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Journal of Generalized Lie Theory and Applications
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Classi cations of some classes of Zinbiel algebras

J. Q. ADASHEV, A. Kh. KHUDOYBERDIYEV, and B. A. OMIROV

Institute of Mathematics and Information Technologies of Academy of Sciences, 29 F. Hodjaev Street, 100125 Tashkent, Uzbekistan E-mails: [email protected], [email protected], [email protected]

Received June 19, 2009 Revised October 06, 2009

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Abstract

In this work, the nul- liform and liform Zinbiel algebras are described up to iso- morphism. Moreover, the classi cation of complex Zinbiel algebras dimensions  3 is extended up to dimension 4.

Introduction

One of the important objects of the modern theory of nonassociative algebras is Lie algebras. Active investigations in the theory of Lie algebras lead to the appearance of some general- izations of these algebras such as Mal'cev algebras, Lie superalgebras, binary Lie algebras, Leibniz algebras, and others.

In the present work, we consider algebras which are dual to Leibniz algebras. Recall that Leibniz algebras were introduced in [6] in the nineties of the last century. They are defined by the following identity:

image

J.-L. Loday in [5] studied categorical properties of Leibniz algebras and considered in this connection a new object { Zinbiel algebra (read Leibniz in the reverse order). Since the category of Zinbiel algebras is Koszul dual to the category of Leibniz algebras, sometimes they are also called dual Leibniz algebras [7].

In works [3, 4], some interesting properties of Zinbiel algebras were obtained. In particular, the nilpotency of an arbitrary complex finite-dimensional Zinbiel algebra was proved in [4]. For the examples of Zinbiel algebras, we refer to works [4, 5, 7].

Since description of all finite-dimensional complex Zinbiel algebras (which are nilpotent) is a boundless problem, it is natural to add certain restrictions for their investigation. One of such restrictions is the condition on the nilindex. Recall that in works [2, 8] some descriptions of nilpotent Leibniz algebras and Lie algebras are given.

At the beginning of the study of any class of algebras, it is important to describe up to isomorphism algebras of lower dimensions, because such description gives examples for to establish or reject certain conjectures. In this way, in [4], the classification of complex Zinbiel algebras of dimensions ≤ 3 is given. Applying some general results obtained for finite-dimensional Zinbiel algebras, we extend the classification of complex Zinbiel algebras up to dimension 4. It should be noted that this classification shows that associative algebras play the crucial role in the classification of four-dimensional algebras, which are defined by the multilinear identity of degree 3.

Classification of complex nul-filiform Zinbiel algebras

Definition 2.1. An algebra A over a field F is called Zinbiel algebra if for any x; y; z ∈ A the identity

image (2.1)

holds.

For a given Zinbiel algebra A, we define the following sequence:

image

Definition 2.2. A Zinbiel algebra A is called nilpotent if there exists s ∈ N such that As = 0. The minimal number s satisfying this property is called index of nilpotency or nilindex of the algebra A.

It is not diffcult to see that the index of nilpotency of an arbitrary n-dimensional nilpotent algebra does not exceed the number n + 1.

Definition 2.3. An n-dimensional Zinbiel algebra A is called nul-filiform if dim Ai = image

It is evident that the last definition is equivalent to the fact that algebra A has maximal index of nilpotency.

Theorem 2.4. An arbitrary n-dimensional nul-filiform Zinbiel algebra is isomorphic to the following algebra:

image (2.2)

where omitted products are equal to zero and image is a basis of the algebra, the symbols Cts are binomial coeffcients defined asimage

Proof. Let A be an n-dimensional nul-filiform Zinbiel algebra and let image be a basis of the algebra A such that image for some elementsimage of algebra A, we have

image

where image Note thatimage Indeed, in the opposite case, image

Similarly, for image we have

image

where image (otherwiseimageimage Continuing this process, we obtain that elements

image

are distinct from zero. It is not diffcult to check the linear independence of these elements. Hence, we can choose the elements image as a basis of algebra A. We have by construction

image (2.3)

We prove equality (2.2) by induction on j for every i.

