Описание локальных дифференцирований на йордановых алгебрах размерности пять

Автор: Арзикулов Ф.Н., Нуриддинов О.О.

Журнал: Владикавказский математический журнал @vmj-ru

Статья в выпуске: 4 т.26, 2024 года.

Бесплатный доступ

В данной статье мы исследуем локальные дифференцирования на конечномерных йордановых алгебрах. Теорема Глисона - Кахане - Желазко, являющаяся фундаментальным вкладом в теорию банаховых алгебр, утверждает, что всякий унитальный линейный функционал F на комплексной унитальной банаховой алгебре A такой, что F(a) принадлежит спектру σ(a) для каждого a∈A, является мультипликативным. В современной терминологии это эквивалентно следующему условию: любой унитальный линейный локальный гомоморфизм из унитальной комплексной банаховой алгебры A в C мультипликативен. Напомним, что линейное отображение T из банаховой алгебры A в банахову алгебру B называется локальным гомоморфизмом, если для каждого a в A существует гомоморфизм Φa:A→B, зависящий от a, такой, что T(a)=Φa(a). Аналогичное понятие было введено и изучено для характеризации дифференцирований на операторных алгебрах. А именно, понятие локального дифференцирования было введено в 1990 г. Р. Кэдисоном и Д. Ларсоном, а также независимо А. Суруром. Р. Кадисон дал описание всех непрерывных локальных дифференцирований алгебры фон Неймана со значениями в ее двойственном банаховом бимодуле. Б. Джонсон обобщил результат Р. Кадисона и доказал, что каждое локальное дифференцирование C∗-алгебры со значениями в ее банаховом бимодуле является дифференцированием. Известно, что каждое локальное дифференцирование JB-алгебры является дифференцированием. В частности, каждое локальное дифференцирование на конечномерной полупростой йордановой алгебре является дифференцированием. В настоящей статье мы исследуем дифференцирования и локальные дифференцирования на пятимерных нильпотентных неассоциативных йордановых алгебрах. Описание локальных дифференцирований нильпотентных йордановых алгебр является открытой проблемой. Мы даем описание локальных дифференцирований на пятимерных нильпотентных неассоциативных йордановых алгебрах над алгебраически замкнутым полем характеристики ≠2, 3. Приводится также критерий того, что линейный оператор на йордановой алгебре размерности пять является локальным дифференцированием.

Еще

Йорданова алгебра, дифференцирование, локальное дифференцирование, нильпотентная Йорданова алгебра

Короткий адрес: https://sciup.org/143183730

IDR: 143183730 | DOI: 10.46698/y5752-5645-6737-n

Текст научной статьи Описание локальных дифференцирований на йордановых алгебрах размерности пять

In the present paper we study local derivations of Jordan algebras.

The Gleason–Kahane–Zelazko theorem, which is a fundamental contribution in the theory of Banach algebras, asserts that every unital linear functional F on a complex unital Banach algebra A , such that F(a) belongs to the spectrum a(a) of a for every a G A, is multiplicative (cf. [1, 2]). In modern terminology this is equivalent to the following condition: every unital linear local homomorphism from a unital complex Banach algebra A into C is multiplicative.

We recall that a linear map T from a Banach algebra A into a Banach algebra B is said to be a local homomorphism if, for every a in A , there exists a homomorphism Ф _а : A ^ B , depending on a , such that T(a) = Ф _a ( a ) .

A similar notion was introduced and studied to give a characterization of derivations on operator algebras. Namely, the concept of local derivations was introduced by R. Kadison [3] and D. Larson, A. Sourour [4] independently in 1990. A linear map V from an algebra A into itself is a local derivation if, for each a in A , there is a derivation D a on A with D a ( a ) = V( a ) .

R. Kadison proves that each continuous local derivation from a von Neumann algebra into its dual Banach bemodule is a derivation. B. Jonson [5] extends the above result by proving that every local derivation from a C ^∗ -algebra into its Banach bimodule is a derivation. Based on these results, many authors have studied local derivations on operator algebras.
2. Local Derivations on Five-Dimensional Nilpotent Non-Associative Jordan Algebras

In [6] the first author of the present paper and N. Umrzaqov have made one of the first contributions to the theory of local mappings in the case of Jordan algebras. They proved that a linear local Jordan multiplier of the Jordan algebra of symmetric matrices over an arbitrary field of characteristics =2 is a Jordan multiplier operator.

By Theorem 5.4 in [7] every local derivation on a JB-algebra is a derivation. In particular, every local derivation on a finite dimensional semisimple Jordan algebra is a derivation. In the present paper we investigate derivations and local derivations on five-dimensional nilpotent non-associative Jordan algebras.

In the following, we will work over algebraically closed fields F of characteristic = 2 , 3 and, furthermore, all Jordan algebras are assumed to be of finite dimension over F. In 2016, in [8] a method to construct nilpotent Jordan algebras using central extension of lower dimension algebras was presented. The authors showed that any nilpotent algebra can be obtained via this methods and provided a list of 35 single non-associative Jordan algebras together with 6 families of algebras depending on parameters in F. The list of these algebras is given in Table 3 of [9] (using the notations [8]).

The description of local derivations of nilpotent Jordan algebras is an open problem. In the present paper we give the description of local derivations on five-dimensional nilpotent non-associative Jordan algebras over F. We also give a criterion of a linear operator on Jordan algebras of dimension five to be a local derivation. We develop a technique for constructing a local derivation on an arbitrary low-dimension algebra, which is not a derivation.

Let J be a Jordan algebra of dimension five over an algebraically closed field F of characteristic = 2 with a basis { e i , e 2 , е з , e 4 , e 5 } . Let x be an element in J . Then x = x 1 e 1 + x ₂ e ₂ + х з е з + x ₄ e ₄ + X 5 e 5 , for some elements x 1 , x ₂ , X 3 , x ₄ , X 5 in F. Throughout of the paper let x = ( x i , X 2 , X 3 , X 4 , X 5 ) ^Tr . Conversely, if v = ( x i , X 2 , X 3 , X 4 , X 5 ) ^Tr is a column vector with x i , x 2 , x 3 , x 4 , x 5 in F, then, throughout of the paper, by v we will denote the element x 1 e 1 + x ₂ e ₂ + x 3 e 3 + x ₄ e ₄ + x 5 e 5 , i. e., v = x 1 e 1 + x ₂ e ₂ + x 3 e 3 + x ₄ e ₄ + x 5 e 5 .

Recall that a linear mapping D on a Jordan algebra J , satisfying, for each pair x , y of elements in J , D(xy) = D(x)y + xD ( y ) , is called a derivation, and a linear mapping V : J ^ J is called a local derivation if for every x G J there exists a derivation D : J ^ J , such that V( x ) = D ( x ) .

Let T : J ^ J be a linear operator. Then T(x) = Ax,x G J, for the matrix A = (aij)5j=i of the linear operator T, where x = xiei + x2e2 + x3e3 + x4e4 + x5e5. For example, if J = J5 (see table 1) and T is a derivation on J, then T has the following form

T ( x ) = ax 1 e 1 + (2 ax 2 + вх 4 ) е ₂ + YX 3 e 3

+ ^ - ^e x 3 +

(a + 2 Y^)

e 4 + (5x i + 6x 3 + &X 4 + (2 a + y ) x 5 ) e 5

with respect to the basis { e 1 , e 2 , e 3 , e 4 , e 5 }, for some α, β, γ, δ, θ, σ in F. This form of a derivation can be directly calculated, and we will omit the calculations of the forms of derivations on the Jordan algebras. Throughout this paper, let E ij be a linear mapping of the algebra J with basis { e i ,... ,6 5 } , which maps the basis element e j to e i and the rest basis elements to zero. Then, for example, the vector space of derivations of the algebra J 5 admits a basis of the following form:

^{ E 1 , 1 + 2E 2,2 + E 4 , 4 + ² E 5 , 5 , ² ^E 3 , 3 + E 4 , 4 + ² ^E 5 , 5 , E 2 , 4 ^- E 4 , 3 , E 5,1 E 5 , 3 , E 5 , 4 ^} -

Our principal tool for the description of local derivations on Jordan algebras of dimension five is the form of derivations, depending on a basis of these Jordan algebras. Our main goal in this paper to justify table 1 below and prove the theorems corresponding to this table.

