Derivations and Centroids of Four Dimensional Associative Algebras

In this paper, we focus on derivations and centroids of four dimensional associative algebras. Using an existing classification result of low dimensional associative algebras, we describe the derivations and centroids of four dimensional associative algebras. We also identify algebra(s) that belong to the characteristically nilpotent class among the algebras of four dimensional associative algebras.


Introduction
The interest in the study of derivations of algebras goes back to a paper by Jacobson [1]. There, Jacobson proved that any Lie algebra over a field of characteristic zero which has non degenerate derivations is nilpotent. In the same paper, he asked for the converse. Dixmier and Lister [2] have given a negative answer to the converse of Jacobson's hypothesis by constructing an example of a nilpotent Lie algebra all of whose derivations are nilpotent (hence degenerate). Lie algebras whose derivations are nilpotent have been called characteristically nilpotent. The result of Dixmier and Lister in [2] is assumed to be the origin of the theory of characteristically nilpotent Lie algebras. A few years later in 1959, Leger and Togo [3] published a paper showing the importance of the characteristically nilpotent Lie algebras. The results of Leger and Togo have been extended for the class of non associative algebras.
The theory of characteristically nilpotent Lie algebras constitutes an independent research object since 1955. Until then, most studies about Lie algebras were oriented to the classical aspects of the theory, such as semi-simple and reductive Lie algebras (see [4]).
The structural theory of finite dimensional associative algebras have been treated by [5]. Many interesting results related to the problem have appeared since then. Further works in this field can be found in [6], [7], [8], [9] and [10]. Further development of the theory of associative algebras was in 80-s of the last century when many open problems, remaining on unsolved since 30-s, have been solved Most classification problems of finite dimensional associative algebras have been studied for certain property(s) of associative algebras while the complete classification of associative algebras in general is still an open problem.
Centroids of algebras play important role in the classification problems and in different areas of structure theory of algebras. The centroid of a Lie algebra is known to be a field and this fact plays an important role in the classification problem of finite dimensional extended affine Lie algebras over arbitrary field of characteristic zero (see [11]). Benkarta and Neher studied extended affine and root graded Lie algebras in [11]. Melville in [12] studied the centroids of nilpotent Lie algebra. Centroids and derivations of associative alge-bras in dimension less than 4 has been treated in [13] and provided an impetus to further the study in higher dimension.
In this study, we concentrate on the derivations and centroids of associative algebras in dimension four. Using the classification result of associative algebras, we give a description of derivations and centroids of associative algebras in dimension 4.

Preliminaries
Definition 1. An algebra A over a field K is a vector space over K equipped with a bilinear map Definition 2. An associative algebra A is a vector space over a field K equipped with bilinear map Ψ : A × A :−→ A satisfying the associative law: Definition 3. A Lie algebra L over a field K is an algebra satisfying the following conditions: Definition 4. Let (A 1 , ·) and (A 2 , •) be two associative algebras over a field K. A homomorphism between A 1 and A 2 is a K-linear mapping f : The set of all homomorphism from A 1 to A 2 is denoted by Hom K (A 1 , A 2 ). If A 1 = A 2 = A, then it is an associative algebra with respect to composition operation and denoted by Hom K (A). The linear mapping associated with Hom K (A) is called an endomorphism. A bijective homomorphism is called isomorphism and the corresponding algebras are said to be isomorphic.
Example 5. Let V be an n-dimensional vector space over a field K. The set of all endomorphisms, End(V ), forms a vector space. The multiplication of two elements f, g ∈ End(V ) is defined by This product turns End(V ) into an associative algebra.
Example 6. Let the product of two elements in Example 5 above be defined by The set of all derivations of an associative algebra A is denoted by Der(A). The following fact can easily be proven.
For a ∈ A, let L a and R a be the elements of Hom(A) defined by: Theorem 9. Let D ∈ Hom K (A). Then the following conditions are equivalent : Proof. The proof of equivalence of 1-3 is straightforward using the definition of derivation.
is said to be centralizer of H in A.
It must however be noted that Z A (A) = Z(A), the center of A.
The set of all central derivations is denoted by C(A).
In the foregoing, we give a few earlier results on some properties of centroids of associative algebras which show the relationship between derivation and centroid. The proof of some of the facts given below can be found in [13]. Proposition 15 ( [13]). Let A be an associative algebra over a field K. Then for any d ∈ Der(A) and φ ∈ Γ(A): is a central derivation of A.

Procedure for finding derivations
Let A be an n-dimensional associative algebra and d be its derivation. Fix a basis {e 1 , e 2 , ..., e n } of A. Particularly, d•L e i (y) = L d(e i ) (y)+L e i •d(y) for basis vectors e i i = 1, 2, . . . , n.
we have An element d of the derivations being a linear transformation of the vector space A is represented in a matrix form (d ij ) i,j=1,2,··· ,n i.e., The last equations along with structure of A give constraints for elements of the matrix d. Solving the system of equations, we can find the matrix As mention earlier, we provide the classification results of 4-dimensional associative algebra from [10] which we use in our study. Note that As m n denotes m th isomorphism class of associative algebra in dimension n.

Four-dimensional Associative algebras
Theorem 16. Any four-dimensional complex associative algebra can be included in one of the following isomorphism classes of algebras: In what follows, the following notations are introduced: 1. IC: isomorphism classes of algebras. Since the classification of four-dimensional associative algebra is already known from Theorem 16, Table 1 shows the derivations four-dimensional associative algebra as given below: Theorem 17. The derivation of four-dimensional associative algebras has the following form:  As 54 As Proof. From Theorem 16, we provide the proof only for one case to illustrate the approach used, the other cases can be carried out similarly with little or no modification(s). Let us consider As is a basis of Der(A) and dimDer(A) = 6. The derivation of the remaining parts of dimension four algebras can be carried out in a similar manner as shown above.
1. There is one class of characteristically nilpotent associative algebras in the list of isomorphism classes of four dimensional associative algebras.
2. The dimensions of the derivation algebras in this case vary between zero and nine.

Description of centroids of four dimensional associative algebras
In this section, we give the description of centroids of four-dimensional complex associative algebras. Let a ji e j , i = 1, 2, ..., 4. must satisfy the following systems of equations: It is observed that if the structure constants {γ k ij } of the associative algebra A are given, then, in order to describe its centroid one has to solve the system of equations above with respect to the a ij , i, j = 1, 2, ..., 4. Again, we use classification results of fourdimensional complex associative algebras from [10] to compute the centroids of four dimensional complex associative algebras.