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TABIIY USULDA GRADUIRLANGAN FILIFORM LEYBNITS
ALGEBRALARINING KVAZI-DIFFERENSIALLASHLARI
Musayev Sardor Xabibulla o’g’li
FAN VA TEXNOLOGIYALAR UNIVERSITETI
“Aniq fanlar” kafedrasi o‘qituvchisi
ANNOTATSIYA
Ushbu maqolada tabiiy usulda graduirlangan filiform Leybnits algebralarining
ro‘yxatini keltiramiz va bu algebralarning oddiy differensiallashi, kvazi-
differensiallashini hisoblaymiz va ularning umumiy ko‘rinishini topamiz.
Kalit so‘zlar: Leybnits algebralari,
tabiiy usulda graduirlangan filiform Leybnits
algebralari
, d
ifferensiallash, kvazi-differensiallash.
ABSTRACT
In this article, we list naturally graded filiform Leibniz algebras and calculate
simple differentiation, quasi-differentiation of these algebras, and find their general
representation.
Key words:
Leibniz algebras, naturally graded filiform Leibniz algebras,
differentiation, quasi-differentiation.
AННОТАЦИЯ
В данной статье мы перечисляем естественно градуированные филиформные
алгебры
Лейбница,
вычисляем
простое
дифференцирование,
квазидифференцирование этих алгебр и находим их общее представление.
Ключевые слова:
Алгебры Лейбница, естественно градуированные
филиформные алгебры Лейбница, дифференцирование, квазидифференцирование.
KIRISH
Hozirgi kunda Li algebralarning umumlashmasi hisoblangan Leybnits
algebralari sinfi jadal suratda o‘rganilmoqda. Ta‘kidlash joizki, Leybnits ayniyatini
qanoatlantiruvchi algebralar birinchi bo‘lib 1965-yilda A.Bloxning ishida D-algebralar
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nomi bilan kiritilgan edi. Lekin, D-algebralarni o‘rganishga unchalik e‘tibor
berilmagan bo‘lib, faqatgina J.L. Lode va T.Pirashvilining ishlaridan keyingina Leybnits
algebralari jadal suratda o‘rganila boshlandi va hozirgi kunga kelib bu algebralarga
bag‘ishlangan bir qator maqolalar chop qilindi Leybnits algebralari o‘tgan asrning 90-
yillarida fransuz matematigi J.L. Lode tomonidan ushbu
[𝑥, [𝑦, 𝑧]] = [[𝑥, 𝑦], 𝑧] − [[𝑥, 𝑧], 𝑦]
Leybnits ayniyati bilan xarakterlanadigan algebra sifatida fanga kiritilgan. 1998-yildan
boshlab Leybnits algebrasining strukturaviy nazariyasini Sh.A. Ayupov va B.A.
Omirovlar o‘rgana boshladi. Algebraning o‘lchami qancha kattalashgan sari, uni
tavsiflash shuncha murakkab bo‘ladi. Nilpotent Leybnits algebralari bilan Ayupov Sh.A.,
Omirov B.A., Raximov I.S., Rixsiboev I.M., Xudoyberdiyev A.X. va boshqalar
shug‘ullangan. Katta o‘lchamdagi nilpotent Li algebralarini ham o‘rganish murakkab
bo‘lgani uchun, nilpotent algebralar bir necha sinflarga bo‘linadi. Masalan, nol filiform,
filiform, kvazi filiform va boshqa sinflar.
So‘nggi
yillarda
noassotsiativ
algebralarning
differensiallashlari
va
differensiallashlarning umumlashmalari hisoblangan qator operatorlar keng
o‘rganilmoqda. Xususan, kvazi-differensiallashlar tushunchalari operator algebralaridan
tashqari Li va Leybnits algebralari uchun ham o‘rganildi. Ushbu maqolada kichik
o’lchamli Leybnits algebralarining kvazi-differensiallashlari tushunchasi o‘rganiladi.