In the following table a necessary and sufficient condition for a linear operator to be a derivation on all nilpotent non-associative Jordan algebras of dimension five is listed. Also, it is indicated that, wether each local derivation of these Jordan algebras is a derivation or not. Forms of the matrices of local derivations, which are not a derivation, are also provided in this paper.

The fourth column of the table indicates whether each local derivation of the corresponding Jordan algebra is a derivation or not, i. e., if yes, then sign «+» is put in the appropriate place of the column, if not, then sign « — » is put in this place. All notations of table 1 are taken from [9].

Table 1

Local Derivations on Five-Dimensional Nilpotent Non-Associative Jordan Algebras

^J	Multiplication table	A basis of the vector space of derivations	Is each local derivation a derivation?
J 5	e l = e 2 ,	{ E i , i + 2 E 2 , 2 + E 4 , 4 + 2 E g , g , 2 Е з , з + E 4 , 4 + 2 E g , g ,	-
	e ² = е 2 е з = e g	E 2 , 4 — E 4 , 3 , E g , 1 , E g , 3 , E g,4 }
J 6	e l = e 2 ,	{ E i , i + 2E- 22 + e 4 , 4 + 2 E g,g , E i , 2 — Е з , 4 ,	-
	e i e 4 = е 2 е з = e g	Е з , 3 + E 4 , 4 + E g,g , E 1 , 4 + 2 E- 2,g , E i , 5 , Е з , 5 , E 4 , g }
J 7	e i e 2 = e 3 ,	{ E i , i + Е з,з + E g,g , E 2,2 + Е з,з + E g,g , E 4,4 + E g , g ,	-
	e 3 e 4 = e g ,	^E 1 , g , E 2 , g , E 4 , g ^}
J 8	e i e 2 = e 3 ,	{ E i , i + Е з , з + E 4,4 + 2 E g , g , E 2,2 + E 33 — E 4,4 ,	-
	e 3 e 4 = e l = e g	^E 1 , g , E 2 , g , E 4 , g ^}
J 9	e i e 2 = e 3 ,	{ E i , i + E 2,2 + E 3 , 3 + 2 E g , g , E i , 5 ,	+
	e 3 e 4 = e g ,	^E 2 , g , ^E 4 , g ^}
	e l = e 2 = e g
J 10	e i e 2 = e 3 , e i e 3 =	{ E 1 , 1 + E 2 , 2 + 2 E 3 , 3 + 2 E 4,4 + 3 E g , g ,	+
	= e 2 e 3 = e 4 = e g	E 4 , 3 — E 1 , 4 — E 2 , 4 , E 1 , g , E 2 , g , E 4 , s }
J 11	e i e 2 = e 3 ,	{ E l , 1 + E 3 , 3 + E 4 , 4 + 2 E g , g ,E 2 , 2 + E 3 , 3 + 2 E 4 , 4	-
	e i e 3 = e 4 = e g ,	+ E g , g , E 2 , 3 + E 3,g , E 4 , 3 — E 1 , 4 , E 1 , g , E 2 , g , E 4 , g }

^J	Multiplication	A basis of the	Is each
	table	vector space of	local
		derivations	derivation a derivation?
J 12	e i e 2 = е з , e i e 3 =	{ E l , l + 2 E 2 , 2 + 3 Е з , з + 2 E 4 , 4 + 4 E g , g , Е 2 , з + E 3 , g ,	-
	2 2 = e 2 = e 4 = e g	E 4 , 3 — E l , 4 , E l , g , E 2 , g , E 4 , g }
J 13	e i e 2 = е з , e i e 4 =	{ E l , l + E 2 , 2 + 2 Е з , з + 2 E 4 , 4 + 3 E g , g ,E l , 3 — E i , 4	-
	= е 1 е з = е 2 е з = e g	+ E 3 , g , E 2 , 4 + E 3 , g , E l , g , E 2 , g , E 4 , g }
J 14	e i e 2 = е з ,	{ E l , l + Е з , з + 2 E 4 , 4 + 2 E g , g , E 2 , 2 + Е з , з + E g , g ,	-
	e i e 3 = e 2 e 4 = e g	E 2 , 3 + E 3 , g , E l,4 + E 3 , g , E l , g , E 2,g , E 4 , g }
J 5	e l = e 2 , e i e 2 = е з ,	{ E l , l + 2 E 2 , 2 + 3 Е з , з + 2 E 4 , 4 + 4 E g , g , — E l , 2 — 2 Е 2 , з	-
	e 4 = ae g , e 2 = e i e 3 ,	+ E 4 , 3 — E 3 , g , — aE l , 2 + aE 2 , 3 + E l , 4 + (1 — a ) E 3 , g ,
	= e 2 e 4 = e g , a G F	E l , 3 + 2 E 2 , g , E l , g , E 4 , g }
J 16	e i e 2 = e 3 = e 4 ,	{ E l , l + 2 E 3 , 3 + E 4 , 4 + 2 E g , g , E 2 , 2 + 2 E 3 , 3	-
	e i e 4 = e g ,	+ E 4 , 4 + E g,g , E' 3,2 — E l , 3 , E l , g , E 2 , g , E 3 , g }
J 17	e i e 2 = e 4 ,e 3 = e 4 ,	{ E l , l + 3 E 2 , 2 + 2 E 3 , 3 + 4 E 4 , 4 + 5 E g , g ,	-
	e i e 4 = e 2 e 3 = e g	E'3,2 — E l , 3 + 3 E 2 , 4 + 2 E 4 , g , E l , g , E 2 , g , E 3 , g }
J 18	e i e 2 = e 4 ,e 3 = e 4	{ E l , l + 2 E 2 , 2 + 2 E 3 , 3 + 3 E 4 , 4 + 4 E g , g ,	+
	e i e 4 = e 2 = e g ,	^E l , g , E 2 , g , E 3 , g ^}
J 19	e i e 2 = e 4 , e 3 = e 4	{ E l , l + 2 E 3 , 3 + E 4 , 4 + 2 E g,g , E 2 , 2 + 2 E 3 , 3 + E 4 , 4	+
	e 3 e 4 = e g ,	+ 2 ^E g , g , E l , g ^,E 2 , g ^,E 3 , g ^}
J 20	e i e 2 = e 4 , e 3 = e 4	{ E l , l + 3 E 2 , 2 + 3 E 3 , 3 + 4 E 4 , 4 + 2 E g , g ,	+
	e 3 e 4 = e l = e g ,	^E l , g , E 2 , g , E 3 , g ^}
J 21	e i e 2 = e 3 = e 4 , e 3 e 4 = e l = e 2 = e ₅	^{ E l , g , ^E 2 , g , E 3 , g ^}	+
J 22	^e l = ^e 2 ,	{ E l , l + 2 E 2 , 2 + 2 E 3 , 3 + 3 E 4 , 4 + 4 E g , g ,	+
J	e i e 2 = e 3 = e 4 , e i e 4 = e 2 = e ₅	E l , 4 + 2 E 2 , g , E l , g , E 3 , g }
J	^e 1 = ^e 3 ,	{ E l , l + E 2 , 2 + 2 E 3 , 3 + 2 E 4 , 4 + 3 E g , g ,	-
	e i e 3 = e 2 e 4 = e ₅ ,	E l , 3 + 2 E 3 , g + eE 4 , g , E 2 , 3 + E 2 , 4 + E 4 , g ,
	e i e 2 = e 4 ,	E l , 4 + E 4 , g , E l , g , E 2 , g }
	e 2 e 3 = ee g , в G F
J Y	e l = e 3 , e i e 2 = e 4 ,	{ E l , l + 2 E 3 , 3 + E 4 , 4 + 2 E g , g , E 2 , 2 + E 4 , 4 + E g , g ,	-
	e 2 e 3 = e g , e i e 4 = Ye s ,	E l , 3 + E 4 , g , E l , 4 + 2yE 3 , 5 , E 2 , 4 + YE 4 , g ,
	Y G F * \{ 1 , - 1 }	^E l , g , E 2 , g ^}
J 2 ⁰ 4	e l = e 3 , e i e 2 = e 4 ,	{ E l , l + 2 E 3 , 3 + E 4 , 4 + 2 E g , g , E 2 , 2 + E 4 , 4 + E g , g ,	-
	e 2 e 3 = e g	E l , 3 + E 4 , g , E l , 4 , E 2 , 4 , E l , g , E 2 , g }
- ¹ ₂ J 24	e l = e 3 , e l e 2 = e 4 ,	{ E l , l + 2 E 3 , 3 + E 4 , 4 + 2 E g , g , E 2 , 2 + E 4 , 4 + E g , g ,	-
	e 2 e 3 = e g	E l , 2 + 2 E 3 , 4 , E l , 3 + E 4 , g , E l , 4 — E 3 , g , E 2 , 4 — 2 E 4 , g ,
	e l e 4 = — ^ e g	^E l , g , E 2 , g ^}
J 25	e l = e 3 , e l e 2 = e 4 ,	{ E l , l + E 2 , 2 + 2 E 3 , 3 + 2 E 4 , 4 + 3 E g , g ,	-
	e l e 3 = e l e 4 = e g ,	E l , 2 + 2 E 3 , 4 , E l , 3 + 2 E 3 , g — 2 E 4 , g ,
J 6	e 2 e 3 = — 2 e g	E l , 4 + 2 E 3 , g , E 2 , 4 + E 4 , g , E l , g , E 2 , g }
J 6	e ² = e 3 , e 2 = e 4 ,	{ E l , l + E 2 , 2 + 2 E 3 , 3 + 2 E 4 , 4 + 3 E g , g ,	-
	e 2 e 4 = Se g ,	E l , 3 + 2 E 3 , g , — SE- 2,3 + E i , 4 + 2 E 3 , g ,
	e l e 3 = e l e 4 = e g ,	— E 2 , 3 + E 2 , 4 + 2 SE 4,g , E l , g , E 2 , g }
	S G F *
_J ₂0₆	e ² = e 3 , e 2 = e 4 ,	{ E i , i + E 2 , 2 + 2 E 3 , 3 + 2 E 4 , 4 + 3 E g , g ,	-
	e l e 3 = e l e 4 = e g ,	E l , 3 + 2 E 3 , g , E 2 , 3 — E 2 , 4 , E i , 4
		⁺ ^E 3 , g , ^E l , g , ^E 2 , g ^}
^£,^ ^J 27	e 2 = e 3 , e 2 = e 4 ,	{ E l , l + E 2 , 2 + 2 E 3 , 3 + 2 E 4 , 4 + 3 E g , g ,	-
	e 2 e 3 = e l e 4 = e g ,	E l , 3 — E 2 , 4 + 2 pE 3 , g , E 2 , 3 — pE 2 , 4
	e 2 e 4 = ee g , e l e 3 = pe g ,	+2(1 — ep ) E 4 , g , E l , 4 — eE 2 , 4 + 2 E 3 , g
	e, p G F , ep =1	— 2 e ² E 4 , g ,E l , g ,E 2 , g }
a ^£,^£ ¹ J ₂₇	e ² = e 3 , e 2 = e 4 ,	{ E i , i + E 2 , 2 + 2 E 3 , 3 + 2 E 4 , 4 + 3 E g , g ,	-
	e 2 e 3 = e l e 4 = e g ,	E l , 3 — E 2 , 4 + 2 e ^- ^l E 3 , g — 2 eE 4 , g ,
	e 2 e 4 = ee g , e l e 3 = e ^- ^l e g ,	E 2 , 3 — e ^l E 2 , 4 , E l , 4 — eE 2 , 4 + E 3 , g
	e G F * \{ — 1 }	+ e ² E 4 , g , E l , g , E 2 , g }

