A CHARACTERIZATION OF DERIVATIONS AND AUTOMORPHISMS ON SOME SIMPLE ALGEBRAS Текст научной статьи по специальности «Физика»

URAL MATHEMATICAL JOURNAL, Vol. 8, No. 2, 2022, pp. 46-58

DOI: 10.15826/umj.2022.2.004

A CHARACTERIZATION OF DERIVATIONS AND AUTOMORPHISMS ON SOME SIMPLE ALGEBRAS

Farhodjon Arzikulov

V.I. Romanovskiy Institute of Mathematics, Universitet street 9, Tashkent, 100174, Uzbekistan [email protected]

Furkat Urinboyev

Namangan State University, Uychi street 316, Namangan, 716019, Uzbekistan [email protected]

Shahlo Ergasheva

Kokand State Pedagogical Institute, Turon street 23, Kokand, 150700, Uzbekistan [email protected]

Abstract: In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic 0 without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticom-mutative algebra D and the seven-dimensional central simple commutative algebra C. We prove that every local derivation of these algebras D and C is a derivation, and every 2-local derivation of these algebras D and C is also a derivation. We also prove that every local automorphism of these algebras D and C is an automorphism, and every 2-local automorphism of these algebras D and C is also an automorphism.

Keywords: Simple algebra, Derivation, Local derivation, 2-Local derivation, Automorphism, Local automorphism, 2-Local automorphism, Basis of identities.

1. Introduction

In the present paper, we study local and 2-local derivations and automorphisms of simple finite-dimensional algebras without finite basis of identities, constructed by Kislitsin in [19] and [20]. Kadison in [12] introduced and investigated a notion of local derivations. He proved that each continuous local derivation from a von Neumann algebra into its dual Banach bimodule is a derivation. Semrl introduced a similar notion of 2-local derivations. He proved that any 2-local derivation of the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H is a derivation [24]. After, numerous new results related to the description of local and 2-local derivations of associative algebras have appeared. For example, papers [1, 3, 4, 15, 16, 22] are devoted to local and 2-local derivations of associative algebras.

The study of local and 2-local derivations of nonassociative algebras was initiated in papers [5, 6] of Ayupov and Kudaybergenov (for the case of Lie algebras). They proved that each local and 2-local derivation on a semisimple finite-dimensional Lie algebra are derivations. In [8], examples of 2-local derivations on nilpotent Lie algebras that are not derivations are given. After the cited

works, the study of local and 2-local derivations was continued for Leibniz algebras [7] and Jordan algebras [2]. Local and 2-local automorphisms were also studied in many cases. For example, local and 2-local automorphisms on Lie algebras have been studied in [5, 10].

The variety of Malcev algebras is a generalization of the variety of Lie algebras [23]. It is closely related to other classes of nonassociative structures: it is a proper subvariety of binary Lie algebras, and, under the multiplication ab — ba, an alternative algebra is a Malcev algebra. Moreover, it is connected with various classes of algebraic systems such as Moufang loops, Poisson-Malcev algebras, etc. The study of generalizations of derivations of simple Malcev algebras was initiated by Filippov in [11] and continued in some papers of Kaygorodov and Popov [13, 14].

Now, a linear operator V on A is called a local automorphism if, for every x € A, there exists an automorphism <x of A, depending on x, such that V(x) = <x(x). The concept of local automorphism was introduced by Larson and Sourour [21] 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 Semrl in [24] as 2-local automorphisms. Namely, a map A : A — A (not necessarily linear) is called a 2-local automorphism if, for every x, y € A, there exists an automorphism <x,y : A — A such that A(x) = <x,y(x) and A(y) = <x,y(y). After the work of Semrl, it appeared numerous new results related to the description of local and 2-local automorphisms of algebras (see, for example, [5, 7, 9, 10, 16]).

