Description of three-dimensional evolution algebras

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by  Yolanda Cabrera Casado

DESCRIPTION OF THREE-DIMENSIONAL EVOLUTION ALGEBRAS
YOLANDA CABRERA CASADO, MERCEDES SILES MOLINA, AND M. VICTORIA VELASCO
Abstract. Weclassifythreedimensionalevolutionalgebrasoverafieldhavingcharacteristic
different from 2 and in which there are roots of orders 2, 3 and 7.
7
1 1. Introduction
0
2
The use of non-associative algebras to formulate Mendel’s laws was started by Ethering-

b
ton in his papers [5, 6]. Other genetic algebras (those that model inheritance in genetics)
e
F called evolution algebras emerged to study non-Mendelian genetics. Its theory in the finite-
6 dimensional case was introduced by Tian in [7]. The systematic study of evolution algebras of
arbitrary dimension and of their algebraic properties was started in [1], where the authors an-

]
A alyze evolution subalgebras, ideals, non-degeneracy, simple evolution algebras and irreducible
R evolution algebras. The aim of this paper is to obtain the classification of three-dimensional
. evolution algebras having in mind to apply this classification in a near future in a biological
h
t setting and to detect possible tools to implement in wider classifications.
a
m Two-dimensional evolution algebras over the complex numbers were determined in [2],
although we have found that this classification is incomplete: the algebra A with natural
[
basis {e ,e } such that e2 = e and e2 = e is a two-dimensional evolution algebra not
2 1 2 1 2 2 1
v isomorphic to any of the six types in [2]. We realized of this fact when classifying the three-
9 dimensionalevolutionalgebrasAsuchthatdim(A2) = 2andhavingannihilator1ofdimension
1
1 (see Tables 14-16).
2
7 The three dimensional case is much more complicated, as can be seen in this work, were
0
it is proved that there are 116 types of three-dimensional evolution algebras. All of them
.
1 are classified in Tables 1-24. The matrices appearing in different tables are not isomorphic
0
(in the meaning that they do not generate the same evolution algebra). Matrices in different
7
1 rows of a same table neither are isomorphic. In general, different values of the parameter for
:
v matrices in the same row give non-isomorphic evolution algebras, but in some case this is not
i true. These cases are displayed in Tables 2(cid:48)-23(cid:48).
X
Just after finishing this paper we found the article [4], where one of the aims of the authors
r
a is to classify indecomposable2 nilpotent evolution algebras up to dimension five over alge-
braically closed fields of characteristic not two. The three-dimensional ones can be localized
in our classification and for these, it is not necessary to consider algebraically closed fields.
In this paper we deal with evolution algebras over a field K of characteristic different from
2 and in which every polynomial of the form xn −α, for n = 2,3,7 and α ∈ K has a root in
the field. We denote by φ a seventh root of the unit and by ζ a third root of the unit.
2010 Mathematics Subject Classification. Primary 17D92, 17A60.
Key words and phrases. Genetic algebra, evolution algebra, annihilator, extension property.
1The annihilator of A, ann(A), is defined as the set of those elements x in A such that xA=0.
2Irreducible following [1].
1
2 Y. CABRERA, M. SILES, AND M. V. VELASCO