^J	Multiplication table	A basis of the vector space of derivations	Is each local derivation a derivation?
^— ¹ , ^— ¹ J 27	e l = е з , e 2 = e 4 ,	{ E i , i + E 2 , 2 + 2 Е з , з + 2 E 4 , 4 + 3 E g , g ,	-
	е 2 е з = e i e 4 = e g ,	E 1 , 3 — E 2 , 4 — 2 E 3 , g + 2 E 4 , g , E 2 , 3 + E 2 , 4 ,
	e 2 e 4 = e i e 3 = — e s ,	E 1 , 4 + E 2 , 4 — E 3 , g — E 4 , g , E 1 , g , E 2 , g }
J 28	e 1 = e 2 , e 3 = e s ,	{ E 1 , 1 + 2 E 2 , 2 + 2 E 4 , 4 , Е з , 2 + 2 E g , 4 ,	-
	е 2 е з = e 4	Е з , з + E 4 , 4 + 2 E g , g , E 1 , 4 , Е з , 4 , E 1 , g , E 3 , g }
J 29	e 2 = e 2 , e 2 e 3 = e 4 ,	{ E 1 , 1 + 2 E 2 , 2 + 2 E 4 , 4 + E g , g , E 1 , 2 + E g , 4 ,	-
	e i e 3 = e g	Е з , з + E 4 , 4 + E g , g , E 1 , 4 , Е з , 4 , E 1 , g , E 3 , g }
J 30	e ² = e 2 ,	{ E 1 , 1 + 2 E 2 , 2 + Е з , з + 3 E 4 , 4 + 3 E g , g ,	-
	e 2 e 3 = e 4 ,	E 1 , 2 + 2 Е з , 2 + E g , 4 ,
	e i e 3 = e 3 = e g ,	E 1 , 4 , E 3 , 4 , E 1 , g , E 3 , g ^}
J 31	e 2 = e 2 , e 2 e 3 = e 4 ,	{ E 1 , 1 + 2 E 2 , 2 + 4 E 4 , 4 + 3 E g , g , E 1 , 3 + E g , 4 ,	-
	e i e 2 = e g	Е з , 3 + E 4 , 4 , E 1 , 4 , Е з , 4 , E 1 , g , E 3 , g }
J 32	e ² = e 2 ,	{ E 1 , 1 + 2 E 2 , 2 + 2 Е з , з + 4 E 4 , 4 + 3 E g , g ,	-
	e 2 e 3 = e 4 ,	E 1 , 2 + E 1 , 3 + E g , 4 + 4 E 2 , g ,
	e i e 2 = e i e 3 = e g	E 1 , 4 , E 3 , 4 , ^E 1 , g , E 3 , g ^}
J 33	e 2 = e 2 , e 2 e 3 = e 4 ,	{ E 1 , 1 + 2 E 2 , 2 + 2 Е з , з + 72 E 4 , 4 + 3 E g , g ,	+
	e i e 2 = e 3 = e g ,	E 1 , 4 , E 3 , 4 , ^E 1 , g , E 3 , g ^}
J 34	e 2 = e g , e i e 2 = e 3 ,	{ E 1 , 1 + Е з , з + 2 E 4 , 4 + 2 E g , g , E 2 , 2 + Е з , з + E 4 , 4 ,	-
	e i e 3 = e 4	E 1 , 3 + 2 E g , 4 , E 2 , 3 + Е з , 4 , E 1 , 4 , E 2 , 4 , E 1 , g , E 2 , g }
J 35	e 2 = e g , e i e 2 = e 3 ,	{ E 1 , 1 + Е з , 3 + 2 E 4 , 4 , E 2 , 2 + Е з , з + E 4 , 4	-
	e i e 3 = e 4	+2 E g , g , E 2 , 3 + Е з , 4 , E 1 , 4 , E 2 , 4 , E 1 , g , E 2 , g }
J 36	e i e 2 = e 3 , e i e 3 = e 4 ,	{ E 1 , 1 + Е з , з + 2 E 4 , 4 + E g , g , E 2 , 2 + Е з , з	-
	e 2 e 3 = e g	+ E 4 , 4 + 2 E g , g ,E 1 , 4 , E 2 , 4 , E 1 , g ,E 2 , g }
J 37	e i = e 2 = e g ,	{ E 1 , 1 + E 2 , 2 + 2 Е з , з + 3 E 4 , 4 + 2 E g , g ,	-
	e i e 2 = e 3 , e i e 3 = e 4	E 2 , 3 + Е з , 4 , E 1 , 4 , E 2 , 4 , E 1 , g , E 2 , g }
J 38	e 2 = e 2 e 3 = e g ,	{ E 1 , 1 + 2 E 2 , 2 + 2 Е з , 3 + 2 E 4 , 4 + 2 E gg ,	+
	e i e 2 = e 3 , e i e 3 = e 4	E 1 , 4 , E 2 , 4 , ^E 1 , g , E 2 , g ^}
J 39	e 2 = e g , e i e 2 = e 3 ,	{ E 1 , 1 + 2 E 2 , 2 + 3 Е з , з + 4 E 4 , 4 + 2 E g , g , Е 1 , з	-
	e 2 = e i e 3 = e 4	+2 E g , 4 , E 2 , 3 + Е з , 4 , E 1 , 4 , E 2 , 4 , E 1 , g , E 2 , g }
J 40	e 2 = e 2 e 3 = e g ,	^{ E 1 , 4 , E 2 , 4 , E 1 , g , E 2 , g ^}	+
	e i e 2 = e 3 ,
	e 2 = e i e 3 = e 4
J 1	e 2 = e g , e i e 2 = e 3 ,	{ E 1 , 1 + E 2 , 2 + 2 Е з , з + 3 E 4 , 4 + 2 E g , g ,	-
	e 2 = Ae g ,	2 E 1 , 3 + 2 ^AE 2 , 3 + 2 (1 + ^A ⁾ ^E 3 , 4 ,
	e i e 3 = e 2 e 3 = e 4 ,	E 1 , 4 , E 2 , 4 , ^E 1 , g , E 2 , g ^}
	A € F

