Derivations and Centroids of Four Dimensional Associative Algebras

Transcription

1 arxiv: v1 [math.ra] 9 Jun 2016 Derivations and Centroids of Four Dimensional Associative Algebras A.O. Abdulkareem 1, M.A. Fiidow 2 and I.S. Rakhimov 2,3 1 Department of Mathematics, College of Physical Sciences, Federal University of Agriculture Abeokuta PMB 220, Alabata road, Abeokuta, Ogun State, Nigeria. afeezokareem@gmail.com 2 Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, UPM 300 Serdang, Selangor Darul Ehsan, Malaysia. wilow8@gmail.com 3 Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, UPM 300 Serdang, Selangor Darul Ehsan, Malaysia. risamiddin@gmail.com November 18, 2018 Abstract 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 algebra of four dimensional associative algebras. AMS Subject Classification: 16D70 Key Words and Phrases:Derivation, Centroid, Associative algebras, Characteristically nilpotent. 1

2 1 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 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 []). 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 ofaliealgebrais known tobeafieldandthis factplays 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- 2

3 bras in dimension less than 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. 2 Preliminaries Definition 1. An algebra A over a field K is a vector space over K equipped with a bilinear map λ : A A A. 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: Ψ(Ψ(x,y),z) = Ψ(x,Ψ(y,z)) for all x,y,z A. Definition 3. A Lie algebra L over a field K is an algebra satisfying the following conditions: [x,x] = 0, [[x,y],z]+[[y,z],x]+[[z,x],y] = 0, x, y, z L. Definition. 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 : A 1 A 2 such that f(x y) = f(x) f(y) for all x, y A 1. 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. 3

4 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 (f g)(v) = f(g(v)), for all v V. This product turns End(V) into an associative algebra. Example 6. Let the product of two elements in Example 5 above be defined by [f,g] = f g g f. Then, (End(V),[, ]) is a Lie algebra. Definition 7. A linear transformation d of an associative algebra A is called a derivation if for any x,y L d(xy) = d(x) y +x d(y). The set of all derivations of an associative algebra A is denoted by Der(A). The following fact can easily be proven. Proposition 8. Let A be an algebra. Then Der(A) is a Lie algebra with respect to the bracket [f,g] = f g g f. For a A, let L a and R a be the elements of Hom(A) defined by: L a (x) = a x, R a (x) = x a, x A. Theorem 9. Let D Hom K (A). Then the following conditions are equivalent : 1. d Der(A), 2. [d,l x ] = L d(x), x A, 3. [d,r y ] = R d(y), y A. Proof. The proof of equivalence of 1-3 is straightforward using the definition of derivation. Definition 10. Let A be an arbitrary associative algebra over a field K: The centroid of A, Γ(A) is defined by Γ(A) = {φ End(A) : φ(xy) = φ(x)y = xφ(y)}

5 Definition 11. Let H be a non empty subset of A. The set Z A (H) = {x A : x H = H x = 0}. is said to be centralizer of H in A. It must however be noted that Z A (A) = Z(A), the center of A. Definition 12. Letφ End(A). Ifφ(A) Z(A)andφ(A 2 ) = 0, then φ is called a central derivation. 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 13 ([13]). Let A be an associative algebra over a field K. If φ Γ(A) and d Der(A) then φ d is a derivation of A. Proposition 1 ([13]). Let A be an associative algebra over a field K. Then C(A) = Γ(A) Der(A). Proposition 15 ([13]). Let A be an associative algebra over a field K. Then for any d Der(A) and φ Γ(A): 1. The composition d φ is in Γ(A) if and only if φ d is a central derivation of A; 2. The composition d φ is a derivation of A if and only if [d φ] is a central derivation of A. 3 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 ei (y) = L d(ei )(y)+l ei d(y) for basis vectors e i i = 1,2,...,n. 5

