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Chelyabinsk Physical and Mathematical Journal. 2023. Vol. 8, iss. 2. P. 228-237.
DOI: 10.47475/2500-0101-2023-18206
A CHARACTERIZATION OF LOCAL DERIVATIONS ON LOW-DIMENSION JORDAN ALGEBRAS
F.N. Arzikulov1'2'", O.O. Nuriddinov2b
1V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan 2Andizhan State University, Andizhan, Uzbekistan "[email protected], [email protected]
We consider local derivations on finite-dimensional Jordan algebras. We developed a technique for the description of the vector space of local derivations on an arbitrary low-dimension Jordan algebra. We also give a description of local derivations on some Jordan algebras of dimension four.
Keywords: Jordan algebra, derivation, local derivation, nilpotent element.
Introduction
The present paper is devoted to local derivations on Jordan algebras. The history of local derivations begins with the Gleason — Kahane — Zelazko theorem in [1] and [2], which is a fundamental contribution in the theory of Banach algebras. This theorem 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 E A, is multiplicative. 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 $a : A ^ B, depending on a, such that T(a) = $«(a).
Later, in [3], R. Kadison introduces the concept of local derivation and proves that each continuous local derivation from a von Neumann algebra into its dual Banach bemodule is a derivation. B. Jonson [4] extends the above result by proving that every local derivation from a C*-algebra into its Banach bimodule is a derivation. In particular, Johnson gives an automatic continuity result by proving that local derivations of a C * -algebra A into a Banach A-bimodule X are continuous even if not assumed a priori to be so (cf. [4, Theorem 7.5]). Based on these results, many authors have studied local derivations on operator algebras.
By Theorem 5.4 in [5] every local derivation on a JB-algebra is a derivation. So, in [5] the description of local derivations on JB-algebras is given. In [6] the first and the second authors of the present paper 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 is a Jordan multiplier operator.
In the present paper, we investigate derivations and local derivations on Jordan algebras. 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 E J there exists a derivation D : J ^ J such that V(x) = D(x).
In the following, we will work over an algebraically closed field F of characteristic = 2 and, furthermore, all Jordan algebras are assumed to be of dimension four over F. In [7], the list of Jordan algebras over F of dimension less than or equal to four is provided.
Description of local derivations of Jordan algebras with nilpotent elements and nilpotent Jordan algebras is an open problem. Therefore, we choose for our investigation appropriate Jordan algebras from this list, and give description of local derivations on these Jordan algebras. We also give a criterion of a linear operator on Jordan algebras of dimension four to be a local derivation. We developed a technique for the description of the vector space of local derivations on an arbitrary low-dimension Jordan algebra.
1. Local derivations on Jordan algebras of dimension four
Let J be a Jordan algebra of dimension four with a basis (ei, e2, e3, e4}. Let x be an element in J. Then we can write x = x1e1 + x2e2 + x3e3 + x4e4, for some elements x1, x2, x3, x4 in F. Throughout of the paper let x = (x1,x2,x3,x4)Tr.
Let T : J ^ J be a linear operator. Then T(x) = 4=1^S4=1 bi,jxj)e^ x E J, for the matrix B = (bi,j)4,j=1 of the linear operator T, where x = x1 e1 + x2e2 + x3e3 + x4e4. For example, if J = J62 (see Table) and T is a derivation on J, then T has the form T(x) = a1,1x1 n1 + 2a1,1x2n2 + (a3,1 x1 + 3a1,1 x3)n3 + a1,1x4n4, x E J, with respect to the basis (n1, n2, n3, n4} (see Table), where a1,1, a3,1 in F. This common form of a derivation can be directly calculated, and we will omit the calculations of common forms of derivations on the Jordan algebras.
Local derivations of four-dimensional Jordan algebras
J Multiplication table Common form of a derivation Is each local derivation a derivation?