Remark. Local derivations of Jordan algebras of dimension less than or equal to four are described in [10].

3. Description of Local Derivations on Some Jordan Algebras of Dimension Five

Теорема 1. Each local derivation on the Jordan algebra J 10 is a derivation.

<1 Let V be an arbitrary local derivation on J io . Then V( x ) = ^ 5=i (^ 5=i a i,j X j )e i , x G J io for the matrix A = (a ij ) 5 j=i of the local derivation V , where x = x i e i + X 2 e 2 + х з е з + X 4 e 4 + x s e s . By the definition, for any element x G J io , there exists a derivation D _x such that V( x ) = D x x = Ax . By Table 1

D x ( x ) = a x x _i e _i + a x x ₂ e ₂ + (2 a ^x x ₃ + e x x ₄ ) e ₃

- β ^x x 1

e x x 2 + 2 a x x 4

) e 4 + ( ^ x

x _i + Y x x ₂ + 0^xx ₄ + 3 a ^x x ₅ ) e ₅ ,

(3 . 1)

where α ^x , β ^x , γ ^x , δ ^x and θ ^x in F, depending on x .

If we take x — e i , then V( e i ) — D _e 1 ( e i ) — a e ¹ e i — в ^е 1 e 4 + i ^e 1 e 5 . At the same time, V( e i ) — a i , i e i + a 2 , i e 2 + a i , i e i + a 4 , i e 4 + a ₅ , i e ₅ . Hence, a i , i — a e ¹ , a 2 , i — 0 , a i , i — 0 ,

^a 4 , i —

в^е1 , а 5Д — i ^e ¹ . Similarly we get a _i , ₂ — 0 , a ₂ , ₂ — a e ² , a i , ₂ — 0 , a ₄ , ₂ — —,

e e ² , a 52 — Y e ²

if we take x — e ₂ . Also, we similarly get a i , | — 0 , a ₂ , i — 0 , a i , i — 2 a ^e ^з , a ₄ _i i — 0 , a 5 , i — 0 , a i , 4 — 0 , a 2 , 4 — 0 , a i , 4 — — в ^е 4 , a 4 , 4 — ^з a e ⁴ , a 5 , 4 — ^ e ⁴ , a i , 5 — 0 , a 2 , 5 — 0 , a i , 5 — 0 , a 4 , 5 — 0 , a 5 , 5 — 3 a e ⁵ .

Now, since V is linear we have V( e i + e 2 ) — V( e i ) + V( e 2 ) . From this it follows that

a e ¹ ⁺ e ² e i + a e ¹ ^+e 2 e 2 + (-в e ¹ ⁺ e ² - — a e ¹ e 1 + a e ² e ₂ + (— e e ¹

в ^е ¹ ^+е 2 ) e 4 + ( A ^e ¹ ⁺ e ² + Y e ¹ ⁺ e ² ) e 5

— в e ² ) e 4 + ( i ^e ¹ + Y e ² ) e 5 .

Hence, a e ¹ ^+e 2 — a e ¹ , a e ¹ ^+e 2 — a e ² and a e ¹ — a e ²

Similarly, from the equality V(ei + ei) — V(ei) + V(ei) we get ae1+e3ei + 2ae1+eзei — вe1+e3e4 + ie1+e3 e5 — ae1 ei + 2aeзei — вe1 e4 + ie1 e5. Hence, ae1+e3 — ae1, ae1+e3 — ae3 and ae1 — ae3

Similarly, from the equality V( e i + e 5 ) — V( e i ) + V( e 5 ) we get 2 а ^e ^з ^+e 5 e i + 3 a ^e 3 ^+e 5 e 5 — | a ^e 3 e i + 3 a ^e 5 e 5 . Hence, a ^e 3 ^+e 5 — a ^e 3 , a ^e 3 ^+e 5 — a ^e 5 and a ^e 3 — a ^e 5 .

Now, since V is linear we have V( e i + e 4 ) — V( e i ) + V( e 4 ) . From this it follows that

V( e i + e 4 ) — a e ¹ ^+e 4 e i + в ^е ¹ ^+е 4 е з + (-в e ¹ ⁺ e ⁴ + | a ^e ¹ ^+e ⁴^ + ( ^ e i + e ⁴ ₊ 0 e ¹ ^+e 4 ) e 5 — a e ¹ e i + в ^е 4 е з + (—в^ + | a e ⁴ e 4^

e 4

^e 5 ^.

Hence,

a e ¹ ^+e 4 — a e ¹ , e e ¹ ⁺ e ⁴ — в ^е 4 ,

в ^е ¹ ^+е 4 + 2 a e ¹ ^+e 4 —

- в ^е ¹ + | a e ⁴ ,

i. e.,

в ^е 1 + 2 a e ⁴

в ^е 4 + 2 a e ¹

(3 . 2)

Similarly, в ^е 1 — e e ²

by V( e ₂ + e ₄ ) — V( e ₂ ) + V( e ₄ ) , we have -в e ² + 2 a e ⁴ by (3.2). Thus, we have

— в ^е 4 + 2 a e ¹ . Hence,

V(x) — Ax — ae1 xiei + ae1 x2e2 + (|ae1 xi + ве4 x4)ei + ^—ве1 xi — ве1 x2 + |ae4x4^ e4 + (^e1 xi + Ye2x2 + ^e4x4 + 3ae1 xs)e5, x G Jio, and V(x) — Ax, x G J10. Now, since V is linear we have V(e1 + e3 + e4) — V(e1) + V(e3) + V(e4). From this it follows that ae1+e3+e4 — ae1, 2ae1+е3+е4 + в e1+e3+e4 — 2ae1 + в e4,

в ^е ¹ ^+е ^з ^+е 4 ₊ | a e ¹ ^+e 3 ^+e ⁴ — — в ^е 1 + | a e ⁴ .

— в e ¹ + 2 a e ⁴ , i.e., a e ⁴ — a e ¹ + | ( в ^е 1 — в ^е 4 ) . So, we

Hence, в e ¹ ^+e 3 ⁺ e ⁴ — в e ⁴ , -в e ⁴ + 2 a e ¹ have

V( x ) — a ^e ¹ x _i e _i + a ^e ¹ x ₂ e ₂ + (| a ^e ¹ x ₃ + в ^e ⁴ x ₄ ) e _i + ^ — в ^e ¹ x _i

β ^e ¹ x 2

+ ^| a e ¹ + в ^е 1 — в^^ X 4^ e 4 + ( i ^e 1 x i + Y e ² X 2 + ^ e ⁴ X 4 + 3 a ^e 1 X 5 ) e 5 , x G J io .