6 we have (d L ei L ei d)(y) = L d(ei )(y) 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., d(e i ) = n d ji e j, i = 1,2,,n j=1 (d L ei L ei d)(y) = d L ei L ei d = n d ji L ej (y) j=1 n d ji L ej i = 1,2,,n (1) j=1 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 d = (d ij ) = d 11 d 12 d 1n.... d n1 d n2 d nn. As mention earlier, we provide the classification results of -dimensional associative algebra from [10] which we use in our study. Note that As m n denotes mth isomorphism class ofassociative algebra indimension n. 3.1 Four-dimensional Associative algebras Theorem 16. Any four-dimensional complex associative algebra can be included in one of the following isomorphism classes of algebras: As 1 : e 1e 1 = e 3, e 2 e 2 = e ; As 2 : e 1e 2 = e 3, e 2 e 1 = e ; As 3 : e 1 e 2 = e, e 3 e 1 = e ; As : e 1e 1 = e, e 2 e 2 = e 2, e 2 e 3 = e 3 As 5 : e 1e 1 = e, e 2 e 2 = e 2, e 3 e 2 = e 3 ; 6

7 As 6 : e 1e 2 = e 3, e 2 e 1 = e, e 2 e 2 = e 3 ; As 7 : e 1e 2 = e 3, e 2 e 1 = e 3, e 2 e 2 = e ; As 8 : e 1 e 2 = e, e 2 e 1 = e, e 3 e 3 = e ; As 9 (α) : e 1e 2 = e, e 2 e 1 = 1+α 1 α e, e 2 e 2 = e 3 ; As 10 : e 1 e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e ; As 11 : e 1 e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e e 1 = e ; As 12 : e 1e 1 = e 1, e 1 e 2 = e 2, e 1 e = e, e 2 e 3 = e ; As 13 : e 1e 1 = e 1, e 1 e 2 = e 2, e 3 e 3 = e 3, e 3 e = e ; As 1 : e 1 e 1 = e 1, e 1 e = e, e 2 e 1 = e 2, e 2 e = e 3 ; As 15 : e 1e 1 = e 1, e 1 e = e, e 2 e 1 = e 2, e 3 e 1 = e 3 ; As 16 : e 1e 1 = e 1, e 2 e 1 = e 2, e 3 e 1 = e 3, e e 1 = e ; As 17 : e 1 e 1 = e 1, e 2 e 1 = e 2, e 3 e 2 = e, e e 1 = e ; As 18 : e 1e 1 = e 1, e 2 e 1 = e 2, e 3 e 3 = e 3, e e 3 = e ; As 19 : e 1e 1 = e 1, e 2 e 2 = e 2, e 2 e = e, e 3 e 1 = e 3 ; As 20 : e 1e 1 = e 1, e 2 e 2 = e 2, e 3 e 3 = e 3, e e = e ; As 21 : e 1 e 1 = e 3, e 1 e 3 = e, e 2 e 2 = e, e 3 e 1 = e ; As 22 : e 1e 1 = e, e 1 e 2 = e 3, e 2 e 1 = e 3, e 2 e 2 = 2e 3 +e ; As 23 (µ) : e 1e 1 = e, e 1 e 2 = e 3, e 2 e 1 = µe, e 2 e 2 = e 3 ; As 2 : e 1 e 1 = e, e 1 e 2 = e, e 2 e 1 = e, e 3 e 3 = e ; As 25 : e 1e 1 = e, e 1 e = e 3, e 2 e 1 = e 3, e e 1 = e 3 ; As 26 : e 1e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e, e 2 e 1 = e 2 ; As 27 : e 1e 1 = e 1, e 1 e 2 = e 2, e 1 e = e, e 3 e 1 = e 3, e e 1 = e ; As 28 : e 1 e 1 = e 1, e 1 e 2 = e 2, e 2 e 1 = e 2, e 3 e 1 = e 3, e e 1 = e ; As 29 : e 1e 1 = e 1, e 1 e 3 = e 3, e 1 e = e, e 2 e 2 = e 2, e 3 e 2 = e 3 ; As 30 : e 1e 1 = e 1, e 1 e 3 = e 3, e 2 e 2 = e 2, e 2 e = e, e e 1 = e ; As 31 : e 1 e 1 = e 1, e 2 e 2 = e 2, e 2 e 3 = e 3, e 3 e 1 = e 3, e e 1 = e ; As 32 : e 1e 1 = e 1, e 2 e 2 = e 2, e 2 e 3 = e 3, e 3 e 1 = e 3, e e 2 = e ; As 33 : e 1e 1 = e 1, e 2 e 2 = e 2, e 3 e 2 = e 3, e e 3 = e 3, e e = e ; As 3 : e 1e 1 = e 1, e 2 e 2 = e 2, e 3 e 3 = e 3, e 3 e = e, e e 3 = e ; As 35 : e 1 e 1 = e, e 1 e 2 = λe, e 2 e 1 = λe, e 2 e 2 = e, e 3 e 3 = e ; As 36 : e 1e 1 = e, e 1 e = e 3, e 2 e 1 = e 3, e 2 e 2 = e 3, e e 1 = e 3 ; 7

