EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 1, 2021, 278-300
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Properties of nilpotent evolution algebras with no
maximal nilindex

Ahmad Alarafeen1, Izzat Qaralleh2,∗, Azhana Ahmad1

1 School of Mathematical Sciences, Universiti Sains Malaysia 11800 USM, Penang, Malaysia
2 Department of Mathematics, Faculty of Science, Tafila Technical University, Tafila, Jordan

Abstract. As a system of abstract algebra, evolution algebras are commutative and non-associative
algebras. There is no deep structure theorem for general non-associative algebras. However, there
are deep structure theorem and classification theorem for evolution algebras because it has been
introduced concepts of dynamical systems to evolution algebras. Recently, in [25], it has been
studied some properties of nilpotent evolution algebra with maximal index (dim E2 = dim E− 1).
This paper is devoted to studying nilpotent finite-dimensional evolution algebras E with dim E2 =
dim E − 2. We describe Lie algebras related to the evolution of algebras. Moreover, this result
allowed us to characterize all local and 2-local derivations of the considered evolution algebras. All
automorphisms and local automorphisms of the nilpotent evolution algebras are found.

2020 Mathematics Subject Classifications: 6S10, 82B26, 12J12, 39A70, 47H10, 60K35

Key Words and Phrases: Evolution algebra, derivation, local derivation, automorphism, local
automorphism

1. Introduction

The departure point of a new type of evolution algebra has been introduced by [34].
This algebra is motivated by some evolution laws of genetics. The study of evolution
algebras serves as a foundation of a new research area in algebra and the theory of dynamic
systems. Many related open problems have to be addressed to develop research in this
area (for further details, we refer to [33]).

We note that evolution algebras are not defined by identities, and therefore they do
not form a type of non-associative algebras, such as Lie, Jordan, or alternative algebras.
Thus, to investigate such algebras, a different approach has to be used (see [7, 9, 12]).

In [12], the relationships among nil, right nilpotent evolution algebras, which are de-
fined by an upper triangular matrix of structural constants, have been found. A further

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i1.3912

Email addresses: ahmadalarfeen@gmail.com (A. Alarafeen),
izzat math@yahoo.com (I. Qaralleh), azhana@usm.my (A. Ahmad)

http://www.ejpam.com 278 c© 2021 EJPAM All rights reserved.



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 279

problem which has been addressed in [10, 15, 18, 19, 27] is the classification of low-
dimensional evolution algebras. Nevertheless, a full classification of nilpotent evolution
algebras is a tricky task. [20] have investigated certain properties of nilpotent evolution
algebras with maximal nilindex. In the current study, we analyze some propensities of
nilpotent evolution algebras whose index of nilpotency is 2n−2 + 1.

The derivation of non-associative algebra forms the Lie algebra, which is considered as
one of the important tools for studying its structure. Extensive work has been conducted
on the subject of derivations of genetic algebras ( [13], [17], [20], [28],[16],[1]). since the
multiplication is trivial then set of all until is invertible In fact, [7, 14] have investigated
several properties of derivations of n-dimensional complex evolution algebras,depending
on the rank of the appropriate matrices. Recently, many paper have been devoted to
study the derivation of evolution algebras see for instance [2, 26, 30, 31]. Other properties
of evolution algebra have been investigated in [5, 6, 8, 9, 11, 23, 29]. In [25], it has been
study the properties of the nilpotent finite-dimensional evolution algebras with maximal
nil index such as derivation, local derivation, automorphism, and local automorphism.

In the present study, we explicitly describe the space of derivations of evolution algebras
with nilindex 2(n−2) + 1, which allows us to study further properties of the evolution alge-
bras.Moreover, we describe all local and 2-local derivations of the considered algebra.We
stress that the notions of local automorphism and local derivation were introduced and
investigated independently by Kadison [22] and Larson and Sourour [24]. Subsequently,
P. Šemrl [32] introduced the concepts of 2-local automorphisms and 2-local derivations.
The preceding studies have led to a series of works devoted to description of mappings
which are close to automorphisms and derivations of C∗-algebras and operator algebras.
For details and the survey, we refer to the work of [3, 4].

The paper is organized as follows. Section 2 provides preliminary information about
evolution algebras. Derivations of non-associative algebras form the Lie algebra; thus, so,
in section 3 we describe the Lie algebra associated with evolution algebras whose nilindex
is 2(n−2)+1. Furthermore,based on results in section 3, section 4 describes local and 2-local
derivations of the considered evolution algebras. In section 5, we find all automorphisms
and local automorphisms of the nilpotent evolution algebras with nilindex 2(n−2) + 1 .

2. Evolution algebras

Recall the definition of evolution algebras. Let E be a vector space over a field K. In
the follows, we always assume that K has characteristic zero. The vector space E is called
evolution algebra w.r.t. natural basis {e1, e2, ...} if a multiplication rule · on E satisfies

ei · ej = 0, i 6= j,

ei · ei =
∑
k

aikek, i ≥ 1.

From the preceding definition, it follows that evolution algebras are commutative
(therefore, flexible).



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 280

We denote by A = (aij)
n
i,j=1 the matrix of the structural constants of the finite-

dimensional evolution algebra E. Obviously, rankA = dim(E · E). Thus, for finite-
dimensional evolution algebra, the rank of the matrix does not depend on choice of natural
basis. In the following, for convenience, we write uv instead u ·v for any u,v ∈ E and we
write E2 instead of E ·E.

A linear map ψ : E1 → E2 is called a homomorphism of evolution algebras if ψ(uv) =
ψ(u)ψ(v) for any u,v ∈ E1. Moreover, if ψ is bijective, then it is called an isomorphism.
In this case, the last relation is denoted by E1

∼= E2.
For an evolution algebra E, we introduce the following sequence, k ≥ 1

Ek =
k−1∑
i=1

EiEk−i. (1)

As E is commutative algebra, we obtain

Ek =

bk/2c∑
i=1

EiEk−i,

where bxc denotes the integer part of x.

Definition 1. An evolution algebra E is called nilpotent if some m ∈ N such that Em = 0.
The smallest m such that Em = 0 is called the index of nilpotency.

Theorem 1. [12] An n-dimensional evolution algebra E is nilpotent iff it admits a natural
basis such that the matrix of the structural constants corresponding to E on this basis is
represented in the form

à =



0 ã12 ã13
... ã1n

0 0 ã23
... ã2n

...
...

...
. . .

...

0 0 0
... ãn−1,n

0 0 0
... 0


.