Since the local derivation V was chosen arbitrarily, we have it has the following form

V( x ) = ax 1 e 1 + ax ₂ e ₂ + (2 ах з + вx ₄ ) e з + ^- вх1 — вх 2 + ^ а + в

— Y^ X 4^ e 4

+(5xi + YX2 + ^Х4 + 3ах5)e5, x G J10, where α, β, γ, δ, θ, ν belong to F.

Let V( x ) = Ax , x G J 10 . We suppose that, for each a G J io , there exists a derivation of the form (3.1) with a matrix A _a , such that

V( a ) = Aa = A^a.

We find the form of the matrix A, for which the map, defined by the matrix A is a local derivation. For this propose we consider the following system of linear equations a1aa = a1a;

a ₂ a _a = a ₂ a ;

< 2а з а а + а 4 в а = 2a ₃ a + a 4 V ; (3 . 3)

—а 1 в а — а 2 в а + a 43 а а = —а 1 в — а 2 в + a 4 ( 2 а + в — v);

^aiba + a2Ya + a4^ + а53аа = ai^ + a2Y + "-А + 3a5a with respect to the variables аа, ва, Yа, ^а, ^а, for each a G Jio. Here, а, в, Y, ^, $, v are elements from F. Note that, if the left part of any equation of the system (3.3) is equal to zero, then the right part of this equation is also equal to zero.

Now, suppose that a3 =0, a4 = 0, ai =0 and a2 = 0. Then we have аа = а;

₍ ва = v ;

—a i v = —a 1 в + а 4 (в — v );

a i d а + a 2 Y a + a4^ = a i ^ + a 2 Y + "-A-

This system has a solution for any a 4 = 0 , a 3 = 0 , a i = 0 , a 2 = 0 and a 5 if and only if в = v .

Thus, we have

V( x ) = аx ₁ e ₁ + ах ₂ е ₂ + (2 ах з +

вх 4 ) е з + ^— вx 1 — вх 2 + I ах 4^

e 4

+( 5x i + YX 2 + #X 4 + 3 ах 5 ) e 5 , x G J io .

But this coincides with the form (3.1) of a derivation on J 10 . This completes the proof. ⊲

Similarly we can prove the following theorem.

Теорема 2. Each local derivation on the algebras J 9 , J 18 , J 19 , J 20 , J 21 , J 22 , J 33 , J 38 , J 40 is a derivation.

4. Criterions of Linear Operators on Jordan Algebras of Dimension Five to be a Local Derivation

Theorem 3. A linear operator V on the Jordan algebra J 5 is a local derivation if and only if this linear operator belongs to the subspace with the following basis

LocDer⁽ ^J 5 ) = |E i _] i + 2E 2,2 + ² E 4 , 4 , E 2 , 4

—

E 4,3 , E 3,3 + 2 E 4 , 4 , E 5 , i , E 5 , 3 , E 5 , 4 , E 5 , 5 ^ • (⁴ - l)

<1 Let J 5 be the Jordan algebra over the field F taken from Table 1 with the basis { e i , e 2 , e 3 , e 4 , e 5 } .

First we prove that the matrix of each local derivation on J 5 has the matrix form (4.1). Let V be an arbitrary local derivation on J 5 . Then V( x ) = S i ₌i (^ 5₌i a i,j X j ) e i , x € J 5 , for the matrix A = (a ij ) 5 j=i of the local derivation V , where x = x i e i + X 2 e 2 + х з е з + X 4 e 4 + X 5 e 5 . By the definition, for any element x € J 5 , there exists a derivation D _x , such that V( x ) = D x x = Ax . By Table 1

D x ( x ) = Axx = a x x i e i + (2 a x x 2 + в ^x X 4 ) e 2 + Y x x 3 e 3 + ( - в ^x x з

( 1 \ \

+ ( 2ax + ^ yxj x4 ) e4 + (^xxi + ^xx3 + axx4 + (2ax + y x)x5)e5, x € J5, where ax, вх, Yx, ^x, ^x, $x and £x in F, depending on x.

Using equalities V(ei) = Dei(ei) = Aeiei, i = 1, 2, 3,4, 5 we get ai,i = ae1, a2,i a3,i = 0, a4,i = 0, a5,i = ^e1, ai,2 = 0, a2,2 = 2ae2, a3,2 = 0, a4,2 = 0, a5,2 = 0, ai,3

a2,3 = 0, аз,з = ye3, a4,3 = -ве3, a5,3 = ^e3, ai,4 = 0, a2,4 = ве4, аз,4 = 0, a4,4 = 2ae4 + 2ye4, a5,4 = ae4, ai,5 = 0, a2,5 = 0, аз,5 = 0, a4,5 = 0, a5,5 = 2ae5 + ye5.

Now, since V is linear we have V( e i + e 2 ) = V( e i ) + V( e 2 ) . From this it follows that

V( e i + e 2 ) = a e ¹ ^+e 2 e i + 2 a e ¹ ^+e 2 e 2 + 5 e ¹ ^+e 2 e 5 = a e ¹ e i + 2 a e ² e 2 + ^ ' e ₅ .

Hence, a ^e 1 ^+e 2 = a ^e 1 , a ^e 1 ^+e 2 = a ^e 2 , i e ¹ ^+e 2 = 5 ^e 1 , and a ^e 1 = a ^e 2 . Thus, we have

V( x ) = Ax = a ^e 1 x _i e _i + (2 a ^e 1 x ₂ + в ^e ⁴ x ₄ ) e ₂ + Y e ³ x ₃ e ₃

+( - в ^e ^з x 3 + ^2 a e ⁴ + 2 y e ⁴^ x 4^ e 4 + ^^e 1 x i + ^ ^e 3 x 3 + a e ⁴ x 4 + (2 a ^e 5 + Y e ⁵ ) x 5^ e 5 , x € J 5 .

Now, since V is linear we have V( e i + e 3 + e 4 ) = V( e i ) + V( e 3 ) + V( e 4 ) . From this it follows that

V( e i + e 3 + e 4 ) = a ^e 1 ^+e 3 ^+e ⁴ e i + в e ¹ ^+e 3 ^+e ⁴ e 2 + Y ^e 1 ^+e 3 ^+e ⁴ e 3

+ - в e i + e 3 + e 4 । 2_a e i + e 3 + e 4 | 1 Y e i + e 3 + e 4^ e 4 । ⁽ j e i + e 3 + e 4 + ^ e i + e 3 + e 4 + ^ e i + e 3 + e 4)^ = a ^e 1 e i + в e ⁴ e 2 + Y e ³ e 3 + ( - в ^e ^з + 2 a e ⁴ + 1 y e ⁴) e 4 + ( 5 ^e 1 + ^ ^e 3 + a e ⁴ ) e 5 .

Hence, -в ^e 3 + 2 a e ⁴ + 2 Y e ⁴

So, we have

- e e ⁴ + 2 a ^e 1 + 2 Y ^e 3 , i. e., 2 a e ⁴ + 2 Y e ⁴ = e e ³

в ^е ⁴ +2 a ^e 1 + 2 y ^e 3 .

V( x ) = Ax = a e ¹ x _i e _i + (2 a ^e 1 x ₂ + в ^e ⁴ x ₄ ) e ₂ + Y e ³ x 3 e 3

+ ( - в ^e ^з x 3 + ^^e 3 - в ^e ⁴ + 2 a ^e ¹ + 2 Y e ³) x 4^ e 4 + ( 5 ^e 1 x i + 0 ^e 3 x 3 + a e ⁴ x 4 + (2 a ^e 5 + y e ⁵ ) x s ) e 5 > x € J 5 .

Since the local derivation ∇ was chosen arbitrarily, we have it has the following form

V( x ) = Ax = ax _i e _i + (2 ax 2 + вx ₄ ) e ₂ + Yx ₃ e ₃

+ ( - vx 3 + (v - в + 2 a + 2 Y^x 4^ e 4 + ( 5x i + ^x 3 + ax 4 + Zx 5 ) e 5 , x € J 5 , where a , в , Y , ^ $ , a , v , Z belong to F.

