# Conservative algebras of $2$-dimensional algebras, III

Farhodjon Arzikulov1,2 1 V.I. Romanovskiy Institute of Mathematics, Namangan
Regional Department, Uzbekistan Academy of Sciences  2 Department of
Mathematics, Andizhan State University, Andizhan, Uzbekistan.
<EMAIL_ADDRESS>and  Nodirbek Umrzaqov3 3 Department of Mathematics,
Andizhan State University, Andizhan, Uzbekistan<EMAIL_ADDRESS>
###### Abstract.

In the present paper we prove that every local and $2$-local derivation on
conservative algebras of $2$-dimensional algebras are derivations. Also, we
prove that every local and $2$-local automorphism on conservative algebras of
$2$-dimensional algebras are automorphisms.

###### Key words and phrases:

Conservative algebra, derivation, local derivation, $2$-local derivation,
automorphism, local automorphism, $2$-local automorphism

###### 2010 Mathematics Subject Classification:

16W25, 46L57, 47B47, 17C65, 17A30

00footnotetext:

##  1\. Introduction

The present paper is devoted to the study of conservative algebras. In 1972
Kantor [12] introduced conservative algebras as a generalization of Jordan
algebras (also, see a good written survey about the study of conservative
algebras [25]).

In 1990 Kantor [14] defined the multiplication $\cdot$ on the set of all
algebras (i.e. all multiplications) on the $n$-dimensional vector space
$V_{n}$ over a field ${\mathbb{F}}$ of characteristic zero as follows: $A\cdot
B=[L^{A}_{e},B]$, where $A$ and $B$ are multiplications and $e\in V_{n}$ is
some fixed vector. If $n>1$, then the algebra $W(n)$ does not belong to any
well-known class of algebras (such as associative, Lie, Jordan, or Leibniz
algebras). The algebra $W(n)$ is a conservative algebra [12].

In [12] Kantor classified all conservative $2$-dimensional algebras and
defined the class of terminal algebras as algebras satisfying some certain
identity. He proved that every terminal algebra is a conservative algebra and
classified all simple finite-dimensional terminal algebras with left quasi-
unit over an algebraically closed field of characteristic zero [13]. Terminal
algebras were also studied in [18, 19].

In 2017 Kaygorodov and Volkov [16] described automorphisms, one-sided ideals,
and idempotents of $W(2)$. Also a similar problem is solved for the algebra
$W_{2}$ of all commutative algebras on the $2$-dimensional vector space and
for the algebra $S_{2}$ of all commutative algebras with zero multiplication
trace on the $2$-dimensional vector space. The papers [15, 17] are also
devoted to the study of conservative algebras and superalgebras.

Let $\mathcal{A}$ be an algebra. A linear operator $\nabla$ on $\mathcal{A}$
is called a local derivation if for every $x\in\mathcal{A}$ there exists a
derivation $\phi_{x}$ of $\mathcal{A}$, depending on $x$, such that
$\nabla(x)=\phi_{x}(x)$. The history of local derivations had begun from the
paper of Kadison [11]. Kadison introduced the concept of local derivation and
proved that each continuous local derivation from a von Neumann algebra into
its dual Banach bimodule is a derivation.

A similar notion, which characterizes nonlinear generalizations of
derivations, was introduced by Šemrl as $2$-local derivations. In his paper
[26] was proved that a $2$-local derivation of the algebra $B(H)$ of all
bounded linear operators on the infinite-dimensional separable Hilbert space
$H$ is a derivation. After his works, appear numerous new results related to
the description of local and $2$-local derivations of associative algebras
(see, for example, [1], [3], [4], [20], [21], [23]).

The study of local and $2$-local derivations of non-associative algebras was
initiated in some papers of Ayupov and Kudaybergenov (for the case of Lie
algebras, see [5, 6]). In particular, they proved that there are no non-
trivial local and $2$-local derivations on semisimple finite-dimensional Lie
algebras. In [8] examples of $2$-local derivations on nilpotent Lie algebras
which are not derivations, were also given. Later, the study of local and
$2$-local derivations was continued for Leibniz algebras [7], Malcev algebras
and Jordan algebras [2]. Local automorphisms and $2$-local automorphisms, also
were studied in many cases, for example, they were studied on Lie algebras [5,
10].

Now, a linear operator $\nabla$ on $\mathcal{A}$ is called a local
automorphism if for every $x\in\mathcal{A}$ there exists an automorphism
$\phi_{x}$ of $\mathcal{A}$, depending on $x$, such that
$\nabla(x)=\phi_{x}(x)$. The concept of local automorphism was introduced by
Larson and Sourour [22] in 1990. They proved that, invertible local
automorphisms of the algebra of all bounded linear operators on an infinite-
dimensional Banach space $X$ are automorphisms.