Due to Theorem 1, any nilpotent evolution algebra E with dim(E2) = n − 2 has the
following form:

e2
i =


n∑

j=i+1
aijej , i ≤ n− 2;

0, i ∈ {n− 1, n}.
(2)

where aij ∈ K and ai,i+1 6= 0 for any i < n− 1.

Theorem 2. [11] Let E be a nilpotent evolution algebra. Then, E has maximal index of
nilpotency 2(n−2) + 1 if and only if the multiplication table of E is given by (2).



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 281

In the following, we will work with nilpotent evolution algebras with 2(n−2) + 1 index
of nilpotency. Due to the last theorem, we only consider evolution algebras with the
multiplication table given by (2).

Lemma 1. Let E1, E2 be two isomorphic evolution algebras. Then, Der(E1) ∼= Der(E2).

Lemma 2. Let E and E′ be evolution algebras with basis {ei}ni=1 and {fi}ni=1 respectively,
defined by

e2
i =

{
ai,i+1ei+1 + ain−1en−1 + ainen, i < n− 1;
0, i ∈ {n− 1, n}. f2

i =

{
fi+1, i < n− 1;
0, i ∈ {n− 1, n}.

If ai,i+1 6= 0 for every i < n− 1, then E ∼= E′.

Proof. Let ai,i+1 6= 0 for every i < n− 1. If n = 3 after changing the basis e1, e2, e3 to
f1 = e1, f2 = e2

1, and f3 = e3, we immediately get E′.
So. let us suppose n ≥ 4. Then, the linear mapping ϕ : E→ E′ defined by

ϕ :


f1 = e1

f2 = e2
1

fi+1 =
i−1∏
k=1

a2i−k

k,k+1e
2
i , 2 ≤ i < n− 1

fn = en

(3)

is an isomorphism from E to E′.

3. Derivations

In this section, we consider derivations of nilpotent evolution algebras with 2n−2 + 1
index of nilpotency.

Recall that derivation of an evolution algebra E is a linear mapping d : E → E such
that d(uv) = d(u)v + ud(v) for all u,v ∈ E.

We note that for any algebra, the space Der(E) of all derivations is a Lie algebra w.r.t.
the commutator multiplication:

[d1, d2] = d1d2 − d2d1, ∀d1, d2 ∈ Der(E).

For a given structural matrixA = (aij)
n
i,j≥1 of nilpotent evolution algebra E with dim(E2) =

n− 2, we denote
IA = {(i, j) : i+ 1 < j < n− 1, aij 6= 0}. (4)

Theorem 3. Let E be an evolution algebra with structural matrix A = (aij)
n
i,j≥1 in a

natural basis {ei}ni=1. If E is a nilpotent with rankA = n−2, then the following statements
hold



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 282

(i) if IA 6= ∅, then

Der(E) =




0 0 . . . d1n−1 d1n

0 0 . . . d2n−1 d2n
...

...
. . .

...
...

0 0 . . . dn−1n−1 dn−1n

0 0 . . . dnn−1 dnn




where

dn−1,n−1 = −an−2,ndn,n−1;

dn−1,n = −an−2,ndnn;

dim = −
n−i∑
k=1

ai−1,k+idk+i,m, m ∈ {n− 1, n}

(ii) if IA = ∅, then

Der(E) =





α 0 . . . β γ
0 2α . . . d2,n−1 d2n
...

...
. . .

...
...

0 0 . . . dn−2,n−1 dn−2,n

0 0 . . . dn−1,n−1 dn−1,n

0 0 . . . s t


: α, β, γ, s, t ∈ K


where

dn−1,n−1 = 2n−2α− an−2,ns

dn−1,n = (2n−2 − t)an−2,n

di,n−1 = (2i−1 − 2n−2)αai−1,n−1 + (ai−1,n−1an−2,n − ai−1,n)s

di,n = ai−1,n−1dn−1,n + an−2,n(2i−1α− t), 2 ≤ i < n− 1.

Proof. The (i) and (ii) are easy to check for n = 3, 4. Thus, we consider only the
case n > 4. Let d be a derivation. We represent d in a matrix form based on {ei}ni=1 as
follows: d(ei) =

∑n
j=1 dijej . Then, we have djie

2
i + dije

2
j = 0 for all 1 ≤ i < j ≤ n. As

e2
i and e2

j are linearly independent, then dij = dji = 0 for any 1 ≤ i < j < n − 1. If we

take m ∈ {n − 1, n}, then considering that e2
n−1 = e2

n = 0 from dmie
2
i + dime2

m = 0 one
has dmi = 0 for any i < m.

Thus, we have shown the following:

dij = 0, if i 6= j, i ≤ n, j < n− 1. (5)



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 283

On the other hand, we have d(e2
i ) = 2diie

2
i for any i ≤ n. Then, for i = n − 2 using (2),

we obtain d(en−1 + an−2,nen) = 2dn−2,n−2e
2
n−2. Then, we have the following system:

dn−1,n−1 + an−2,ndn,n−1 = 2dn−2,n−2

dn−1,n + an−2,ndn,n = 2an−2,ndn−2,n−2. (6)

Furthermore, we assume that i < n− 2. Then, one finds

d(e2
i ) = d

 n∑
j=i+1

aijej

 =

n∑
j=i+1

aijd(ej)

=

n−2∑
j=i+1

aijdjjej +

n∑
j=i+1

aijdj,n−1en−1 +

n∑
j=i+1

aijdjnen. (7)

On the other hand, from

d(e2
i ) = 2diie

2
i = 2dii

n∑
j=i+1

aijej

with (7), one finds

2dii = di+1,i+1, 1 ≤ i < n− 2 (8)

aijdjj = 2aijdii, i+ 2 ≤ j ≤ n− 2 (9)

n∑
j=i+1

aijdj,n−1 = 2diiai,n−1, 1 ≤ i < n− 2 (10)

n∑
j=i+1

aijdjn = 2diiain, 1 ≤ i < n− 2. (11)

From (8),(9), we can easily derive

djj = 2j−1d11, 2 ≤ j ≤ n− 2 (12)

aijd11 = 0, i+ 2 ≤ j ≤ n− 1. (13)

Now, we consider (10), (11).

di+1,n−1 = 2ai,n−1dii −
n−i∑
k=1

ai−1,k+idk+i,n−1

di+1,n = 2ai,ndii −
n−i∑
k=1

ai−1,k+idk+i,n. (14)



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 284

Thus, from (5),(6),(12),(13) and (14), we conclude that d is a derivation of evolution
algebra given by (2) if and only if

dij = dn−1,i = dni = 0, 1 ≤ i 6= j ≤ n− 2 (15)

djj = 2j−1d11, 2 ≤ j ≤ n− 2 (16)

aijd11 = 0, i+ 2 ≤ j ≤ n− 2 (17)

di+1,n−1 = 2ai,n−1dii −
n−i∑
k=1

ai−1,k+idk+i,n−1 (18)

di+1,n = 2ai,ndii −
n−i∑
k=1

ai−1,k+idk+i,n. (19)

Case IA 6= ∅. In this case, we have ai0j0 6= 0 for a pair (i0, j0) that satisfies i0 + 2 ≤
j0 < n−1. Then, from (17), one finds d11 = 0. Plugging this fact into (16),(18), and (19),
we obtain

Der(E) =




0 0 . . . d1n−1 d1n

0 0 . . . d2n−1 d2n
...