Let V( x ) = Ax , x G J 5 . We suppose that for each a G J 5 there exists a derivation of the form (4.2) with a matrix A _a , such that

V( a ) = Aa = A x a.

We find the form of the matrix A, for which the map, defined by the matrix A is a local derivation. For this propose we consider the following system of linear equations a1 aa = a1a;

2 a 2 a a + a 4 e a = 2 a 2 a + a ₄ в ;

< аз Y a = a 3 Y ;

^{— а} 3 ^в а + ^a 4 ⁽² ^a a + 2 Y a ) = ^—a 3 ^v + ^a 4 ⁽ ^{v —} ^в + ² ^a + 2 Y );

^ai^a + аз^а + a4^a + a5(2aa + Ya) = ai^ + аз^ + a4a + a5Z with respect to the variables aa, вa, Ya, ^a, 0a, aa, for each a G J5. Here, a, в, Y, ^ 0, a, ν , µ, ζ are elements from F. Note that, if the left part of any equation of the last system is equal to zero, then the right part of this equation is also equal to zero.

Now, suppose that аз =0, a4 = 0. Then, if ai = 0, we have aa = a;

e a = ² 0 4 a + в - ² 0 4 a ;

⁽ Y a = Y ; ⁽⁴ - ³⁾

- ^e a = ^-v + a 3 ( ^{v - в} + ² ^a + 2 ^Y ) ^- a 3 (² ^a + 1 ^Y ) ^;

^a i ^ a + a 3 0 a + a 4 ^ a + a 5 (2 a a + Y a ) = a i 5 + a 3 0 + a 4 ^ + a 5 Z-

From (4.3) it follows that, if в = v then, in this case, the system of linear equations (4.3) has a solution. Thus we have aiaa = aia;

2 a 2 a _a + a ₄ в _a = 2 a 2 a + a ₄ в ;

⁽ a 3 Y a = a 3 Y ; ⁽⁴ - ⁴⁾

^—a 3 ^в a + ^a 4 ⁽² ^a a + 2 Y a ) = ^—a 3 ^в + ^a 4 ⁽² ^a + 2 Y );

^a i ^ a + a 3 0 a + a 4 ^ a + a 5 (2 a a + Y a ) = a i 5 + a 3 0 + a 4 ^ + a 5 Z.

Now we prove that, for each element a G J 5 , the system of linear equations (4.4) has a solution.

If a 3 = 0 , a 4 = 0 and a i = 0 , then, if a 2 = 0 , we have

Д = в;

₍ Y a = Y ;

a _a = a ;

a 3 0 a + a 4 CT a = a 3 0 + a 4 CT + a 5 Z - + a 5 (2 a + y )

and, in this case, the system of linear equations (4.4) has a solution. Else, if a 2 = 0 , then we have

2 a 2 a _a + a ₄ в _a = 2 a 2 a + a ₄ в ;

( Ya = y;

—a з в a + a 4 2 a a = —a з в + a 4 2 a ;

a i ^ a + a 3 0 a + a 4 CT a + a 5 (2 a a + Y ) = a i ^ + a 3 0 + a 4 ^ + a 5 Z.

Hence, aa = a;

x Ya = y;

⁽ в а = в ;

a i ^ a + а з ^ а + a 4 ^ a = a i S + а з в + a 4 ^ + a 5 Z - a 5 (2 a + y ) •

The last equation of this system has at least two variables with a nonzero coefficient. So, in this case, the system of linear equations (4.4) also has a solution.

Now, it is rests to consider the case аз = 0, a4 = 0. In this case we have a1aa = a1a;

a ₂ a a = a ₂ a;

< Y a = Y ; в а = в;

^a i ^ a + а з ^ а + a 5 2 a a = a i 8 + а з ^ + a 4 ^ + a 5 Z - a 5 Y-

Similarly, for any a 1 , a 2 and a 5 from F this system has a solution.

Thus, for each element a G J 5 , the system of linear equations (4.4) has a solution. Hence, the linear operator of the form (4.1) is a local derivation. Since the local derivation V was chosen arbitrarily and V is defined by derivations of the form (4.2) we have that each local derivation on the Jordan algebra J 5 has the form (4.1).

Now, we prove that, if a linear map V : J 5 ^ J 5 has the form (4.1), then V is a local derivation on J 5 . For this propose, it is sufficient to prove that for each a G J 5 there exists a derivation of the form (4.2), such that

V( a ) = Aa = A X a.

But, this is equivalent to proving the existence of a solution of the system of linear equations (4.4), what was done in the previous part of the present proof. This ends the proof. ⊲

Similarly we can prove the following theorem.

- ¹

Теорема 4. A linear operator on the Jordan algebras J 7 , J 8 , J 13 , J 14 , J ₂ ⁰ ₄ , J ₂₄ ² , J 25 , J ₂ ⁰ ₆ , J ₂ ^- ₇ ^1,-1 , J 35 , J 36 , J 37 , is a local derivation if and only if this linear operator belongs to the subspace with the following basis, respectively:

• LocDer( J 7 ) = {E i,i + E 33 E 2,2 + E 3 , 3 ,E 4 , 4 ,E 5 , 1 ,E 5 , 2 , E 54 E ₅,₅ } ,
• LocDer( J 8 ) = { E_u + E 3 , 3 + E 4 , 4 ,E 2 , 2 + E 3 , 3 - E ₄,₄ ,E ₅,1 ,E ₅,₂ ,E ₅,₄ ,E ₅,₅ } ,
• LocDer( J 13 ) = {E i,i + E 2,2 + 2E 33 + 2E 44 E ₃,1 - E^E^E s^ E^ , E 53 E 54 E ?^} ,
• ^LocDer( ^J 14 ) = ^{ E 1 , 1 , E 2 , 2 + E 3 , 3 ^, E 3 , 2 + E 5 , 1 ^, E 3 , 3 + ²E 4 , 4 ^, E 5 , 2 ^, E 5 , 3 ^, E 5 , 5 ^} ,
• LocDer( j 24 ) = { E 1 , 1 ,E 2 , 2 ,E 3 , 1 ,E 3 , 3 ,E 4 , 1 ,E 4 , 2 ,E 4 , 4 ,E 5 , 1 ,E 5 , 2 ,E 5 , 4 ,E 5 , 5 } ,
• LocDer( J 2 - ² ) = { E 1 , 1 , E 2 , 1 , E ₂,₂ ,E ₃,1 ,E ₃,₃ ,E ₄,1 ,E ₄,₂ , E 4 , 3 , E 4 , 4 , E 5 , 1 , E 5 , 2 ,E 5 , 3 ,E 5 , 4 ,E 5 , 5 } ,
• LocDer( J 25 ) = { E 1 , 1 , E 2 , 1 ,E 2 , 2 + 2E 3 , 3 ,E 3 , 1 ,E 4 , 1 ,E 4 , 2 , E 43 E 44 E ₅,1 ,E ₅,2 ,E ₅,3 ,E ₅,4 ,E ₅,₅ } ,
• LocDer( J 26 ) = { E 1 , 1 + E 2 , 2 ,E 3 , 1 ,E 3 , 2 - E 4 , 2 , 2E 33 + 2 E₄4 + 3E 5 , 5 ,E 4 , 1 ,E 5 , 1 ,E 5 , 2 , £ 5 , 3 } ,
• LocDer( j 27¹ ^, ^- 1 ) = { E 1 , 1 + E 22 E^E^ , 2E 33 + 2E₄ 4 + 3E ₅,₅ , E^E^ , E 51 E 5,₂ , -E 5,3 + E 5 , 4 ^} ,
• LocDe^r( J 35 ) — ^{ ^E 1 , 1 , E 2 , 2 , E 3 , 2 , E 3 , 3 , E 4 , 1 , E 4 , 2 , E 4 , 3 , E 4 , 4 , E 5 , 1 , E 5 , 2 , E 5 , 5 ^} ,
• ^LocDer( J 36 ) ^— ^{ E 1 , 1 + E 3 , 3 , E 2 , 2 + E 3 , 3 , E 4 , 1 , E 4 , 2 , E 4 , 4 , E 5 , 1 , E 5 , 2 , E 5 , 5 ^} ,
• LocDer( J 37 ) — { E 1 , 1 + E ₂,₂ + 2E 33 + 3E 4 , 4 + 2E 55 , E 32 , E 4 , 1 , E 42 , E 43 , E 51 ^5,2} .