In the present paper, we continue the study of derivations and automorphisms of simple algebras. We study derivations and automorphisms of simple algebras, which do not belong to well-known classes of algebras (commutative, associative, alternative, Lie, Jordan, etc.). The simple finite-dimensional algebras without finite basis of identities, constructed by Kislitsin are such algebras. Namely, we prove that any local derivation (automorphism) of the simple finite-dimensional algebras without finite basis of identities, constructed by Kislitsin in [19] and [20], is a derivation (an automorphism, respectively), and every 2-local derivation (automorphism) of these algebras is also a derivation (an automorphism, respectively). Note that central simple finite-dimensional algebras which has no finite basis of identities were considered in the works [17] and [18] of Isaev and Kislitsin.

2. A simple finite-dimensional algebra without finite basis of identities

Let D = (e, vi, v2, en, ei2, e22,p)F be an algebra over a field F of characteristic 0 whose nonzero products of basis elements from

{e,vi,v2 ,eii,ei2 ,e22,p} (2.1)

are defined by the rules

vieij = — eij vi = vj, v2p = —pv2 = e, v^e = —evi = vi, eij e = —eeij = eij, pe = —ep = p.

Then D is a simple anticommutative algebra without finite basis of identities [20]. Let a be an element in D. Then we can write

a = aie + a2vi + a3v2 + a4eii + a5ei2 + a6e22 + azp for some elements ai, a2, a3, a4, a5, a6, and a7 in F. Throughout the paper, let

a = (ai, a2, a3, a4, a5, a6, a7 )T.

Conversely, if v = (a1, 07)T is a column vector with ai,02,03,04,05, a6, and 07 in F,

then, throughout the paper, we will denote by V the element

i.e.,

0ie + 02Vi + 03V2 + 04611 + 05612 + 06e22 + 07^;

V = 01e + 02V1 + 03V2 + 04611 + 05612 + 06622 + 07^. Let A be an algebra. A linear map D: A ^ A is called a derivation if

D(xy) = D(x)y + xD(y)

for any two elements x, y € A.

Our principal tool for the description of local and 2-local derivations of D is the following proposition.

Proposition 1. A linear map D: D ^ D is a derivation if and only if the matrix of D in the standard basis (2.1) has the following form:

/ 0 0 0 a2,2 00 00 00 00 00

0 0

0 0

0 0

«2,2 + «5,5 0 0

0 0 0

00

0 05,5 00

0 0 0 0 0 0

\

0 0 0 («2,2 + 05,5)/

Here the action of D corresponds to multiplying the matrix by a column on the right.

Proof. The proof is carried out by checking the derivation property on the algebra D. Let A = (ai,j)Jj=i be the matrix of the derivation D. Then

Avie-ij = —Ae-ijVi = Avj, AV2P = —Apv2 = Ae, Avjë- = —AevJ = Avj, Aëïjë- = —Aœïj = Aëïj, Ape. = —Aëp = Ap.

On the other hand,

Hence,

Avie-ij — Avie-ij + ViAe.

i ij

ij

Avj = Avie-ij + ViAe.

ij

So,

Âvï = ÂW[e 11 + viÀën, ai , 2e + a2 ,2Vi + 03 , 2V2 + 04 , 2eii + 05 , 2ei2 + a6 , 2622 + 07, 2P = (0i,2e + 02,2Vi + 03,2^2 + 04,26!! + 05,2ei2 + 06,2622 + a7,2p)eii +vi(0i,4e + a2,4Vi + a3,4V2 + a4,4eii + a5,4ei2 + a6,4e22 + a7,4p)

0

if i = 1, j = 1, and

ai,2e + a2,2vi + a3,2v2 + a4,2eii + a5,2ei2 + a6,2e22 + a7,2P = —ai,2eii + a2,2vi + ai,4vi + a4,4 vi + a5,4v2.

This implies that

ai,2 = 0, ai,4 + a4,4 = 0, a3,2 = a5,4, a4,2 = —ai,2 = 0, a5,2 = 0, a6,2 = 0, a7,2 = 0.

In addition, if i = 1 and j = 2, then

Av2 = AvTe-12 + v\Aei2

and

ai,3e + a2,3vi + a3,3v2 + a4,3en + a5,3ei2 + a6,3e22 + a7,3p = (ai,2e + a2,2vi + a3,2 v2 + a4,2eii + a5,2ei2 + a6,2e22 + a7,2p)ei2 +vi(ai,5e + a2,5vi + a3,5v2 + a4,5en + a5,5ei2 + a6,5e22 + a7,5p) = —ai,2ei2 + a2,2v2 + ai,5vi + a4,5vi + a5,5v2.

