2Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current Challenges, Innovations and Prospects
KICHIK O'LCHAMLI KVAZI-FILIFORM SIMMETRIK LEYBNITS
ALGEBRALARINING TASNIFI
I. B. Chorieva
V.I.Romanovskiy nomidagi Matematika instituti tayanch doktoranti.
O'zbekiston.
Noassotsiativ algebralar nazariyasi bugungi kunda tez suratlarda o'rganilib kelinayotgan sohalardan biri bo'lib, Li algebralarining umumlashmasi bo'lgan Leybnits algebralari sinfi alohida ahamiyat kasb etadi. Leybnits algebralari barcha chapdan yoki o'ngdan ko'paytirish operatorlari differentsiallash bo'ladigan algebra sifatida fanga kiritilgan bo'lib, ular mos ravishda chap va o'ng Leybnits algebralari deb nomlanadi. Chap va o'ng Leybnits algebralari simmetrik xususiyatlarga ega bo'lganligi uchun odatda ulardan biri biri o'rganiladi. Bir vaqtning o'zida ham chap ham o'ng Leybnits algebrasi bo'ladigan algebralar esa simmetrik Leybnits algebrasi deb ataladi. Bugungi kunda simmetrik Leybnits algebralarini tasniflashning bir qator usullari mavjud bo'lib, ulardan biri 2014-yilda S.Benayadi va S.Hidri [3] tomonidan taklif qilingan usul hisoblanadi. Bu usul barcha simmetrik Leybnits algebralarini Li algebralari yordamida qurish mumkinligiga asoslanadi. Biz ushbu ishda S.Benayadi va S.Hidri usulidan foydalangan holda ba'zi kichik o'lchamli qvazi-filiform simmetrik Leybnits algebralarining tasnifini keltiramiz.
1-ta'rif: F maydon ustida berilgan (L,[-,-]) algebradan olingan ixtiyoriy
x,y,z elementlar uchun quyidagi ayniyatlar bajarilsa, u holda (L,[-,-]) Li algebrasi deyiladi:
[ x y ] = -[y, x], [ x,[ y, z ]] + [ z,[^ y ]] + [ y,[z, x ]] = 0.
2-ta'rif: F maydon ustida berilgan (L,•) algebradan olingan ixtiyoriy x,y,z
elementlar uchun quyidagi ayniyatlar bajarilsa, u holda (L,•) simmetrik Leybnits
algebrasi deyiladi:
x •( y • z) = (x • y)• z + y •(x • z),(x • y)• z = x •(y • z ) + (x • z)• y.
Berilgan L simmetrik Leybnits algebrasi uchun quyidagi quyi markaziy qatorni aniqlab olamiz:
L1 = L,Li =[Li-1,L], i > 2.
3-ta'rif: Agar n -o'lchamli L simmetrik Leybnits algebrasi uchun
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Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current
dim ( L ) = n — i — 2,1 < i < n — 2 shart o'rinli bo'lsa, u holda L algebra kvazi-
filiform simmetrik Leybnits algebrasi deyiladi: Quyida biz kichik o'lchamli kvazi-filiform Li algebralarining tasnifini keltiramiz [2], [4]:
K i ei+1 , [e7 , e2 ] = e5 , [e7 , e3 ] = ^ [e2 , e4 ] = ^ [e2 , e5 ] = e6 , 2 < i < 5
[e2,3 ] = É?4 + é?7.
[^ ei ] = i [el, e9 ] = e7 , [e2 , e5 ] = e9 , [e2 , e6 ] = 2e7 , [e2 , e7 ] = 3e8,2 < i < 5 [e3 , e4 ] = —e9 , [e3 , e5 ] = —e7 , [e3 , e9 ] = —3e8.