Kichik o’lchamli Leybnits algebralarining kvazi-differensiallashlari va ularning xossalari
aniqlanadi.
Ta’rif 1.
𝐹
maydonda berilgan
(𝐿, [−, −])
algebraning ixtiyoriy
𝑥, 𝑦, 𝑧
elementlari
uchun quyidagi Leybnits ayniyati o‘rinli bo‘lsa:
[𝑥, [𝑦, 𝑧]] = [[𝑥, 𝑦], 𝑧] − [[𝑥, 𝑧], 𝑦],
u holda
(𝐿, [−, −])
algebra Leybnits algebrasi deb ataladi.
Ta’rif 2.
Aytaylik,
𝑑: 𝐿 → 𝐿
chiziqli akslantirish bo‘lsin. Agar
(𝐿, [−, −])
Leybnits
algebrasining ixtiyoriy elementlari uchun quyidagi tenglik bajarilsa:
𝑑([𝑥, 𝑦]) = [𝑑(𝑥), 𝑦] + [𝑥, 𝑑(𝑦)],
u holda
𝑑
chiziqli akslantirish
𝐿
Leybnits algebrasining differensiallashi deyiladi.
Barcha differensiallashlar to‘plamini
𝐷𝑒𝑟(𝐿)
kabi belgilaymiz.
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Ta’rif 3.
Agar
𝐷 ∈ 𝐸𝑛𝑑(𝐿)
akslantirish uchun,
∃𝐷′, 𝐷
′′
∈ 𝐸𝑛𝑑(𝐿)
akslantirishlar topilib,
∀𝑥, 𝑦 ∈ 𝐿
elementlar uchun quyidagi ayniyat bajarilsa,
[𝐷(𝑥), 𝑦] + [𝑥, 𝐷
′
(𝑦)] = 𝐷
′′
([𝑥, 𝑦])
u holda
𝐷
akslantirishga
𝐿
Leybnits algebrasining
umumlashgan differensillashi
deyiladi.
Ta’rif 4.
Agar
𝐷 ∈ 𝐸𝑛𝑑(𝐿)
akslantirish uchun,
∃𝐷
′
∈ 𝐸𝑛𝑑(𝐿)
akslantirish topilib,
∀𝑥, 𝑦 ∈ 𝐿
elementlar uchun quyidagi ayniyat bajarilsa,
[𝐷(𝑥), 𝑦] + [𝑥, 𝐷(𝑦)] = 𝐷
′
([𝑥, 𝑦])
u holda
𝐷
akslantirishga
𝐿
Leybnits algebrasining
kvazi-differensillashi
deyiladi.
𝐿
Leybnits algebrasining barcha umumlashgan va kvazi differensiallashlari
to‘plami mos ravishda
𝐺𝐷𝑒𝑟(𝐿)
va
𝑄𝐷𝑒𝑟(𝐿)
kabi belgilanadi. Ta‘kidlash joizki,
ixtiyoriy differensiallash kvazi differensiallash bo‘ladi. Biroq, kvazi-differensiallashlar
oddiy differensiallash bo‘lmasligi mumkin.
Endi algebraning sentroidi, kvazi-sentoidi va sentral differensiallashlari
tushunchalarini aniqlaymiz.
Ta’rif 5.
𝐿
Leybnits algebrasining
∀𝑥, 𝑦 ∈ 𝐿
elementlari uchun quyidagi,
[𝐷(𝑥), 𝑦] = [𝑥, 𝐷(𝑦)] = 𝐷([𝑥, 𝑦])
ayniyatni bajaradigan
𝐷 ∈ 𝐸𝑛𝑑(𝐿)
akslantirishlarga
𝐿
Leybnits algebrasining
sentroidi
deyiladi. Barcha sentroidlar to‘plamini
𝐶(𝐿)
bilan belgilanadi.
Ta’rif 6.
