6Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
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DESCRIPTION OF LOCAL DERIVATIONS ON LOW-DIMENSIONAL JORDAN ALGEBRAS
O. O. Nuriddinov
Andijon davlat universiteti. O'zbekiston.
The present paper is devoted to local derivations on Jordan algebras. The Gleason - Kahane - Zelazko theorem is a basis of the concept of local derivations. This theorem is a fundamental contribution in the theory of Banach algebras. The concept of local derivations was introduced and studied by R. Kadison. He proved that each continuous local derivation from a von Neumann algebra into its dual Banach bemodule is a derivation. Later, B. Jonson extended the above result by proving that every local derivation from a C* — algebra into its Banach bimodule is a derivation. In particular, Johnson proved 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.
It is known that every local derivation on a JB —algebra is a derivation. In [2] Nuriddinov (the author of the present paper) gived the description of local derivations on five - dimensional nilpotent associative Jordan algebras. Also, in [3] Nuriddinov and Urinbayev gived the description of local derivations on four - dimensional nilpotent noncommutative Jordan algebras over C. Arzikulov and Nuriddinov gived the description of local derivations on five-dimensional nilpotent nonassociative Jordan algebras in [4].
In the present paper I give the description of local derivations on Jordan algebras of dimension less than or equal to four.
Let J be a Jordan algebra of dimension four over an algebraically closed field ¥ of characteristic ^ 2 with a basis [e1 ,e2 ,e3 ,e4 }. Let x be an element in J. Then x = x1 e1+ x2 e2+ x3 e3 + x4 e4 , for some elements x1 ,x2 ,x3 , x4 in F. Throughout of the paper let x = (x1 ,x2 ,x3 ,x4)Tr. Conversely, if v = (x1 ,x2 ,x3 , x4)Tr is a column vector with x1 ,x2 ,x3 , x4 in F, then, throughout of the paper, by b v we will denote the element x1 e1 +x2 e2+ x3 e3 + x4 e4, i.e., v = x1 e1 + x2 e2 + x3 e3 + x4 e4.
Definition. 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 is called a local derivation if for every x E J there
exists a derivation D-.J^J such that V(x) = Dx(x).
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Let J45 be the Jordan algebra with a basis [e1 ,n1 ,n2 ,n3] defined in [1]. In this algebra, the multiplication table is defined as follows:
1
222 n2=n2= n2 ,e1 n1 = — n1
Theorem. Each local derivation of the Jordan algebra J45 is a derivation. Proof. Let V be a local derivation on J45 . Then
4 4/4 \
V(X) = X bl-iXiei + X (X bijXj I Ui-! ,XE^45 j=1 i=2\j=1 /
for the matrix B = (pi )._ 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 J45 there exists a derivation Dx such that V(x) = Dx(x). (x) =
axx2n1 + (Pxx1 + 2axx3 + Yxx4)n2 + axx4n3, x E $45
for some elements ax , and yx in F, depending on x.
If we take x = e1, then V(e1) = D6i(e1) = fiein2. On the other hand, V(e1) = + b2,1^1 + b31U2 + b41U3. Hence, b1,1 = 0, b^ = 0, b3A = pei, b4A = 0. Similarly we get b12 = 0, b22 = an1, b32 = 0, b42 = 0, if we take x = n1. Also, we analogously get b13 = 0, b2 3 = 0, b3 3 = 2a n2 , b43 = 0, b14 = 0, b2 4 = 0, b3A = yn3, b4A = an3 .
Now, since V is linear we have V(n1 + n2) = V(n1) + V(n2). From this it follows that
V(n1 + n2) =
an1+n2n1 + 2an1+n2n2 = an1n1 + 2an2n2 Hence an1+n2 = an1 , an1+n2 = an2 and an1 = an2.