A similar notion, which characterizes non-linear generalizations of
automorphisms, was introduced by Šemrl in [26] as $2$-local automorphisms.
Namely, a map $\Delta:\mathcal{A}\to\mathcal{A}$ (not necessarily linear) is
called a $2$-local automorphism, if for every $x,y\in\mathcal{A}$ there exists
an automorphism $\phi_{x,y}:\mathcal{A}\to\mathcal{A}$ such that
$\Delta(x)=\phi_{x,y}(x)$ and $\Delta(y)=\phi_{x,y}(y)$. After the work of
Šemrl, it appeared numerous new results related to the description of local
and $2$-local automorphisms of algebras (see, for example, [5], [7], [9],
[10], [21]).

In the present paper, we continue the study of derivations, local and
$2$-local derivations of conservative algebras of $2$-dimensional algebras. We
prove that every local and $2$-local derivation of the conservative algebras
of $2$-dimensional algebras are derivations. In the present paper, we continue
the study of automorphisms, local and $2$-local automorphisms in the case of
conservative algebras of $2$-dimensional algebras. We prove that every local
and $2$-local automorphism of the conservative algebras of $2$-dimensional
algebras are automorphisms.

##  2\. Preliminaries

Throughout this paper ${\mathbb{F}}$ is some fixed field of characteristic
zero. A multiplication on $2$-dimensional vector space is defined by a
$2\times 2\times 2$ matrix. Their classification was given in many papers
(see, for example, [24]). Let consider the space $W(2)$ of all multiplications
on the $2$-dimensional space $V_{2}$ with a basis $v_{1}$, $v_{2}$. The
definition of the multiplication $\cdot$ on the algebra $W(2)$ is defined as
follows: we fix the vector $v_{1}\in V_{2}$ and define

$(A\cdot B)(x,y)=A(v_{1},B(x,y))-B(A(v_{1},x),y)-B(x,A(v_{1},y))$

for $x$, $y\in V_{2}$ and $A$, $B\in W(2)$. The algebra $W(2)$ is conservative
[14]. Let consider the multiplications $\alpha^{k}_{i,j}$ ($i$, $j$, $k=1,2$)
on $V_{2}$ defined by the formula
$\alpha^{k}_{i,j}(v_{t},v_{l})=\delta_{it}\delta_{jl}v_{k}$ for all $t$,
$l\in\\{1,2\\}$. It is easy to see that $\\{\alpha^{k}_{i,j}|i,j,k=1,2\\}$ is
a basis of the algebra $W(2)$. The multiplication table of $W(2)$ in this
basis is given in [15]. In this work we use another basis for the algebra
$W(2)$ (from [16]). Let introduce the notation

$e_{1}=\alpha^{1}_{11}-\alpha^{2}_{12}-\alpha^{2}_{21},\,\,e_{2}=\alpha^{2}_{11},\,\,e_{3}=\alpha^{2}_{22}-\alpha^{1}_{12}-\alpha^{1}_{21},\,\,e_{4}=\alpha^{1}_{22},\,\,e_{5}=2\alpha^{1}_{11}+\alpha^{2}_{12}+\alpha^{2}_{21},$
$e_{6}=2a^{2}_{22}+\alpha^{1}_{12}+\alpha^{1}_{21},\,\,e_{7}=\alpha^{1}_{12}-\alpha^{1}_{21},\,\,e_{8}=\alpha^{2}_{12}-\alpha^{2}_{21}.$

It is easy to see that the multiplication table of $W(2)$ in the basis
$e_{1},\dots,e_{8}$ is the following.

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $2e_{1}$ | $e_{3}$ | $0$ | $-e_{5}$ | $e_{8}$ | $0$
$e_{3}$ | $-2e_{3}$ | $-e_{1}$ | $-3e_{4}$ | $0$ | $e_{6}$ | $0$ | $0$ | $-e_{7}$
$e_{4}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{5}$ | $-2e_{1}$ | $-3e_{2}$ | $-e_{3}$ | $0$ | $-2e_{5}$ | $-e_{6}$ | $-e_{7}$ | $-2e_{8}$
$e_{6}$ | $2e_{3}$ | $e_{1}$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{7}$ | $2e_{3}$ | $e_{1}$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{8}$ | $0$ | $e_{2}$ | $-e_{3}$ | $-2e_{4}$ | $0$ | $-e_{6}$ | $-e_{7}$ | $0$

The subalgebra generated by the elements $e_{1},\dots,e_{6}$ is the
conservative (and, moreover, terminal) algebra $W_{2}$ of commutative
$2$-dimensional algebras. The subalgebra generated by the elements
$e_{1},\dots,e_{4}$ is the conservative (and, moreover, terminal) algebra
$S_{2}$ of all commutative $2$-dimensional algebras with zero multiplication
trace [15].

Let $\mathcal{A}$ be an algebra. A linear map $D:\mathcal{A}\to\mathcal{A}$ is
called a derivation, if $D(xy)=D(x)y+xD(y)$ for any two elements $x$,
$y\in\mathcal{A}$.