This implies that

ai,3 = 0, a2,3 = ai,5 + a4,5, a3,3 = a2,2 + a5,5, a4,3 = 0, a5,3 = —ai,2, a6,3 = 0, a7,3 = 0. Besides, if i = 2 and j = 2, then

Av2 = AT^e-22 + v2Ae^

and

ai,3e + a2,3vi + a3,3v2 + a4,3en + a5,3ei2 + a6,3e22 + a7,3p = (ai,3e + a2,3vi + a3,3 v2 + a4,3en + a5,3ei2 + a6,3e22 + a7,3p)e22 +v2(ai,6e + a2,6vi + a3,6v2 + a4,6en + a5,6ei2 + a6,6e22 + a7,6p) = —ai,3e22 + a3,3v2 + ai,6v2 + a6,6v2 + a7,6e.

This implies that

ai,3 = a7,6 = 0, a2,3 = 0, ai,6 + a6,6 = 0, a4,3 = 0, a5,3 = 0, a6,3 = —ai,3 = 0, a7,3 = 0.

Similarly, we have

a3,3 = —a7,7, a2,i = 0, ai,7 = 0, a6,7 = 0, a4,i = 0, a5,i = 0, a6,i = 0,

«1,2 = 0, «4,1 = 0, «5,1 = 0, «1,3 = 0, «7,1 = 0, «6,1 = 0,

«1,4 = 0, «2,1 = 0, «1,1 = 0, «1,5 = 0, «1,6 = 0, «3,1 = 0, «1,7 = 0,

«6,2 = 0, «7,2 = 0, «4,3 = 0, «3,6 = 0, «3,7 = 0, «3,5 = 0, «2,7 = 0,

«3,4 = 0, «1,2 = 0, «3,2 = 0, «4,7 = 0, «5,7 = 0, «2,6 = 0, «3,4 = 0, «2,4 = 0,

«2,5 = 0, «1,2 = 0, «5,6 = 0, «4,6 = 0, «1,3 = 0, «2,3 = 0, «6,5 = 0, «7,5 = 0,

a6,4 = 0, a7,4 = 0.

As a result, we get the matrix from Proposition 1. The proof is complete. □

Let A be an algebra. A linear map V: A — A is called a local derivation if, for any element x € A, there exists a derivation D: A — A such that V(x) = D(x).

Theorem 1. Each local derivation on the simple algebra D is a derivation.

Proof. Let V be a local derivation on D, and let A = (a^j)7j=1 be the matrix of V. Then

V(vi) = a^vi = a2,2Vi, V(v2) = (a2f2 + a525)"2 = a3,3"2, V(ei,2) = aS]s2 ei,2 = as,sei,2, V(p) = -(a|2 + ap,5)p = az,7P,

and the remaining components of the matrix A are equal to zero. At the same time,

V(vi + V2 + ei,2 + p) = V(vi) + V(v2) + V(ei,2) + V(p) (2.2)

and

V7/ I, , \ vi+V2+ei,2+P I i vi +V2+ei,2+P I vi+v2+6i,2+P\

V(vi + V2 + ei,2 + p) = a22 ' vi + (a22 ' + a5,5 ' )v2

I vi+V2+ei,2+P / vi+v2+6i,2+P , vi+v2+6i,2+P\

+a5,5 ' ei,2 - (a2,2 ' + «5,5 ' )P-

By 2.2, we have

Hence,

vi+V2 +ei,2+p I / vi+V2+ei,2+P I vi+v2+6i,2+P\ a2,2 ' vi + (a2,2 ' + a5,5 ' ) v2

vi +v2+ei,2+p /Vi +V2+ei,2 +P , vi +V2 +6i,2+P\

+a5,5 ei,2 - (a2,2 + a5,5 )p

= aVi2Vi + № + «525)v2 + 4!52ei,2 - (ap,2 + a5,5)P"

avi+v2+ei,2+p = m «vi +V2+ei,2+P + avi+v2+6i,2+P = av2 + av2

«2,2 = «2,2, «2,2 + «5,5 = «2,2 + «5,5,

avi+v2+ei,2+p = _ei,2 avi+v2+6i,2+P , avi+v2+6i,2+P = ap , p

«5,5 = «5,5 , «2,2 + «5,5 = «2,2 + a5,5"