In Section 2 we introduce the essential definitions. For every arbitrary finite dimensional
algebra, fix a basis B = {e | i = 1,...,n}. The product of this algebra, relative to the
i
basis B is determined by the matrices of the multiplication operators, M (λ ) (see (1)).
B ei
The relationship under change of basis is also established. In the particular case of evolution
algebras Theorem 2.2 shows this connection.
We start Section 3 by analyzing the action of the group S (cid:111) (K×)3 on M (K). The
3 3
orbits of this action will completely determine the non-isomorphic evolution algebras A when
dim(A2) = 3 and in some cases when dim(A2) = 2.
We have divided our study into four cases depending on the dimension of A2, which can
be 0, 1, 2 or 3. The first case is trivial. The study of the third and of the fourth ones is
made by taking into account which are the possible matrices P that appear as change of basis
matrices. It happens that for dimension 3, as we have said, the only matrices are those in
S (cid:111)(K×)3.
3
When the dimension of A2 is 2, there exists three groups of cases (four in fact, but two of
them are essentially the same). Let B = {e ,e ,e } be a natural basis of A such that {e2,e2}
1 2 3 1 2
is a basis of A2 and e2 = c e2+c e2 for some c ,c ∈ K. The first case happens when c c (cid:54)= 0.
3 1 1 2 2 1 2 1 2
Then, P ∈ S (cid:111)(K×)3. The second group of cases arises when c = 0 and c (cid:54)= 0. Then, the
3 1 2
matrix P is id , (2,3),3 or the matrix Q given in Case 2 (when dim(A2 = 2)). The third one
3
appears when case happens when c ,c = 0. In this case the matrix P is id or the matrices
1 2 3
Q(cid:48) and Q(cid:48)(cid:48) given in Case 4 (when dim(A2 = 2)).
For P ∈ S (cid:111)(K×)3, we classify taking into account: the dimension of the annihilator of
3
A, the number of non-zero entries in the structure matrix (which remains invariant, as it is
proved in Proposition 3.2), and if the algebra A satisfies Property (2LI)4.
For P ∈ {id ,(2,3),Q}, we obtain a first classification, given in the different Figures. Then
3
we compare which matrices produce isomorphic algebras and eliminating redundancies we
get the matrices given in the set S that appears in Theorem 3.5. Again, some of these
matrices give isomorphic evolution algebras. In order to classify them, we take into account
that the number of non-zero entries of the matrices in S remains invariant under the action
of the matrix P (see Remark 3.7). Note that the resulting matrices correspond to evolution
algebras with zero annihilator and do not satisfy Property (2LI).
For P ∈ {id ,Q(cid:48),Q(cid:48)(cid:48)} we classify taking into account that the third column of the structure
3
matrix has three zero entries (the dimension of the annihilator is one and, consequently, they
do not satisfy Property (2LI)) and the number of zeros in the first and the second row remains
invariant under change of basis matrices (see Remark 3.8).
For dim(A2) = 3 we classify by the number of non-zero entries in the structure matrix.
In the case dim(A2) = 1 it is not efficient to tackle the problem of the classification by
obtainingthepossiblechangeofbasismatrices,althoughforcompletenesswehavedetermined
them in Appendix A. This is because we follow a different pattern. The key point for this
study will be the extension property 5 ((EP) for short). We have classified taking into account