J49 2 n2 = n2n3 = ni, ein = in¿, i = 2, 3 (ßxi + ax2)ni + (7x3 + (7 - 1 a)x4)n2 + (a - y)X4n3 +
J62 2 2 ni = n| = n2, nin2 = n ai,i xini + 2ai,iX2n2 + (03,1x1 +3ai,iX3 + a3,4X4)n3 + ai,iX4 n4 +
J63 n2 = n2, n4 = -n2 - n3, nin2 = n2n4 = n3 (ai,ixi — 3 ai,iX4)ni + (02,iXi +2ai,iX2 + ( — 5 ai,i — 02,i )x4)n2 +(03,ixi + 2a2,iX2 + § ai,iX3 +a3,4X4)n3 + ( — 3 ai,ixi + ai,iX4)n4
J64 n2 = n2, n| = -n2, nin2 = n2n4 = n3 (ai,ixi + ai,4X4)ni + (a2,ixi +2ai,iX2 — Ö2,i)x4)n2 + («3,iXi +2a2,iX2 + (3ai,i + ai,4)x3 +a3,4X4)n3 + (ai,4Xi + ai,iX4)n4
J65 n2 = n2, nin2 = n2n4 = n3 ai,ixini + 2ai,iX2n2 + (03,iXi + 03,3X3 + Ö3,4X4)n3 + ((—3ai,i + a3,3)xi + (—2ai,i +Ö3,3)x4)n4
Our principal tool for the description of local derivations on Jordan algebras of dimension four is the common form of derivations, depending on a basis of these Jordan algebras. Our main goal in this paper to justify Table and prove the theorems corresponding to these tables. In Table a necessary and sufficient condition for a linear operator to be a derivation on some Jordan algebras of dimension four is listed. Also, it is indicated that, wether each local derivation of these Jordan algebras is a derivation or not. The third column of the tables 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 are taken from [8].
2. Description of local derivations on some Jordan algebras of dimension four
We prove the appropriate statements for Table, i. e. for four-dimensional Jordan algebras.
Theorem 1. Each local derivation of the Jordan algebras J49 and J62 is a derivation.
Proof. Let J49 be the Jordan algebra over the field F with the basis [e\, n1, n2, n3}. Let V be a local derivation on J49. Then
V(x) = I bij Xj I ei + ( xA Ui-i, x E J49
\j=1 J i=2 \j=1 J
for the matrix B = (bitj)'4,j=1 of the local derivation V, where x = x1 e1+x2n1 +x3n2+x4n3 and x1, x2, x3, x4 E F.
By the definition for any element x E J49 there exists a derivation Dx such that V(x) = Dx(x). By the form of a derivation we get
Dx(x) = (pxxi + axx2)ni + Yxx3 + (^fx — 1 ax^ x^ n2 + (ax — jx)x4U3
for some elements ax, Px and Yx in F, depending on x.
Now, by the equality V(n1 + n3) = V(ni_) + V(n3), we have ani+n3 = ani, Yni +n3 — 1 ani+n3 = Yn3 — 2an3, ani +n3 — 1ni+n3 = an3 — Yn3. Hence, ani+n3 = an3 and ani = an3. Thus,
/ 0 0 0 0 \
pei ani 0 0
B
0 0 Yn2 Yn'3 — 2 ani
\ 0 0 0 ani —Yn3 J
The equality D(a) = V(a) = Ba, where a = (a1,a2,a3,a4)Tr, we can rewrite as the following system of linear equations
+ a2a = a1pni + a2ani, a3Y + a4(Y — 2 a) = a3Yn2 + a4(Yn'3 — | ani), a4(a — y) = a4(ani — Yn3),
with the parameters ai, i = 1, 2, 3, 4. We rewrite
a2a + = a2ani + ei,
— 1 a4a + (a3 + a4)Y = a3Yn2 + a4(ia'3 — | ani),
a4a — a4Y = a4(ani — Yn3).
Note, if a4 = 0, then the system of linear equations has a solution for any a1, a2, a3. Now, suppose a4 = 0. Then the system of linear equations has the following form
a^ + a2a = a1pni + a2ani, —a + 2( ^ + 1)y = 2 ^ Yn2 + 2y n3 — ani, . a — y = ani — Yn3.
Hence,
a1(5 + a2a = a^"1 + a2a",
(2 + !)y = 2 y"2 + Y "3, .
a — y = a"1 — y"3 .
If 2a3 + 1 = 0, then 2a3y"2 + Y"3 mast be equal to 0 since V is a local derivation, i.e., the system of linear equations must have a solution for every a in J49. Hence,
Y"3. Thus, by the form of a derivation, we
Y
»2
2f4Y"2 + Y"3 = 0, if a4 = —2as, i.e., get the statement of the theorem.
The appropriate statement of the theorem for the algebra J62 is similarly proved. The proof is complete. □
Let V be a local derivation on J63. Then V(x) = Y1 4=1(S4=:l bjjxj)nj, x E J63 for the matrix B = (6^-)4,j=1 of the local derivation V, where x = x1n1 + x2n2 + x3n3 + x4n4 and xL, x2, x3, x4 E F.