8 As 37 : e 1e 2 = e, e 1 e 3 = e, e 2 e 1 = e, e 2 e 2 = e, e 3 e 1 = e ; As 38 : e 1e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e, e 2 e 1 = e 2, e 3 e 1 = e 3 ; As 39 : e 1 e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e, e 3 e 1 = e 3, e 3 e 2 = e ; As 0 : e 1 e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 2 e 1 = e 2, e 3 e 1 = e 3, e e 1 = e ; As 1 : e 1e 1 = e 1, e 1 e 2 = e 2, e 2 e 1 = e 2, e 3 e 3 = e 3, e 3 e = e, e e 3 = e ; As 2 : e 1e 1 = e 1, e 1 e 3 = e 3, e 1 e = e, e 2 e 2 = e 2, e 3 e 1 = e 3, e e 2 = e ; As 3 : e 1 e 1 = e 1, e 1 e 3 = e 3, e 2 e 1 = e 2, e 2 e 3 = e, e 3 e 1 = e 3, e e 1 = e ; As : e 1e 1 = e 1, e 1 e 3 = e 3, e 2 e 2 = e 2, e 2 e = e, e 3 e 1 = e 3, e e 1 = e ; As 5 : e 1e 1 = e 1, e 1 e = e, e 2 e 2 = e 2, e 2 e 3 = e 3, e 3 e 1 = e 3, e e 2 = e ; As 6 : e 1 e 1 = e 1, e 2 e 2 = e 2, e 2 e 3 = e 3, e 2 e = e, e 3 e 1 = e 3, e e 1 = e ; As 7 : e 1 e 1 = e 1, e 2 e 2 = e 2, e 2 e 3 = e 3, e 2 e = e, e 3 e 2 = e 3, e e 2 = e ; As 8 : e 1e 1 = e 2, e 1 e 2 = e 3, e 1 e 3 = e, e 2 e 1 = e 3, e 2 e 2 = e, e 3 e 1 = e ; As 9 : e 1e = e 2, e 2 e 3 = e 2, e 3 e 1 = e 1, e 3 e 2 = e 2, e 3 e 3 = e 3, e e 3 = e ; As 50 : e 1 e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e, e 2 e 1 = e 2, e 2 e 2 = e, e e 1 = e ; As 51 : e 1e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e, e 2 e 1 = e 2, e 3 e 1 = e 3, e e 1 = e ; As 52 : e 1e 1 = e 1, e 1 e 2 = e 2, e 1 e = e, e 2 e 1 = e 2, e 2 e 2 = e, e 3 e 1 = e 3, e e 1 = e ; As 53 : e 1 e 1 = e 1, e 2 e 2 = e 2, e 2 e 3 = e 3, e 2 e = e, e 3 e 2 = e 3, e 3 e 3 = e, e e 2 = e ; As 5 : e 1 e 1 = e 1, e 1 e 2 = e 2, e 2 e 3 = e 1, e 2 e = e 2, e 3 e 1 = e 3, e 3 e 2 = e, e e 3 = e 3, e e = e ; As 55 : e 1e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e, e 2 e 1 = e 2, e 2 e 2 = e, e 3 e 1 = e 3, e e 1 = e ; As 56 : e 1e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e, e 2 e 1 = e 2,e 2 e 3 = e, e 3 e 1 = e 3, e 3 e 2 = e, e e 1 = e ; As 57 : e 1 e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e, e 2 e 1 = e 2,e 2 e 2 = e 3, e 2 e 3 = e, e 3 e 1 = e 3, e 3 e 2 = e, e e 1 = e ; As 58 : e 1e 1 = e 1, e 1 e 2 = e 2, e 1 e 3 = e 3, e 1 e = e, e 2 e 1 = e 2, e 2 e 2 = e, e 2 e 3 = e, e 3 e 1 = e 3, e 3 e 2 = e, e e 1 = e ; for all α C\{1} and λ,µ C. In what follows, the following notations are introduced: 1. IC: isomorphism classes of algebras. 2. Dim: Dimensions of the algebra of derivations. 8