the following properties: whether or not A2 has the extension property, the dimension of the
3The matrix obtained from the identity matrix, id , when exchanging the second and the third rows
3
4For any basis {e ,e ,e } the ideal A2 has dimension two and it is generated by {e2,e2}, for every i,j ∈
1 2 3 i j
{1,2,3} with i(cid:54)=j.
5There is a natural basis of A2 that can be enlarged to a natural basis of A
DESCRIPTION OF THREE-DIMENSIONAL EVOLUTION ALGEBRAS 3
annihilator of A, and whether or not the evolution algebra A has a principal6 two-dimensional
evolution ideal which is degenerate7 as an evolution algebra (PD2EI for short).
The classification of three-dimensional evolution algebras is achieved in Theorem 3.5. We
summarize the cases in the tables that follow.
A2hasEP dim(ann(A)) AhasaPD2EI Number
No 0 Yes 1
No 1 Yes 1
Yes 2 No 1
Yes 1 No 1
Yes 0 No 1
Yes 2 Yes 1
Yes 1 Yes 1
dim(A2)=1
Non-zeroentries
dim(ann(A)) *Non-zeroentriesinS AhasProperty(2LI) Number
**Non-zeroentriesinrows1and2
1 1** No 2
1 2** No 4
1 3** No 2
1 4** No 2
0 4* No 3
0 5* No 6
0 6* No 3
0 7* No 6
0 8* No 3
0 9* No 3
0 4 Yes 4
0 5 Yes 3
0 6 Yes 7
0 7 Yes 6
0 8 Yes 2
0 9 Yes 1
dim(A2)=2
Non-zeroentries Number
3 3
4 6
5 16
6 15
7 8
8 2
9 1
dim(A2)=3
2. Product and change of basis
In this section we study the product in an arbitrary algebra by considering the matrices
associatedtotheproductbyanyelementinafixedbasis. Wespecializetothecaseofevolution
6Principal means that it is generated as an ideal by one element.
7An evolution algebra is non-degenerate if e2 (cid:54)= 0 for any element e in any basis (see [1, Definition 2.16
and Corollary 2.19]). Otherwise we say that it is degenerate.
4 Y. CABRERA, M. SILES, AND M. V. VELASCO
algebras and obtain the relationship for two structure matrices of the same evolution algebra
relative to different basis.
2.1. The product of an algebra. Let A be a K-algebra. Assume that B = {e | i ∈ Λ} is a
i
(cid:88)
basis of A, and let {ω } ⊆ K be the structure constants, i.e. e e = ω e and ω
kij i,j,k∈Λ i j kij k kij
k∈Λ
is zero for almost all k. Since in this paper we will deal only with finite dimensional evolution
algebras, we will assume that Λ is finite and has cardinal n.
For any element a ∈ A the following map defines the left multiplication operator by a,
denoted as λ :
a
λ : A → A
a
x (cid:55)→ ax
Then, for every i ∈ Λ we have
 ω ··· ω 
1i1 1in
MB(λei) =  ... ... ... ,
ω ··· ω
ni1 nin
where for any linear map T : A → A we write M (T) to denote the matrix in M (K)
B Λ
associated to T relative to the basis B.
Let A be an algebra and let B = {e | i ∈ Λ} be a basis of A. For arbitrary elements
i
(cid:80) (cid:80)
x = α e and y = β e in A the product xy is as follows:
i∈Λ i i i∈Λ i i
(cid:32) (cid:33)(cid:32) (cid:33) (cid:32) (cid:33)
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
xy = α e β e = α β e e = α β ω e = α β ω e .
i i j j i j i j i j kij k i j kij k
i∈Λ j∈Λ i,j∈Λ i,j∈Λ k∈Λ k,i,j∈Λ
Denote by ξ (x) the coordinates of an element x in A relative to the basis B, written by
B
columns. Then:
(cid:32) (cid:33) ω ··· ω α β 
111 11n 1 1
ξB(xy) = ξB (cid:88) αiβjωkijek =  ... ... ...  ... 
k,i,j∈Λ ω ··· ω α β
n11 n1n 1 n
+ ···
ω ··· ω α β 
1n1 1nn n 1
+  ... ... ...  ... .
ω ··· ω α β
nn1 nnn n n
That is,
α β 
i 1
(cid:88) .
(1) ξB(xy) = MB(λei) .. .
i∈Λ α β
i n
An evolution algebra over a field K is a K-algebra A provided with a basis B = {e | i ∈ Λ}
i