Similarly, from the equality V(n1 + n3) = V(n1) + V(n3) we get an± = an2 . Since the local derivation V was chosen arbitrarily, we have it has the following
form
V(x) = an1x2n1 + (fie1x1 + 2an1x3 + Yn3x4)n2 + an1x4n3, x E 045
where a, p, y belong to F.
Let V(x) = Bx, x E J45. We suppose that for each a E J45 there exists a
derivation of the form a matrix Aa such that
V(a) = Aa = Aaa.
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
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a2<xa —
a^ + 2a.3aa + 04ya = a^ + 20.3a + 04y
&4&Q = 0.4ft
with respect to the variables aa , ya for each a E J45. Here, a, ft, y are elements
from F. Note that, if the left part of any equation of the system is equal to zero, then
the right part of this equation is also equal to zero.
Now, suppose that a1 ^ 0, o2^0 and a3 = 0. Then we have
aa = a a4
Pa = — (Y — Ya). a1
This system has a solution for any a1 ^ 0,a2 ^ 0, a3 = 0.
Now, suppose that a ^ 0. Then we have
aa = a
a1 2a3
Ya= — (P — Pa)+—-(a — aa) + Y
O4 O4
This system has a solution. Thus we have
V(x) = ax2e1 + (ftx1 + 2ax3 + yx4)n2 + ax4n3 But this coincides with the form (1.1) of a derivation on J45 . This completes the proof.
Let J63 be the Jordan algebra defined in [1] with a basis {e1,n1,n2,n3} such
that
22 n2 = n2 = n2 , n1n2 = n3
Theorem. A linear operator V on the Jordan algebra J63 is a local derivation if
and only if this linear operator belongs to the following subspace
1 11
L0CDer(J63) = [En —^1,4 + 2E2i2 — ^2,4 ^E4,1 + E4,4 ,
E2,1 — E31 E32 Eз з, E3 4}.
Proof. Let J63 be the Jordan algebra over the field F taken with the basis [n1, n2, n3, n4}. First we prove that the matrix of each local derivation on J63 has the matrix form.
Let V be an arbitrary local derivation on J63. Then
44
VM = ai,ixi jei,xEj63
i=1\j=1 /
for the matrix A = (a^j ). of the local derivation V, where x = x1n1 + x2n2 +
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By the definition, for any element x E J63 , there exists a derivation Dx such that V(x) = Dxx = Ax.
1
2 ~-X"4 I ■ 1 ■ Ifx--!----X-~2 • I 2
8 \ / 1 - axX3 + SxX4) n3 + (-where ax, fix, yx and Sx in F, depending on x.
Using equalities V(ni) = Dn.(n[) = Annv i = 1,2,3,4 we get a11 = an±,
Dx(x) — Axx — (axx± -1axx4) n± + (ßxx± + laxx2 + (-^ax - ßx)) +
+ (jxx1 + 2ßxx2 + 8 VxX3 + Sxx4) n3 + (-^ axx1 + VxXi) U4 ,
a2,1 = ßn1,a3,1 = Yn1,a4,1 = 1an1,a1,2 = °,a2,2 = 2an2 ,a3,2 = 2ßn2,a4,2 =
^ ~~ ' "1,2
8^1 1
^ a1,3 = ^ a2,3 = ^ a3,3 =3an3 , a4,3 = ^ a1,4 = "an4, a2,4 = — ~an4 — ßn4,
3 n4
ß "№4 1^4
a3,4 = ^n4,a4,4 = an4 .
Now, since V is linear we have V(n1 + n4) = V(n1) + V(n4). From this it follows that
V(n1 + n4) = (an1+n4 — 1an1+n4) n1 + (^n1+n4 — 1an1+n4 — Pn1+n4) n2 + + (Yn1+n4 + 8n1+n4)n3 + (-1an1+n4 + an1+n4)n4 = (an1 — 1an4)n1 +
+ (,ni- + ant--nt)n2 + irni++t)n3 + (-),ani+a^t)ni. Hence an1 = an4 and @n1 = pn4.