By the definition for any element x E J63 there exists a derivation such that V(x) = Dx(x). By the form of a derivation we get
Dii(x) —
( «1,1 0 0 - 3 «1,1 \ í X1 \
«2,1 2 «1,1 0 - 3 «1,1 - «1,1 X2
«3,1 2«1,1 8 1 3 «1,1 «1,4 X3
\ - 3 ai,1 0 0 al,1 J \ X4 )
Similarly to the above algebras we get the following equality
B
( a21 a1,1 0 0 -1 «24 3 a1,1 \
21 a22,11 2a2,21 0 1 24 24 - 3«1,1 - «2,1
21 a32,11 2a22,21 8 «23 3 «1,1 24 a3,4
V -1 an1 3 a1,1 0 0 24 a12,41 I
Now, by the equality V(ni + ^4) = V(ni) + V(^) we have a"11+"4 — 3a"11+"4 =
_ 1 a»4
«1,1 3 «1,1,
_I/7 21+24 I „"4+»4 o a1 1 + «11
»1 +»4 X1,1
»1 X1,1
^1,1 »4 «1,1 •
-1 a»1 3 «1,1
3
+ a»41. Hence, «»,1 - 1 a24 — -1 «»1 + <4,
On the other hand, the equality V(nL + n4) = V(nL) + V(n4) gives us the following
"1 1 "4 "4 : "i+"4_ "1 : "4 "4
a2,i — 3 ai,i — a2,i , — 3ai,i = a2,i — 3ai,i — a2,i.
equalities a»11+ra4 - 1 a»^»4 - a»11+ra4 - ~21 ^24 ~24 ^21+24 _ ^24
Since a»11+ra4 — a»41, we have a»11
»4
a2/L •
Now V(ni + n + m) = V(ni) + VM + V(m) gives us a"\+"2+"4 + 2a"\+"2+"4
1 „21+22+24 3 «1,1
»1 +»2 +»4 _ 1 „ »1 +»2 +»4 «1,1 3 «1,1
_ +»2 +»4 _ „»1 i O1nn4 _ nn4 5n»1 +»2 +»4
«2,1 — «2,1 + 2a1,1 3 «1,1 «2,1, 3 «1,1 „21 1 "4 1 "1+22+24 I "1+22+24
«1,1 3 «1,1 3 «1,1 +«1,1
2an21 - 3an41, and
1„ 24
— 1n 21 I „24 „21+22+24 _ 3 «1,1 + «1,1, «1,1 —
'3U1,1 3U1,1
a?! — a^. So, 5a2^22 +24 — |aft — 2a221 - |aft and a221 — aft. Thus, we get
B
( 21 a1211 0 0 _ 1 an1 3 «11 \
a21 a21 2a1211 0 _1 an1 - an1 3 a11 a21
21 a3211 2a2i 8 an3 3 «11 24 a34
V -1 an1 3 a11 0 0 21 a1211 I
We consider the following system of linear equations
1 21 1 2 1 a1«1,1 — 3 a4a1,1 — a1«11 — 3 a4a11,
a1«2,1 + 2a2«1,1 + a4(-1 «1,1 - «2,1)
a1a211 + 2a2a211 + a4(-1 a211 - a211),
a1«3,1 + 2a2«2,1 + 8 a3«1,1 + a4«3,4 — alanll + 2a2an2l + 3 a3a21! + a4an4
—1 a1an + a4«1,1 — — 1 a 1a21 + a4aft.
We rewrite this system as follows
(ai — "I a4)ai,i — (ai — 1 h (—1 ai + a4)a\" — (—" ai + a4)an\
(2a2 — 1 a^ai" + (ai — a4 )a2,i — (2a2 — | a4)ar^i + (ai — a4)a'.
1
2,1,
1 a3ai,i + 2a2a2,i + aia3,i + a4 a3,4 — 3 asa^i + 2a2a<22"i + ai a3Li + a4a3i4. The subsystem
(ai — "I a4)ai,i — (ai — 1 a^a™1", (—i ai + a4)aii — (—i ai + a4)a'lii,
(2a2 — i a4)aii + (ai — a4)a2,i — (2a2 — | a^aH + (ai — a4)anii,
always has a solution. So, if at least one of ai, a4 distinct from zero, then our system of linear equations has a solution.
