 2 2 6 2 4 2

µ4 1 0 2 2 2 10 3 3 4

µ5 1 03(α = 0)

2(α 6= 0)

3α = 0)

2(α 6= 0)

2 83(α = 0)

2(α 6= 0)

33(α = 0)

2(α 6= 0)

µ6 1 0 2 2 2 8 3 4 3

µ7 2 1 3 2 2 6 3 3 3

µ8 2 1 1 1 1 7 3 3 4

µ9 3 2 3 1 1 4 3 3 3

µ10 2 1 2 1 1 4 3 3 3

µ11 2 1 3 2 1 5 3 3 3

µ12 2 1 22(α = 1)

1(α 6= 1)

14(α = 1)

3(α 6= 1)

2 3 2

µ13 2 1 2 2 1 5 3 3 3

µ14 2 1 2 1 1 4 2 3 2

µ15 2 0 2 2 2 6 2 3 2

µ16 2 0 2 2 2 5 3 3 3

µ17 2 0 23(α = 1)

2(α 6= 1)

2 5 2 3 2

µ18 1 02(α2 = −1)

1(α2 6= −1)

2(α2 = −1)

1(α2 6= −1)

1 5 14(α = 0)

3(α 6= 0)

1

µ19 1 0 1 1 1 5 2 3 2

µ20 1 0 1 1 1 5 2 3 2

µ21 2 0 2 2 2 6 3 3 3

µ22 2 0 2 3 2 7 3 3 3

µ23 2 03(α = −1)

2(α 6= −1)

2 2 7 3 3 3

µ24 1 0 1 1 1 7 2 3 3

µ25 2 0 2 2 2 7 3 3 3

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Using the corollary of the proposition 1 and the values of invariants in the above table, we can

conclude existence of the possible degenerations and below we present this list:

µ2 → µ1,

µ3 → µ1, µ2, µ6,

µ4 → µ1,

µ5(α 6= 0)→ µ1, µ2, µ4,

µ5(α = 0)→ µ1, µ2,

µ6 → µ1, µ2,

µ7 → µ1, µ2, µ5(α = 0), µ23(α = −1),

µ8 → µ1, µ4,

µ9 → µ1, µ2, µ5(α = 0), µ7, µ11, µ23(α = −1),

µ10 → µ1, µ2, µ4, µ5(α = 0), µ6, µ7, µ11, µ13, µ16, µ21, µ22, µ23, µ25,

µ11 → µ1, µ2, µ5(α = 0), µ7, µ23(α = −1),

µ12(α = 1)→ µ1, µ2, µ3, µ4, µ5, µ6, µ7, µ11, µ13, µ15, µ16, µ17, µ21, µ22, µ23, µ25,

µ12(α 6= 1)→ µ1, µ2, µ3, µ4, µ5, µ6, µ7, µ10, µ11, µ12(α = 1), µ13, µ14, µ15, µ16,

µ17, µ21, µ22, µ23, µ25,

µ13 → µ1, µ2, µ4, µ5(α = 0), µ6, µ7, µ21, µ22, µ23, µ25,

µ14 → µ1, µ2, µ3, µ4, µ5, µ6, µ7, µ11, µ13, µ15, µ16, µ17, µ21, µ22, µ23, µ25,

µ15 → µ1, µ2, µ4, µ5, µ6, µ22, µ23, µ25,

µ16 → µ1, µ2, µ4, µ5(α = 0), µ6, µ21, µ22, µ23, µ25,

µ17(α = 1)→ µ1, µ2, µ5(α = 0), µ22,

µ17(α 6= 1)→ µ1, µ2, µ3, µ4, µ5, µ6, µ15, µ21, µ22, µ23, µ25,

µ18(α2 = −1)→ µ1, µ2, µ4, µ5, µ6,

µ18(α2 6= −1)→ µ1, µ2, µ4, µ5, µ6, µ24,

µ19 → µ1, µ2, µ4, µ5, µ6, µ24,

µ20 → µ1, µ2, µ4, µ5, µ6, µ24,

µ21 → µ1, µ2, µ4, µ5(α = 0), µ6, µ22, µ23, µ25,

µ22 → µ1, µ2, µ5(α = 0),

µ23(α = −1)→ µ1, µ2, µ5(α = 0),

µ23(α 6= −1)→ µ1, µ2, µ4, µ5(α = 0), µ6,

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µ24 → µ1, µ2, µ4, µ5(α = 0), µ6,

µ25 → µ1, µ2, µ4, µ5(α = 0), µ6,

The following algebras do not take appear on the right side of this list: µ8, µ9, µ12 (α 6= 1), µ18(α),

µ19, µ20. This means that the algebras µ8, µ9, µ19, µ20 are rigid and the algebras µ12 (α 6= 1), µ18(α)

form rigid families.

It should be noted that non rigidity of µ2 and µ5(α = 0) follow from proposition 2 and 3.