...
. . .

...
...

0 0 . . . dn−1n−1 dn−1n

0 0 . . . dnn−1 dnn




where

dn−1,n−1 = −an−2,ndn,n−1;

dn−1,n = −an−2,ndnn;

dim = −
n−i∑
k=1

ai−1,k+idk+i,m, m ∈ {n− 1, n}.

Case IA = ∅. In this case, (17) is true for any d11 ∈ K. Thus, from (15),(16), (18),
and (19), we conclude that

Der(E) =





α 0 . . . β γ
0 2α . . . d2,n−1 d2n
...

...
. . .

...
...

0 0 . . . dn−2,n−1 dn−2,n

0 0 . . . dn−1,n−1 dn−1,n

0 0 . . . s t


: α, β, s, t ∈ K


where

dn−1,n−1 = 2n−2α− an−2,ns



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 285

dn−1,n = (2n−2 − t)an−2,n

di,n−1 = (2i−1 − 2n−2)αai−1,n−1 + (ai−1,n−1an−2,n − ai−1,n)s

di,n = ai−1,n−1dn−1,n + an−2,n(2i−1α− t), 2 ≤ i < n− 1.

The proof is complete.

Remark 1. i. In [25], It has been considered the nilpotent evolution algebras with
maximal nil index and they found 1 ≤ dimDer(E) ≤ 2.

ii. From the proved theorem, we infer that 1 ≤ dimDer(E) ≤ 5. This type of result can
be proved using the work of Jacobson [21]. However, the advantage of Theorem 3 is
that it fully describes the structure of the derivations on a natural basis.

Corollary 1. Lie algebras

E =





α 0
... β γ

0 2α
... 0 0

...
...

. . .
...

...

0 0
... 2n−2α 0

0 0
... s t


: α, β, γ, s, t ∈ K


and

E′ =





α 0
... β γ

0 2α
... d2,n−1 d2n

...
...

. . .
...

...

0 0
... dn−1,n−1 dn−1,n

0 0
... s t


: α, β, γ, s, t ∈ K


are isomorphic for any di,n−1, di,n ∈ K, i = 2, n− 1.

Remark 2. We stress that isomorphisms of Lie algebras do not imply isomorphism of the
corresponding evolution algebras (see lemma 1).

4. Local and 2-local derivations for evolution algebras

The results of section 3 allow us to describe local and 2-local derivations of nilpotent
evolution algebra. In this section, we want to fully describe local and 2-local derivations
of nilpotent evolution algebras with 2n−2 + 1 index of nilpotency.

Recall that a linear mapping ∆ on E is called local derivation if for every u ∈ E, a
derivation du exists such that ∆(u) = du(u). A mapping (not necessary linear) D : E→ E
is called 2-local derivation of algebra E if for every u,v ∈ E there exists a derivation du,v
of E such that D(u) = du,v(u) and D(v) = du,v(v). Therefore, it is natural to find all
local derivations of E.



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 286

Theorem 4. Let E be an n-dimensional nilpotent evolution algebra with 2(n−2) + 1 index
of nilpotency. Then, the following statements hold:

(i) If n = 3, then the space of all local derivations has the following form:
 α β γ

0 δ 0
0 s t

 : α, β, γ, δ, s, t ∈ K

 . (20)

(ii) If n > 3, then every local derivation of E is a derivation.

Proof. (i) Let n = 3. Due to Lemma 2, we may assume that an evolution algebra E is
given by e2

1 = e2 and e2
2 = e2

3 = 0. Take an arbitrary linear map ∆ on E, i.e.,

∆(u) = (∆11u1+∆21u2+∆31u3)e1+(∆12u1+∆22u2+∆32u3)e2+(∆13u1+∆23u2+∆33u3)e3,

∀u = u1e1 + u2e2 + u3e3.

If ∆ is a local derivation, then for any u, there exist αu, βu, su, and tu such that

∆11u1 + ∆21u2 + ∆31u3 = αuu1

∆12u1 + ∆22u2 + ∆32u3 = βuu1 + 2αuu2 + suu3

∆13u1 + ∆23u2 + ∆33u3 = γuu1 + tuu3;

From the first equation, we get ∆21 = ∆31 = 0. If we take u such that u1 = u3 = 0,
then from the second equation, we immediately find ∆22 ∈ {0, 2∆11}. we find that ∆ is a
derivation of E if ∆22 = 2∆11.

Suppose ∆22 = 0 and ∆11 6= 0. Then, for every u, we can find that derivation du
satisfies ∆(u) = du(u) as follows:

du =



 ∆11 ∆12 − 2∆11u2
u1

∆13

0 2∆11 0
0 ∆32 ∆33

 , if u1, u3 6= 0, 0 0 0
0 0 0
0 0 0

 , if u1 = u3 = 0.

This result means that a linear mapping defined by

∆ =

 α β γ
0 0 0
0 s t

 : α, β, γ, s, t ∈ K (21)

is a local derivation of E. Finally, as every derivation of algebra E is local derivation and
due to (21), one obtains (20).