such that e e = 0 whenever i (cid:54)= j. Such a basis B is called a natural basis. Now, the structure
i j
DESCRIPTION OF THREE-DIMENSIONAL EVOLUTION ALGEBRAS 5
constants of A relative to B are the scalars ω ∈ K such that e2 := e e = (cid:80) ω e . The
ki i i i ki k
k∈Λ
matrix M := (ω ) is said to be the structure matrix of A relative to B.
B ki
For any finite dimensional evolution algebra A with a natural basis B we have
(cid:88)
M = M (λ ).
B B ei
i∈Λ
In case of A being an evolution algebra and B = {e | i ∈ Λ} a natural basis of A, the
i
structure constants satisfy that ω = 0 for every i,j,k ∈ Λ with i (cid:54)= j. If we denote
kij
ω = ω we obtain that:
kii ki
ω 0 ··· 0α β  0 ··· 0 ω α β 
11 1 1 1n n 1
ξB(xy) =  ... ... ... ... ... +...+... ... ... ...  ... 
ω 0 ··· 0 α β 0 ··· 0 ω α β
n1 1 n nn n n
ω ··· ω α β 
11 1n 1 1
=  ... ... ...  ... ,
ω ··· ω α β
n1 nn n n
because for every i ∈ Λ the matrix M (λ ) has zero entries except at most in its ith column.
B ei
Summarizing,
α β 
1 1
.
(2) ξB(xy) = MB .. .
α β
n n
Definition 2.1. Let A be an algebra and B = {e | i ∈ Λ} a basis of A. For arbitrary
i
(cid:88) (cid:88)
elements x = α e and y = β e in A, we define
i i i i
i∈Λ i∈Λ
(cid:32) (cid:33) (cid:32) (cid:33)
(cid:88) (cid:88) (cid:88)
x• y = α e • β e := α β e .
B i i B i i i i i
i∈Λ i∈Λ i∈Λ
Now, in the case of an evolution algebra we may write (2) as follows.
(3) ξ (xy) = M (ξ (x)• ξ (y)),
B B B B B
where, by abuse of notation, we write • to multiply two matrices, by identifying the matrices
B
with the corresponding vectors and multiplying them as in Definition 2.1.
2.2. Change of basis. First, we study the matrix of the product of a finite dimensional
arbitrary algebra under change of basis. Then we fix our attention in evolution algebras.
Let B = {e | i ∈ Λ} and B(cid:48) = {f | j ∈ Λ} be two bases of an algebra A. Suppose that
i j
the relation between these bases is given by
(cid:88) (cid:88)
e = q f and f = p e ,
i ki k i ki k
k∈Λ k∈Λ
6 Y. CABRERA, M. SILES, AND M. V. VELASCO
where {p } and {q } are subsets of K such that P := (q ) and P := (p ) are
ki k,i∈Λ ki k,i∈Λ BB(cid:48) ki B(cid:48)B ki
the change of basis matrices. Assume that the structure constants of A relative to B and to
B(cid:48) are, respectively, {(cid:36) } and {ω } . Then, for every i,j ∈ Λ:
kij i,j,k∈Λ kij i,j,k∈Λ
(cid:32) (cid:33)(cid:32) (cid:33)
(cid:88) (cid:88) (cid:88) (cid:88)
f f = p e p e = p p e e = p p (cid:36) e
i j ki k tj t ki tj k t ki tj mkt m
k∈Λ t∈Λ k,t∈Λ k,t,m∈Λ
(cid:32) (cid:33)
(cid:88) (cid:88) (cid:88) (cid:88)
= p p (cid:36) q f = (p p (cid:36) q ) f = ω f .
ki tj mkt lm l ki tj mkt lm l lij l
k,t,m,l∈Λ l∈Λ k,t,m∈Λ l∈Λ
(cid:80)
Therefore, (p p (cid:36) q ) = ω .
k,t,m∈Λ ki tj mkt lm lij
Our next aim is to express every ω in terms of certain matrices. To find such matrices,
lij
write:
ω = p p (cid:36) q +...+p p (cid:36) q
lij 1i 1j 111 l1 1i 1j n11 ln
.
.
.
+p p (cid:36) q +...+p p (cid:36) q
1i nj 11n l1 1i nj n1n ln
.
.
.
+p p (cid:36) q +...+p p (cid:36) q
ni 1j 1n1 l1 ni 1j nn1 ln
.
.
.
+p p (cid:36) q +...+p p (cid:36) q .
ni nj 1nn l1 ni nj nnn ln
In terms of matrices,
(cid:36) ··· (cid:36) p p 
111 11n 1i 1j
ωlij = (cid:0)ql1 ··· qln(cid:1) ... ... ...  ... 
(cid:36) ··· (cid:36) p p
n11 n1n 1i nj
+ ···
(cid:36) ··· (cid:36) p p 
1n1 1nn ni 1j
+ (cid:0)ql1 ··· qln(cid:1) ... ... ...  ... .
(cid:36) ··· (cid:36) p p
nn1 nnn ni nj