Our main tool for study of local and $2$-local derivations of the algebras
$S_{2}$, $W_{2}$ and $W(2)$ is the following lemma [15, Theorem 6], where the
matrix of a derivation is calculated in the new basis $e_{1},\dots,e_{8}$.

######  Lemma 2.1.

A linear map $D:W(2)\to W(2)$ is a derivation if and only if the matrix of $D$
has the following matrix form:

$\left(\begin{array}[]{cccccccc}0&\alpha&0&0&0&0&0&0\\\
0&-\beta&0&0&0&0&0&0\\\ 2\alpha&0&\beta&0&0&0&0&0\\\
0&0&3\alpha&2\beta&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&-\alpha&\beta&0&0\\\
0&0&0&0&0&0&\beta&\alpha\\\ 0&0&0&0&0&0&0&0\\\ \end{array}\right),$ (2.1)

where $\alpha$, $\beta$ are elements in ${\mathbb{F}}$.

Now, we give a characterization of automorphisms on conservative algebras of
$2$-dimensional algebras.

Let $\mathcal{A}$ be an algebra. A bijective linear map
$\phi:\mathcal{A}\to\mathcal{A}$ is called an automorphism, if
$\phi(xy)=\phi(x)\phi(y)$ for any elements $x$, $y\in\mathcal{A}$.

Our principal tool for study of local and $2$-local automorphisms of the
algebras $S_{2}$, $W_{2}$ and $W(2)$ is the following lemma, which was proved
in [16, Theorem 11].

######  Lemma 2.2.

A linear map $\phi:W(2)\to W(2)$ is an automorphism if and only if the matrix
of $\phi$ has the following matrix form:

$\left(\begin{array}[]{cccccccc}1&a&0&0&0&0&0&0\\\
0&\frac{1}{b}&0&0&0&0&0&0\\\ 2ab&a^{2}b&b&0&0&0&0&0\\\
3a^{2}b^{2}&a^{3}b^{2}&3ab^{2}&b^{2}&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&0&0&0&-ab&b&0&0\\\ 0&0&0&0&0&0&b&ab\\\ 0&0&0&0&0&0&0&1\\\
\end{array}\right),$ (2.2)

where $a$, $b$ are elements in ${\mathbb{F}}$ and $b\neq 0$.

##  3\. Local derivations of conservative algebras of $2$-dimensional algebras

In this section we give a characterization of derivations on conservative
algebras of $2$-dimensional algebras.

Let $\mathcal{A}$ be an algebra. A linear map
$\nabla:\mathcal{A}\to\mathcal{A}$ is called a local derivation, if for any
element $x\in\mathcal{A}$ there exists a derivation
$D_{x}:\mathcal{A}\to\mathcal{A}$ such that $\nabla(x)=D_{x}(x)$.

######  Theorem 3.1.

Every local derivation of the algebra $W(2)$ is a derivation.

###### Proof.

Let $\nabla$ be an arbitrary local derivation of $W(2)$ and write

$\nabla(x)=B\bar{x},x\in W(2),$

where $B=(b_{i,j})_{i,j=1}^{8}$,
$\bar{x}=(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8})$ is the vector
corresponding to $x$. Then for every $x\in W(2)$ there exist elements $a_{x}$,
$b_{x}$ in ${\mathbb{F}}$ such that

$B\bar{x}=\left(\begin{array}[]{cccccccc}0&a_{x}&0&0&0&0&0&0\\\
0&-b_{x}&0&0&0&0&0&0\\\ 2a_{x}&0&b_{x}&0&0&0&0&0\\\
0&0&3a_{x}&2b_{x}&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&-a_{x}&b_{x}&0&0\\\
0&0&0&0&0&0&b_{x}&a_{x}\\\ 0&0&0&0&0&0&0&0\\\
\end{array}\right)\left(\begin{array}[]{c}x_{1}\\\ x_{2}\\\ x_{3}\\\ x_{4}\\\
x_{5}\\\ x_{6}\\\ x_{7}\\\ x_{8}\\\ \end{array}\right).$

