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**Received** June 19, 2009 **Revised** October 06, 2009

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

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:

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.

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

(2.1)

holds.

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

**Definition 2.2.** A Zinbiel algebra *A* is called nilpotent if there exists s ∈ N such that *A*^{s} = 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 A^{i} =

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:*

(2.2)

*where omitted products are equal to zero and* *is a basis of the algebra, the
symbols C ^{t}_{s} are binomial coeffcients defined as*

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

where Note that Indeed, in the opposite case,

Similarly, for we have

where (otherwise Continuing this process, we obtain that elements

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

(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

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

Suppose that the equality is true for all 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:

We denote the algebra from Theorem 2.4 as *NF _{n}*. It is not diffcult to see that the

In this section, we classify filiform Zinbiel algebras.

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

**Definition 3.2.** Given a filiform Zinbiel algebra A, put and Then 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:*

*where omitted products are equal to zero and* *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 of the algebra A such that

Introduce the following notations:

Consider the possible cases.

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

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

Apply identity (2.1) in the following multiplications:

i.e., we have

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

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

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

(3.1)

The equality

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

*Case* 2. Let Then In the case of taking we obtain the conditions of Case 1. Therefore, we need to consider only the
case of By the following change of basis:

we can suppose α_{2} = 1.

The products

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* *in an arbitrary n-dimensional* *complex filiform Zinbiel algebra such that the multiplication of the algebra has the following
form*:

(3.2)

*where*

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

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

can be obtained from and identity (2.1).

By the similar process, we obtain

So, we have the products

Now define the products and put

Taking the change

we can suppose

Identity (2.1) implies

Therefore,

Consider the product

from which we obtain for some γ.

Similar to the proof of Theorem 3.3, we can obtain that for 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*:

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

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

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

where The relations

imply the following restrictions:

From these restrictions, we get

Consider the product

On the other hand,

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

Consider the product

On the other hand,

Comparing the coeffcients, we have

Now consider the following cases.

*Case* 1. Let Then and If then and we have
algebra then taking we geti.e., the algebra is obtained.

*Case* 2. Let Then putting we get and algebra 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 -graded algebras of that variety can be defined, which is called superalgebras. In the same way, we define a notion of Zinbiel super- algebra by the following identity:

where and

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.

Since an arbitrary finite-dimensional complex Zinbiel algebra is nilpotent, then for an arbi-
trary four-dimensional Zinbiel algebra A the condition A^{5} = 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*:

**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 *A*^{2}; dim *A*^{3}; dim *A*^{4}):

(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 *A*_{1}.

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

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

where

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

Note that the algebra defined by multiplication

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

*Case* 2. Let Then If then taking where we have Case 1. It remains to consider the case Denote Then we can write

Consider the products

If we replace basic elements as follows:

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

(omitted products are equal to zero).

Let us check the isomorphism inside this family.

Consider the general change of generators of the basic elements:

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

Consider the following cases.

*Case* 2.1. Let Then and

Taking and we obtain i.e., we have the algebra *A*_{4}.

*Case* 2.2. Let Then putting

we get i.e., the algebra *A*_{5} 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 *A*^{3} = 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.

- Albeverio S, Omirov BA, Rakhimov IS (2006) Classification of 4-dimensional nilpotent complex Leibniz algebras. Extracta Math 21, 197-210.
- Ayupov ShA, Omirov BA (2001) On some classes of nilpotent Leibniz algebras. Sibirsk. Mat Zh 42: 18-29.
- Dzhumadil'daev AS (2005) Identities for multiplications derived by Leibniz and Zinbiel multiplications. Abstracts of short communications of International conference \Operator algebras and quantum theory of probability" Tashkent pp: 76-77.
- Dzhumadil'daev AS, Tulenbaev KM (2005) Nilpotency of Zinbiel algebras. J Dyn Control Syst 11: 195-213.
- Loday JL (1995) Cup-product for Leibniz cohomology and dual Leibniz algebras. Math Scand 77: 189-196.
- Loday JL, Pirashvili T (1993) Universal enveloping algebras of Leibniz algebras and cohomology. Math Ann 296: 139-158.
- Loday JL, Frabetti A, Chapoton F, Goichot F (2001) Dialgebras and Related Operads. Lecture Notes in Math, Springer-Verlag, Berlin.
- Vergne M (1970) Cohomologie des algebres de Lie nilpotentes. Application a l'etude de la variete desalgebres de Lie nilpotentes. Bull Soc Math France 98: 81-116

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