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 287

(ii) Let ∆ be a non-zero local derivation given by matrix (∆ij)
n
i,j≥1. Assume that

IA 6= ∅. Then, due to ∆(ei) = dei(ei), for any i ≤ n, we immediately obtain

∆1,n−1 = d
(e1)

1,n−1, ∆1n = d
(e1)

1,n ; (22)

∆im = d
(ei)

im , m ∈ {n− 1, n}, 2 ≤ i < n− 1; (23)

∆n−1,n−1 = d
(en−1)

n−1,n−1, ∆n−1,n = d
(en−1)

n−1,n ; (24)

∆n,n−1 = d(en)

n,n−1, ∆n,n = d(en)
nn ; (25)

∆ij = 0, otherwise. (26)

Taking u =
∑n−2

k=2 ek. we obtain ∆im = −
∑n−i

k=1 ai−1,k+i∆k+i,m. We consider that, v =
en−1 + an−2,nen, and then a derivation dv exists such that ∆(v) = dv(v). Thus,

(−an−2,nd
(en−1)
n,n−1 + an−2,nd

(en)
n,n−1)en−1 + (−an−2,nd

(en−1)
n,n + an−2,nd

(en)
nn )en

= (−an−2,nd
(u)
n,n−1 + an−2,nd

(u)
n,n−1)en−1 + (−an−2,nd

(u)
n,n + an−2,nd

(u)
nn )en.

This result implies d
(en−1)
n,n−1 = d

(en)
n,n−1, d

(en−1)
n,n = d

(en)
nn . Therefore, one has ∆n−1,n−1 =

−an−1,n∆n,n−1. Suppose that IA = ∅.We establish that ∆ ∈ Der(E) for any local deriva-
tion ∆. Due to Lemmas 2 and 1, to show every local derivation can be a derivation we
need to check only for evolution algebra E′ (see Lemma 2).

As ∆(ei) = dei(ei), for any i ≤ n, we can easily find

∆ii = d
(ei)
ii , i ≤ n− 1

∆1,n−1 = d
(e1)
1,n−1

∆1n = d
(e1)
1n

∆n,n−1 = d
(en)
n,n−1

∆n,n = d
(en)
n,n

∆ij = 0, otherwise.

(27)

Taking u =
∑n−2

k=1 ek, we obtain

∆ii = 2i−1∆11, i < n− 1. (28)

Consider v = e2 + en−1. Then, a derivation dv exists such that ∆(v) = dv(v). Due to
the assumption (ii) of Theorem 3, we have

∆22e2 + ∆n−1,n−1en = 2d
(v)
11 e2 + 2n−2d

(v)
11 en.

This result implies

2d
(v)
11 = ∆22

2n−2d
(v)
11 = ∆n−1,n−1.



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 288

Adding the preceding equations into (28), we obtain ∆ii = 2i−1∆11. Then, using (27),
one finds

∆ii = d
(e1)
ii , i ≤ n− 1

∆1,n−1 = d
(e1)
1,n−1

∆1n = d
(e1)
1n

∆n,n−1 = d
(en)
n,n−1

∆n,n = d
(en)
n,n

∆ij = 0, otherwise.

Thus, due to Theorem 3, we conclude that ∆ is a derivation. The proof is complete.

Remark 3. In [25], it was proven that if n > 2 then all local derivation is derivation, in
the above theorem we find the if n > 3 then all local derivation is derivation.

Theorem 5. Every 2-local derivation of nilpotent evolution algebras with 2n−2 + 1 index
of nilpotency is a derivation.

Proof. Let D be a non-zero 2-local derivation of E. Denote Γ1 = {u ∈ E : u1 6=
0}, Γ2 = {u ∈ E : un 6= 0}.

Case IA = ∅. By definition, functionals αu,v, βu,v, γu,v, su,v and tu,v exist such that

D(u) =
n−2∑
k=1

2k−1αu,vukek+(
βu,vu1 + (Kn−1αu,v −Mn−1su,v)un−1 + su,vun +

n−2∑
i=2

(Kiαu,v +Misu,v)ui

)
en−1

+

(
γu,vu1 + (Ln−1αu,v −Nn−1tu,v)un−1 + tu,vun +

n−2∑
i=2

(Liαu,v +Nitu,v)ui

)
en

D(v) =
n−2∑
k=1

2k−1αu,vvkek+(
βu,vv1 + (Kn−1αu,v −Mn−1su,v) vn−1 + su,vvn +

n−2∑
i=2

(Kiαu,v +Misu,v) vi

)
en−1

+

(
γu,vv1 + (Ln−1αu,v −Nn−1tu,v) vn−1 + tu,vvn +

n−2∑
i=2

(Liαu,v +Nitu,v) vi

)
en

(29)

where u =
∑n

k=1 ukek and v =
∑n

k=1 vkek. Take an arbitrary non-zero u ∈ E. Then, for



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 289

any v,v′ ∈ E from the preceding equations, we find

n−2∑
k=1

2k−1αu,vukek+(
βu,vu1 + (Kn−1αu,v −Mn−1su,v)un−1 + su,vun +

n−2∑
i=2

(Kiαu,v +Misu,v)ui

)
en−1

+

(
γu,vu1 + (Ln−1αu,v −Nn−1tu,v)un−1 + tu,vun +

n−2∑
i=2

(Liαu,v +Nitu,v)ui

)
en

=
n−2∑
k=1

2k−1αu,v′ukek+(
βu,v′u1 +

(
Kn−1αu,v′ −Mn−1su,v′

)
un−1 + su,v′un +

n−2∑
i=2

(
Kiαu,v′ +Misu,v′

)
ui

)
en−1

+

(
γu,v′u1 +

(
Ln−1αu,v′ −Nn−1tu,v′

)
un−1 + tu,v′un +

n−2∑
i=2

(
Liαu,v′ +Nitu,v′

)
ui

)
en,

(30)

which is equivalent to

αu,vuk = αu,v′uk, k = 1, n− 2(
βu,vu1 + (Kn−1αu,v −Mn−1su,v)un−1 + su,vun +

n−2∑
i=2

(Kiαu,v +Misu,v)ui

)
=(

βu,v′u1 +
(
Kn−1αu,v′ −Mn−1su,v′

)
un−1 + su,v′un +

n−2∑
i=2

(
Kiαu,v′ +Misu,v′

)
ui

)
(
γu,vu1 + (Ln−1αu,v −Nn−1tu,v)un−1 + tu,vun +

n−2∑
i=2

(Liαu,v +Nitu,v)ui

)
=(

γu,v′u1 +
(
Ln−1αu,v′ −Nn−1tu,v′

)
un−1 + tu,v′un +

n−2∑
i=2

(
Liαu,v′ +Nitu,v′

)
ui

)
.

