Lie guruhlari va Lie algebralari lug'ati - Glossary of Lie groups and Lie algebras
Vikipediya lug'ati
Bu lug'at da qo'llanilgan terminologiya uchun matematik nazariyalari Yolg'on guruhlar va Yolg'on algebralar . Lie guruhlari va Lie algebralarining vakillik nazariyasidagi mavzular uchun qarang Vakillik nazariyasining lug'ati . Boshqa variantlar yo'qligi sababli, lug'at, shuningdek, ba'zi bir umumlashtirishlarni o'z ichiga oladi kvant guruhi .
Izohlar :
Lug'at davomida, ( ⋅ , ⋅ ) { displaystyle ( cdot, cdot)} ichki mahsulot Evklidlar makonining E va ⟨ ⋅ , ⋅ ⟩ { displaystyle langle cdot, cdot rangle} ⟨ β , a ⟩ = ( β , a ) ( a , a ) ∀ a , β ∈ E . { displaystyle langle beta, alfa rangle = { frac {( beta, alfa)} {( alfa, alfa)}}}, forall alfa, beta in E.} A
abeliya 1. An abelyan Lie guruhi abel guruhi bo'lgan Lie guruhi. 2. An abeliyan algebra bu yolg'on algebra [ x , y ] = 0 { displaystyle [x, y] = 0} x , y { displaystyle x, y} qo'shma 1. An Yolg'on guruhining qo'shma vakili : E'lon : G → GL ( g ) { displaystyle operatorname {Ad}: G dan operatorname {GL} ({ mathfrak {g}})} shu kabi E'lon ( g ) { displaystyle operatorname {Ad} (g)} v g : G → G , x ↦ g x g − 1 { displaystyle c_ {g}: G dan G, x mapsto gxg ^ {- 1}} 2. An Lie algebrasining qo'shma tasviri Lie algebra tasviridir reklama : g → g l ( g ) { displaystyle { textrm {ad}}: { mathfrak {g}} to { mathfrak {gl}} ({ mathfrak {g}})} reklama ( x ) y = [ x , y ] { displaystyle { textrm {ad}} (x) y = [x, y]} Ado Ado teoremasi : Har qanday sonli o'lchovli Lie algebrasi subalgebra uchun izomorfdir g l V { displaystyle { mathfrak {gl}} _ {V}} afine 1. An afine Lie algebra Kac-Moody algebrasining ma'lum bir turi. 2. An affin Veyl guruhi . analitik 1. An analitik kichik guruh B
B 1. (B, N) juftlik Borel 1. Armand Borel (1923 - 2003), shveytsariyalik matematik 2. A Borel kichik guruhi . 3. A Borel subalgebra maksimal echiladigan subalgebra. 4. Borel-Bott-Vayl teoremasi Bruhat 1. Bruhat parchalanishi C
Kartan 1. Élie Cartan (1869 - 1951), frantsuz matematikasi 2. A Cartan subalgebra h { displaystyle { mathfrak {h}}} g { displaystyle { mathfrak {g}}} N g ( h ) = h { displaystyle N _ { mathfrak {g}} ({ mathfrak {h}}) = { mathfrak {h}}} 3. Erituvchanlik uchun karton mezon : Yolg'on algebra g { displaystyle { mathfrak {g}}} iff κ ( g , [ g , g ] ) = 0 { displaystyle kappa ({ mathfrak {g}}, [{ mathfrak {g}}, { mathfrak {g}}]) = 0} 4. Yarim soddaligi uchun karton mezon : (1) Agar κ ( ⋅ , ⋅ ) { displaystyle kappa ( cdot, cdot)} g { displaystyle { mathfrak {g}}} g { displaystyle { mathfrak {g}}} F { displaystyle F} κ ( ⋅ , ⋅ ) { displaystyle kappa ( cdot, cdot)} 5. The Kartan matritsasi ildiz tizimining Φ { displaystyle Phi} ( ⟨ a men , a j ⟩ ) men , j = 1 n { displaystyle ( langle alpha _ {i}, alfa _ {j} rangle) _ {i, j = 1} ^ {n}} Δ = { a 1 … a n } { displaystyle Delta = { alfa _ {1} ldots alpha _ {n} }} Φ { displaystyle Phi} 6. Cartan kichik guruhi 7. Karton parchalanishi Casimir Casimir o'zgarmas , universal o'ralgan algebraning taniqli elementi.Klibsh-Gordan koeffitsientlari Klibsh-Gordan koeffitsientlari markaz 2. Ichki to'plamning markazlashtiruvchisi X { displaystyle X} g { displaystyle { mathfrak {g}}} C g ( X ) := { x ∈ g | [ x , X ] = { 0 } } { displaystyle C _ { mathfrak {g}} (X): = {x in { mathfrak {g}} | [x, X] = {0 } }} markaz 1. Yolg'on guruhining markazi bu markaz guruhning. 2. Yolg'on algebra markazi o'zi markazlashtiruvchisi: Z ( L ) := { x ∈ g | [ x , g ] = 0 } { displaystyle Z (L): = {x in { mathfrak {g}} | [x, { mathfrak {g}}] = 0 }} markaziy seriyalar 1. A tushayotgan markaziy qator (yoki pastki markaziy qator) - bu Lie algebrasining ideallar ketma-ketligi L { displaystyle L} C 0 ( L ) = L , C 1 ( L ) = [ L , L ] , C n + 1 ( L ) = [ L , C n ( L ) ] { displaystyle C ^ {0} (L) = L, , C ^ {1} (L) = [L, L], , C ^ {n + 1} (L) = [L, C ^ { n} (L)]} 2. An ortib borayotgan markaziy qator (yoki yuqori markaziy qator) - bu Lie algebra ideallari ketma-ketligi L { displaystyle L} C 0 ( L ) = { 0 } , C 1 ( L ) = Z ( L ) { displaystyle C_ {0} (L) = {0 }, , C_ {1} (L) = Z (L)} C n + 1 ( L ) = π n − 1 ( Z ( L / C n ( L ) ) ) { displaystyle C_ {n + 1} (L) = pi _ {n} ^ {- 1} (Z (L / C_ {n} (L)))} π men { displaystyle pi _ {i}} L → L / C n ( L ) { displaystyle L dan L / C_ {n} (L)} gacha Chevalley 1. Klod Chevalley (1909 - 1984), frantsuz matematikasi 2. A Chevalley asoslari a asos tomonidan qurilgan Klod Chevalley hamma mol-mulk bilan tuzilish konstantalari butun sonlar. Chevalley ushbu asoslardan analoglarini qurish uchun foydalangan Yolg'on guruhlar ustida cheklangan maydonlar , deb nomlangan Chevalley guruhlari . murakkab aks ettirish guruhi murakkab aks ettirish guruhi coroot coroot Kokseter 1. H. S. M. Kokseter (1907 - 2003), Britaniyada tug'ilgan Kanadalik geometr 2. Kokseter guruhi 3. Kokseter raqami D.
olingan algebra 1. The yolg'on algebra algebra g { displaystyle { mathfrak {g}}} [ g , g ] { displaystyle [{ mathfrak {g}}, { mathfrak {g}}]} 2. Olingan qator - bu Lie algebra ideallari ketma-ketligi g { displaystyle { mathfrak {g}}} D. 0 g = g , D. n g = D. n − 1 g { displaystyle D ^ {0} { mathfrak {g}} = { mathfrak {g}}, D ^ {n} { mathfrak {g}} = D ^ {n-1} { mathfrak {g} }} Dinkin 1. Evgeniy Borisovich Dinkin (1924 - 2014), sovet va amerikalik matematik 2. Dynkin diagrammalari
Dynkin diagrammalari . E
kengaytma Aniq ketma-ketlik 0 → g ′ → g → g ″ → 0 { displaystyle 0 to { mathfrak {g}} ' to { mathfrak {g}} to { mathfrak {g}} ^ {' '} to 0} g { displaystyle { mathfrak {g}}} Yolg'on algebra kengaytmasi ning g ″ { displaystyle { mathfrak {g}} ^ {''}} g ′ { displaystyle { mathfrak {g}} '} eksponent xarita The eksponent xarita Yolg'on guruhi uchun G bilan g { displaystyle { mathfrak {g}}} g → G { displaystyle { mathfrak {g}} dan G} gacha eksponent E6 , E7 , E7½ , E8 , En , Favqulodda yolg'on algebra F