In other words

$\left\\{\begin{array}[]{ll}b_{1,1}x_{1}+b_{1,2}x_{2}+b_{1,3}x_{3}+b_{1,4}x_{4}+b_{1,5}x_{5}+b_{1,6}x_{6}+b_{1,7}x_{7}+b_{1,8}x_{8}=a_{x}x_{2}\hbox{;}\\\
b_{2,1}x_{1}+b_{2,2}x_{2}+b_{2,3}x_{3}+b_{2,4}x_{4}+b_{2,5}x_{5}+b_{2,6}x_{6}+b_{2,7}x_{7}+b_{2,8}x_{8}=-b_{x}x_{2}\hbox{;}\\\
b_{3,1}x_{1}+b_{3,2}x_{2}+b_{3,3}x_{3}+b_{3,4}x_{4}+b_{3,5}x_{5}+b_{3,6}x_{6}+b_{3,7}x_{7}+b_{3,8}x_{8}=2a_{x}x_{1}+b_{x}x_{3}\hbox{;}\\\
b_{4,1}x_{1}+b_{4,2}x_{2}+b_{4,3}x_{3}+b_{4,4}x_{4}+b_{4,5}x_{5}+b_{4,6}x_{6}+b_{4,7}x_{7}+b_{4,8}x_{8}=3a_{x}x_{3}+2b_{x}x_{4}\hbox{;}\\\
b_{5,1}x_{1}+b_{5,2}x_{2}+b_{5,3}x_{3}+b_{5,4}x_{4}+b_{5,5}x_{5}+b_{5,6}x_{6}+b_{5,7}x_{7}+b_{5,8}x_{8}=0\hbox{;}\\\
b_{6,1}x_{1}+b_{6,2}x_{2}+b_{6,3}x_{3}+b_{6,4}x_{4}+b_{6,5}x_{5}+b_{6,6}x_{6}+b_{6,7}x_{7}+b_{6,8}x_{8}=-a_{x}x_{5}+b_{x}x_{6}\hbox{;}\\\
b_{7,1}x_{1}+b_{7,2}x_{2}+b_{7,3}x_{3}+b_{7,4}x_{4}+b_{7,5}x_{5}+b_{7,6}x_{6}+b_{7,7}x_{7}+b_{7,8}x_{8}=b_{x}x_{7}+a_{x}x_{8}\hbox{;}\\\
b_{8,1}x_{1}+b_{8,2}x_{2}+b_{8,3}x_{3}+b_{8,4}x_{4}+b_{8,5}x_{5}+b_{8,6}x_{6}+b_{8,7}x_{7}+b_{8,8}x_{8}=0\hbox{.}\end{array}\right.$

Taking $x=(1,0,0,0,0,0,0,0)$, $x=(0,0,1,0,0,0,0,0)$, $x=(0,0,0,1,0,0,0,0)$,
etc, from this it follows that

$b_{1,1}=b_{1,3}=b_{1,4}=b_{1,5}=b_{1,6}=b_{1,7}=b_{1,8}=$
$=b_{2,1}=b_{2,3}=b_{2,4}=b_{2,5}=b_{2,6}=b_{2,7}=b_{2,8}$
$=b_{3,2}=b_{3,4}=b_{3,5}=b_{3,6}=b_{3,7}=b_{3,8}$
$=b_{4,1}=b_{4,2}=b_{4,5}=b_{4,6}=b_{4,7}=b_{4,8}$
$=b_{5,1}=b_{5,2}=b_{5,3}=b_{5,4}=b_{5,5}=b_{5,6}=b_{5,7}=b_{5,8}$
$=b_{6,1}=b_{6,2}=b_{6,3}=b_{6,4}=b_{6,7}=b_{6,8}$
$=b_{7,1}=b_{7,2}=b_{7,3}=b_{7,4}=b_{7,5}=b_{7,6}$
$=b_{8,1}=b_{8,2}=b_{8,3}=b_{8,4}=b_{8,5}=b_{8,6}=b_{8,7}=b_{8,8}=0.$

Then for every $x\in W(2)$ there exist elements $a_{x}$, $b_{x}$ in
${\mathbb{F}}$ such that

$\left\\{\begin{array}[]{ll}b_{1,2}x_{2}=a_{x}x_{2}\hbox{;}\\\
b_{2,2}x_{2}=-b_{x}x_{2}\hbox{;}\\\
b_{3,1}x_{1}+b_{3,3}x_{3}=2a_{x}x_{1}+b_{x}x_{3}\hbox{;}\\\
b_{4,3}x_{3}+b_{4,4}x_{4}=3a_{x}x_{3}+2b_{x}x_{4}\hbox{;}\\\
b_{6,5}x_{5}+b_{6,6}x_{6}=-a_{x}x_{5}+b_{x}x_{6}\hbox{;}\\\
b_{7,7}x_{7}+b_{7,8}x_{8}=b_{x}x_{7}+a_{x}x_{8}\hbox{.}\end{array}\right.$
(3.1)

Using $1$-th and $3$-th equalities of system (3.1) we get

$\left\\{\begin{array}[]{ll}2b_{1,2}x_{1}x_{2}=2a_{x}x_{1}x_{2}\hbox{;}\\\
b_{3,1}x_{1}x_{2}+b_{3,3}x_{2}x_{3}=2a_{x}x_{1}x_{2}+b_{x}x_{2}x_{3}\hbox{.}\end{array}\right.$

and

$(b_{3,1}-2b_{1,2})x_{1}x_{2}+b_{3,3}x_{2}x_{3}=b_{x}x_{2}x_{3}.$

Hence, $b_{3,1}=2b_{1,2}$. Similarly, using equalities of (3.1) we get

$b_{4,3}=3b_{1,2},b_{2,2}=-b_{3,3},b_{4,4}=-2b_{2,2}.$

Using $1$-th and $5$-th equalities of system (3.1) we get

$\left\\{\begin{array}[]{ll}b_{1,2}x_{2}x_{5}=a_{x}x_{2}x_{5}\hbox{;}\\\
b_{6,5}x_{5}x_{2}+b_{6,6}x_{6}x_{2}=-a_{x}x_{5}x_{2}+b_{x}x_{6}x_{2}\hbox{.}\end{array}\right.$

and

$(b_{6,5}+b_{1,2})x_{2}x_{5}+b_{6,6}x_{6}x_{2}=b_{x}x_{6}x_{2}.$

Hence, $b_{6,5}=-b_{1,2}$.