As, u 6= 0, we obtain αu,v = αu,v′ for any v,v′ ∈ E. This result means that

αu,v =: αu. (31)

Moreover, if u ∈ Γ, then one finds

βu,v =: βu, γu,v =: γu. (32)

Now, if u ∈ Γ2, then one finds

su,v =: su, tu,v =: tu. (33)



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 290

Taking (31),(32),(33) into (36), we conclude that mapping D can be defined as

D(u) =



n−2∑
k=1

2k−1αuukek+(
βuu1 + (Kn−1αu −Mn−1su)un−1 + suun +

n−2∑
i=2

(Kiαu +Misu)ui

)
en−1

+

(
γuu1 + (Ln−1αu −Nn−1tu)un−1 + tuun +

n−2∑
i=2

(Liαu +Nitu)ui

)
en

if u ∈ Γ1 ∪ Γ2

n−2∑
k=2

2k−1αuukek +

(
(Kn−1αu −Mn−1su)un−1 +

n−2∑
i=2

(Kiαu +Misu)ui

)
en−1

+

(
(Ln−1αu −Nn−1tu)un−1 +

n−2∑
i=2

(Liαu +Nitu)ui

)
en if u 6∈ Γ1 ∪ Γ2.

(34)
Then, for any u′,v′ ∈ E, we can find derivation d given by

dij =



2i−1α, if 1 ≤ i = j < n− 1
β, if i = 1, j = n− 1
γ, if i = 1, j = n
s, ifi = n, j = n− 1
t, ifi = n, j = n
Kn−1α−Mn−1s, if i = n− 1, j = n− 1
Ln−1α−Nn−1t, if i = n− 1, j = n
Kiα+Nis, if 2 ≤ i ≤ n− 2, j = n− 1
Liα+Nit, if 2 ≤ i ≤ n− 2, j = n
0, otherwise

such that D(u′) = d(u′), D(v′) = d(v′). Then, from (34) one obtains

αu′ = αv′ = α for any u′,v′ ∈ E. (35)

This result means that functional αu is a constant.
To complete the proof, we show βu, γu, su, and tu are constants for any u ∈ E. We

consider non-zero points u,v ∈ Γ1∪Γ2. Using the first equality of (34) and noting (35) by
definition of 2-local derivation, we obtains βu = βv, γu = γv, su = sv, and tu = tv. This
result means that βu, γu, su and tu do not depend on u, i.e., βu = β, γu = γ, su = s and
tu = t for any u ∈ Γ1 ∪Γ2. Placing this result and (35) into (34) yields D, which has the
following form:



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 291

D(u) =
n−2∑
k=1

2k−1αukek+(
βu1 + (Kn−1α−Mn−1s)un−1 + sun +

n−2∑
i=2

(Kiα+Mis)ui

)
en−1

+

(
γu1 + (Ln−1α−Nn−1t)un−1 + tun +

n−2∑
i=2

(Liα+Nit)ui

)
en.

(36)

Due to Theorem 3 (ii), D is a derivation.
Case IA 6= ∅. By definition, there exist functionals βu,v, γu,v, su,v and tu,v such that

D(u) =

(
βu,vu1 + su,vun + su,v

n−2∑
i=2

diui

)
en−1 +

(
γu,vu1 + tu,vun + tu,v

n−2∑
i=2

diui

)
en

D(v) =

(
βu,vv1 + su,vvn + su,v

n−2∑
i=2

divi

)
en−1 +

(
γu,vv1 + tu,vvn + tu,v

n−2∑
i=2

divi

)
en

(37)

where u =
∑n

k=1 ukek and v =
∑n

k=1 vkek.
Take arbitrary u ∈ Γ1 ∪ Γ2. Then from the first equation of (37) we obtain βu,v =

βu,v′ , su,v = su,v′ , tu,v = tu,v′ for any v,v′ ∈ E. This result means that βu,v, tu,v, and
tu,v do not depend on v, i.e., βu,v = βu, su,v = su, tu,v = tu, ∀u ∈ Γ1 ∪ Γ2. On the other
hand, from the second equation of (37) we obtains βu,v = βv, su,v = sv and tu,v = tv for
any v ∈ Γ1 ∪ Γ2. These facts yield that βu =: β, su =: s and tu =: t for any u,v ∈ E.
Consequently, we have

D(u) =

(
βu1 + sun + s

n−2∑
i=2

diui

)
en−1 +

(
γu1 + tun + t

n−2∑
i=2

diui

)
en.

Due to Theorem 3 (i), we obtain D ∈ Der(E).

5. Automorphisms and local automorphisms

Recall that by an automorphism of an evolution algebra E, we mean an isomorphism
of E into itself. The set of all automorphisms is denoted by Aut(E). It is known that
Aut(E) is a group. In this section, to describe Aut(E) of nilpotent evolution algebras with
maximal index of nilpotency.

If IA 6= ∅, then by η we denote the largest common divisor of all numbers 2j−1 − 2i

where (i, j) ∈ IA, i.e.,
η = LCD(i,j)∈IA(2j−1 − 2i). (38)

Theorem 6. Let E be an n-dimensional nilpotent evolution algebra with 2n−2 + 1 index
of nilpotency and A = (aij)

n
i,j=1 be its structural matrix in a natural basis {ei}ni=1. Then,

the following statements hold:



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 292

(i) if IA 6= ∅ then

Aut(E) =





α 0 . . . 0 β γ
0 α2 . . . 0 ϕ2,n−1 ϕ2,n
...

...
. . .

...
...

0 0 . . . α2n−2
ϕn−2,n−1 ϕn−2,n

0 0 . . . 0 ϕn−1,n−1 ϕn−1,n

0 0 . . . 0 s t


: α, β, γ, s, t ∈ K, αη = 1


where η is defined as (38), and ϕin−1, ϕin is given by the following recurrence for-
mula.

ϕn−1,n−1 = α2n−2 − an−2,ns,

ϕn−1,n = an−2,n

(
α2n−2 − t

)
,

ϕi,n−1 = ai−1,n−1α
2i−1 −

n−i∑
j=i+1

ai−1,jϕj,n−1, 1 < i < n− 1,

ϕi,n = ai−1,nα
2i−1 −

n−i∑
j=i+1

ai−1,jϕj,n, 1 < i < n− 1.

(ii) if IA = ∅ then

Aut(E) =





α 0 . . . 0 β γ
0 α2 . . . 0 ϕ2,n−1 ϕ2,n
...

...
. . .

...
...

0 0 . . . α2n−2
ϕn−2,n−1 ϕn−2,n

0 0 . . . 0 ϕn−1,n−1 ϕn−1,n

0 0 . . . 0 s t


: α, β, γ, s, t ∈ K, α 6= 0


where ϕin−1, ϕin is given by the following recurrence formula:

ϕn−1,n−1 = α2n−2 − an−2,ns,

ϕn−1,n = an−2,n

(
α2n−2 − t

)
,

ϕi,n−1 = ai−1,n−1

(
α2i−1 − ϕn−1,n−1

)
− ai−1,ns, 1 < i < n− 1,

ϕi,n = ai−1,n

(
α2i−1 − t

)
− ai−1,n−1ϕn−1,n, 1 < i < n− 1.