𝐿
Leybnits algebrasining
∀𝑥, 𝑦 ∈ 𝐿
elementlari uchun quyidagi,
[𝐷(𝑥), 𝑦] = [𝑥, 𝐷(𝑦)]
ayniyatni bajaradigan
𝐷 ∈ 𝐸𝑛𝑑(𝐿)
akslantirishlarga
𝐿
Leybnits algebrasining
kvazi-
sentroidi
deyiladi. Barcha kvazi-sentroidlar to‘plamini
𝑄𝐶(𝐿)
bilan belgilanadi.
Ta’rif 7.
𝐿
Leybnits algebrasining
∀𝑥, 𝑦 ∈ 𝐿
elementlari uchun quyidagi,
[𝐷(𝑥), 𝑦] = [𝑥, 𝐷(𝑦)] = 𝐷([𝑥, 𝑦]) = 0
ayniyatni bajaradigan
𝐷 ∈ 𝐸𝑛𝑑(𝐿)
akslantirishlarga
𝐿
Leybnits algebrasining
sentral
differensiallashi
deyiladi. Barcha sentral differensiallashlar to‘plamini
𝑍𝐷𝑒𝑟(𝐿)
bilan
belgilanadi. Ma‘lumki,
𝑍𝐷𝑒𝑟(𝐿) ⊆ 𝐷𝑒𝑟(𝐿) ⊆ 𝑄𝐷𝑒𝑟(𝐿) ⊆ 𝐺𝐷𝑒𝑟(𝐿) ⊆ 𝐸𝑛𝑑(𝐿)
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munosabat o‘rinli bo‘ladi. Shuningdek,
𝐶(𝐿) ⊆ 𝑄𝐶(𝐿) ⊆ 𝑄𝐷𝑒𝑟(𝐿)
munosabat ham o‘rinli bo‘ladi.
NATIJALAR:
Ma‘lumki, har qanday n o‘lchamli tabiiy usulda graduirlangan filiform Leybnits
algebralari quyidagi o‘zaro izomorf bo‘lmagan algebralardan biriga izomorf bo‘ladi:
𝑭
𝒏
𝟏
: [𝒆
𝟏
, 𝒆
𝟏
] = 𝒆
𝟑
, [𝒆
𝒊
, 𝒆
𝟏
] = 𝒆
𝒊+𝟏
, 𝟐 ≤ 𝒊 ≤ 𝒏 − 𝟏
𝑭
𝒏
𝟐
: [𝒆
𝟏
, 𝒆
𝟏
] = 𝒆
𝟑
, [𝒆
𝒊
, 𝒆
𝟏
] = 𝒆
𝒊+𝟏
, 𝟑 ≤ 𝒊 ≤ 𝒏 − 𝟏
Ma‘lumki, bu algebralarning oddiy differensiallashlar fazosining matritsalari
quyidagicha bo‘ladi:
𝑫𝒆𝒓(𝑭
𝒏
𝟏
)
=
(
𝒅
𝟏,𝟏
𝒅
𝟏,𝟐
𝒅
𝟏,𝟑
. . .
𝒅
𝟏,𝒏−𝟏
𝒅
𝟏,𝒏
𝟎
𝒅
𝟏,𝟏
+ 𝒅
𝟏,𝟐
𝒅
𝟏,𝟑
. . .
𝒅
𝟏,𝒏−𝟏
𝒅
𝟐,𝒏
𝟎
𝟎
𝟐𝒅
𝟏,𝟏
+ 𝒅
𝟏,𝟐
. . .
𝒅
𝟏,𝒏−𝟐
𝒅
𝟏,𝒏−𝟏
. . .
. . .
. . .
. . .
. . .
. . .
𝟎
𝟎
𝟎
. . .
(𝒏 − 𝟐)𝒅
𝟏,𝟏
+ 𝒅
𝟏,𝟐
𝒅
𝟏,𝟑
𝟎
𝟎
𝟎
. . .
𝟎
(𝒏 − 𝟏)𝒅
𝟏,𝟏
+ 𝒅
𝟏,𝟐
)
,
𝑫𝒆𝒓(𝑭
𝒏
𝟐
) =
(
𝒅
𝟏,𝟏
𝒅
𝟏,𝟐
𝒅
𝟏,𝟑
. . .