Using $2$-th and $5$-th equalities of system (3.1) we get

$\left\\{\begin{array}[]{ll}b_{2,2}x_{2}x_{6}=-b_{x}x_{2}x_{6}\hbox{;}\\\
b_{6,5}x_{5}x_{2}+b_{6,6}x_{6}x_{2}=-a_{x}x_{5}x_{2}+b_{x}x_{6}x_{2}\hbox{.}\end{array}\right.$

and

$b_{6,5}x_{5}x_{2}+(b_{6,6}+b_{2,2})x_{6}x_{2}=-a_{x}x_{5}x_{2}.$

Hence, $b_{6,6}=-b_{2,2}$.

Using $1$-th and $6$-th equalities of system (3.1) we get

$\left\\{\begin{array}[]{ll}b_{1,2}x_{2}x_{8}=a_{x}x_{2}x_{8}\hbox{;}\\\
b_{7,7}x_{7}x_{2}+b_{7,8}x_{8}x_{2}=b_{x}x_{7}x_{2}+a_{x}x_{8}x_{2}\hbox{.}\end{array}\right.$

and

$b_{7,7}x_{7}x_{2}+(b_{7,8}-b_{1,2})x_{8}x_{2}=b_{x}x_{7}x_{2}.$

Hence, $b_{7,8}=b_{1,2}$.

Using $2$-th and $6$-th equalities of system (3.1) we get

$\left\\{\begin{array}[]{ll}b_{2,2}x_{2}x_{7}=-b_{x}x_{2}x_{7}\hbox{;}\\\
b_{7,7}x_{7}x_{2}+b_{7,8}x_{8}x_{2}=b_{x}x_{7}x_{2}+a_{x}x_{8}x_{2}\hbox{.}\end{array}\right.$

and

$(b_{7,7}+b_{2,2})x_{7}x_{2}+b_{7,8}x_{8}x_{2}=a_{x}x_{8}x_{2}.$

Hence, $b_{7,7}=-b_{2,2}$.

These equalities show that the matrix of the linear map $\nabla$ is of the
form (2.1). Therefore, by lemma 2.1 $\nabla$ is a derivation. This completes
the proof. ∎

Since a derivation on $W(2)$ is invariant on the subalgebras $S_{2}$ and
$W_{2}$, we have the following corollary.

######  Corollary 3.2.

Every local derivation of the algebras $S_{2}$ and $W_{2}$ is a derivation.

##  4\. $2$-Local derivations of conservative algebras of $2$-dimensional
algebras

In this section we give another characterization of derivations on
conservative algebras of $2$-dimensional algebras.

A (not necessary linear) map $\Delta:\mathcal{A}\to\mathcal{A}$ is called a
$2$-local derivation, if for any elements $x$, $y\in\mathcal{A}$ there exists
a derivation $D_{x,y}:\mathcal{A}\to\mathcal{A}$ such that
$\Delta(x)=D_{x,y}(x)$, $\Delta(y)=D_{x,y}(y)$.

######  Theorem 4.1.

Every $2$-local derivation of the algebras $S_{2}$, $W_{2}$ and $W(2)$ is a
derivation.

###### Proof.

We will prove that every $2$-local derivation of $W(2)$ is a derivation.

Let $\Delta$ be an arbitrary $2$-local derivation of $W(2)$. T hen, by the
definition, for every element $a\in W(2)$, there exists a derivation
$D_{a,e_{2}}$ of $W(2)$ such that

$\Delta(a)=D_{a,e_{2}}(a),\,\,\,\Delta(e_{2})=D_{a,e_{2}}(e_{2}).$

By lemma 2.1, the matrix $A^{a,e_{2}}$ of the derivation $D_{a,e_{2}}$ has the
following matrix form:

$A^{a,e_{2}}=\left(\begin{array}[]{cccccccc}0&\alpha_{a,e_{2}}&0&0&0&0&0&0\\\
0&-\beta_{a,e_{2}}&0&0&0&0&0&0\\\
2\alpha_{a,e_{2}}&0&\beta_{a,e_{2}}&0&0&0&0&0\\\
0&0&3\alpha_{a,e_{2}}&2\beta_{a,e_{2}}&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&-\alpha_{a,e_{2}}&\beta_{a,e_{2}}&0&0\\\
0&0&0&0&0&0&\beta_{a,e_{2}}&\alpha_{a,e_{2}}\\\ 0&0&0&0&0&0&0&0\\\
\end{array}\right).$