Proof. Let ϕ be a linear mapping on E. Now, we represent ϕ on the basis elements as
follows:

ϕ(ei) =
n∑
j=1

ϕijej , 1 ≤ i ≤ n.

We want to describe matrix (ϕij)
n
i,j=1 when ϕ is an automorphism of E. Suppose that ϕ

is an automorphism. Then, we have

ϕ(ei)ϕ(ej) = 0, i 6= j

ϕ(e2
i ) = [ϕ(ei)]

2, 1 ≤ i ≤ n



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 293

which is equivalent to the followings:

n−1∑
k=1

ϕikϕjke
2
k = 0, i 6= j (39)

n∑
j=i+1

aij

n∑
k=1

ϕjkek =
n−1∑
k=1

ϕ2
ike

2
k, i ≤ n− 2 (40)

an−1,n(

n∑
k=1

ϕnkek) =

n−1∑
k=1

ϕ2
n−1,ke

2
k, (41)

n−1∑
k=1

ϕ2
nke

2
k = 0. (42)

The linear independence of {e2
1, e

2
2, · · · , e2

n−2} together with (39),(42) implies

ϕikϕjk = 0, i 6= j, k ≤ n− 2 (43)

ϕn−1,k = ϕnk = 0, k ≤ n− 2. (44)

We find that ϕn−2,n−2 6= 0. Plugging (44) into (41), we find

ϕn−2,n−2 = ϕ2
n−3,n−3

ϕn−3,k = 0, k ≤ n− 3
(45)

Inserting e2
l =

∑n
j=l+1 aljej , l ≤ n− 2 into (40), we obtain

n∑
j=i+1

aijϕjl =
l−1∑
j=1

ajlϕ
2
ij , i ≤ n− 2, l ≥ 2 (46)

n∑
j=i+1

aijϕj1 = 0, i ≤ n− 2 (47)

We claim:
ϕil = 0, l + 1 ≤ i
ϕj+1,j+1 = ϕ2

jj , j ≤ n− 2.
(48)

Let us prove the last relations by induction. Due to (44),(45), the first step is satisfied.
We take an arbitrary i0 > 1, and assume that for any i > i0, assertion (48) holds.

We must prove that ϕi0l = 0 for any l ≤ i0 − 1 and ϕi0i0 = ϕ2
i0−1,i0−1. Rewriting (46)

for i = i0 > 1, we find
n∑

j=i0+1

ai0jϕjl =
l−1∑
j=1

ajlϕ
2
i0j , l ≥ 2 (49)



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 294

If j > i0, then due to the assumption, we have ϕjl = 0 for any l ≤ i0. As, for any l ≤ i0
the left side of (49) is equals to zero. Thus,

l−1∑
j=1

ϕ2
i0jajl = 0, 2 ≤ l ≤ i0. (50)

If l = 2 then from (50) we obtain ϕi01 = 0. Suppose that ϕi0,l = 0 for every l < l0 ≤ i0.
Then this fact together with (50) for l = l0 implies ϕi0,l0 = 0. Thus, we have shown that
ϕi0,l = 0 for every l ≤ i0. From the arbitrary-ness of i0 > 1, we conclude that

ϕil = 0, l + 1 < i. (51)

On the other hand, rewriting (46) for l = i+ 1 and keeping in mind (51), we obtain

ϕi+1,i+1 = ϕ2
ii, i ≤ n− 2.

The last equality yields ϕi+1,i+1 = ϕ2
ii for every i ≤ n− 2. This together with (45) implies

ϕii = ϕ2i−1

11 6= 0, i ≤ n− 2. (52)

Thus, from (51) and (52), it follows (48).
Plugging (51) into (43), we obtain

ϕij = 0, i < j < n− 1. (53)

Let us consider (46) for l > i+ 1. Then, for every i ≤ n− 2, we obtain

ailϕll = ailϕ
2
ii, i+ 1 < l < n− 1 (54)

n∑
j=i+1

aijϕj,n−1 = ai,n−1ϕ
2
ii, l = n− 1. (55)

n∑
j=i+1

aijϕjn = ainϕ
2
ii, l = n. (56)

From (56) with (52) we obtain a recurrence formula for ϕin−1, ϕin as follows:

ϕn−1,n−1 = α2n−2 − an−2,ns,

ϕn−1,n = an−2,n

(
α2n−2 − t

)
,

ϕi,n−1 = ai−1,n−1α
2i−1 −

n−i∑
j=i+1

ai−1,jϕj,n−1, 1 < i < n− 1,

ϕi,n = ai−1,nα
2i−1 −

n−i∑
j=i+1

ai−1,jϕj,n, 1 < i < n− 1.

(57)



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 295

Hence, we infer that ϕ is an automorphism of evolution algebra (2) if and only if the
followings holds:

ϕij = 0, i 6= j, j < n− 1

ϕii = ϕ2i−1

11 , i ≤ n− 2

ailϕll = ailϕ
2
ii, i+ 1 < l < n− 1

ϕn−1,n−1 = α2n−2 − an−2,ns,

ϕn−1,n = an−2,n

(
α2n−2 − t

)
,

ϕi,n−1 = ai−1,n−1α
2i−1 −

n−i∑
j=i+1

ai−1,jϕj,n−1, 1 < i < n− 1,

ϕi,n = ai−1,nα
2i−1 −

n−i∑
j=i+1

ai−1,jϕj,n, 1 < i < n− 1.

(58)

Now let us consider two cases w.r.t.IA.
Case IA 6= ∅. For the sake of convenience, we denote ϕ11 = α 6= 0. Then, from (58)

one obtains

ϕij = ϕji = 0, i 6= j, j < n− 1

ϕii = α2i−1
, i ≤ n− 2

α2l−1−2i = 1, (i, l) ∈ IA
ϕ1,n−1 = γ, ϕ1n = β, ϕn,n−1 = s, ϕn,n = t

ϕn−1,n−1 = α2n−2 − an−2,ns,

ϕn−1,n = an−2,n

(
α2n−2 − t

)
,

ϕi,n−1 = ai−1,n−1α
2i−1 −

n−i∑
j=i+1

ai−1,jϕj,n−1, 1 < i < n− 1,

ϕi,n = ai−1,nα
2i−1 −

n−i∑
j=i+1

ai−1,jϕj,n, 1 < i < n− 1.

where α, β, γ, s, t ∈ K, and αη = 1, which implies the assertion.
Case IA = ∅. For the automorphism ϕ, we have

ϕij = ϕji = 0, i 6= j, j < n− 1

ϕii = α2i−1
, 1 ≤ i ≤ n− 2

ϕ1,n−1 = γ, ϕ1n = β, ϕn,n−1 = s, ϕn,n = t

ϕn−1,n−1 = α2n−2 − an−2,ns,

ϕn−1,n = an−2,n

(
α2n−2 − t

)
,

ϕi,n−1 = ai−1,n−1

(
α2i−1 − ϕn−1,n−1

)
− ai−1,ns, 1 < i < n− 1,

ϕi,n = ai−1,n

(
α2i−1 − t

)
− ai−1,n−1ϕn−1,n, 1 < i < n− 1.



