On extensions of Leibniz algebras

This paper is dedicated to the study extensions of Leibniz algebras using the annihilator approach. The extensions methods have been used earlier to classify certain classes of algebras. In the paper we first review and adjust theoretical background of the method for Leibniz algebras then apply it to classify four-dimensional Leibniz algebras over a field K. We obtain complete classification of four-dimensional nilpotent Leibniz algebras. The main idea of the method is to transfer the “base change” action to an action of automorphism group of the algebras of smaller dimension on cocycles constructed by the annihilator extensions. The method can be used to classify low-dimensional Leibniz algebras over other finite fields as well.


Introduction
This paper describe an approach for the classification of Leibniz algebras over finite fields, which is equivalent to the Skjelbred-Sund method (see [2,3]) for the classification of Lie lagebras. The approach is principally based on the actions of group automorphisms on Grassmannian of subspaces of the 2 nd cohomology groups of algebras of smaller dimension. This approach was also used in the classifications of nilpotent cases of Leibniz, Jordan and Malcev algebras over different fields (see [4,5,6,7,8]).

Extension of Leibniz algebra via annihilator
In this section we introduced the concept of an annihilator extension of Leibniz algebras. In fact, this is the definition of right Leibniz algebra. Herein and after the notation L will be used for a Leibniz algebras over a field K. We define a series of ideals of L by setting One has Note that Ann(L) of a nilpotent algebra L is nontrivial.
Proof. Considering ψ an isomorphism, it implies [ψ(a), ψ(L k 1 )] = 0 if and only if [a, L k 1 ] = 0. Furthermore, we have ψ(L k 1 ) = L k 2 for any k ∈ N. Therefore, for all k ∈ N, we get: Definition 2.4. Let suppose that {L i } be a sequence of algebra L and {ψ i } be the sequence of its homomorphisms from L i to L i+1 , then is called exact if one has im ψ i = ker ψ i+1 for each i.
Definition 2.5. Let L 1 , L 2 , and L 3 be Leibniz algebras. The algebra L 2 is called an extension of L 3 by L 1 if there exist homomorphisms ψ 1 : L 1 −→ L 2 and ψ 2 : L 2 −→ L 3 such that the following sequence Definition 2.6. The sequence is called an annihilator extension if the kernel of ψ 2 is contained in the annihilator of L 2 . That is, ker ψ 2 ⊂ Ann(L 2 ).
It is easy to see from the definition above that the annihilator extension of the algebra L 1 must be abelian. In the next section we construct such an extension by using 2-cocycle on L 3 .

Leibniz algebra cocycles
We introduce in this part the concept of 2-cocycle for algebras L and give a few of its properties. Consider a vector space Obviously L θ is an algebra with respect to the multiplication [·, ·].
The following lemma can be easily proven.
Lemma 3.1. The algebra L θ is a Leibniz algebra if and only if θ is a Leibniz 2-cocycle.
A particular type of 2-cocycles called a coboundary, define as follows.
Definition 3.2. Let L be a Leibniz algebra and V a vector space over a field K. Furthermore, is named as a coboundary. The entire set of all coboundaries is denoted by B 2 (L, V ).
Obviously BL 2 (L, V ) is a subspace of ZL 2 (L, V ). Using the notion of 2-cocycles and 2coboundaries, one defines the second cohomology group of a Leibniz algebra L by V as follows: Suppose that L be a Leibniz algebra and θ ∈ ZL 2 (L, V ). The set is named the radical of θ.
When constructing a Leibniz algebra as L θ = L⊕V , we restrict θ such that Ann(L θ ) = V . In this way we can avoid constructing the same Leibniz algebra as annihilator extension of different Leibniz algebras.
Suppose that L is a algebra with a basis {e 1 , e 2 , e 3 , · · ·, e n }. Then, by ∆ i,j we describe the bilinear form ∆ i,j : L × L −→ K by ∆ i,j (e l , e m ) = 1, wherever {i, j} = {l, m} and 0 otherwise.

Analogue of the Skjelbred-Sund theorem
There is an action of Aut(L) on ZL 2 (L, K) as follows: let φ ∈ Aut(L) and θ ∈ ZL 2 (L, K) then Let assume that G m (HL 2 (L, K)) be the Grassmanian of subspaces of dimension m in HL 2 (L, K).
It follows that, φ · T ∈ G m (HL 2 (L, K)). Let express the orbit of T ∈ G m (HL 2 (L, K)) under the action of group automorphism Aut(L) on G m (HL 2 (L, K)) as Orb(T ).
We have the following lemmas.
Lemma 4.1. Let T 1 and T 2 be two elements of G m (HL 2 (L, K)) defined by As a consequence of Lemma 4.1 above, we define the subspace  In the next section we apply this theorem, to get an algorithm to construct all nilpotent Leibniz algebras of dimension n over finite fields given those algebras of dimension n − m in the following way:

Application
In this part, we apply the analogous of Skjelbred-Sund method (Theorem 4.3) to classify 4dimensional non-Lie nilpotent Leibniz algebras over K = Z 3 . We make use the list of 3dimensional non-Lie Leibniz algebras over Z 3 in [1]. We denote L ij to represent j th algebra, of dimension i.