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Journal of Generalized Lie Theory and Applications
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On Derivations of Some Classes of Leibniz Algebras

Isamiddin S. Rakhimov1,2 and Al-Hossain Al-Nashri1

1Department of Mathematics, Faculty of Science, University of Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

2Institute for Mathematical Research (INSPEM), University of Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia Address correspondence to Isamiddin S. Rakhimov, [email protected]

Received date: May 23, 2012; Revised date: September 10, 2012; Published date: September 25, 2012

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Abstract

In this paper, we describe the derivations of complex n-dimensional naturally graded filiform Leibniz algebras NGF1, NGF2, and NGF3.We show that the dimension of the derivation algebras of NGF1 and NGF2 equals n+1 and n+2, respectively, while the dimension of the derivation algebra of NGF3 is equal to 2n−1. The second part of the paper deals with the description of the derivations of complex n-dimensional filiform non Lie Leibniz algebras, obtained from naturally graded non Lie filiform Leibniz algebras. It is well known that this class is split into two classes denoted by FLbn and SLbn. Here we found that for L ∈ FLbn, we have n−1≤dimDer(L)≤n+1 and for algebras L from SLbn, the inequality n−1 ≤ dimDer(L) ≤ n+2 holds true.

Introduction

A graded algebra is an algebra endowed with a gradation which is compatible with the algebra bracket. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra. Lie algebra sl2 of trace-free 2×2 matrices is graded by the generators:

equation

These satisfy the relations equation Hence, with

equation

the decomposition equation presents sl2 as a graded Lie algebra.

It is well-known that the natural gradation of nilpotent Lie and Leibniz algebras is very helpful in investigation of their structural properties. This technique is more effective when the length of the natural gradation is sufficiently large. In the case when it is maximal the algebra is called filiform. For applications of this technique, for instance, see [12] and Goze et al. [4] (for Lie algebras) and [1,2,7] (for Leibniz algebras) cases. In [12] Vergne introduced the concept of naturally graded filiform Lie algebras as those admitting a gradation associated with the lower central series. In that paper, she also classified them, up to isomorphism. Apart from that, several authors have studied algebras which admit a connected gradation of maximal length (i.e., the length is exactly the dimension of the algebra). So, Khakimdjanov started this study in [6], Reyes, in [3], continued this research by giving an induction classification method, and finally, Millionschikov in [9] gave the full list of these algebras (over an arbitrary field of zero characteristic).

Recall that an algebra L over a field K is called Leibniz algebra if it satisfies the following Leibniz identity:

equation

where [·, ·] denotes the multiplication in L (first the Leibniz algebras have been introduced in [8]). It is not difficult to see that the class of Leibniz algebras is “non-antisymmetric” generalization of the class of Lie algebras. In this paper, we are dealing with the derivations of some classes of complex Leibniz algebras.

The outline of the paper is as follows. Section 2 contains preliminary results on Leibniz algebras which we will use in the paper. The main results of the paper are in Section 3. The first part of this section deals with the description of derivations of naturally graded Leibniz algebras. In the second part (Section 3.2) we study derivations of filiform Leibniz algebras arising from naturally graded non Lie filiform Leibniz algebras. It is known that the last is split into two disjoint subclasses [2]. In this paper, we denote these classes by FLbn and SLbn. We show that according to dimensions of the derivation algebras each class is split into subclasses as follows:

equation

where Fi and Sj are subclasses of FLbn and SLbn, respectively, with the derivation algebras’ dimensions i and j.

Further all algebras considered are over the field of complex numbers equation and omitted products of basis vectors are supposed to be zero.

Preliminaries

This section contains definitions and results which will be needed throughout the paper.

Let L be a Leibniz algebra. We put

equation

Definition 1 A Leibniz algebra L is said to be nilpotent if there exists s ∈ equation such that

equation

Definition 2 An n-dimensional Leibniz algebra L is said to be filiform if

Obviously, a filiform Leibniz algebra is nilpotent.

Definition 3 A linear transformation d of a Leibniz algebra L is called a derivation if equation

equation

The set of all derivations of an algebra L is denoted by Der(L). By Lbn we denote the set of all n-dimensional filiform Leibniz algebras, appearing from naturally graded non Lie filiform Leibniz algebras. For Lie algebras the study of derivations has been initiated in [5]. The derivations of naturally graded filiform Leibniz algebras were first considered by Omirov in [10]. In the following theorem, we declare the results of the papers [2,12].

Theorem 1. Any complex n-dimensional naturally graded filiform Leibniz algebra is isomorphic to one of the following pairwise non isomorphic algebras:

equation

where α ∈ {0,1} for even n and α = 0 for odd n.

Here is a result of the papers [2,11] on decomposition of Lbn into two disjoint classes.

Theorem 2. Any complex n-dimensional filiform Leibniz algebra L, obtained from naturally graded non Lie filiform Leibniz algebra, admits a basis e1,e2, . . . , en such that the table of L has one of the following forms:

equation
equation

We denote algebras from FLbn and SLbn by L(α45, . . . , αn−1,θ) and L(β34, . . . , βn−1,γ), respectively.