Using identities (2.1), (2.3), we can prove by induction the equality

image

i.e., equality (2.2) is true for j = 1 and every i.

Suppose that the equality is true for all image and every i.

Let us prove the equality (2.2) for j = k and every i. Using the inductive hypothesis and the following chain of equalities:

image

We denote the algebra from Theorem 2.4 as NFn. It is not diffcult to see that the n- dimensional Zinbiel algebra is one generated if and only if it is isomorphic to the algebra NFn.

Classification of complex filiform Zinbiel algebras

In this section, we classify filiform Zinbiel algebras.

Definition 3.1. An n-dimensional Zinbiel algebra A is said to be filiform if dim imageimage

Definition 3.2. Given a filiform Zinbiel algebra A, put image and image Thenimage and we obtain the graded algebra grA. If an algebra B is isomorphic to grA, then we say that the algebra B is naturally graded.

In the following theorem, the classification of complex naturally graded filiform Zinbiel algebras is represented.

Theorem 3.3. An arbitrary n-dimensional (n ≥ 5) naturally graded complex filiform Zinbiel algebra is isomorphic to the following algebra:

image

where omitted products are equal to zero and image is a basis of the algebra.

Proof. Let A be a Zinbiel algebra satisfying conditions of the theorem. Similar to the work [8], we choose a basis image of the algebra A such thatimageimage

Introduce the following notations:

image

Consider the possible cases.

Case 1. Let image Then without loss of generality we can supposeimage Change the basis as follows: image We can suppose α1 = 1. Then the space spanned on the vectors image forms a nul-filiform Zinbiel algebra of the dimension n - 1. From the proof of Theorem 2.4, we can conclude imageimage

Let us show that the omitted products are equal to zero.

Apply identity (2.1) in the following multiplications:

image

i.e., we have image

Taking the change of the basic elements by the following way:

image

it is not diffcult to see that image Moreover, the other products are not changed, i.e., we can suppose image

Using identity (2.1) and the method of mathematical induction, it is easy to prove

image (3.1)

The equality

image

can be proved by induction and using (2.1), (3.1).

Case 2. Let image Thenimage In the case ofimage takingimage we obtain the conditions of Case 1. Therefore, we need to consider only the case of image By the following change of basis:

image

we can suppose α2 = 1.

The products

image

image

deduce the contradiction to the existence of algebra in this case.

The following proposition allows to extract a "convenient" basis in an arbitrary complex filiform Zinbiel algebra. Such basis in the literature is called adapted [8].

Proposition 3.4. There exists a basis image in an arbitrary n-dimensional image complex filiform Zinbiel algebra such that the multiplication of the algebra has the following form:

image (3.2)

image

where image

Proof. By Theorem 3.3 we have that any n-dimensional complex filiform Zinbiel algebra is isomorphic to the algebra of the form image where

image

Similarly to the proof of Theorem 3.3, it is not diffcult to establish that the multiplications

image

can be obtained from image and identity (2.1).

By the similar process, we obtain

image

So, we have the products

image

Now define the productsimage and put

image

Taking the change

image

we can suppose image

Identity (2.1) implies

image

Therefore, image

Consider the product

image

from which we obtain image for some γ.

Similar to the proof of Theorem 3.3, we can obtain that image forimage which complete the proof of the proposition.

The classification of complex filiform Zinbiel algebras is given in the following theorem.

Theorem 3.5. An arbitrary n-dimensional (n ≥ 5) complex filiform Zinbiel algebra is iso- morphic to one of the following pairwise nonisomorphic algebras:

image

Proof. From Proposition 3.4, we have the multiplication (multiplication (3.2)) of n- dimensional complex filiform Zinbiel algebra, namely,

image

Let us check the isomorphism inside this family of algebras. Consider the general change of the generators of the basic elements in the form

image

where image Then for the remainder elements of the new basis we have

image

where image The relations

image

imply the following restrictions:

image

From these restrictions, we get

image

Consider the product

image

On the other hand,

image

Comparing the coeffcients at the basic element en-1, we obtain

image

Consider the product

image

On the other hand,

image

Comparing the coeffcients, we have

image

Now consider the following cases.