9 3. m 2 = d 11 +d 21,. m 3 = d 33 d 11, 5. m = 2 1 α d 12, 6. m 5 = d 22 +d 33, 7. m 6 = d 33 d 22. 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: Table 1: Derivations of four-dimensional associative algebras IC Derivation Dim IC Derivation Dim 0 d d 31 d 32 2d As 2 0 d d 31 d 32 m d 1 d 2 0 2d 22 d 1 d 2 0 m 2 As 1 As 3 As 5 As 7 d 21 d d 21 0 d 22 0 d 1 d 2 d 3 m 2 0 d 32 d 33 0 d d 11 d 11 d d d 31 d 32 m 2 0 d 1 d 2 0 2d 22 6 As As 6 7 As 8 0 d 32 d 33 0 d d 11 0 d d 31 d 32 2d 11 0 d 1 d 2 0 2d 11 d 11 d d 21 m d 33 0 d 1 d 2 d 3 2d 33 Continued to the next page 5 7 9

10 IC Derivation Dim IC Derivation Dim d 11 d As 9 (α) 0 d d 31 d 32 2d As 10 d 21 d 22 d 23 d 2 d 31 d 32 d 33 d 3 12 d 1 d 2 m m 2 d 1 d 2 d 3 d As 11 As 13 As 15 As 17 As 19 As 21 As 23 (µ) As 25 d 21 d 22 d 23 0 d 31 d 32 d 33 0 d d d 21 d d 3 d d 21 d 22 d 23 0 d 31 d 32 d 33 0 d d d 21 d d 33 0 d 1 d 2 d 21 m 5 d 31 0 d d 2 0 d 3 d 21 2 d d 31 d 21 2d 11 0 d 1 d 2 2d 31 3d 11 0 d d 31 d 32 2d 11 0 d 1 d 2 0 2d 11 d 21 2d d 31 d 32 3d 11 m 7 d d 11 8 As 12 As 1 8 As 16 5 As 18 As 20 5 As 22 5 As 2 5 As 26 d 21 d d 33 0 d 1 d 2 d 21 m 5 d 21 d d 32 d 33 d 21 d m 6 d 21 d 22 d 23 d 2 d 31 d 32 d 33 d 3 d 1 d 2 d 3 d d 21 d d 3 d d 31 d 32 2d 11 0 d 1 d 2 0 2d 11 d 21 d d 11 0 d 1 d 2 d 3 2d 11 0 d d 31 0 d 33 d 3 d 1 0 d 3 d Continued to the next page