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 296

where α, β, γ, s, t ∈ K, which implies the assertion.
The proof is complete.

5.1. Local automorphisms of evolution algebras

In the previous section, we have been able to find the set of all automorphisms of
evolution algebra (2). Now, we show that every local automorphism is an automorphism
if evolution algebra is defined by (2) with n > 3. Recall that a linear mapping ψ from
E to E is called local automorphism if for every u ∈ E there exists an automorphism
ϕu ∈ Aut(E) such that ψ(u) = ϕu(u).

Theorem 7. Let E be an n-dimensional nilpotent evolution algebra with 2n−2 + 1 index
of nilpotency. Then, the following statements hold:

(i) If n = 3, then the set of all local automorphisms has the following form:
 α β γ

0 l2 0
0 s T

 : α, β, γ, l, s, t ∈ K, αl 6= 0

 . (59)

(ii) If n > 3, then every local automorphism of E is an automorphism.

Proof. (i) Let n = 3. Due to Lemma 2, we may assume that an evolution algebra E is
given by e2

1 = e2 and e2
2 = e2

3 = 0. Take an arbitrary linear map ψ on E, i.e.,

ψ(u) = (ψ11u1 + ψ21u2 + ψ31u3)e1 + (ψ12u1 + ψ22u2 + ψ32u3)e2 + (ψ13u1 + ψ23u2 + ψ33u3)e3

∀u = u1e1 + u2e2 + u3e3.

If ψ is a local automorphism, then for any u, there exist αu, βu, γu, su and tu, such
that

ψ11u1 + ψ21u2 + ψ31u3 = αuu1

ψ12u1 + ψ22u2 + ψ32u3 = βuu1 + α2
uu2 + suu3

ψ13u1 + ψ23u2 + ψ33u3 = γuu1 + tuu3.

From the first equation, we obtain ψ21 = ψ31 = 0, and from the third equation, we
have ψ23 = 0. If we take u such that u1 = u3 = 0, then from the second equation, we
immediately find ψ22 = α2

u. It yields that if ψ is a local automorphism, it has the following
form: 

 α β γ
0 l2 0
0 s T

 : α, β, γ, l, s, t ∈ K, αl 6= 0

 (60)



A. Alarafeen, I. Qaralleh, A. Ahmad / Eur. J. Pure Appl. Math, 14 (1) (2021), 278-300 297

We show that (60) is indeed a local automorphism of (2). In fact, for any u ∈ E, we may
take an automorphism ϕu of (2) as follows:

ϕu =



 α β + (l2−α2)u2
u1

γ

0 α2 0
0 s t

 , if u1u3 6= 0

 l 0 0
0 l2 0
0 0 0

 , if u1 = u3 = 0

From this, one can check that ψ(u) = ϕu(u).
(ii) Let n > 3. Let ψ be a local automorphism for (2). By definition of local auto-

morphism, for every u ∈ E we have ψ(u) = ϕu(u), where ϕu is an automorphism. Then,
theorem 6 implies ψij = 0 for every i 6= j, j < n − 1. On the other hand, taking u = ei,
i ≤ n, we conclude that the local automorphism ψ has the following form:

ψ =



αe1 0 0
... 0 βe1 γe1

0 α2
e2 0

... 0 ϕ
(e2)
2,n−1 ϕ

(e2)
2n

...
...

...
. . .

...
...

0 0 0
... α2n−2

en−2
ϕ

(en−2)
n−2,n−1 ϕ

(en−2)
n−2,n

0 0 0
... 0 ϕ

(en−1)
n−1,n−1 ϕ

(en−1)
n−1,n

0 0 0
... 0 sen ten


Now, we take arbitrary v =

∑n
i=1 viei. Then, from ψ(v) = ϕv(v), we obtain

α2i−1

ei vi = α2i−1

v vi, i < n− 1 (61)

βe1v1 + senvn +

n−1∑
k=2

ϕ
(ek)
k,n−1vk = βvv1 + svvn +

n−1∑
k=2

ϕ
(v)
k,n−1vk (62)

γe1v1 + tenvn +
n−1∑
k=2

ϕ
(ek)
k,n vk = βvv1 + tvvn +

n−1∑
k=2

ϕ
(v)
k,nvk. (63)

From (61) we find

α2i−1

ei = α2i−1

e1 , i < n− 1 (64)

Consequently, ϕ
(ek)
k,n−1 = ϕ

(e1)
k,n−1 and ϕ

(ek)
kn = ϕ

(e1)
kn for any k < n. Based on this fact, the

following is obtained from (62) and (63),

γe1v1 + senvn = γvv1 + svvn, βe1v1 + tenvn = βvv1 + svvn. (65)

Finally, taking v′ = e2 + en, we obtain γe1 = γv, βe1 = βv, sen = sv, and ten = tv, for
any v ∈ E.



REFERENCES 298

Thus, we conclude that local automorphism ψ = (ϕij) has the following form:

ϕij =



α2i−1

e1 , i = j > n− 1

γe1 , i = 1, j = n− 1

βe1 , i = 1, j = n

ϕ
(e1)
i,n−1, i > 1, j = n− 1

ϕ
(e1)
in , i > 1, j = n

se1 , i = n, j = n− 1

te1 , i = n, j = n

0, otherwise.

Thus, theorem 6 implies that the local automorphism ψ is an automorphism. The proof
is complete.

Remark 4. In [25], it was proven that if n > 2 then all local automorphism is auto-
morphism, in the above theorem we find that if n > 3 then all local automorphism is
automorphism.

References

[1] Ahmad Alarafeen, Izzat Qaralleh, and Azhana Ahmad. Derivation of five-dimensional
lotka-voltera algebra. arXiv preprint arXiv:1912.08594, 2019.