Suppose that ai — 0 and a4 — 0. Then we get
2a2anx\,
2a2ai,i g
3 a3 ai,i I — 3^
If additionally a2 — 0, then this system has a solution. Else, if a2 — 0, then we get
3 a3 ai,i + 2a2a2,i — 3 a3al3i + 2a2ali.
a1,1
n1 1 n,11 ,
„ _ 4 a3 n3 1 12 4 a3 ni
a2,i — 3 aa2 ai,i + a2,i — 3 aa2 ai,i,
i.e., in this case our system also has a solution. But, may be al^ — ali what does not allow to V be a derivation.
Thus, in all cases the present system of linear equations has a solution. Hence, the linear mapping V, defined by the matrix
( n1 an,11 0 0 3 ai,i \
n1 a11 2an,11 0 i ni ni — 3 ai,i — a2,i
n1 a1i 2an2i g an3 3 ai,i a3,4
V — i nni 3 ai,i 0 0 ni ani
is a local derivation. So, in particular, if a!l\ — a1\, then V is a local derivation, which is not a derivation. Thus, we have the following theorem.
Theorem 2. A linear operator V : J63 ^ J63 is a local derivation if and only if the matrix of V has the following form,
/ a1,1 0 0 — 3 ai,i \
a2,1 2a1,1 0 — 3 ai,i — a2,i
a3,1 a3,2 a3,3 a3,4
V — 3 ai,i 0 0 a1,1 )
Example 1. By the arguments above, in the case F — R, the linear operator V(x) — 2x2n3, x E J63, where x — xini + x2n2 + x3n3 + x4n4 and xi, x2, x3, x4 E F, is a local derivation, which is not a derivation.
Let V be a local derivation on J64. Then V(x) — Y14,=iQ24j=i bi,jxj)ni, x E J64 for the matrix B — (bi,j)4,j=i of the local derivation V, where x — xini + x2n2 + x3n3 + x4n4 and x1, x2, x3, x4 E F.
By the definition for any element x E J64 there exists a derivation Dx such that V(x) = Dx(x). By the form of a derivation we get
/ ax,1 0 0 ax,4 \ x1
aX2,l 2aX,i 0 -ah X2
ah 2aX,i 3ax i + q>2 4 aX,4 x3
V ax,4 0 0 ax,1 / x4
Dx{x) — Axx
Similarly to the above algebras we get the following equality
B
(
\
ni aih
ni
ni ni
a
1,4
0
2ani
2an2
0
2,1
0 0
3a~n i + ar 4 0
„ni a1,4
ni a2,i
a3,4
-.m
a
1,1
\
/
ni 2,1
Now, we take the equality V(ni+n4) = V(n)+V(n4). Then we get a — dn\. Hence, al3'1i
ni+ni 2,1
a
ni+ni 2,1
-.ni 12,l-
The equality D(a) — V(a) equations
Ba we can represent as the following system of linear
aiaii + a4ai,4 — aia^i +
aia2,i + 2a2 ai,i — = aian\ + 2a2an i a4a2.i j
aia3,i + 2a2 a2,i + a3(3ai,i + ai,4) + a4a3,4 = = aian3\ + 2a2a1%i + a:i (3a3i + a^) + a4an3%
aiai,4 + a4ai,i = aiani4 + a4an4ij with the parameters a^, i = 1, 2, 3, 4. We rewrite this system in the following form
aiaii + a4ai,4 = aian^ + a4a344, a4ai,i + aiai,4 = aian^ + a4an4i,
2a2ai,i + (ai — a4)a2,i = 2a2a32i + (ai — a4)al3^ij
3a3ai i + a3ai 4 + 2a2a2 i + aia3 i + a4a3 4 =
= aiali + 2a2al32i + a3(3a3^i + a3%) + a4al%. Suppose ai = 0. Then, if a1 = a4, then the system has the following form
aiai ,i + aiai ,4 = aianii + ai an44, 2a2ai ,i = 2a2a3ij
3a3ai,i + a3ai,4 + 2a2a2,i + aia3 i + a4a3,4 =
= aia33^i + 2a2a3\ + a3(3a3\ + a334) + a4a33
(1)
ni
.4'
Since aia3,i = 0, a4a3,4 = 0 the system has a solution. Else, if ai = a4, then the subsystem of linear equations
aiai,i + a4ai,4 = aianii + a4an44, a4aii + aiai,4 = aia3'4 + a4a34
has a solution. Since (ai — a4)a2,i = 0 we have the subsystem
aiaii + a4ai,4 = aia^i + a4a3^^, a4ai,i + aiai,4 = aian^ + a4an4i,
2a2ai,i + (ai — a4)a2,i = 2a2a32i + (ai — a4)a!3^i
also has a solution for any a2, a1 = 0 and a4 = a1. Hence, by a1a3,1 = 0 our system of linear equations has a solution.