This implies that

+ «v2 = «vi + a6i,2 _P + «P = «vi + a6i,2 *2,2 + «5,5 = «2,2 + «5,5 , «2,2 + «5,5 = «2,2 + «5,5

and

A =

j 0 0 0 0 0 0 0 \

0 «2,2 0 0 0 0 0

0 0 .11 , „ei,2 «2,2 + «5,5 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 a61,2 «5,5 0 0

0 0 0 0 0 0 0

V 0 0 0 0 0 0 -«2 + 4i52 ) /

Hence, by Proposition 1, V is a derivation. This completes the proof.

□

We give another characterization of derivations on the algebra D in the following theorem. Let A be an algebra. A (not necessary linear) map A : A ^ A is called a 2-local derivation if, for all elements x,y € A, there exists a derivation : A ^ A such that A(x) = (x) and

A(y) = (y).

Theorem 2. Each 2-local derivation on the simple algebra D is a derivation.

Proof. Suppose that A is a 2-local derivation on D and, for elements 0, b € D, Da,b is a derivation on D such that Da,b(0) = A(0) and Da,b(b) = A(b). Let Aa,b = (0^)7.,=;! be the matrix of Da,5.

Let

0 = A16 + A2V1 + A3V2 + A461,1 + A561,2 + A662,2 + A7P be an arbitrary element from D. For every v € D, there exists a derivation D2,a such that

Then from

it follows that

Hence,

A(v) = Dv>0(u), A(a) = Dv>0(a). (vi) = Dv1,o(vi), v € D,

a2,S! V1 = a2,S! v1-

vi,v _ vi,o

^2,2 = a2,2 •

o —

Therefore,

A(0) = Dv1,a(0) = A2V1 + + 0515")A3V2 + 0515^561,2 - (0212" + a51;,a)A7p.

Similarly, from

D22,2 (V2) = D^2,a(V2), V € D,

it follows that

A(0) = Dv2,a(0) = 022^aA2V1 + + 0525v)A3V2 + 0525^561,2 - (0222" + 0525")A7P. Similarly, we have

A(0) = Dei,2,«(0) = 02l22,aA2V1 + (02l22,a + 05152,")A3V2 + 05l52,vA5e1,2 - (02l22,a + 05152,")A7P, A(0) = Dp,a(0) = a2,aA2Vl + + a5),a)AзV2 + 0^561,2 - (02,2 + a5,5)A7p.

Hence,

A(0) = D21 ,„(0) = D22,a(0) = D61,2,0(0) = Dp,^) = A2V1 + (0222™ + 0525™)A3V2 + 05152,zA5e1,2 - (a2,,2 + 0p,5)A7P

for any V,w,z,t € D. Note that the components in the last sum do not depend on the element 0. Therefore, the map A is linear and it is a local derivation. The linear operator A has the following matrix:

I 0 0

n vi,v 0 a2,2

A =

0 0 0 0

00

0 0

a2,2 + a5,5 0 0 0 0

0 0 0 0

0a 00

ei,2,z 5,5

0 0 0 0 0 0

0 0 0 0 0 0

\

0 0 0 -(a2,2 + ap,5) ;

From A(v2 + p) = A(v2) + A(p), we get

/ «,V2+P I a,v2+p\ +P I ,.a,v2+P\„ _ / v2,w , V2,w\ / p,t , _P,t\„

(a2,2 + a5,5 )v2 - (a2,2 + a5,5 )p = (a2,2 + a5,5 )v2 - (a2,2 + a5,5)p

Hence,

a,V2+p | a,V2+p

V2,w I V2,w _ p,t | p,t

a^ o + a^ 5 — ^2 2 + a

a2,Y " + a5,5 - "2,2 "T" "5,5 "2,2

From A(vi + V2 + ei,2) — A(vi) + A(v2) + A(ei,2), we get

5,5

(2.3)

a

2,2

vi ,v X2,2 ,

a

",vi+v2+ei,2 . a,vi+V2+ei,2

2,2

a

+ a5,5 __«,vi+v2+ei,2 5,5

— a

V2,w I V2,w a2,2 + a5,5 ,

_ei,2,z

5,5

Hence,

V2,w I V2,w _ vi,v . ei,2,z

a2,2 + a5 5 = a2 2 + a5 5

5,5

By (2.3), we also have

P,t I p,t vi ,v | '

a2 2 + a5 5 = a2 2 + a.