bepul algebra F F4 asosiy Uchun "Veylning asosiy kamerasi ", qarang #Veyl . G
G G2 umumlashtirilgan 1. uchun "Umumiy karton matritsasi ", qarang #Kartan . 2. uchun "Umumlashtirilgan Kac-Moody algebra ", qarang # Kac - Moody algebra . 3. uchun "Umumlashtirilgan Verma moduli ", qarang #Verma . H
homomorfizm 1. A Yolg'on guruhi gomomorfizmi guruh homomorfizmi bo'lib, u ham silliq xaritadir. 2. A Yolg'on algebra homomorfizmi chiziqli xarita ϕ : g 1 → g 2 { displaystyle phi: { mathfrak {g}} _ {1} to { mathfrak {g}} _ {2}} ϕ ( [ x , y ] ) = [ ϕ ( x ) , ϕ ( y ) ] ∀ x , y ∈ g 1 . { displaystyle phi ([x, y]) = [ phi (x), phi (y)] , for all x, y in { mathfrak {g}} _ {1}.} Xarish-Chandra 1. Xarish-Chandra , (1923 - 1983), hind amerikalik matematik va fizik 2. Xarish-Chandra gomomorfizmi eng yuqori 1. The eng katta vazn teoremasi , eng yuqori og'irliklarni ko'rsatib, kamaytirilmaydigan tasavvurlarni tasniflang. 2. eng yuqori vazn 3. eng og'ir vaznli modul Men
ideal An ideal yolg'on algebra g { displaystyle { mathfrak {g}}} g ′ { displaystyle { mathfrak {g '}}} [ g ′ , g ] ⊆ g ′ . { displaystyle [{ mathfrak {g '}}, { mathfrak {g}}] subseteq { mathfrak {g'}}.} indeks Yolg'on algebra ko'rsatkichi o'zgarmas konveks konus An o'zgarmas konveks konus ichki avtomorfizmlar ostida o'zgarmas bo'lgan, ulangan Lie guruhining Lie algebrasidagi yopiq konveks konusdir. Ivasava parchalanishi Ivasava parchalanishi J
Jakobining o'ziga xosligi 1. Karl Gustav Yakob Jakobi
Karl Gustav Yakob Jakobi (1804 - 1851), nemis matematikasi. 2. Ikkilik amal berilgan [ , ] : V 2 → V { displaystyle [,]: V ^ {2} dan V} gacha Jakobining o'ziga xosligi davlatlar: [[x , y ], z ] + [[y , z ], x ] + [[z , x ], y ] = 0. K
Kac-Moody algebra Kac-Moody algebra Qotillik 1. Vilgelm o'ldirish (1847 - 1923), nemis matematikasi. 2. The Qotillik shakli yolg'on algebra bo'yicha g { displaystyle { mathfrak {g}}} κ ( x , y ) := Tr ( reklama x reklama y ) ∀ x , y ∈ g { displaystyle kappa (x, y): = { textrm {Tr}} ({ textrm {ad}} , x , { textrm {ad}} , y) for all x, y { mathfrak {g}}} da Kirillov Kirillov belgilar formulasi L
Langlendlar Langlandlarning parchalanishi Langlands dual Yolg'on 1. Sofus yolg'on
Sofus yolg'on (1842 - 1899), a Norvegiyalik matematik 2. A Yolg'on guruh silliq manifoldning mos keladigan tuzilishiga ega bo'lgan guruhdir. 