Case 1. Let image Then image andimage If image then image and we have algebra image then takingimage we getimagei.e., the algebra image is obtained.

Case 2. Let image Then puttingimage we getimage and algebraimage is obtained.

Note that the obtained algebras are not pairwise isomorphic.

Comparing the description of complex filiform Leibniz algebras [2] and the result of The- orem 3.5, we can note how the class of complex filiform Zinbiel algebras is "thinner". So, al- though Zinbiel algebras and Leibniz algebras are Koszul dual, they are quantitatively strongly distinguished even in the class of filiform algebras.

It is known that for any variety of algebras image-graded algebras of that variety can be defined, which is called superalgebras. In the same way, we define a notion of Zinbiel super- algebra image by the following identity:

image

where image andimage

It should be noted that the proof on the nilpotency of finite-dimensional complex Zinbiel algebras can be extended to the proof of solvability of the finite-dimensional complex Zinbiel superalgebras.

Moreover, from the obtained classification of complex filiform Zinbiel algebras, we give the following conjecture.

Conjecture 3.6. Finite-dimensional complex -graded Zinbiel algebra (Zinbiel superalge- bra) is nilpotent.

Classification of four-dimensional complex Zinbiel algebras

Since an arbitrary finite-dimensional complex Zinbiel algebra is nilpotent, then for an arbi- trary four-dimensional Zinbiel algebra A the condition A5 = 0 holds.

Take into account that the direct sum of nilpotent Zinbiel algebras is nilpotent; further we will not consider split algebras case.

Theorem 4.1. An arbitrary four-dimensional complex nonsplit Zinbiel algebra is isomorphic to the one of the following pairwise nonisomorphic algebras:

image

 

Proof. Note that the result of [1, Proposition 3.1] also holds for Zinbiel algebras. Therefore, we have the following possible cases for invariant sequence (dim A2; dim A3; dim A4):

(3; 2; 1); (2; 1; 0); (2; 0; 0); (1; 0; 0); (0; 0; 0):

Obviously, a Zinbiel algebra with the condition (3; 2; 1) is nul-filiform. Using Theorem 2.4, we obtain the algebra A1.

Consider an algebra with the invariant sequence (2; 1; 0) (this algebra is filiform).

Let image be a basis of algebra A satisfying the conditions imageimage Then we can suppose that

image

where image

Case 1. Let image Then by arguments analogous to the arguments in the proofs of Theorems 3.3 and 3.5 we obtain the following algebras:

image

image

Note that the algebra defined by multiplication

image

is split. So, in this case, we have the algebras image

Case 2. Let image Thenimage If image then taking image whereimage we have Case 1. It remains to consider the caseimage Denoteimageimage Then we can write

image

Consider the products

image

If we replace basic elements as follows:

image

we obtain image i.e., the multiplication in the algebra has the following form:

image

(omitted products are equal to zero).

Let us check the isomorphism inside this family.

Consider the general change of generators of the basic elements:

image

where image Expressing basic elementsimage via basic elementsimage and analyzing the relations of the family in new basis, we obtain the following restrictions:

image

Consider the following cases.

Case 2.1. Let image Then image and

image

Taking image andimage we obtain image i.e., we have the algebra A4.

Case 2.2. Let image Then putting

image

we get image i.e., the algebra A5 is obtained.

Note that algebras with the conditions (2; 0; 0), (1; 0; 0), (0; 0; 0) are associative algebras. Therefore, we can use the classification of four-dimensional algebras of Leibniz [1], i.e., choose algebras with the condition A3 = 0.

Note that the problem of classification of complex five-dimensional Zinbiel algebras is open and the solution of this problem is equivalent to the classification of such associative algebras, which is still not obtained.

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