[ei,ei] = ei+1,[e9,e2 ] = 2e7 , [e9 , e3 ] = 2e8 , [e2 , e6 ] = 2e7 , [e2 , e7 ] = e8 , 2 < ■ < 7
L
'7,3
L1
9,5
L2
L3
9,5
9,5
[e3, e5 ] = —ee7, [e2 , e5 ] = e6 + e9 , [e4 , e5 ] = —2e8.
[el, ei] = e+A^ e7 ] = e8 , [ei , e9 ] = e7 , [e2 , e5 ] = e9 , [e2 , e6 ] = 2e7 , 2 < ■ < 5 [^ e4 ] = —e9,[e3, e5 ] = —'e6 ] = 2e8 , [e4 , e5 ] = —3e8.
Bizga ( L, •) algebra berilgan bo'lsin. U holda quyidagi ko'paytmalarni aniqlab olamiz:
[ x y ]=1 (x • y—y •x ),
X o
1-tasdiq. [1] Quyidagi tasdiqlar ekvivalent:
1) ( L, •) simmetrik Leybnits algebrasi.
2) Quyidagi shartlar bajariladi:
(a) ( L,[—,—]) Li algebrasi.
(b)Ixtiyoriy x,y<=L uchun xoy ko'paytma (l,[-,-]) algebraning markaziga tegishli bo'ladi.
(c) Ixtiyoriy x,y,zeL uchun ([x, j]) ° z = 0 va(io>')o; = 0.
Bu tasdiqdan kelib chiqadiki, ( L, •) simmetrik Leybnits algebrasini ( L,[—,—])
Li algebrasi va ixtiyoriy ( L,[—,—]) Li algebrasidan olingan x, y, z
elementlar uchun, œ([ x, y ], z ) = œ(œ( x, y ), z ) = 0
tenglikni qanoatlantiruvchi œ : L x L ^ Z (L) (bu erda Z (L ) Li algebrasining
markazi) simmetrik bichiqli forma orqali berish mumkin. Ya'ni berilgan Li algebrasi va undagi simmetrik bichiqli forma orqali simmetrik Leybnits algebrasining
ko'paytmasi quyidagicha aniqlanadi:
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259
Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current
Cha'Les.hnnoeon.and^en
x • y = [x,y] + co(x,y).
2-tasdiq: L7 3, L!95, L295 va L395 Li algebralari orqali hosil qilingan simmetrik Leybnits algebralari sinfi mos ravishda quyidagicha bo'ladi:
S
(a,ß,7) .
S
1 (« ,Д , ) .
9,5
ei • e = ei+1 e • ei = — ei+1 e7 • e2 = e5 , e2 • e7 = -е5,Э — i — 5, e7 • e3 = e6 , e3 • e7 = —e6 , e2 • e4 = e5 , e4 • e2 = — e5 , e2 • e5 = e6 , e5 • e2 = —e6 , e2 • e3 = e4 + e7 , e3 • e2 = — e4 — e7 , ei • ei = «É?^ ei • e2 = e3 + ße^ e2 • ei = —e3 + ße6, e2 • e2 = Ye6-ei • ei = ei+ ei * ei = —ei+ ei • e9 = É7 , e9 • ei = —e7,3 — i — 5, e2 • e5 = e9 , e5 • e2 = —e9 , e3 • e4 = —e9 , e4 • e3 = e9 , e2 • e6 2e7 , e6 • e2 2e7 , e2 • e7 3e8 , e7 • e2 3é8 ,
e e^ , ^^ e e^f, e ^^ 3 ^^, ^^ 3 eg,
ei • ei = C^ie8 , ei • e2 = e3 + ßie8 , e2 ' ei = —e3 + ßie8 , e2 • e2 = ^"ie8. ei • ei ez+1, ei " ei ez+1 , e2 ' e7 e8 , e7 • e2 e8 , 3 — i — 7,
ç<2 («2,ß2,2 ) .