3. A Yolg'on algebra vektor maydoni g { displaystyle { mathfrak {g}}} F { displaystyle F} Yolg'on qavs yoki qisqartirish. qavs ), bu quyidagi shartlarni qondiradi: ∀ a , b ∈ F , x , y , z ∈ g { displaystyle forall a, b in F, x, y, z in { mathfrak {g}}} [ a x + b y , z ] = a [ x , z ] + b [ y , z ] { displaystyle [ax + by, z] = a [x, z] + b [y, z]} bilinmaslik ) [ x , x ] = 0 { displaystyle [x, x] = 0} o'zgaruvchan ) [ [ x , y ] , z ] + [ [ y , z ] , x ] + [ [ z , x ] , y ] = 0 { displaystyle [[x, y], z] + [[y, z], x] + [[z, x], y] = 0} Jakobining o'ziga xosligi ) 4. Yolg'on guruhi - Yolg'on algebra yozishmalari 5. Yolg'on teoremasi Ruxsat bering g { displaystyle { mathfrak {g}}} hal etiladigan Lie algebra ustida algebraik yopiq maydon xarakterli 0 { displaystyle 0} V { displaystyle V} vakillik ning g { displaystyle { mathfrak {g}}} V { displaystyle V} xususiy vektor ning barcha elementlari uchun g { displaystyle { mathfrak {g}}} 6. Compact Lie guruhi . 7. Semisimple Lie guruhi ; qarang #emisimple . Levi Levi parchalanishi N
nolpotent 1. A nilpotent Lie guruhi . 2. A nilpotent yolg'on algebra bu yolg'on algebra nolpotent ideal sifatida; ya'ni ba'zi kuch nolga teng: [ g , [ g , [ g , … , [ g , g ] … ] ] ] = 0 { displaystyle [{ mathfrak {g}}, [{ mathfrak {g}}, [{ mathfrak {g}}, dots, [{ mathfrak {g}}, { mathfrak {g}}] nuqta]]] = 0} 3. A nilpotent element yarim semple Lie algebra[1] x shundayki qo'shma endomorfizm a d x { displaystyle ad_ {x}} 4. A nilpotent konus normalizator Subspace normalizatori K { displaystyle K} g { displaystyle { mathfrak {g}}} N g ( K ) := { x ∈ g | [ x , K ] ⊆ K } { displaystyle N _ { mathfrak {g}} (K): = {x in { mathfrak {g}} | [x, K] subseteq K }} M
maksimal 1. uchun "maksimal ixcham kichik guruh ", qarang # ixcham . 2. uchun "maksimal torus ", qarang #torus . P
parabolik 1. Parabolik kichik guruh . 2. Parabolik subalgebra . ijobiy Uchun "ijobiy ildiz ", qarang #ijobiy . Q
kvant kvant guruhi .kvantlangan kvantlangan zarf algebra .R
radikal 1. The Yolg'on guruhining radikal . 2. The Lie algebra radikalidir g { displaystyle { mathfrak {g}}} g { displaystyle { mathfrak {g}}} haqiqiy haqiqiy shakl .reduktiv 1. A reduktiv guruh . 2. A reduktiv Lie algebra . aks ettirish A aks ettirish guruhi , aks ettirish natijasida hosil bo'lgan guruh. muntazam 1. A Lie algebrasining oddiy elementi . 2. Ildiz tizimiga nisbatan muntazam element.Ruxsat bering Φ { displaystyle Phi} γ ∈ E { displaystyle gamma in E} ( γ , a ) ≠ 0 ∀ γ ∈ Φ { displaystyle ( gamma, alpha) neq 0 , forall gamma in Phi} Oddiy ildizlarning har bir to'plami uchun Δ { displaystyle Delta} Φ { displaystyle Phi} γ ∈ E { displaystyle gamma in E} ( γ , a ) > 0 ∀ γ ∈ Δ { displaystyle ( gamma, alpha)> 0 , forall gamma in Delta} γ { displaystyle gamma} Δ ( γ ) { displaystyle Delta ( gamma)} Δ = Δ ( γ ) { displaystyle Delta = Delta ( gamma)} Φ + ( γ ) = { a ∈ Φ | ( a , γ ) > 0 } { displaystyle Phi ^ {+} ( gamma) = { alpha in Phi | ( alpha, gamma)> 0 }} a { displaystyle alpha} Φ + ( γ ) { displaystyle Phi ^ {+} ( gamma)} a = a ′ + a ″ { displaystyle alpha = alfa '+ alfa' '} a ′ , a ″ ∈ Φ + ( γ ) { displaystyle alpha ', alfa' ' in Phi ^ {+} ( gamma)} Δ ( γ ) { displaystyle Delta ( gamma)} Φ + ( γ ) { displaystyle Phi ^ {+} ( gamma)} ildiz 1. yarim semple Lie algebra