Main results

3.1 Derivations of graded Leibniz algebras

In this section, we study the derivations of NGFi, i = 1,2, 3. In each case, we give a basis of the derivation algebra. Let d be represented by a matrix equation on the basis {e1,e2, . . . , en}. We describe the matrix D.

Theorem 3. The dimension of the derivation algebras of NGF1, NGF2, and NGF3 are equal to n+1, n+2, and 2n−1, respectively.

Proof. Let us start from NGF1. We take equation,we have

equation equation

Therefore,

equation   (3.1)

From [e2,e1] = e3, we find

equation

Hence,

equation   (3.2)

Comparing (3.1) and (3.2), we obtain

equation

According to the table of multiplication of NGF1, one has [e3,e1] = e4. Thus

equation

Therefore,

equation

For k ≥ 5 one can find

equation   (3.3)

Indeed, it is true for k = 4. Suppose that it is true for k and show that it is the case for k+1. Considering ek+1 = [ek,e1] we have

equation

Hence, we get

equation

In fact, en = [en−1,e1], therefore

equation

We substitute k by n−1 in (3.3) and obtain equation Therefore,

equation

That is,

equation

The matrix of d on the basis {e1,e2,e3, . . . , en} has the following form:

equation

Consider the following system of vectors:

equation

where Eij is the matrix with zero entries except for the element aij = 1. It is easy to see that the set {v1,v2,v3, . . . , vn+1} presents a basis of Der(NGF1), therefore, dimDer(NGF1) = n+1.

Next, we describe the derivation algebra of NGF2. equation

equation

If one uses [e2,e1] = 0, then


equation

Therefore,

equation

Because of [e3,e1] = e4, we find that

equation

Similarly,

equation

Then the matrix of d has the form

equation

where

equation

From the view of D it is easy to conclude that dimDer(NGF2) = n+2.

Let us now consider the derivation algebra of NGF3. We take equation where j = 1, 2. Then due to [e2,e1] = e3 one has

equation

Hence,

equation

Consider e4 = [e3,e1], then

equation

Therefore,

equation

Similarly,

equation

Then the matrix of derivations has the form

equation

where

equation

Thus, the dimension of Der(NGF3) is 2n−1.

3.2 Derivations of filiform Leibniz algebras

Now we study the derivations of classes from Theorem 2.

Theorem 4. The dimensions of the derivation algebras of FLbn are equal to n−1, n or n+1.

Proof. Depending on constraints for the structure constants α45, . . . , αn−1 and θ, we have the following distribution for dimensions of the derivation algebras of elements from FLbn:

equation

We will treat only one case, where equation The other cases are similar. Put equation equation where j = 1, 2. Then owing to [e1,e1] = e3, one has

equation

Due to [e2,e1] = e3, we have

equation
equation

Comparing the last two expressions for d(e3), we obtain

equation

From [e3,e1] = e4, one has

equation

Let us consider [e4,e1] = e5. Then,

equation

Similarly,

equation   (3.4)

From en = [en−1,e1] we get

equation

The substitution k by n−1 in (3.4) gives equation and then

equation

As a result one has

equation   (3.5)

On the other hand,

equation

Notice that α4 = 0, therefore

equation

This implies that

equation

We substitute k by n−2 in (3.4) to obtain

equation

Then

equation

Thus,

equation   (3.6)

Comparing (3.5) and (3.6), we obtain

equation   (3.7)

The matrix of d has the form equation where

equation

Hence, in this case the dimension of Der(L) for L ∈ FLbn is n.

Now we describe the derivation algebra of elements from SLbn.

Theorem 5. The dimensions of the derivation algebras for elements of SLbn vary between n−1 and n+2.

Proof. Similarly to the case of FLbn for the class SLbn, we have the distribution for dimension of derivation algebra as follows:

equation

Let equation where j = 1, 2. Since [e1,e1] = e3, we have

equation

Thus,

equation

From [e2,e1] = 0 we get

equation

Therefore, we obtain

equation

Consider [e3,e1] = e4. Then

equation

Thus,

equation

Take [e4,e1] = e5. Then

equation

Hence,

equation

Similarly,

equation   (3.8)

It is clear that this relation is true for k ≥ 5. Consider equation

equation

We substitute k by n−1 in (3.8), and obtain

equation

Thus,

equation

On the other hand,

equation

Then

equation

And then

equation

Thus,

equation   (3.10)

Comparing (3.9) and (3.10), we obtain

equation   (3.11)

From (3.11) we get

equation

The matrix of d has the form equation

equation

The dimension of the derivation algebra of L ∈ SLbn is n−1.

The other cases are treated similarly.

And at last, if equation then the dimension of the derivation algebra of L is n+2, which is immediate from the case of NGF2.

Acknowledgment

The work was supported by FRGS Project 01-12-10-978FR MOHE (Malaysia).

References

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