This is equivalent to:
DESCRIPTION OF THREE-DIMENSIONAL EVOLUTION ALGEBRAS 7
q ··· q (cid:36) ··· (cid:36) p p ··· p p 
11 1n 111 11n 1i 11 1i 1n
MB(cid:48)(λfi) =  ... ... ...  ... ... ...  ... ... ... 
q ··· q (cid:36) ··· (cid:36) p p ··· p p
n1 nn n11 n1n 1i n1 1i nn
+ ···
q ··· q (cid:36) ··· (cid:36) p p ··· p p 
11 1n 1n1 1nn ni 11 ni 1n
+  ... ... ...  ... ... ...  ... ... ... 
q ··· q (cid:36) ··· (cid:36) p p ··· p p
n1 nn nn1 nnn ni n1 ni nn
(cid:32) (cid:33)
(cid:88)
= P−1 M (λ )p P .
B(cid:48)B B ek ki B(cid:48)B
k
We finish the section by asserting the relationship among two structure matrices associated
to the same evolution algebra relative to different bases. We include the proof of Theorem
2.2 for completeness. The ideas we have used can be found in [7, Section 3.2.2.].
Theorem 2.2. Let A be an evolution algebra and let B = {e ,...,e } be a natural basis of
1 n
A with structure matrix M = (ω ). Then:
B ij
(i) If B(cid:48) = {f ,...,f } is a natural basis of A and P = (p ) is the change of basis matrix
1 n ij
(cid:88)
P , i.e., f = p e , for every i, then |P| =(cid:54) 0 and
B(cid:48)B i ji j
j
ω ··· ω p  p  0
11 1n 1i 1j
(4)  ... ... ...  ... •B  ...  = ... for every i (cid:54)= j.
ω ··· ω p p 0
n1 nn ni nj
Moreover,
p ··· p −1ω ··· ω p2 ··· p2 
11 1n 11 1n 11 1n
(5) MB(cid:48) =  ... ... ...   ... ... ...  ... ... ...  = P−1MBP(2),
p ··· p ω ··· ω p2 ··· p2
n1 nn n1 nn n1 nn
where P(2) = (p2 ).
ij
(ii) Assume that P = (p ) ∈ M (K) has non-zero determinant and satisfies the relations
ij n
in (4). Define B(cid:48) = {f ,...,f }, where f = (cid:80) p e , for every i. Then, B(cid:48) is a natural
1 n i j ji j
basis and (5) is satisfied.
Proof. (i). Clearly, since B and B(cid:48) are two bases of A then |P| =(cid:54) 0. Besides, since B and B(cid:48)
are natural bases, by (2) we have:
ω ··· ω p  p  0
11 1n 1i 1j
ξB(fifj) =  ... ... ...  ... •B  ... = ...
ω ··· ω p p 0
n1 nn ni nj
and
8 Y. CABRERA, M. SILES, AND M. V. VELASCO
ω ··· ω p2 
11 1n 1i
ξB(fi2) =  ... ... ...  ... 
ω ··· ω p2
n1 nn ni
for every i,j, being i (cid:54)= j.
On the other hand, if M = ((cid:36) ), for every i (cid:54)= j we obtain:
B(cid:48) ij
p ··· p −1ω ··· ω p2  (cid:36) 
11 1n 11 1n 1i 1i
ξB(cid:48)(fi2) =  ... ... ...   ... ... ...  ...  =  ... 
p ··· p ω ··· ω p2 (cid:36)
n1 nn n1 nn ni ni
and consequently
p ··· p −1ω ··· ω p2 ··· p2 
11 1n 11 1n 11 1n
MB(cid:48) =  ... ... ...   ... ... ...  ... ... ...  = P−1MBP(2).
p ··· p ω ··· ω p2 ··· p2
n1 nn n1 nn n1 nn
(ii). AssumethatP = (p )hasnonzerodeterminant. ThenB(cid:48),definedasinthestatement,
ij
is a basis of A. Moreover, if (4) is satisfied, then B(cid:48) is a natural basis as follows by (2). (cid:3)
The formula (4) can be rewritten in a more condensed way. Concretely (see [7]),
(6) M (P ∗P) = 0,
B
where P ∗P = (c ) ∈ M (K), being c = p p for every pair (i,j) with i < j
k(i,j) n×n(n−1) k(i,j) ki kj
2
and i,j ∈ {1,...,n}.
3. Three-dimensional evolution algebras
The aim of this section is to determine the three-dimensional evolution algebras over a field
K having characteristic different from two and such that for any α ∈ K and n = 2,3,7, the
equation xn = α has a solution. For our purposes, we divide our study in different cases,
depending on the dimension of A2.
3.1. Action of S (cid:111)(K×)3 on M (K).
3 3
Let K be a field. By K× we denote K\{0}. For every α,β,γ ∈ K×, we define the matrices:
     