ildizi :Ruxsat bering g { displaystyle { mathfrak {g}}} h { displaystyle { mathfrak {h}}} g { displaystyle { mathfrak {g}}} a ∈ h ∗ { displaystyle alpha in { mathfrak {h}} ^ {*}} g a := { x ∈ g | [ h , x ] = a ( h ) x ∀ h ∈ h } { displaystyle { mathfrak {g _ { alpha}}}: = {x in { mathfrak {g}} | [h, x] = alfa (h) x , forall h in { mathfrak {h}} }} a { displaystyle alpha} g { displaystyle { mathfrak {g}}} g a ≠ { 0 } { displaystyle { mathfrak {g _ { alpha}}} neq {0 }} Barcha ildizlarning to'plami bilan belgilanadi Φ { displaystyle Phi} 2. Ildiz tizimi Ichki to‘plam Φ { displaystyle Phi} E { displaystyle E} Φ { displaystyle Phi} oraliq ( Φ ) = E { displaystyle { textrm {span}} ( Phi) = E} 0 ∉ Φ { displaystyle 0 notin Phi} Barcha uchun a ∈ Φ { displaystyle alpha in Phi} v ∈ R { displaystyle c in mathbb {R}} v a ∈ Φ { displaystyle c alpha in Phi} v = ± 1 { displaystyle c = pm 1} Barcha uchun a , β ∈ Φ { displaystyle alfa, beta in Phi} ⟨ a , β ⟩ { displaystyle langle alpha, beta rangle} Barcha uchun a , β ∈ Φ { displaystyle alfa, beta in Phi} S a ( β ) ∈ Φ { displaystyle S _ { alpha} ( beta) in Phi} S a { displaystyle S _ { alpha}} a { displaystyle alpha} S a ( x ) = x − ⟨ x , a ⟩ a { displaystyle S _ { alpha} (x) = x- langle x, alpha rangle alpha} 3. Ildiz ma'lumotlari 4. Ildiz tizimining ijobiy ildizi Φ { displaystyle Phi} Δ { displaystyle Delta} Φ { displaystyle Phi} Δ { displaystyle Delta} 5. Ildiz tizimining salbiy ildizi Φ { displaystyle Phi} Δ { displaystyle Delta} Φ { displaystyle Phi} Δ { displaystyle Delta} 6. uzun ildiz 7. qisqa ildiz 8. ildiz tizimiga teskari: Ildiz tizimi berilgan Φ { displaystyle Phi} a v = 2 a ( a , a ) { displaystyle alfa ^ {v} = { frac {2 alfa} {( alfa, alfa)}}} Φ v = { a v | a ∈ Φ } { displaystyle Phi ^ {v} = { alfa ^ {v} | alfa in Phi }} Φ v { displaystyle Phi ^ {v}} Φ { displaystyle Phi} 9. ildiz tizimining asosi: "oddiy ildizlar to'plami" ning sinonimi 10. ikkilamchi ildiz tizimi: "ildiz tizimiga teskari" so'zining sinonimi S
Serre Serr teoremasi (cheklangan qisqartirilgan) ildiz tizimi berilganligini ta'kidlaydi Φ { displaystyle Phi} Φ { displaystyle Phi} oddiy 1. A oddiy Lie guruhi abeliya bo'lmagan, noan'anaviy ravishda bog'langan oddiy kichik guruhlarga ega bo'lmagan birlashtirilgan Lie guruhi. 2. A oddiy algebra yolg'on algebra, u abelian emas va faqat ikkita idealga ega, o'zi va { 0 } { displaystyle {0 }} 3. shunchaki bog'langan guruh (oddiy Lie guruhi, shunchaki Dynkin diagrammasi ko'p qirrali bo'lmagan holda bog'langan). 4. oddiy ildiz . Ichki to‘plam Δ { displaystyle Delta} Φ { displaystyle Phi} Δ { displaystyle Delta} E { displaystyle E} Ning har bir elementi Φ { displaystyle Phi} Δ { displaystyle Delta} 5. Oddiy Lie algebralarining tasnifi Klassik yolg'on algebralari :