11 IC Derivation Dim IC Derivation Dim d 21 d d 31 0 d 33 d 3 5 As 28 0 d d 31 0 d 33 d d d 1 0 d 3 d As 27 As 29 As 31 As 33 As 35 As 37 As 39 As 1 As 3 d 31 d 31 d 33 0 d d d 31 d 31 d 33 0 d d 0 d 32 d 33 d 32 d 11 d d 21 d d 1 d 2 d 3 2d 11 d 21 d d 21 d 11 0 d 1 d 2 d 3 2d 11 d 21 d d 33 0 d 1 d 2 d 21 m 5 0 d d d 21 d d 33 0 d 1 d 2 d 21 m 5 As 30 As 32 2 As 3 5 As 36 5 As 38 5 As 0 2 As 2 5 As d 31 0 d 33 0 d 1 d 1 0 d d 31 d 31 d d 2 0 d d d d 31 d 32 0 m 7 d 1 d d 22 d d 32 d 33 0 d d 0 d 22 d d 32 d 33 0 d d 0 0 d 33 0 d 1 d 1 0 d 0 0 d 33 0 d 1 d 1 0 d Continued to the next page

12 IC Derivation Dim IC Derivation Dim As 5 d 31 d 31 d 33 0 As 6 d 31 d 31 d 33 d 3 6 d 1 d 1 0 d d 1 d 1 d 3 d As 7 As 9 As 51 As 53 As 55 As d 33 d d 3 d d 11 0 d 13 0 d 21 d 22 0 d d 21 m 1 0 d 22 d 23 d 2 0 d 32 d 33 d 3 0 d 2 d 3 d 0 0 d d 3 2d 33 0 d d 32 d d 2 d 3 2d 22 0 d d 32 2d d 2 2d 32 3d 22 As 8 As 50 9 As 52 2 As 5 5 As 56 3 As 58 d 21 2d d 31 2d 21 3d 11 0 d 1 2d 31 3d 21 d 11 0 d d 31 0 d d 2 0 2d 22 0 d d 31 0 d d 2 0 2d 22 0 d 12 d 21 0 d 21 d 22 0 d 21 d 12 0 d 22 d 12 0 d 12 d d 22 d d 32 d d 2 d 3 m 5 0 d d 32 d d 2 d 3 2d 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 1. Applying the system of equation (1), we get d 12 = d 13 = d 1 = d 21 = d 23 = d 2 = d 3 = d 3 = 0, d 33 = 2d 11,d = 2d 22 12

13 Hence, thederivationforas 1 aregivenasfollowsd = So,d 1 = d = ,d 2 =, d 5 = ,d 3 =, d 6 = 0 d d 31 d 32 2d 11 0 d 1 d 2 0 2d 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. Remark 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 {e 1,e 2,e 3,e } be a basis of a - dimensional associative algebra A, then e i e j = γije k k. The coefficients, {γij k}, of the above linear combinations are called C3 the structure constants of A on the basis {e 1,e 2,e 3,e }. An element φ of the centroid Γ(A), being a linear transformation of the vector space A, is represented in a matrix form (a ij ) i,j=1,2,...,, i.e. φ(e i ) = a ji e j, i = 1,2,...,. must satisfy the following systems j=1 13 k=1

14 of equations: (γij t a kt a ti γtj k ) = 0 (2) k=1 (γij k a kt a tj γit k ) = 0 (3) k=1 It is observed that if the structure constants {γij k } 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,...,. Again, we use classification results of fourdimensional complex associative algebras from [10] to compute the centroids of four dimensional complex associative algebras. Theorem 19. The centroids of four dimensional complex associative algebras are given as follows: Table 2: Centroids of four-dimensional associative algebras IC Centroid Dim IC Centroid Dim Γ(A) Γ(A) 0 a As 2 a 31 a 32 a a 1 a 2 0 a 11 As 1 As 3 As 5 a 1 a 2 a 3 a 11 0 a a 22 0 a a 11 As 3 As 6 0 a a 22 0 a a 11 a 31 a 32 a 11 0 a 1 a 2 0 a 11 Continued to the next page 3 5 1

