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

^{2}Institute 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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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 sl_{2} of trace-free 2×2 matrices is graded by the generators:

These satisfy the relations Hence, with

the decomposition presents sl_{2} 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:

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 FLb_{n} and SLb_{n}. We show that according
to dimensions of the derivation algebras each class is split into subclasses as follows:

where F_{i} and S_{j} are subclasses of FLb_{n} and SLb_{n}, respectively, with the derivation algebras’ dimensions *i* and *j*.

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

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

Let *L* be a Leibniz algebra. We put

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

**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

The set of all derivations of an algebra *L* is denoted by Der(*L*). By Lb* _{n}* 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:

*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 Lb* _{n}* 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 e_{1},e_{2}, . . . , e_{n} such that the table of L has one of the following forms:

We denote algebras from FLb* _{n}* and SLb

3.1 Derivations of graded Leibniz algebras

In this section, we study the derivations of NGF*i*, *i* = 1,2, 3. In each case, we give a basis of the derivation algebra.
Let *d* be represented by a matrix on the basis {*e _{1},e_{2}, . . . , e_{n}*}. We describe the matrix
D.

**Theorem 3.** The dimension of the derivation algebras of NGF_{1}, NGF_{2}, and NGF_{3} are equal to n+1, n+2, and
2_{n−1}, respectively.

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

Therefore,

(3.1)From [*e _{2}*,

Hence,

(3.2)Comparing (3.1) and (3.2), we obtain

According to the table of multiplication of NGF1, one has *[e _{3},e_{1}] = e_{4}*. Thus

Therefore,

For k ≥ 5 one can find

(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 e_{k+1} =
[*e _{k},e_{1}*] we have

Hence, we get

In fact, *e _{n} = [e_{n−1},e_{1}]*, therefore

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

That is,

The matrix of d on the basis *{e _{1},e_{2},e_{3}, . . . , e_{n}}* has the following form:

Consider the following system of vectors:

where E_{ij} is the matrix with zero entries except for the element *a _{ij}* = 1. It is easy to see that the set
{

Next, we describe the derivation algebra of NGF_{2}.

If one uses [*e _{2},e_{1}*] = 0, then

Therefore,

Because of [*e _{3},e_{1}*] =

Similarly,

Then the matrix of d has the form

where

From the view of *D* it is easy to conclude that dimDer(NGF_{2}) = *n*+2.

Let us now consider the derivation algebra of NGF_{3}. We take where *j* = 1, 2. Then due to *[e _{2},e_{1}] = e_{3}* one has

Hence,

Consider *e _{4} = [e_{3},e_{1}]*, then

Therefore,

Similarly,

Then the matrix of derivations has the form

where

Thus, the dimension of Der(NGF_{3}) is 2*n*−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 FLb_{n} are equal to n−1, n or n+1.

*Proof*. Depending on constraints for the structure constants α_{4},α_{5}, . . . , α_{n−1} and θ, we have the following distribution
for dimensions of the derivation algebras of elements from FLb* _{n}*:

We will treat only one case, where The other cases are similar. Put where *j* = 1, 2. Then owing to *[e _{1},e_{1}] = e_{3},* one has

Due to *[e _{2},e_{1}] = e_{3}*, we have

Comparing the last two expressions for d(e_{3}), we obtain

From [*e _{3}*,

Let us consider [*e _{4}*,

Similarly,

(3.4)From *e _{n} = [e_{n−1},e_{1}]* we get

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

As a result one has

(3.5)On the other hand,

Notice that α_{4} = 0, therefore

This implies that

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

Then

Thus,

(3.6)Comparing (3.5) and (3.6), we obtain

(3.7)The matrix of d has the form where

Hence, in this case the dimension of Der(*L*) for L ∈ FLb* _{n}* is

Now we describe the derivation algebra of elements from SLb* _{n}*.

**Theorem 5.** The dimensions of the derivation algebras for elements of SLb_{n} vary between n−1 and n+2.

*Proof*. Similarly to the case of FLb* _{n}* for the class SLb

Let where *j* = 1, 2. Since [*e _{1}*,

Thus,

From [*e _{2}*,

Therefore, we obtain

Consider [*e _{3}*,

Thus,

Take [*e _{4}*,

Hence,

Similarly,

(3.8)It is clear that this relation is true for k ≥ 5. Consider

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

Thus,

On the other hand,

Then

And then

Thus,

(3.10)Comparing (3.9) and (3.10), we obtain

(3.11)From (3.11) we get

The matrix of d has the form

The dimension of the derivation algebra of L ∈ SLb* _{n}* is

The other cases are treated similarly.

And at last, if then the dimension of the derivation
algebra of *L* is *n*+2, which is immediate from the case of NGF_{2}.

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

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- Goze M, Hakimjanov Y (1994) Sur les alg`ebres de Lie nilpotentes admettant un tore de derivations. Manuscripta Math 84:115–124.
- Jacobson N (1955) A note on automorphisms and derivations of Lie algebras. Proc Amer Math Soc 6: 281–283.
- Khakimdjanov Y (2000) Varieties of Lie algebra laws, in Handbook of Algebra. North-Holland, Amsterdam 2: 509–541.
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- Rakhimov IS, Bekbaev UD (2010) On isomorphisms and invariants of finite dimensional complex filiform Leibniz algebras. Comm Algebra 38: 4705–4738.
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