Maxsus chiziqli algebra A l ( l ≥ 1 ) { displaystyle A_ {l} (l geq 1)} l 2 + 2 l { displaystyle l ^ {2} + 2l} s l ( l + 1 , F ) = { x ∈ g l ( l + 1 , F ) | T r ( x ) = 0 } { displaystyle { mathfrak {sl}} (l + 1, F) = {x in { mathfrak {gl}} (l + 1, F) | Tr (x) = 0 }} izsiz matritsalar)Ortogonal algebra B l ( l ≥ 1 ) { displaystyle B_ {l} (l geq 1)} 2 l 2 + l { displaystyle 2l ^ {2} + l} o ( 2 l + 1 , F ) = { x ∈ g l ( 2 l + 1 , F ) | s x = − x t s , s = ( 1 0 0 0 0 Men l 0 Men l 0 ) } { displaystyle { mathfrak {o}} (2l + 1, F) = {x in { mathfrak {gl}} (2l + 1, F) | sx = -x ^ {t} s, s = { begin {pmatrix} 1 & 0 & 0 0 & 0 & I_ {l} 0 & I_ {l} & 0 end {pmatrix}} }} Simpektik algebra C l ( l ≥ 2 ) { displaystyle C_ {l} (l geq 2)} 2 l 2 − l { displaystyle 2l ^ {2} -l} s p ( 2 l , F ) = { x ∈ g l ( 2 l , F ) | s x = − x t s , s = ( 0 Men l − Men l 0 ) } { displaystyle { mathfrak {sp}} (2l, F) = {x in { mathfrak {gl}} (2l, F) | sx = -x ^ {t} s, s = { begin { pmatrix} 0 va I_ {l} - I_ {l} & 0 end {pmatrix}} }} Ortogonal algebra D. l ( l ≥ 1 ) { displaystyle D_ {l} (l geq 1)} 2 l 2 + l { displaystyle 2l ^ {2} + l} o ( 2 l , F ) = { x ∈ g l ( 2 l , F ) | s x = − x t s , s = ( 0 Men l Men l 0 ) } { displaystyle { mathfrak {o}} (2l, F) = {x in { mathfrak {gl}} (2l, F) | sx = -x ^ {t} s, s = { begin { pmatrix} 0 va I_ {l} I_ {l} & 0 end {pmatrix}} }}
Favqulodda yolg'on algebralari :
yarim oddiy 1. A semisimple Lie group 2. A yarim semple Lie algebra nolga teng bo'lmagan abeliyan idealga ega bo'lmagan noldan iborat Lie algebraidir. 3. A yarim oddiy element Lie algebra yarim semplidir hal etiladigan 1. A hal qilinadigan Yolg'on guruhi 2. A hal etiladigan Lie algebra yolg'on algebra g { displaystyle { mathfrak {g}}} D. n g = 0 { displaystyle D ^ {n} { mathfrak {g}} = 0} n ≥ 0 { displaystyle n geq 0} D. g = [ g , g ] { displaystyle D { mathfrak {g}} = [{ mathfrak {g}}, { mathfrak {g}}]} g { displaystyle { mathfrak {g}}} Split Stiefel Stifel diagrammasi ixcham bog'langan Lie guruhining.subalgebra Subspace g ′ { displaystyle { mathfrak {g '}}} g { displaystyle { mathfrak {g}}} g { displaystyle { mathfrak {g}}} [ g ′ , g ′ ] ⊆ g ′ . { displaystyle [{ mathfrak {g '}}, { mathfrak {g'}}] subseteq { mathfrak {g '}}.} T
Ko'krak Ko'krak qafasi .toral 1. toral Lie algebra 2. maksimal toral subalgebra U
V
V
Veyl 1. Hermann Veyl (1885 - 1955), nemis matematikasi 2. A Veyl xonasi tarkibidagi to‘ldiruvchining bog‘langan tarkibiy qismlaridan biridir V , ildiz vektorlariga ortogonal bo'lgan giperplanlar olib tashlanganida, ildiz tizimi aniqlangan haqiqiy vektor maydoni. 3. The Weyl belgilar formulasi oddiy Lie guruhlarining qisqartirilmaydigan murakkab tasavvurlari belgilarini yopiq shaklda beradi. 4. Veyl guruhi : Ildiz tizimining Weyl guruhi Φ { displaystyle Phi} E { displaystyle E} Φ { displaystyle Phi} Adabiyotlar