Suppose «1 = 0. Then the system has the following form
__«4
«4«1,4 — «4«i 4,
__«4
«4«1,1 — «4«i i,
2«2«i,i — «4«2,i = 2a2an2i — «4«2'1i,
3a3a1,1 + «3 ai,4 + 2«2«21 + «4«3,4 = 2«2«H21 + «^«ft + «ft) + «4«H
«4
4-
In this case, if «4 = 0, then we have 2a2a1,1 = 2a2an,21,
3«3«i,i + «3«i,4 + 2«2«2,i = 2«2«H,2L + a3(3an31 + «ft).
If we consider the cases «2 = 0 and «2 = 0 separatively, then we easily see that this system of linear equations has a solution. Now, if «4 = 0, then we have
„«4 «1,4,
«4
«1,4 «1,1 = «1
2«2«1,1 — «4«2,1 = 2«2«ft — «4«nii, 3«3«i i + «3«1 4 + 2«2«2 1 + «4«3 4 :
2«2«n21 + «3 (3««31 + «ft) + «4«n
1,4,
and this system also has a solution for any «2, «3 and «4 = 0.
Thus, in all cases, the system of linear equations (1) has a solution. Therefore, the map, generated by the matrix
/ «1 «1,1 0 0 «4 «1,4 \
«2,1 2«ft 0 ««1 «2,1
««1 «3,1 2««,2i 3«« i + «« 4 ««4 «3,4
V «1 «1,4 0 0 «4 ««,41 /
is a local derivation by the form of a derivation. At the same time, if, for example, «ft = ««21, then this local derivation (i.e., V) is not a derivation again by the form of a derivation. So, we have the following theorem.
Theorem 3. A linear operator V : J64 ^ J64 is a local derivation, if and only if the matrix of V has the following form,
«1,4 —«2,1 «3,4 «4,4
«1,1 «2,1 «3,1 «1,4
0
«2,2 «3,2 0
0 0
«3,3 0
/
Example 2. By the arguments above, in the case F = R, the linear operator V(x) = 2x2n2, x E J64, where x = x1n1 + x2n2 + x3n3 + x4n4, is a local derivation, which is not a derivation.
Let V be a local derivation on J65. Then V(x) = 4=1^S4=1 Xj)nj, x E J65 for the matrix B = (6^-)4,j=1 of the local derivation V, where x = x1n1 + x2n2 + x3n3 + x4n4 and x1, x2, x3, x4 E F.
By the definition for any element x E J65 there exists a derivation such that V(x) = Dx(x). By the form of a derivation we get
Dx(x) — Axx
/ «x,i 0 0 0 \ x1
0 2«X,i 0 0 x2
«f,i 0 «3,3 «3,4 x3
\ — 3«x,i + «x,3 0 0 — 2«1,1 + «3,3 J x4
By the equalities V(n») = Dni (n»), Dni (n») = Anin», i = 1, 2, 3, 4 we have
B
/ n1 «M 0 0 0 \
0 2<21 0 0
ani «3,1 0 «n3 «3,3 «n4 «3,4
V 3«n1 1 «ni 0«1,1 + «3,3 0 0 2«n4 + «n4 2«1,1 + «3,3 I
Now, by the equality V(n + n2) = V(ni) + V(n2) we have aft = a32
The equality D(a) = V(a) = Ba we can rewrite as the following system of linear equations
_ ni
®lQl,1 — ®iQi i, 2a2a1,1 = 2a2a")11,
aia3,i + a3;3 + a4a3>4 = aianii + a3an>3 + a4an"4,
ai(—3aM + a3,3) + a4(—2aM + «3,3) = ai(—3ani + a"^) + a4(—2an4i + a^), with the parameters a», i = 1, 2, 3, 4. We rewrite
_ ni
aiai,i — aiai 1,
_ ni
a2ai,i — a2aii,
(—3a1 — 2a4)a1,1 + (a1 + a4)a3,3 = a1(-3ani1 + ani3) + a4(-2a"41 + «","3),
aia3,i + a3a3,3 + a4a3,4 = aianii + a3an>3 + a4an>4.