Theorem 5. A linear operator on the Jordan algebra J 11 is a local derivation if and only if this linear operator belongs to the subspace with the following basis:

^{LocDer( J}n ) ^— { ^E i , i ^, E 2 , 2 , E 3 , 2 , ^E 3 , 3 , E 34 E 4,1 ,E 44 E 5 , 1 ^,E 5,2 , E 5 , 3 , E 5 , 4 , E 5 , 5 ^} .

<1 Let V be an arbitrary local derivation on J 11 . Then V( x ) — ^ ⁵ ₌1 (^ 5₌1 a i,j x j )e i , x G J 11 for the matrix A — ( a ij ) ⁵ j ₌1 of the local derivation V , where x — x i e i + x 2 e 2 + x 3 e 3 + x 4 e 4 + x 5 e 5 . By the definition, for any element x G J 11 , there exists a derivation D _x , such that V( x ) — D x x — A a . By Table 1

D x ( x ) — A x x — a x x 1 e 1 + e x x 2 e 2 + ( Y x x 2 + ( a x + в ^x ) x з + ^ ^x x 4 ) e 3

+ ( - E'x. + (ax + |ex^x4^64 + (0xx1 + Vхx2 + Yxx3 + Pxx4 + (2ax + ex)x5)e5, where αx , βx , γx, δx , θx, σx and ρ in F, depending on x.

If we take x — e 1 , then, by V( x ) — Ax — D x x — A x x , we get a 1 , 1 — a e ¹ , 0 2 , 1 — 0 , 0 3 , 1 — 0 , 0 4 , 1 — — 5 ^e 1 , 0 5 , 1 — 0 ^e 1 . Similarly we compute the other components of A using x — e ₂ , е з , e ₄ , e 5 and have

V( x ) — Ax — a e ¹ x ₁ e ₁ + в e ² x ₂ e ₂ + ( y e ² x ₂ + ( a e ³ + e e ³ ) x ₃ + 5 ^e 4 x ₄ ) e ₃

+ ^ — 5 ^e 1 x 1 + ^a ^e 4 + ^ e e ⁴^ x 4^ e 4 + ( 0 ^e 1 x 1 + v e ² x 2 + y e ³ x 3 + p ^e 4 x 4 + (2 a ^e 5 + в ^e ⁵ ) x 5 ) e 5 .

Since the local derivation ∇ was chosen arbitrarily, we have it has the following form

V( x ) — Ax — ax ₁ e ₁ + вх ₂ е ₂ + ( yx ₂ + Ax ₃ + 5x ₄ ) e ₃ + ( ^x ₁ + vx ₄ ) e ₄ + (0x 1 + vx ₂ + £x ₃ + px ₄ + wx s ) e 5 -

(4 . 6)

where α , β , γ , δ , θ , σ , ξ , ρ , ω , ν , µ , λ belong to F.

Let V( x ) — Ax , x G J 11 . We suppose that for each a G J 11 there exists a derivation of the form (4.5) with a matrix A _a , such that

V( a ) — Aa — A a a.

We find the form of the matrix A, for which the map, defined by the matrix A is a local derivation. For this propose we consider the following system of linear equations a1aa — a1a;

а 2 в а — а 2 в ;

⁽ a 2 Y a + а з ( « _а + в а ) + O 4 ^ a — O 2 Y + 0 3 ^ + « 4 5 ⁽⁴ - ⁷⁾

— a 1 5 a + a 4 ( a a + 2 в а ) — « 1 ^ + « 4 V ;

_ia 1 0 _fl + a 2 V a + a 3 Y a + « 4 P a + O 5 (2 « a + в а ) — « 1 # + « 2 Y + « з € + « 4 Р + « 5 ^

with respect to the variables α _a , β _a , γ _a , δ _a , θ _a , σ _a , ρ _a for each a ∈ J 11 . Note that, if the left part of any equation of the system (4.7) is equal to zero, then the right part of this equation is also equal to zero.

Now, if a i = 0 , a 2 = 0 , then

a _a = a;
в а = в;
* Ya = V o - 0 2 A o ;

⁶ a — ^A o ^;

a l ^ a + a 2 ^ a + a 4 P a = ^vo - ^a 3 (Vo - a4 ^A o ) , a 2

where

Ao = -^ - a4V + a4 (a + 1 в^ , Цо = Y + a3A + a46 - — (a + в), a1 ai \ 2 J a2 a2 a2

V o = a i # + a 2 Y + a< + a 4 P + a 5 W - a 5 (2 a + в ) -

This system has a solution for any a i = 0 , a 2 = 0 , а з , a 4 and a 5 from F and gives a solution of the system (4.7).

If ai =0, a2 = 0, then the system (4.7) has the following form aa = a;

а з в _а + a 4 6 a = а з А + a ₄ 6 - а з а ;

—a i 6 a + 2 a 4 в а — aiv + a 4 V — a 4 a ;

a i ^ a + a 3 Y a + a 4 P a + а 5 в а = a i # + а з € + a 4 P + a 5 ^ - 2 a 5 a.

We have aa = a;

(a 3 + a 42 01) в a = Ц - a 4 A o ;

6 a = A o + 2 a i в a ;

^ai^a + a3Ya + a4Pa + a5 вa = Vo, where

a 4 a 4

A o — —д — — v + —a, ^o — a 3 A + a 4 O — a 3 a, a 1 a 1

v _o = a _i # + a 3 ^ + a ₄ p + a 5 W — 2 a 5 a.

Since a i =0 this system has a solution for any a 3 , a 4 and a 5 from F and also gives a solution of the system (4.7).

If ai =0, a2 = 0, then the system (4.7) has the following form вa = в;

a 2 Y a + a 3 a a + a 4 6 a = a 2 Y + a 3 A + a 4 6 - a з в ;

a ₄ a _a = a ₄ v — 2 a ₄ в ;

Q2° a + a 3 Y a + a 4 P a + 2 a 5 a a = a 2 Y + a 3 ^ + a 4 p + a 5 w - a 5 в.

If a i = 0 , a 2 = 0 and a 4 = 0 , then

' в a = в ;

^{( Y} a = a 2 ^a a + ^A o ^;

4a2aa + (a3" + 2a5) aa = Vo — a3 Ao, where Ao = y + OtA — a4в, ^o = a2Y + аз£ + a5ш - а5в. Since a2 = 0 this system has a solution for any a3, a4 and a5 from F and also gives a solution of the system (4.7).

If a i = 0 , a 2 = 0 and a 4 = 0 , then

Д = в;

a 2 Y a + a 4 $ a = V o — а з А о ;

^a a — ^A o ;

a,20a + a3Ya + a4Pa = Цо — 2a5Ao, where Ao = v — 2в, Цо = a2Y + аз£ + a4P + а5Ш — а5в, vo = a2Y + азА + a45 — азв. Since a2 = 0 and a4 = 0, this system has a solution for any аз and a5 from F and also gives a solution of the system (4.7).

Now, if a1 = 0, a2 = 0, then а3(аа + ea) + a45a = a3A + a45;

< a 4 ( a a + 2 e a) = a 4 V ;

₄a 3 Y a + a 4 P a + a 5 (2 a a + e a ) = а з £ + a 4 p + a 5 U.

If ai = 0, a2 = 0, аз = 0 and a4 = 0, then aa + ea + a4 ^a = A + 04 ^; a3 a3

’ ^2a a + ^e a = ² ^v ^;

a3Ya + a4Pa = аз£ + a4p + a5U — a52v, i. e.,

'— a a + 0 3 5 a = ^A + 0 3 5 — 2 V ;

* ^2a a + ^e a = ² ^v ^;

₄a3 Y a + a 4 P a = а з £ + a 4 p + a 5 U — a 5 2v.