For the Leibniz algebras which can not be excluded from rigidity class by these invariance arguments

we construct an appropriate family of matrices gt ∈ GL4(C(t)) in order to show their non rigidity:

µ14 → µ3, gt(e1) = t−1e1, gt(e2) = e3, gt(e3) = t−2e2, gt(e4) = e4,

µ16 → µ4, gt(e1) = e3, gt(e2) = t−1e2, gt(e3) = −t−1e1, gt(e4) = t−1e1 + e4,

µ14(α 6= 0) → µ, gt(e1) = t−1e1, gt(e2) = αt−1e2, gt(e3) = t−2e3, gt(e4) = t−2e3 + t−1e4, but

µ ∼= µ5(α 6= 0),

µ19 → µ6, gt(e1) = e2 − t−1e4, gt(e2) = t−1e4, gt(e3) = e3, gt(e4) = e1,

µ12(α = 1)→ µ7, gt(e1) = e3, gt(e2) = t−1e4, gt(e3) = e1, gt(e4) = e2,

µ12(α = 0) → µ10, gt(e1) = t−1e2, gt(e2) = e1 + (t−1 − 1)e2 + (t−1 + 1)e3, gt(e3) = −t−1e3,

gt(e4) = t−1e4,

µ10 → µ11, gt(e1) = te1, gt(e2) = te2, gt(e3) = t2e3, gt(e4) = t3e4,

µ12(α 6= 1) → µ12(α = 1), gt(e1) = te1 + (t2 − t)e2 + t3e3 + (t4 − t3)e4, gt(e2) = t2e2, gt(e3) =

t3e3 + (t4 − t3)e4, gt(e4) = t4e4,

µ10 → µ13, gt(e1) = t−1e1, gt(e2) = t−2e2, gt(e3) = t−2e3, gt(e4) = t−3e4,

µ12(α = 1) → µ15, gt(e1) =√

2t−1e1, gt(e2) = 1√2t−1e1 − 1√

2t−1e2, gt(e3) = 2t−2e4, gt(e4) =

−t−2e3 + t−2e4,

µ10 → µ16, gt(e1) = it−1e2, gt(e2) = −it−1e1, gt(e3) = t−2e3, gt(e4) = t−2e4,

µ12(α)→ µ17(α), gt(e1) = t−1e2, gt(e2) = −t−1e1, gt(e3) = −t−2e4, gt(e4) = t−2e3,

µ10 → µ21, gt(e1) = t−3e1, gt(e2) = t−2e2, gt(e3) = t−5e4, gt(e4) = t−5e3,

µ12(α = 1)→ µ22, gt(e1) = t−1e1, gt(e2) = e2, gt(e3) = t−1e4, gt(e4) = e3,

µ7 → µ23(α = −1), gt(e1) = e1 + t−2e3, gt(e2) = e3 + t−1e4, gt(e3) = t−1e2, gt(e4) = e4,

µ14(γ) → µ23(γ), gt(e1) = t−2e2, gt(e2) = −t−3e1 + t−4e3, gt(e3) = t−4e3, gt(e4) = t−5e4, where

γ = (1 + α)/(1− α),

µ20 → µ24, gt(e1) = t−1e1, gt(e2) = e2, gt(e3) = t−12 e3, gt(e4) = t−1e4,

µ15 → µ25, gt(e1) = t−1e1, gt(e2) = e2, gt(e3) = t−1e3, gt(e4) = 2t−1e3 + e4.

12

As to the family of algebras λ14(α), its non rigidity can be obtained from the following argument.

We consider the following closed subset A of NL4(C) :

A = {λ ∈ NL4(C) : λ4 = 0}.

It is easy to see that the set of non Lie filiform Leibniz algebras B forms an open subset in A. We

know [7], that any element λ(α, β, γ) of B can be given in a basis {e1, e2, e3, e4} by the laws

e1e1 = e3, e1e2 = αe4, e2e1 = γe3, e2e2 = βe4, e3e1 = e4.

Let us consider the family of algebras λ(α, β, γ), where γ 6= 0. This family belongs to the orbit of the

algebra λ(α, β, 1), it follows that B ⊆ Orb(λ(α, β, 1)). By looking at our degeneration considerations

above we can conclude that the closure of Orb(λ(α, β, 1)) is contained in the closure of the orbit λ12(α),

that is B ⊆ Orb(λ12(α)). Therefore Orb(λ14(α)) is in Orb(λ12(α)).

Main theorem. The following algebras µ8, µ9, µ19, µ20 are rigid and µ12 (α 6= 1), µ18(α) are

rigid family of algebras in NL4(C). As a consequence, NL4(C) consists of six irreducible components

with the respective dimensions: dimOrb(µ8) = 9, dimOrb(µ9) = 12, dimOrb(µ12(α 6= 1)) = 13,

dimOrb(µ18(α)) = 11, dimOrb(µ19) = 11, dimOrb(µ20) = 11. Thus we have dimNL4(C) = 13.

Acknowledgments. The authors would like to convey their sincere thanks to prof. Sh.A. Ayupov

for his comprehensive assistance and support in the completion of this paper. The second and third

named authors would like to acknowledge the hospitality of the ”Institut fur Angewandte Mathematik”,

Universitat Bonn (Germany). This work is supported in part by the DFG 436 USB 113/4 project

(Germany) and the Fundamental Science Foundation of Uzbekistan.

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