Let $v$ be an arbitrary element in $W(2)$. Then there exists a derivation
$D_{v,e_{2}}$ of $W(2)$ such that

$\Delta(v)=D_{v,e_{2}}(v),\,\,\,\Delta(e_{2})=D_{v,e_{2}}(e_{2}).$

By lemma 2.1, the matrix $A^{v,e_{2}}$ of the derivation $D_{v,e_{2}}$ has the
following matrix form:

$A^{v,e_{2}}=\left(\begin{array}[]{cccccccc}0&\alpha_{v,e_{2}}&0&0&0&0&0&0\\\
0&-\beta_{v,e_{2}}&0&0&0&0&0&0\\\
2\alpha_{v,e_{2}}&0&\beta_{v,e_{2}}&0&0&0&0&0\\\
0&0&3\alpha_{v,e_{2}}&2\beta_{v,e_{2}}&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&-\alpha_{v,e_{2}}&\beta_{v,e_{2}}&0&0\\\
0&0&0&0&0&0&\beta_{v,e_{2}}&\alpha_{v,e_{2}}\\\ 0&0&0&0&0&0&0&0\\\
\end{array}\right).$

Since $\Delta(e_{2})=D_{a,e_{2}}(e_{2})=D_{v,e_{2}}(e_{2})$, we have

$\alpha_{a,e_{2}}=\alpha_{v,e_{2}},\beta_{a,e_{2}}=\beta_{v,e_{2}},$

that it

$D_{v,e_{2}}=D_{a,e_{2}}.$

Therefore, for any element $a$ of the algebra $W(2)$

$\Delta(a)=D_{v,e_{2}}(a),$

that it $D_{v,e_{2}}$ does not depend on $a$. Hence, $\Delta$ is a derivation
by lemma 2.1.

The cases of the algebras $S_{2}$ and $W_{2}$ are also similarly proved. This
ends the proof. ∎

##  5\. 2-Local automorphisms of conservative algebras of 2-dimensional
algebras

A (not necessary linear) map $\Delta:\mathcal{A}\to\mathcal{A}$ is called a
$2$-local automorphism, if for any elements $x$, $y\in\mathcal{A}$ there
exists an automorphism $\phi_{x,y}:\mathcal{A}\to\mathcal{A}$ such that
$\Delta(x)=\phi_{x,y}(x)$, $\Delta(y)=\phi_{x,y}(y)$.

######  Theorem 5.1.

Every $2$-local automorphism of the algebras $S_{2}$, $W_{2}$ and $W(2)$ is an
automorphism.

###### Proof.

We prove that every $2$-local automorphism of $W(2)$ is an automorphism.

Let $\Delta$ be an arbitrary $2$-local automorphism of $W(2)$. Then, by the
definition, for every element $x\in W(2)$,

$x=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}+x_{4}e_{4}+x_{5}e_{5}+x_{6}e_{6}+x_{7}e_{7}+x_{8}e_{8},$

there exist elements $a_{x,e_{2}}$, $b_{x,e_{2}}$ such that

$A_{x,e_{2}}=\left(\begin{array}[]{cccccccc}1&a_{x,e_{2}}&0&0&0&0&0&0\\\
0&\frac{1}{b_{x,e_{2}}}&0&0&0&0&0&0\\\
2a_{x,e_{2}}b_{x,e_{2}}&a_{x,e_{2}}^{2}b_{x,e_{2}}&b_{x,e_{2}}&0&0&0&0&0\\\
3a_{x,e_{2}}^{2}b_{x,e_{2}}^{2}&a_{x,e_{2}}^{3}b_{x,e_{2}}^{2}&3a_{x,e_{2}}b_{x,e_{2}}^{2}&b_{x,e_{2}}^{2}&0&0&0&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&-a_{x,e_{2}}b_{x,e_{2}}&b_{x,e_{2}}&0&0\\\
0&0&0&0&0&0&b_{x,e_{2}}&a_{x,e_{2}}b_{x,e_{2}}\\\ 0&0&0&0&0&0&0&1\\\
\end{array}\right),$

$\Delta(x)=A_{x,e_{2}}\bar{x}$, where
$\bar{x}=(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8})$ is the vector
corresponding to $x$, and

$\Delta(e_{2})=A_{x,e_{2}}e_{2}=(a_{x,e_{2}},\frac{1}{b_{x,e_{2}}},a_{x,e_{2}}^{2}b_{x,e_{2}},a_{x,e_{2}}^{3}b_{x,e_{2}}^{2},0,0,0,0).$

Since the element $x$ was chosen arbitrarily, we have

$\Delta(e_{2})=(a_{x,e_{2}},\frac{1}{b_{x,e_{2}}},a_{x,e_{2}}^{2}b_{x,e_{2}},a_{x,e_{2}}^{3}b_{x,e_{2}}^{2},0,0,0,0)$
$=(a_{y,e_{2}},\frac{1}{b_{y,e_{2}}},a_{y,e_{2}}^{2}b_{y,e_{2}},a_{y,e_{2}}^{3}b_{y,e_{2}}^{2},0,0,0,0),$

for each pair $x$, $y$ of elements in $W(2)$. Hence,
$a_{x,e_{2}}=a_{y,e_{2}}$, $b_{x,e_{2}}=b_{y,e_{2}}$. Therefore

$\Delta(x)=A_{y,e_{2}}x$

for any $x\in W(2)$ and the matrix $A_{y,e_{2}}$ does not depend on $x$. Thus,
by lemma 2.2 $\Delta$ is an automorphism.