𝒅
𝟏,𝒏−𝟏
𝒅
𝟏,𝒏
𝟎
𝒅
𝟐,𝟐
𝟎
. . .
𝟎
𝒅
𝟐,𝒏
𝟎
𝟎
𝟐𝒅
𝟏,𝟏
+ 𝒅
𝟏,𝟐
. . .
𝒅
𝟏,𝒏−𝟐
𝒅
𝟏,𝒏−𝟏
. . .
. . .
. . .
. . .
. . .
. . .
𝟎
𝟎
𝟎
. . . (𝒏 − 𝟐)𝒅
𝟏,𝟏
+ 𝒅
𝟏,𝟐
𝒅
𝟏,𝟑
𝟎
𝟎
𝟎
. . .
𝟎
(𝒏 − 𝟏)𝒅
𝟏,𝟏
+ 𝒅
𝟏,𝟐
)
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Teorema 1.
𝑭
𝒏
𝟏
tabiiy usulda graduirlangan filiform Leybnits algebrasining
𝑸𝑫𝒆𝒓(𝑭
𝒏
𝟏
)
kvazi-differensiallashlar fazosining matritsasi quyidagi ko‘rinishda bo‘ladi:
𝑸𝑫𝒆𝒓(𝑭
𝒏
𝟏
) =
(
𝒅
𝟏,𝟏
𝒅
𝟏,𝟐
𝒅
𝟏,𝟑
. . .
𝒅
𝟏,𝒏−𝟏
𝒅
𝟏,𝒏
𝟎
𝒅
𝟏,𝟏
+ 𝒅
𝟏,𝟐
𝒅
𝟏,𝟑
. . .
𝒅
𝟏,𝒏−𝟏
𝒅
𝟐,𝒏
𝟎
𝒅
𝟑,𝟐
𝒅
𝟑,𝟑
. . .
𝒅
𝟑,𝒏−𝟏
𝒅
𝟑,𝒏
. . .
. . .
. . .
. . .
. . .
. . .
𝟎
𝒅
𝒏−𝟏,𝟐
𝒅
𝒏−𝟏,𝟑
. . . 𝒅
𝒏−𝟏,𝒏−𝟏
𝒅
𝒏−𝟏,𝒏
𝟎
𝟎
𝟎
. . .
𝟎
𝒅
𝒏,𝒏
)
Endi
𝐹
𝑛
2
: [𝑒
1
, 𝑒
1
] = 𝑒
3
, [𝑒
𝑖
, 𝑒
1
] = 𝑒
𝑖+1
, 3 ≤ 𝑖 ≤ 𝑛 − 1
algebraning kvazi
differensiallashini ko‘ramiz:
Teorema 2.
𝑭
𝒏
𝟐
tabiiy usulda graduirlangan filiform Leybnits algebrasining
𝑸𝑫𝒆𝒓(𝑭
𝒏
𝟐
)
kvazi-differensiallashlar fazosining matritsasi quyidagi ko‘rinishda bo‘ladi:
𝑸𝑫𝒆𝒓(𝑭
𝒏
𝟐
) =
(
𝒅
𝟏,𝟏
𝒅
𝟏,𝟐
𝒅
𝟏,𝟑
. . .
𝒅
𝟏,𝒏−𝟏
𝒅
𝟏,𝒏
𝟎
𝒅
𝟐,𝟐
𝟎
. . .
𝟎
𝒅
𝟐,𝒏
𝟎
𝒅
𝟑,𝟐
𝒅
𝟑,𝟑
. . .
𝒅
𝟑,𝒏−𝟏
𝒅
𝟑,𝒏
. . .
. . .
. . .
. . .
. . .
. . .
𝟎
𝒅
𝒏−𝟏,𝟐
𝒅
𝒏−𝟏,𝟑
. . . 𝒅
𝒏−𝟏,𝒏−𝟏
𝒅
𝒏−𝟏,𝒏
𝟎
𝒅
𝒏,𝟐
𝟎
. . .