15 IC Centroid Dim IC Centroid Dim Γ(A) Γ(A) a 11 a As 7 a 31 a 32 a 11 a 12 6 As 8 a 1 a 2 0 a 11 As 9 (α) As 11 As 13 As 15 As 17 As 19 As 21 As 23 (µ) a 31 a 32 a 11 0 a 1 a 2 0 a a a 33 0 a a 22 a 31 0 a 11 0 a 1 a 2 a 31 a 11 a 31 a 32 a 11 0 a 1 a 2 0 a 11 5 As 10 1 As 12 2 As 1 1 As 16 1 As 18 2 As 20 As 22 5 As 2 a 1 a 2 a 3 a 11 a 21 a a 31 0 a 11 0 a a a a 33 0 a a a a 31 a 32 a 11 0 a 1 a 2 0 a 11 a 1 a 2 a 3 a 11 Continued to the next page

16 IC Centroid Dim IC Centroid Dim Γ(A) Γ(A) a 21 a 22 0 a 2 a 31 a 32 a As 26 a 21 a a 31 0 a 11 0 a a 11 As 25 As 27 As 29 As 31 As 33 As 35 As 37 As 39 As 1 As 3 a a 11 0 a a a 22 a 1 a 2 a 3 a 11 a 31 0 a a 31 0 a 11 a 21 a 11 0 a a a 3 a 33 a 31 0 a a 31 0 a 11 2 As 28 1 As 30 1 As 32 2 As 3 As 36 1 As 38 2 As 0 As 2 2 As a a 11 a 21 a a a a a 33 a 31 a 32 a 11 a 1 a 1 a 2 a 3 a 11 a 21 a a 31 0 a 11 0 a 21 a 11 0 a 2 a 31 0 a 11 a 3 0 a 31 0 a 11 a 31 0 a 11 0 a 31 0 a Continued to the next page

17 IC Centroid Dim IC Centroid Dim Γ(A) Γ(A) 1 As 6 1 As 5 As 7 As 9 As 51 As 53 As 55 As 57 0 a a 32 a a 2 0 a 22 0 a 11 a 23 0 a 21 a a 1 a 21 0 a 11 0 a a 32 a a 2 a 32 a 22 a 21 a a 31 0 a 11 0 a 1 a 21 0 a 11 a a 11 As 8 3 As 50 3 As 52 As 5 As 56 2 As 58 a 21 a a 31 a 21 a a 31 a 21 a 11 a 21 a a 31 0 a 11 0 a a 11 a 21 a a 1 a 21 0 a 11 a 21 a a 31 0 a 11 0 a 1 a 31 a 21 a 11 a a Proof. Let us consider As 1, the structure constants are given as follows γ11 3 = 1,γ 22 = 1 others being zero. From the systems of equations (2) and (3), we have a 12 = a 13 = a 1 = a 21 = a 23 = a 2 = a 31 = a 32 = a 3 = a 1 = a 2 = a 3 = 0 and a 33 = a 11 = a. Therefore, we obtain the centroids of 17

18 As 1 in matrix form as follows; Γ(As 1 ) = 0 a Now let us consider As 2. The structure constants are given as follows γ12 3 = 1,γ 21 = 1. Using the system of equations (2) and (3), we get a 12 = a 13 = a 1 = a 21 = a 23 = a 2 = a 3 = a 3 = 0 and a 33 = a 11 = a 22 = a. Therefore, Γ(As 2 ) = a 31 a 32 a 11 0 a 1 a 2 0 a 11. Let us take the class As 3 with structure constants γ 11 = 1,γ 31 = 1 and others are zero. Applying (2) and (3), we get a 12 = a 13 = a 1 = a 21 = a 23 = a 2 = a 31 = a 32 = a 3 = 0 and a 33 = a 11 = a 22 = a. Therefore, Γ(As 3 ) = a 1 a 2 a 3 a 11. Using the same approach, the centroids of associative algebras in dimension four can be computed for the remaining algebras. 5 Acknowledgement Part of this study was presented at the regional fundamental science congress, 201(RFSC201) organised by Universiti Putra Malaysia. The authors would like to thank Assoc. Prof., Dr. M. R. K. Ariffin for his financial support (which facilitated first author participation in RFSC201) through the Putra Grant with vote number

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