The cases of the algebras $S_{2}$ and $W_{2}$ are also similarly proved. The
proof is complete. ∎

##  6\. Local automorphisms of conservative algebras of 2-dimensional algebras

Let $\mathcal{A}$ be an algebra. A linear map
$\nabla:\mathcal{A}\to\mathcal{A}$ is called a local automorphism, if for any
element $x\in\mathcal{A}$ there exists an automorphism
$\phi_{x}:\mathcal{A}\to\mathcal{A}$ such that $\nabla(x)=\phi_{x}(x)$.

######  Theorem 6.1.

Every local automorphism of the algebras $S_{2}$, $W_{2}$ and $W(2)$ is an
automorphism.

###### Proof.

We prove that every local automorphism of $W(2)$ is an automorphism.

Let $\nabla$ be an arbitrary local automorphism of $W(2)$ and $B$ be its
matrix, i.e.,

$\nabla(x)=B\bar{x},x\in W(2),$

where $\bar{x}$ is the vector corresponding to $x$. Then, by the definition,
for every element $x\in W(2)$,

$x=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}+x_{4}e_{4}+x_{5}e_{5}+x_{6}e_{6}+x_{7}e_{7}+x_{8}e_{8},$

there exist elements $a_{x}$, $b_{x}$ such that

$A_{x}=\left(\begin{array}[]{cccccccc}1&a_{x}&0&0&0&0&0&0\\\
0&\frac{1}{b_{x}}&0&0&0&0&0&0\\\ 2a_{x}b_{x}&a_{x}^{2}b_{x}&b_{x}&0&0&0&0&0\\\
3a_{x}^{2}b_{x}^{2}&a_{x}^{3}b_{x}^{2}&3a_{x}b_{x}^{2}&b_{x}^{2}&0&0&0&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&-a_{x}b_{x}&b_{x}&0&0\\\
0&0&0&0&0&0&b_{x}&a_{x}b_{x}\\\ 0&0&0&0&0&0&0&1\\\ \end{array}\right)$

and

$\nabla(x)=B\bar{x}=A_{x}\bar{x}.$

Using these equalities and by choosing subsequently $x=e_{1}$, $x=e_{2}$,
$\dots,x=e_{8}$ we get

$B=\left(\begin{array}[]{cccccccc}1&a_{e_{2}}&0&0&0&0&0&0\\\
0&\frac{1}{b_{e_{2}}}&0&0&0&0&0&0\\\
2a_{e_{1}}b_{e_{1}}&a_{e_{2}}^{2}b_{e_{2}}&b_{e_{3}}&0&0&0&0&0\\\
3a_{e_{1}}^{2}b_{e_{1}}^{2}&a_{e_{2}}^{3}b_{e_{2}}^{2}&3a_{e_{3}}b_{e_{3}}^{2}&b_{e_{4}}^{2}&0&0&0&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&-a_{e_{5}}b_{e_{5}}&b_{e_{6}}&0&0\\\
0&0&0&0&0&0&b_{e_{7}}&a_{e_{8}}b_{e_{8}}\\\ 0&0&0&0&0&0&0&1\\\
\end{array}\right).$

Since $\nabla(e_{6}+e_{7})=\nabla(e_{6})+\nabla(e_{7})$, we have

$b_{e_{6}+e_{7}}=b_{e_{6}},b_{e_{6}+e_{7}}=b_{e_{7}}.$

Hence,

$b_{e_{6}}=b_{e_{7}}.$

Similarly to this equality we get $b_{e_{3}}=b_{e_{6}}$ and
$b_{e_{6}}=b_{e_{2}}\neq 0$. Hence,

$b_{e_{2}}=b_{e_{3}}=b_{e_{6}}=b_{e_{7}}.$ (6.1)

Since $\nabla(e_{5}+e_{8})=\nabla(e_{5})+\nabla(e_{8})$, we have

$a_{e_{5}+e_{8}}b_{e_{5}+e_{8}}=a_{e_{5}}b_{e_{5}},\,\,a_{e_{5}+e_{8}}b_{e_{5}+e_{8}}=a_{e_{8}}b_{e_{8}}.$

From this it follows that

$a_{e_{5}}b_{e_{5}}=a_{e_{8}}b_{e_{8}}.$

Similarly to this equality we get $a_{e_{1}}b_{e_{1}}=a_{e_{8}}b_{e_{8}}$.
Hence,