2 2

,P,t

vi, v

5,5

2,2

ei,2,z 5,5 .

Thus,

A =

0 0 0 0 0 0

0 vi ,v a2,2 0 0 0 0

0 0 vi,v , ei,2,z a2,2 + a5,5 0 0 0

0 0 0 0 0 0

0 0 0 0 ei,2,z a5,5 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0

vi ,v

0 (a2,2 + a

ei,2,z 5,5

)

Therefore, by Proposition 1, A is a derivation. This completes the proof.

□

Let A be an algebra. A linear bijective map $ : A ^ A is called an automorphism if $(xy) = $(x)$(y) for any two elements € A.

Our principal tool for the description of local and 2-local automorphisms of D is the following proposition.

Proposition 2. A linear map $: D ^ D is an automorphism if and only if the matrix of $ in the standard basis (2.1) has the following form :

10

0 a2,2 00

0 0 0 0

a2,2a5,5 0 0 0 1 0 0

0 0 0 0

0 a5,5 0

001 000

0 0 0 0 0 0

«2,2«5,5 /

where a2,2 and a5,5 are nonzero elements from F. Here the action of $ corresponds to multiplying the matrix by a column on the right.

i

Proof. Let B = (bij)7j=i be the matrix of the automorphism Then there exists a derivation D such that

B = eA,

where A is the matrix of D. It is known that

A2 A3

eA = E + A + ^r + ^r +

2! 3!

where E is the unit matrix. Hence,

A2 A3

B = E + A + — + — + 2! 3!

By (2.4) and Proposition 1, B is equal to

( 1 0 o YT * 0 0

OO "'2,2

i=o IT

0

0 0 0

0 0

ECO (02,2+05,5)'

i=0 i\ 0

0 0 0

0 0 0 1

0 0

0 0 0 0

^OO "5,5 =0 0

0

0 0 0 0

n V00 O

u ¿^¿=0 ü u

0 0 0 0

0 0

(2.4)

n Y^OO (-l)'(02,2+05,5)' /

u 0 i\ /

1 0 0 0 0 0 0 \

0 e«2,2 0 0 0 0 0

0 0 ea2,2+05,5 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 e05,5 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 e (02,2+05,5) /

The latter matrix gives the desired form. This completes the proof.

□

Let A be an algebra. A linear map V : A ^ A is called a local automorphism if, for every element x € A, there exists an automorphism : A ^ A such that V(x) = 0x(x).

Theorem 3. Each local automorphism on the simple algebra D is an automorphism.

Then

Proof. Let V be a local automorphism on D, and let A = (ajj)7 j=i be the matrix of V.

V(vi) = a2x2vi = a2,2Vi, V(v2) = a222a525V2 = 03,3^2,

1

V(ei,2) = 05,5" ei,2 = 05,561,2) V(p) = ———p = a7Jp

a2,2a5,5

and the remaining components of the matrix A are equal to zero. At the same time, V(vi + "2 + ei,2 + p) = V(vi) + V("2) + V(ei,2) + V(p)

(2.5)

1

and

w/ II i \ vi+V2+ei,2+p . vi+V2+ei,2+p vi+V2 +6i,2+p .