Since а з = 0 and a 4 =0 this system has a solution for any а з = 0 , а 4 = 0 and a 5 from F and also gives a solution of the system (4.7).

If a i = 0 , a 2 = 0 , а з = 0 and a 4 = 0 then

' 5 a = 5 ;

' a_a + 2 в = V ;

.a = p + §“ — 2? v and, if ai =0, a2 = 0, аз = 0 and a4 = 0, then

J ^a a + ^e a = ^A ^;

(a з Y a + a 5 a a = а з £ + a 5 ^ — a 5 A.

Both of these systems have a solution in the appropriate cases and they also give a solution of the system (4.7). We have considered all cases of values of parameters a 1 , a 2 , a 3 , a 4 and a 5 .

Thus, for each element a G J ii , the system of linear equations (4.7) has a solution.

Hence, the linear operator of the form (4.6) is a local derivation on J 11 . Since the local derivation V was chosen arbitrarily and V is defined by derivations of the form (4.5) we have that each local derivation on the Jordan algebra J 11 has the form (4.6).

Now, we prove that, if a linear map V : J 11 ^ J 11 has the form (4.6), then V is a local derivation on J 11 . For this propose, it is sufficient to prove that for each a G J 11 there exists a derivation of the form (4.5) with a matrix A a , such that

V( a ) = Aa = A ^ a.

But, this is equivalent to proving the existence of a solution of the system of linear equations (4.7), what was done in the previous part of the present proof. This completes the proof. ⊲

Similarly, we can prove the following theorem.

Теорема 6. A linear operator on the Jordan algebras J 6 , J 12 , J ₁ ^α ₅ , J 16 , J 17 , J ₂ ^β ₃ , J ₂ ^γ ₄ , J ₂ ^δ ₆ , J ₂ ^ε ₇ ^,ϕ , J 28 , J 29 , J 30 , J 31 , J 32 , J 34 , J 39 , J ₄ ^λ ₁ is a local derivation if and only if this linear operator belongs to the subspace with the following basis, respectively:

• LocDer( J 6 ) = { E i , i ,E 2 , i ,E 2 , 2 , E 33 E 4,i , E 43 E 4,4^5,1^5,2^5,3^54 E 5,5 },
• LocDer( J 12 ) = { E i , i + 2E 2,2 + 3E₃ 3 + 2E₄ 4 ,E₃ ,₂ , Е з , 4 - E^E^E^E^E^, E 5,5 },
• LocDer( J^) = { E i , i , E 2 , i ,E 2 , 2 + E 44 Е з , 1 , E 32 , E 33 E ₃,4 ,E 4,i ,E 5i ,E 5₂ ,E 5₃ , E 54 E 5 , 5 } ,
• ^LocDer( ^J 16 ) = ^{ ^E 1 , 1 + 2 ^E 3,3 + ^E 4,4 ^,E 2,2 + 2 ^E 3,3 + ^E 4,4 ^, E 2,3 ^— ^E 3,1 ^{, E} 5,1 ^{, E} 5,2 ^{, E} 5,3 ^{, E} 5,5 ^} ,
• LocDer( J 17 ) = { E 1 , 1 + 3E 2 , 2 + 2Е з , з + 4E ₄,₄ ,E ₂,3 , -E 31 + 3E 4,2 , E54 E54 E54 E ₅,₄ , E 5 , 5 } ,
• LocDer( j 23 ) = { E 1 , 1 + Е 2 , 2 ,Е з , 1 ,Е з , 2 + kE 4,2 ,E 3,3 + E 4 , 4 ,E 4 , i ,E 5 , i ,E 5 , 2 ,E 5 , 3 ,E 5 , 4 ,E 5 , 5 } , k G F ,
• LocDer( j 2 ^Y 4 ) = { Е 1 , 1 ,Е 2 , 2 ,Е з , 1 ,Е з , з ,Е 4 , 1 ,Е 4 , 2 ,E 4 , 4 ,E 5 , i ,E 5 , 2 ,E 5 , 3 ,E 5 , 4 ,E 5 , 5 } ,
• LocDer( J 26 ) = { E 1 , 1 + Е 2 , 2 ,Е з , 1 ,Е з , 2 , Е з,з + E 4,4 ,E 4,1 ,E 4,2 ,E 5,1 ,E 5,2 , Е 5 , з , E 5 , 4 , E 5 , 5 } ,
• LocDer( J 28 ) = { E 1 , 1 + 2E 2,2 , E 23 , E 33 , E 4,i , E 43 , E 44 E 4,5 , E54 Е 5 , з ,Е 5 , 5 } ,
• LocDerJTO = { E 1 , 1 + Е 2 , 2 ,Е з , 1 ,Е з , 2 ,E 3,3 + E 4 , 4 ,E 4 , i ,E 4 , 2 ,E 5 , i ,E 5 , 2 ,E 5 , 3 ,E 5 , 4 ,E 5 , 5 } , if (^8,V) = (l^, 1)>
• LocDerJTO = { E 1 , 1 + E 2,2 + 2Е з , з + 2E 4 , 4 + З Е 5 , 5 ,Е з , 1 - Е 4 , 2 ,Е з , 2 - E 4 , 2 ,E 4 , i -
E4,2, E5,i,E5,2,E5,3 - E5,4}, if 8 =1, ^ =1,

• LocDer( J 29 ) = { E 1 , 1 , Е 2 , 1 ,Е 2 , 2 ,Е з , з + E 43 E 4 , i , E 4,4 , E 4 , 5 ,E 5 , i ,E 5 , 3 ,E 5 , 5 } ,
• LocDer( J 30 ) = { E 1 , 1 + Е з , з ,Е 2 , 1 , 2E 2,2 + 3E 4 , 4 + 2E ₅5 , E^, E 4 , i , E 4 , 3 , E 4 , 5 , E 5 , i , £ 5 , 3 } ,
• LocDer( J 31 ) = { E 1 , 1 + 2E 2,2 + 3E 5 , 5 ,E 3 , 1 ,E 3 , 3 ,E 4 , 1 ,E 4 , 3 , E 4,4 , E 4,5 , Е 5 , 1 ,Е 5 , з } ,
• LocDer( j 32 ) = { E 1 , 1 ,E 2 , 1 + E 31 2E 2,2 + 2E 3 , 3 + 4E 44 - 3 E^E^E^E^ + №5 ,2 , 4 E 4 , 5 + ³ ^E 5,5 ^{, E} 5,1 ^, E 5,3 ^} ,
• LocDer( J 34 ) = {E 1,1 , E 24 Е_зл, E 32 , E ₃,₃ , E^ i ,E 4,2 , E 43 E 44 E 4,5 , E 5,i , E 5,2 , E 5,5 },
• LocDer( J 39 ) = { E 1 , 1 + 2E 2,2 , E34 E343E 3,3 + 2E ₅5 , E 4,i , E 4,2, E 43 E 4,4 , E 45 Е54,Е 5,2 }.
• LocDer( J 4 ^A i ) = { E i , i + E ₂,₂ +2E 3,3 + 2E 54 E 34 E 4,i , E 4,2 , E 44 E 44 E 4,5 ,E 5i ,E 5₂ }, if X = 0,
• LocDer( J 4 ^A i ) = {E i,i + E 2,2 + 2E 33 + 3E 4,4 + 2E ₅5 , 2 E 3,i - 2 E 3,2 , E 4,i , E 4,2 , E 45 E 51 E 5,2 }, if X = - 1,
• LocDer( J ₄ ^A i ) = { E 1 , 1 +E 2,2 , E 3 , i , E 3,2 , 2E 33 +3E 4,4 +2E 5,5 , E 44 E 42 -E 4,3 +E 4,5 , E 5 , i , E 5,2 }, if X = - 3 ,
• LocDer( J 4 ^A i ) = { E i , i + E 2,2 , E34 E34 E34 E 4,i , E 4,2 , E 4,3 , E 4,4 , E 4,5 , E 5 , i , E 5,2 , E 5,5 },

if X G {- 1,0,3 } .

Acknowledgements. The authors would like to thank the anonymous referees for their helpful comments and suggestions.

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