𝟎
𝒅
𝒏,𝒏
)
FOYDALANILGAN ADABIYOTLAR RO‘YXATI: (REFERENCES)
1.
Musayev, S. X. o'g'li . (2024). KICHIK O'LCHAMLI LEYBNITS
ALGEBRALARINING
KVAZI-DIFFERENSIYALASHLARI
VA
ULARNING
XOSSALARI. Educational research in universal sciences, 3(3), 112–119.
https://doi.org/10.5281/zenodo.10836664
2.
Musayev S. X. NOL-FILIFORM LEYBNITS ALGEBRALARINING KVAZI-
DIFFERENSIALLASHLARI.
Mathematics,
mechanics
and
intellectual
technologies tashkent-2023. 227-228.
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3. Musayev S.X. LEYBNITS ALGEBRALARINING KVAZI-
DIFFERENSIALLASHLARI VA ULARNING XOSSALARI.
Operator algebralar,
noassotsiativ tuzilmalar va turdosh masalalar. Sentabr-2022. 118-120 betlar.
4.
Musayev S. X. Elementary properties of centroid, quasicentroid and central
derivations of Leibniz algebras. ABSTRACTS OF COMMUNICATIONS:
INTERNATIONAL CONFERENCE LIMIT THEOREMS OF PROBABILITY
THEORY AND MATHEMATICAL STATISTICS. September 26-28, 2022 Tashkent,
Uzbekistan.
5.
Musayev, S. (2024). QUASI-DERIVATIONS OF LOW-DIMENSIONAL LEIBNIZ
ALGEBRAS AND THEIR PROPERTIES. В INTERNATIONAL BULLETIN OF
APPLIED SCIENCE AND TECHNOLOGY (Т. 4, Выпуск 6, сс. 81–86). Zenodo.
https://doi.org/10.5281/zenodo.11560646
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Musayev S. X.
UCH O‘LCHAMLI NILPOTENT LEYBNITS ALGEBRALARINING
KVAZIDIFFERENSIALLASHLARI VA ULARNING XOSSALARI. RAQAMLI
TEXNOLOGIYALAR ASOSIDA TA’LIM JARAYONINI TAKOMILLASHTIRISH.
28-mart, 2024-yil.
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QUASI-DERIVATIONS OF SMALL SIZE LEIBNITZ ALGEBRAS
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JARAYONINI
TAKOMILLASHTIRISH.
28-mart,
2024-yil.
https://doi.org/10.5281/zenodo.11198320
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Musayev S. X.
IKKI O‘LCHAMLI LEYBNITS ALGEBRALARINING
KVAZIDIFFERENSIALLASHLARI VA ULARNING XOSSALARI. RAQAMLI
TEXNOLOGIYALAR ASOSIDA TA’LIM JARAYONINI TAKOMILLASHTIRISH.
28-mart, 2024-yil.
https://doi.org/10.5281/zenodo.11198320
9. .
Musayev S. X.
RAQAMLI TEXNOLOGIYALAR ASOSIDA MATEMATIKANI
O’QITISH. // RAQAMLI TEXNOLOGIYALAR ASOSIDA TA’LIM JARAYONINI
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EDUCATION. RAQAMLI TEXNOLOGIYALAR ASOSIDA TA’LIM JARAYONINI
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OSHIRISHNI TASHKIL ETISH VA UNI BOSHQARISH TEXNOLOGIYALARI. 14-
dekabr, 2023-yil.
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DOI 10.5281/zenodo.10588135
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TABIIY USULDA GRADUIRLANGAN FILIFORM LEYBNITS
ALGEBRALARINING KVAZI-DIFFERENSIALLASHLARI TASNIFI.// RAQAMLI
TEXNOLOGIYALAR ASOSIDA TA’LIM JARAYONINI TAKOMILLASHTIRISH.
28-mart, 2024-yil.
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