V(vi + V2 + ei,2 + p) — a22 vi + a22 , a5 5 , V2+

a

vi+V2+ei,2+p 5,5

ei,2 +

2,2 1

vi+V2+ei,2+p vi+V2+ei,2+p

p.

a

2,2

5,5

By (2.5), we have

a

vi+V2+ei,2+p 2,2

v1 + a2,2

vi+V2+ei,2 +p vi+V2+ei,2+p

a

Hence,

5,5

)v2 + a(

vi +V2 +ei,2+p 5,5

ei,2 +

1

vi+V2+ei,2+p vi +V2+ei,2+p

p

a

2,2

a

v1

ei,2 ,

1

afcvi + a£2a£5u2 + a5i¿-ei,2 + p p P-

a2,2a5,5

a

a

vi+V2+ei,2+p 2,2

vi +V2 +ei,2+p 5,5

=a

v1

2,2,

a

vi+v2+ei,2+p^vi +V2 +ei,2+p = av2 av2

= a2,2a5,5,

2,2

a

5,5

,ei,2 5,5,

a

vi+V2 +ei,2 +p vi+V2+ei,2+p _ p p

This implies that

and

a2 2a5 5 — a0 oa

2,2

VI ei,2

a

5,5

a2,2a5,5.

2,2a5,5

pp

a2,2 a5,5 = a 1 a

vi ei,2

2,2 a5,5

A —

1 0 0 0 0 0 0 \

0 av1 a2,2 0 0 0 0 0

0 0 a2,2a5,5 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 a61,2 a5,5 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

t'l el,2 o2,2o5,5 /

Hence, by Proposition 2, V is an automorphism. This completes the proof.

5,5

□

A (not necessary linear) map A : A ^ A is called a 2-local automorphism if, for all elements x, y € A, there exists an automorphism : A ^ A such that A(x) — (x) and A(y) — (y).

Theorem 4. Each 2-local automorphism on the simple algebra D is an automorphism.

Proof. Suppose that A is a 2-local automorphism on D and, for elements a, b € D, $o,b is

o,b)7 .

an automorphism on D such that $o,b(a) — A(a) and $o,b(b) — A(b). Let Ao,b — (aOf)7^ be the matrix of Then, for all v, z € D, there exists an automorphism $v z such that

A(v) — $v,z (v), A(z) — $v,z (z).

Let Av,z — (aVjz)n-j=1 be the matrix of the automorphism $v,z. Let , ,

a — Aie + A2V1 + A3V2 + A4 ei,i + A5 ei,2 + A6 e2,2 + A/p be an arbitrary element from D. For every v € D, there exists an automorphism $V,o such that

A(v) — $v,o(v), A(a) — $v,o(a).

Then from

it follows that

(vi) = $vi,o(vi), v eD,

a2, 2 V1 = a2, 2 v1-

Hence,

Therefore,

42,2

A2,2

A(a) = $v1;o(a) = Aie + a2 '2v A2V1 + a2 2°a5 15oA3V2 + A4ei,i

+a51¿oA5eií2 + Aee2,2 +

v1,o v1,o a2,2 a5,5

A7P.

Similarly, from

it follows that

$V2,v(V2) = $V2,o(V2), v eD,

A(a) = $v2,o(a) = Aie + a^^vi + a^ a5fí1V A3V2 + A4ei,i

a212 a515

Similarly, we have

A(a) = 12,o(a) = +Aie + a^ ^'^vi + a.

e1,2,o e1,2,o ï2'2 a5,5

1A3V2

+A4ei 1 + ckV^Asei 2 + A6e2,2 +

e1,2,o e1,2,o X2'2 a5,5

A7P,

Hence,

A(a) = $p,o(a) = Ai e + ap^vi + a2'°ap;°A3V2 +A4ei,i + af^Asei^ + Aee2,2 H—PtV

a2,2 a5,5

A(a) = $v1,o(a) = $v2,o(a) = $e12io(a) = $p,o(a) =

Aie + av12v A2 v1 + a,

v2,w v2,w 2,2 a5,5

A3V2 + A4ei'i + a^2'2 A5ei,2 + A6e2,2 +

Pit P't

a2'2a5,5

A7 P

for any t € D. Note that the components in the last sum do not depend on the element a.