$a_{e_{1}}b_{e_{1}}=a_{e_{5}}b_{e_{5}}=a_{e_{8}}b_{e_{8}}.$ (6.2)

Since $\nabla(e_{4}+e_{6})=\nabla(e_{4})+\nabla(e_{6})$, we have

$b_{e_{4}+e_{6}}^{2}=b_{e_{4}}^{2},\,\,b_{e_{4}+e_{6}}^{2}=b_{e_{6}}^{2}.$

From this it follows that

$b_{e_{4}}^{2}=b_{e_{6}}^{2}.$

Hence, by (6.1), we get

$b_{e_{4}}^{2}=b_{e_{2}}^{2}.$ (6.3)

Since $\nabla(e_{2}+e_{8})=\nabla(e_{2})+\nabla(e_{8})$, we have

$a_{e_{2}}=a_{e_{2}+e_{8}},\,\,\,a_{e_{2}+e_{8}}^{2}b_{e_{2}+e_{8}}=a_{e_{2}}^{2}b_{e_{2}},\,\,\,a_{e_{2}+e_{8}}b_{e_{2}+e_{8}}=a_{e_{8}}b_{e_{8}}.$

Hence,

$b_{e_{2}+e_{8}}=b_{e_{2}},\,\,\,a_{e_{2}+e_{8}}b_{e_{2}+e_{8}}=a_{e_{2}}b_{e_{2}}$

and, therefore,

$a_{e_{2}}b_{e_{2}}=a_{e_{8}}b_{e_{8}}.$ (6.4)

Similarly, since $\nabla(e_{2}+e_{3})=\nabla(e_{2})+\nabla(e_{3})$, we have

$a_{e_{2}}=a_{e_{2}+e_{3}},\,\,\,b_{e_{2}}^{-1}=b_{e_{2}+e_{3}}^{-1},\,\,\,a_{e_{2}+e_{3}}^{3}b_{e_{2}+e_{3}}^{2}+3a_{e_{2}+e_{3}}b_{e_{2}+e_{3}}^{2}=a_{e_{2}}^{3}b_{e_{2}}^{2}+3a_{e_{3}}b_{e_{3}}^{2}.$

Hence,

$b_{e_{2}}=b_{e_{2}+e_{3}}$

and by (6.1) and $a_{e_{2}}=a_{e_{2}+e_{3}}$ we get

$a_{e_{2}}^{3}+3a_{e_{2}}=a_{e_{2}}^{3}+3a_{e_{3}}.$

Therefore, $a_{e_{2}}=a_{e_{3}}$ and

$a_{e_{2}}b_{e_{2}}^{2}=a_{e_{3}}b_{e_{3}}^{2}.$ (6.5)

Finally, since $\nabla(e_{1}+e_{8})=\nabla(e_{1})+\nabla(e_{8})$, we have

$a_{e_{1}+e_{8}}b_{e_{1}+e_{8}}=a_{e_{1}}b_{e_{1}},\,\,\,a_{e_{1}+e_{8}}b_{e_{1}+e_{8}}=a_{e_{8}}b_{e_{8}}.$

Hence,

$a_{e_{1}}b_{e_{1}}=a_{e_{8}}b_{e_{8}}.$

By (6.4), from the last equalities it follows that

$a_{e_{1}}b_{e_{1}}=a_{e_{2}}b_{e_{2}},\,\,\,a_{e_{1}}^{2}b_{e_{1}}^{2}=(a_{e_{1}}b_{e_{1}})^{2}=(a_{e_{2}}b_{e_{2}})^{2}=a_{e_{2}}^{2}b_{e_{2}}^{2}.$
(6.6)

By (6.1), (6.2), (6.3), (6.4), (6.5), (6.6) the matrix $B$ has the following
matrix form

$B=\left(\begin{array}[]{cccccccc}1&a_{e_{2}}&0&0&0&0&0&0\\\
0&\frac{1}{b_{e_{2}}}&0&0&0&0&0&0\\\
2a_{e_{2}}b_{e_{2}}&a_{e_{2}}^{2}b_{e_{2}}&b_{e_{2}}&0&0&0&0&0\\\
3a_{e_{2}}^{2}b_{e_{2}}^{2}&a_{e_{2}}^{3}b_{e_{2}}^{2}&3a_{e_{2}}b_{e_{2}}^{2}&b_{e_{2}}^{2}&0&0&0&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&-a_{e_{2}}b_{e_{2}}&b_{e_{2}}&0&0\\\
0&0&0&0&0&0&b_{e_{2}}&a_{e_{2}}b_{e_{2}}\\\ 0&0&0&0&0&0&0&1\\\
\end{array}\right).$

Hence, by lemma 2.2, the local automorphism $\nabla$ is an automorphism.

The cases of the algebras $S_{2}$ and $W_{2}$ are also similarly proved. This
ends the proof. ∎

The authors thank professor Ivan Kaygorodov for detailed reading of this work
and for suggestions which improved the paper.

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