[2] Abdelwahab Alsarayreh, Izzat Qaralleh, and Muhammad Zaini Ahmad. Derivation of
three-dimensional evolution algebra. JP J. Algebra Number Theory Appl, 39:4, 2017.

[3] Shavkat Ayupov, Abror Khudoyberdiyev, and Bakhtiyor Yusupov. Local and 2-
local derivations of solvable leibniz algebras. International Journal of Algebra and
Computation, 2020.

[4] Shavkat Ayupov, Karimbergen Kudaybergenov, and Antonio M Peralta. A survey on
local and 2-local derivations on c∗-and von neumann algebras. Topics in Functional
Analysis and Algebra, Contemporary Mathematics, 672:73–126, 2016.

[5] Julio Becerra, Maŕıa Beltrán, and M Velasco. Pulse processes in networks and evo-
lution algebras. Mathematics, 8(3):387, 2020.

[6] Paula Cadavid, Mary Luz Rodiño Montoya, and Pablo M Rodriguez. The connection
between evolution algebras, random walks and graphs. Journal of Algebra and Its
Applications, 19(02):2050023, 2020.

[7] LM Camacho, JR Gómez, BA Omirov, and RM Turdibaev. The derivations of some
evolution algebras. Linear and Multilinear Algebra, 61(3):309–322, 2013.



REFERENCES 299

[8] LM Camacho, A Kh Khudoyberdiyev, and BA Omirov. On the property of sub-
algebras of evolution algebras. Algebras and Representation Theory, 22(2):281–296,
2019.

[9] Luisa Maŕıa Camacho Santana, José Ramón Gómez Mart́ın, Bakhrom A Omirov,
and RM Turdibaev. Some properties of evolution algebras. Bulletin of the Korean
Mathematical Society, 50 (5), 1481-1494., 2013.

[10] Yolanda Cabrera Casado, Mercedes Siles Molina, and M Victoria Velasco. Classifi-
cation of three-dimensional evolution algebras. Linear Algebra and its Applications,
524:68–108, 2017.

[11] JM Casas, M Ladra, BA Omirov, and UA Rozikov. index and dibaricity of evolution
algebras. Linear Algebra and its Applications, 439(1):90–105, 2013.

[12] José M Casas, Manuel Ladra, Bakhrom A Omirov, and Utkir A Rozikov. On evolution
algebras. In Algebra Colloquium, volume 21, pages 331–342. World Scientific, 2014.

[13] R Costa. On the derivation algebra of zygotic algebras for polyploidy with multiple
alleles. Boletim da Sociedade Brasileira de Matemática-Bulletin/Brazilian Mathemat-
ical Society, 14(1):63–80, 1983.

[14] Hamza Abd El-Qader, Ahmad Termimi Ab Ghani, and Izzat Qaralleh. On evolu-
tion algebras and their derivations. Journal of Mathematics and Computer Scinces,
21(3):213–230, 2020.

[15] Alberto Elduque and Alicia Labra. On nilpotent evolution algebras. Linear Algebra
and its Applications, 505:11–31, 2016.

[16] Rasul Ganikhodzhaev, Farrukh Mukhamedov, Abror Pirnapasov, and Izzat Qar-
alleh. Genetic volterra algebras and their derivations. Communications in Algebra,
46(3):1353–1366, 2018.

[17] Harry Gonshor. Derivations in genetic algebras. Communications in Algebra,
16(8):1525–1542, 1988.

[18] AS Hegazi and Hani Abdelwahab. Five-dimensional nilpotent evolution algebras.
arXiv preprint arXiv:1508.07442, 2015.

[19] AS Hegazi and Hani Abdelwahab. Nilpotent evolution algebras over arbitrary fields.
Linear Algebra and its Applications, 486:345–360, 2015.

[20] P Holgate. The interpretation of derivations in genetic algebras. Linear Algebra and
Its Applications, 85:75–79, 1987.

[21] Nathan Jacobson. A note on automorphisms and derivations of lie algebras. In Nathan
Jacobson Collected Mathematical Papers, pages 251–253. Springer, 1989.



REFERENCES 300

[22] Richard V Kadison. Local derivations. Journal of Algebra, 130(2):494–509, 1990.

[23] A Kh Khudoyberdiyev, Bakhrom A Omirov, and Izzat Qaralleh. Few remarks on
evolution algebras. Journal of Algebra and Its Applications, 14(04):1550053, 2015.

[24] D. R. Larson and A. R. Sourour. Local derivations and local automorphisms of B(x).
In Proc. Sympos. Pure Math. 51, pages 187–194. American Mathematical Society,
1990.

[25] Farrukh Mukhamedov, Otabek Khakimov, Bakhrom Omirov, and Izzat Qaralleh.
Derivations and automorphisms of nilpotent evolution algebras with maximal nilin-
dex. Journal of Algebra and Its Applications, 18(12):1950233, 2019.

[26] Farrukh Mukhamedov, Otabek Khakimov, and Izzat Qaralleh. Classification of nilpo-
tent evolution algebras and extensions of their derivations. Communications in Alge-
bra, pages 1–15, 2020.

[27] Farrukh Mukhamedov, Bakhrom Omirov, and Izzat Qaralleh. Description of three di-
mensional solvable evaluation algebras. In International Conference on Mathematical
Sciences and Statistics 2013, pages 165–174. Springer, 2014.

[28] Farrukh Mukhamedov and Izzat Qaralleh. On derivations of genetic algebras. In
Journal of Physics: Conference Series, volume 553, 2014.

[29] Farrukh Mukhamedov, Izzat Qaralleh, and Abror Pirnapasov. On genetic and evo-
lution algebras in dimension three. International Journal of Algebra, 10(7):327–334,
2016.

[30] Izzat Qaralleh. A description of derivations of a class of nilpotent evolution algebras.
14(2):295–304, 2020.

[31] Izzat Qaralleh and Farrukh Mukhamedov. Volterra evolution algebras and their
graphs. Linear and Multilinear Algebra, pages 1–17, 2019.

[32] Peter Šemrl. Local automorphisms and derivations on b(h). Proceedings of the Amer-
ican Mathematical Society, 125(9):2677–2680, 1997.

[33] Jianjun Paul Tian. Evolution algebras and their applications. Springer, 2007.

[34] Jianjun-Paul Tian and Piotr Vojtĕchovskỳ. Mathematical concepts of evolution al-
gebras in non-mendelian genetics. Quasigroups and Related Systems, 14(1):111–122,
2006.