Therefore, the map A is linear and it is a local automorphism. The linear operator A has the following matrix:

A =

1 0 0 0 0 0 0 \

0 v1 ,v a2'2 0 0 0 0 0

0 0 v2,w v2,w a2,2 a5,5 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 e1,2,z a5,5 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

o2,2o5,5 /

v1 ,v

v1 ,o

1

1

1

From A(v2 + p) = A(v2) + A(p), we get

2,2 5,5 V'2+ a,v2+p a,v2+pP ~ fl2,2 «5,5 + Pit Pit P-a2,2 a5,5 a2,2a5,5

Hence,

a,V2 +p a,V2 +p _ V2,w V2,w _ p,t p,t /„

a2,2 a5,5 = a2,2 a5,5 = a2,2a5,5- (2-u)

From A(vi + V2 + ei,2) = A(vi) + A(v2) + A(ei,2), we get

a,vx+v2+ei,2 _ vi,v a,vi+V2+ei,2 a,vi+V2+ei,2 _ v2,w v2,w

X2,2 = a2,2 , a2,2 a5,5 = a2,2 a5,5

Hence,

By (2.6), we also have

Thus,

a

a,vi+v2 +ei,2 5,5

=a

ei,2,z 5,5 •

V2,w V2,w _ vi,v ei,2,z

a2,2 a5,5 = a a

2,2 5,5

p,t p,t vi,v ei,2,z

f} — n i} n ^

a2,2a5,5

2,2 a5,5

A =

1 0 0 0 0 0 0

0 vi,v a2,2 0 0 0 0 0

0 0 vi,v ei,2,z a2,2 a5,5 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 ei,2 ,z a5,5, 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1 V ,v e a2,2 a5

/

Therefore, by Proposition 2, A is an automorphism. This completes the proof.

□

5

3. A simple central commutative algebra with no finite basis of identities

Let C = (1,v1,v2,e11 ,e12, e22,p)F be an algebra over a field F of characteristic 0, where 1 is unity and nonzero products of basis elements

{1,v1 ,V2,en,e12,e22,p} (3.1)

other than 1 are defined as follows:

V eij = eij Vi = Vj, V2P = pv2 = 1.

Then the algebra C is a simple central commutative algebra with no finite basis of identities [19]. Let a be an element in C. Then we can write

a = a1e + a2V1 + a3 V2 + a4en + a5e12 + a6e22 + azp,

for some elements ai, a2, a3, a4, a5, a6, and a7 in F. Throughout the paper, let

a = (a1, a2, a3, a4, a5, a6, a7 )T.

Conversely, if v = (a1, a2, a3, a4, a5, a6, a7)T is a column vector with ai, a2, a3, a4, a5, a6, and a7 in F, then, throughout the paper, we will denote by v the element

i.e.,

aie + a2Vi + a3V2 + a4eii + a5ei2 + a6e22 + a7p,

? = aie + a2Vi + a3V2 + a4eii + a5ei2 + a6e22 + a7p.

Our principal tool for the description of local and 2-local derivations of C is the following proposition.

Proposition 3. A linear map D : C ^ C is a derivation if and only if the matrix of D in the basis (3.1) has the following form :

0 0 0 0 0 0

0 a2,2 0 0 0 0

0 0 a2,2 + a5,5 0 0 0

0 0 0 0 0 0

0 0 0 0 a5,5 0 0 0 0 0 0 0 \ 0 0 0 0 0 0 -(a2,2 + a5,5) )

0 0 0 0 0 0

\

Here the action of D corresponds to multiplying the matrix by a column on the right. Proof. The proof of this proposition is similar to the proof of Proposition 1.

□

Theorem 5. Each local (2-local) derivation on the simple algebra C is a derivation. Proof. The proof of this theorem is similar to the proofs of Theorems 1 and 2. □

Proposition 4. A linear map $: C — C is an automorphism if and only if the matrix of $ in the standard basis (3.1) has the following form:

/ 1 0 0 0 0 0 0 \

0 a2,2 0 0 0 0 0

0 0 a2'2a5,5 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 a5,5 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

V 12,215,5 /

where a2,2 and a5,5 are nonzero elements from F. Here the action of $ corresponds to multiplying the matrix by a column on the right.

Theorem 6. Each local (2-local) automorphism on the simple algebra C is an automorphism.

Proof. The proof of this theorem is similar to the proofs of Theorems 3 and 4. □

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