α 0 0 1 0 0 1 0 0
Π1(α) := 0 1 0, Π2(β) := 0 β 0, Π3(γ) := 0 1 0.
0 0 1 0 0 1 0 0 γ
It is easy to prove that they commute each other. This implies that
  
α 0 0
 
G = (cid:8)Π1(α)Π2(β)Π3(γ) |α,β,γ ∈ K×(cid:9) = 0 β 0 | α,β,γ ∈ K×
 0 0 γ 
DESCRIPTION OF THREE-DIMENSIONAL EVOLUTION ALGEBRAS 9
 
α 0 0
is an abelian subgroup of GL3(K). We will denote the diagonal matrix 0 β 0 by
0 0 γ
(α,β,γ). With this notation in mind, it is immediate to see that G ∼= K× ×K× ×K× with
product given by (α,β,γ)(α(cid:48),β(cid:48),γ(cid:48)) := (αα(cid:48),ββ(cid:48),γγ(cid:48)).
Now, considerthesymmetric groupS ofallpermutationsofthe set{1,2,3}. Thestandard
3
notation for S is:
3
S = {id,(1,2),(1,3),(2,3),(1,2,3),(1,3,2)},
3
whereidistheidentitymap, (i,j)isthepermutationthatsendstheelementiintotheelement
j and (i,j,k) is the permutation sending i to j, j to k and k to i, for {i,j,k} = {1,2,3}.
We may identify S with the set
3
          
0 1 0 0 0 1 1 0 0 0 1 0 0 0 1
 
(7) id3,1 0 0,0 1 0,0 0 1,0 0 1,1 0 0
 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 
in the following way: id is identified with the identity matrix id , (1,2) with the matrix
3
 
0 1 0
1 0 0
0 0 1
because this matrix appears when permuting the first and the second columns of id , etc.
3
The matrices in (7) are called 3×3 permutation matrices.
From now on, we will consider that S consists of the permutation matrices. This allows
3
to see S as a subgroup of GL (K). Denote by H the subgroup of GL (K) generated by S
3 3 3 3
and (K×)3.
It is not difficult to verify that for every σ ∈ S and every (λ ,λ ,λ ) ∈ (K×)3 its product
3 1 2 3
is as follows:
(λ ,λ ,λ )σ = σ(λ ,λ ,λ ).
1 2 3 σ(1) σ(2) σ(3)
Therefore, we may write
H = {σ(α,β,γ) | σ ∈ S , (α,β,γ) ∈ (K×)3}.
3
The multiplication in H is given by
(8) σ(α ,α ,α )τ(β ,β ,β ) = στ(α ,α ,α )(β ,β ,β )
1 2 3 1 2 3 τ(1) τ(2) τ(3) 1 2 3
= στ(α β ,α β ,α β ).
τ(1) 1 τ(2) 2 τ(3) 3
A semidirect product of S and (K×)3 is defined as S ×(K×)3 with product as in (8). It
3 3
is denoted by
S (cid:111)(K×)3.
3
Notice that S (cid:111)(K×)3 coincides with
3
            