How V(n1 + n2 + n4) = V(n1) + V(n2) + V(n4) gives us
Ln1+n2+n4 1 ^n1+n2+n4) W-1 +
V(n1 +U2 + n4) — (^n1+n7+nA 1 ■ + ( ßn-. +
n1+n2+n4 ' u-n1+n2+n4
1
— a
+
3 u-n1+n2+n4 (-l3a-1
ßn1+n2+n4)n2 + (Yn1
n1+n2+n4
+ 2ßni
+n2+n4
+ 8n1+n2+n4)n3 +
+ ar
^u-n1+n2+n4 ' u-n1+n2+n4
)n4 — (an1 -1an4 )U1 + (ßn1 + 2an2 -
-\an4- ßn4) n2 + (Ynx + 2ßn2 + Ön4)n3 + + an4) n4 .
^ II4 1 II4
Hence an2 = an4. So, we have
V(x) =A* = (an1X1 -1an1x2)n1 + (pn1X1 + 2an1X2 + {-1^n1 -- Pn1) X4) n2 + (Yn.1X1 + 2Pri2X2 +8^n3X3 + 8n4x4) n3 + (-+ + an1x4) n4 ,xE 063 .
Since the local derivation V was chosen arbitrarily, we have it has the following
form
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Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
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ChaLes.hnnoeon.andPru.en
V(x) = Ax = (axi -iax2)n1 + ((3xi + 2ax2 + (— ^ a — ft) x4)n2
+ (yxi + 2<PX2 +1^3 + Sx4)n3 + [—laxi + ax4)n4 ,x e Jra where a, ft, y, S, <p, ^ belong to F.
Let V(x) = Ax, x e J63. We suppose that for each a e J63 there exists a derivation of the form a matrix Aa such that
V(a) — Aa — Aaa.
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
f -i - -i
o-ifta + 2a.2<xa + aA (—^a—fta) — aiP + 2a2a + a4(—1a — ft)
8
aiya + 2a20a + a4Sa — aiY + a2& + a3^ + a4S
i i i i — -aiaa + a4aa — —~aia + a4a
3
with respect to the variables aa, fta, Ya, for each a e J63 . Here a, ft, y, S, y, 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.
This system has a solution for any ai ^ 0 from F and gives a solution of the system.
aa — a
fta—ft
. ^ a± a± a± a± 3a± a± "
If ai — 0,a4 ^ 0 then aa —a
fta — ft
Sa—Hiy+^ + S — ^p—^a
3a4
If at = 0,a2 ^ 0 and a4 = 0 then
1
aa = a
1 a3 a3 / 8 \
a.2\ 3
3
and, if a± = 0,a2 = 0,a3 ^ 0 and a4 = 0 then aa =
8
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Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current _Challenges, Innovations and Prospects
Both of these systems have a solution in the appropriate cases and they also give a solution of the system. We have considered all cases of values of parameters Q-1, &2, ^3 and a4. Thus, for each element a E J63 , the system of linear equations has a solution.
REFERENCES
1. Kashuba, M.E. Martin, Deformations of Jordan algebras of dimension four, Journal of Algebra 399, 277-289 (2014).
2. O. O.Nuriddinov. Description of local derivations on nilpotent associative Jordan algebras of dimension five. Uzbek Mathematical Journal 2022, Volume 66, Issue 4, pp. 113-123 DOI: 10.29229/uzmj.2022-4-14
3. O. Nuriddinov, F Urinboyev. Description of local derivations on nilpotent noncommutative Jordan algebras of dimension four. Uzbek Mathematical Journal 2023, Volume 67, Issue 4,pp.77-86 DOI: 10.29229/uzmj.2023-4-8
4. O.O.Nuriddinov, F.N.Arzikulov. Description of local derivations on jordan algebras of dimension five. Preprint, 2022. 16 p.
5. Department of Mathematics, Andizhan State University, Andizhan, Uzbekistan
6. E-mail address: [email protected]
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