S 9,5 •
e3 • e5 e7 , e5 • e3 e7 , e2 • e6 2e7 , e6 • e2 2é7 ,
2 , 2 , e^ 2 ^^, ^^ 2 ^^,
e
e5 e6 + e9, e5 • e2 e6 e9, e4 • e5 2é8, e5 • e4 2é8'
ei • ei «2e8, ei • e2 e3 + ß2e8 , e2 • ei e3 + ß2e8, e2 • e2 ^2e8.
S
3 («3,ß3,/3) .
9,5
ei • ei
e • e
e1 e9
2 8 2 2
ei+1, ^^ ' ei = —ei+1, ei • e7 = e8, e7 • ei = —e8,3 — i — 7,
: e e • e = —e e • e = —e e • e = e
7 9 i 7 3 4 9 4 3 9
e2 • e5 = e9 , e5 • e2 = —e9, e2 • e6 = 2é7 , e6 • e2 = —2é7 , e3 • e5 = —e7 , e5 • e3 = e7 , e3 • e6 = 2e8, e6 • e3 = —2é8 , e4 • e5 = —3e8, e5 • e4 = 3e8, ei • ei = «3e8, e2 • e2 = Y3e8, ei • e2 = e3 + ß3e8,e2 • ei = —e3 + ß3e8.
1-teorema: Agar S simmetrik Leybnits algebrasi L7 3 orqali hosil qilingan simmetrik Leybnits algebrasi bo'lsa, u holda u o'zaro izomorf bo'lmagan S73(1'ß,r),S7,3(0,1,r),S73(0,0,1) algebralardan biriga izomorf bo'ladi.
2-teorema: Agar S simmetrik Leybnits algebrasi l}9 5 orqali hosil qilingan simmetrik Leybnits algebrasi bo'lsa, u holda u o'zaro izomorf bo'lmagan
oi (1,1,0) Ы (0,1,0) Ç,1 95 , 95 , S о ^ ,S о ç , S
bo'ladi.
3-teorema: Agar S simmetrik Leybnits algebrasi L29 5 orqali hosil qilingan simmetrik Leybnits algebrasi bo'lsa, u holda u o'zaro izomorf bo'lmagan
о 2 (1,ßj,r2 ) çr2 (0,1,r2 ) çr2 (0,0,1) ,S 95 ,S 95
C1 («1,1,1) C1 (1,0,1) C1 (0,0,1) C1 (1,1,0) C1 (0,1,0) C1 (1,0,0) 1 u 1 J U- • • f
S 9,5 1 , S 9,5 , S 9,5 , S 9,5 , S 95 , S 9,5 ' algebralardan biriga izomorf
9,5
algebralardan biriga izomorf bo'ladi.
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Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current _Challenges, Innovations and Prospects
4-teorema: Agar S simmetrik Leybnits algebrasi Z39 5 orqali hosil qilingan simmetrik Leybnits algebrasi bo'lsa, u holda u o'zaro izomorf bo'lmagan
o3 («3,1,1) C3 (1,0,1) C3 (0,0,1) C3 («3,1,0) C3 (0,1,0) c3 (1,0,0) , u , , u- • • r-
S 9,5 3 , S 9,5 , S 9,5 , S 9,5 3 , S 9,5 , S 9,j ; algebralardan biriga izomorf bo'ladi.
REFERENCES
1. Abchir.H., Abid.F., Boucetta.M., A class of Lie racks associated to symmetric Leibniz algebras, Journal of Algebra and Its Applications, 21 (2022) 2250230.
2. Ancochea Bermudez.J.M., Campoamor-Stursberg.R., Garcia Vergnolle.L., Classification of Lie algebras with naturally graded quasi-filiform nilradicals, Journal of Geometry and Physics, 61 (2011) 2168-2186.
3. Benayadi.S., Hidri.S., Quadratic Leibniz algebras, Journal of Lie Theory, 24 (2014) 739-759.
4. Gomez. J.R., Jimenez-Merchan.A., Naturally graded quasi-filiform Lie algebras, Journal of Algebra, 256 (2002) 211-228.
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