α 0 0 0 α 0 0 0 α α 0 0 0 α 0 0 0 α
 
(9) 0 β 0,β 0 0,0 β 0,0 0 β,0 0 β,β 0 0 | α,β,γ ∈K×
 0 0 γ 0 0 γ γ 0 0 0 γ 0 γ 0 0 0 γ 0 
10 Y. CABRERA, M. SILES, AND M. V. VELASCO
Thus, S (cid:111)(K×)3 = {(α,β,γ)σ | α,β,γ ∈ K×, σ ∈ S }.
3 3
We define the action of S (cid:111)(K×)3 on the set M (K) given by:
3 3
   
ω ω ω ω ω ω
11 12 13 σ(1)σ(1) σ(1)σ(2) σ(1)σ(3)
(10) σ ·ω21 ω22 ω23 := ωσ(2)σ(1) ωσ(2)σ(2) ωσ(2)σ(3).
ω ω ω ω ω ω
31 32 33 σ(3)σ(1) σ(3)σ(2) σ(3)σ(3)
 
αω β2ω γ2ω
11 α 12 α 13
 
ω ω ω  
11 12 13  
(11) (α,β,γ)·ω21 ω22 ω23 := αβ2ω21 βω22 γβ2ω23
ω ω ω  
31 32 33  
α2ω β2ω γω
γ 31 γ 32 33
for every σ ∈ S and every (α,β,γ) ∈ (K×)3.
3
For arbitrary P ∈ S (cid:111)(K×)3 and M ∈ M (K), the action of P on M can be formulated
3 3
as follows:
(12) P ·M := P−1MP(2).
Remark 3.1. The action given in (12) has been inspired by Condition (5) in Theorem 2.2.
Notice that any matrix P in S (cid:111)(K×)3 is a change of basis matrix from a natural basis B
3
into another natural basis B(cid:48) and the relationship among the structure matrices M and M
B B(cid:48)
and the matrix P is as given in Condition (5), that is, P−1M P(2) = M(cid:48) . This is the reason
B B
because we define the action of P on M by:
B
P ·M = P−1M P(2).
B B
The result that follows will be very useful in Theorem 3.5.
Proposition 3.2. For any P ∈ S (cid:111)(K×)3 and any M ∈ M (K) we have:
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(i) The number of zero entries in M coincides with the number of zero entries in P ·M.
(ii) The number of zero entries in the main diagonal of M coincides with the number of zero
entries in the main diagonal of P ·M.
(iii) The rank of M and the rank of P ·M coincide.
(iv) Assume that M is the structure matrix of an evolution algebra A relative to a natural
basis B. Assume that A2 = A. If N is the structure matrix of A relative to a natural
basis B(cid:48) then there exists Q ∈ S (cid:111)(K×)3 such that N = Q·M.
3
Proof. Fix an element P in S (cid:111)(K×)3. Then there exist σ ∈ S and (α,β,γ) ∈ (K×)3 such
3 3
that P = σ(α,β,γ). Therefore P · M = (σ(α,β,γ)) · M = σ · ((α,β,γ) · M). Item (i) and
(ii) follows by (10) and (11). Item (iii) is easy to show because P ·M = P−1MP(2) and P is
an invertible matrix. Finally, (iv) follows from the definition of the action and [3, Theorem
4.4]. (cid:3)