Suppose that a1 = 0 and a1 + a4 = 0. Then a4 = 0, —3a1 — 2a4 = 0, and
(2)
n — nni «1,1 — «1,^
2ani — 2an4 — ani + an4 2«1,1 2«1,1 — «3,3 + «3,3.
Clearly, in this case the system (2) has a solution. If a1 = 0 and a1 + a4 = 0, then
ni
«1,1 — «1,^
«3,3 — [«1«n,3 + «4(-2«n,4 + + 2«4«n,1],
aia3,i + a3a3,3 + a4 «3,4 — aia3,i + a3a33 + a4 a^.
Since a1 = 0, i. e., a1a3,1 = 0 we have, in this case, the system (2) has a solution. Now, suppose that ai = 0. Then we get
a2ai,i = a2an>ii,
—2a4a1,1 + a4a3,3 = a4(—2a3'41 + a^g),
a3a3,3 + a4a3,4 = a3an>3 + a4an>4.
In this case, if a4 = 0, then the system (2) is equivalent to the following system
_ ni
a2ai,i — a2aii,
a3a3,3 = a3an,33,
and the system (2) has a solution. Else, if a4 = 0, then the system (2) is equivalent to the following system
ni
«2«1,1 — «2«1 1
-2aM + «3,3 — -2«n,41 + an,43,
«3«3,3 + «4 «3,4 — «3«n3 + «4«n4.
This system has a solution for any a2, a3 and a4 = 0.
Thus, in all cases, the system of linear equations (2) has a solution if 2an\ — 2a3\ = a3\ + a343. Hence, by the form of a derivation, under this condition V is a local derivation with the matrix
/ 11 a11 0 0
0 2ani 0
an1 a3,1 0 a3,3
V -3a1,1 + a1,3 0 0
0 0
\
a3,4 -2al,l + a3,3
14
14
/
At the same time, if, additionally, a3\ = an\, then V is a local derivation, which is not a derivation by the form of a derivation. So, we have the following theorem.
Theorem 4. A linear operator V : matrix of V has the following form,
( ai,i 0 a3,i
\ a4,i
J65 ^ J65 is a local derivation if and only if the
0 0
a3,4 a4,4 )
0 0
2ai,i 0
0 a3,3
0 0
Example 3. By the arguments above, in the case F = R, the linear operator V(x) = x4n4, x E J64, where x = xini + x2n2 + x3n3 + x4n4 and xi, x2, x3, x4 E F, is a local derivation, which is not a derivation.
Remark 1. We note that local derivations of an arbitrary low-dimension algebra can be similarly described using a common form of the matrix of derivations on this algebra. A technique for constructing a local derivation, which is not a derivation, developed by us, can be applied to an arbitrary low-dimension algebra, derivations of which have a matrix of common form.
Acknowledgments. The authors wish to thank I. Kaygorodov for his suggestions and useful remarks.
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Article received 16.04.2022.
Corrections received 14.06.2023.
Челябинский физико-математический журнал. 2023. Т. 8, вып. 2. С. 228-237.
УДК 518.517 Б01: 10.47475/2500-0101-2023-18206
ХАРАКТЕРИЗАЦИЯ ЛОКАЛЬНЫХ ДИФФЕРЕНЦИРОВАНИЙ ЙОРДАНОВЫХ АЛГЕБР МАЛОЙ РАЗМЕРНОСТИ
Ф. Н. Арзикулов1'2'", О. О. Нуриддинов2'6
1 Институт математики имени В. И. Романовского Академии наук Узбекистана, Ташкент,, Узбекистан
2Андижанский государственный университет, Андижан, Узбекистан [email protected], [email protected]
Исследуются локальные дифференцирования на конечномерных йордановых алгебрах. Разработана техника для описания пространства локальных дифференцирований на произвольной йордановой алгебре малой размерности. Дано описание локальных дифференцирований на некоторых йордановых алгебрах размерности четыре.
Ключевые слова: йорданова алгебра, дифференцирование, локальное дифференцирование, нильпотентный элемент.
Поступила в 'редакцию 16.04.2022. После переработки 14.06.2023.
Сведения об авторах
Арзикулов Фарходжон Нематжонович, доктор физико-математических наук, главный научный сотрудник, Институт математики имени В. И. Романовского АН Республики Узбекистан, Ташкент, Узбекистан; Андижанский государственный университет, Андижан, Узбекистан; [email protected].
Нуриддинов Олимжон Одилжонович, аспирант, Андижанский государственный университет, Андижан, Узбекистан; [email protected].