^ Tahririyat uchun eslatma: umumiy Lie algebrasida nilpotent elementning ta'rifi noaniq ko'rinadi. Burbaki, N. (1981), Guruhlar va Algèbres de Lie , Éléments de Mathématique, Hermann Erdmann, Karin & Uayldon, Mark. Yolg'on algebralariga kirish , 1-nashr, Springer, 2006 yil. ISBN 1-84628-040-0Hamfreyz, Jeyms E. Yolg'on algebralari va vakillik nazariyasiga kirish , Ikkinchi bosib chiqarish, qayta ko'rib chiqilgan. Matematikadan aspirantura matnlari, 9. Springer-Verlag, Nyu-York, 1978 yil. ISBN 0-387-90053-5 Jeykobson, Natan , Yolg'on algebralar , 1962 yilgi asl nusxaning respublikasi. Dover Publications, Inc., Nyu-York, 1979 yil. ISBN 0-486-63832-4Kac, Viktor (1990). Cheksiz o'lchovli yolg'on algebralari Kembrij universiteti matbuoti . ISBN 0-521-46693-8 Klaudio Procesi (2007) Yolg'on guruhlari: invariantlar va vakillik orqali yondoshish , Springer, ISBN 9780387260402.Serre, Jan-Per (2000), Algèbres de Lie yarim sodda komplekslar Murakkab Semisimple Lie Algebras ], tarjima qilgan Jons, G. A., Springer, ISBN 978-3-540-67827-4 J.-P. Serre, "Yolg'on algebralari va yolg'on guruhlari", Benjamin (1965) (Frantsuz tilidan tarjima qilingan)
![{ displaystyle [x, y] = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb3ac5611c95c794cec85401f417c5e0146185d)
![{ displaystyle { textrm {ad}} (x) y = [x, y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2513423903809672d4ce6c41ca033e7eb079aef0)
![kappa ({ mathfrak {g}}, [{ mathfrak {g}}, { mathfrak {g}}]) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/611f63096ddec38e37f061f5db1ba352945ccb73)
![C _ {{{{mathfrak {g}}}} (X): = {x in { mathfrak {g}} | [x, X] = {0 } }](https://wikimedia.org/api/rest_v1/media/math/render/svg/69e858f87a76e4e0d979ac0a84f75e9e8a7624f1)
![Z (L): = {x in { mathfrak {g}} | [x, { mathfrak {g}}] = 0 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/17ae3590695b24aa14a0ffd6502a47f991b0c08d)
![C ^ {0} (L) = L, , C ^ {1} (L) = [L, L], , C ^ {{n + 1}} (L) = [L, C ^ {n } (L)]](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6e252dfdace4a46757a4c3606bddd375a7f2a6)
Dynkin diagrammalari![[{ mathfrak {g}}, { mathfrak {g}}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bdae7b3a7c17f4b13b9eea7f88b9a466d2e97aa)
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![[{ mathfrak {g '}}, { mathfrak {g}}] subseteq { mathfrak {g'}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a44bf4ad4def3be0c819d1bee99f67e6dfd4f54)
Karl Gustav Yakob Jakobi
![{ displaystyle [,]: V ^ {2} dan V} gacha](https://wikimedia.org/api/rest_v1/media/math/render/svg/22d03c54c925a62b840232aa1b41de648b0f1ad8)
Sofus yolg'on
![[ax + by, z] = a [x, z] + b [y, z]](https://wikimedia.org/api/rest_v1/media/math/render/svg/7654a168ee84d4b3231619ced5e529fa2eb17ac3)
![[x, x] = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/94ccb967db9ab0773c5aa2e0fc234cd33373f193)
![[[x, y], z] + [[y, z], x] + [[z, x], y] = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/95eabacf905ea58f824b3b3da0bfc319b3be2590)
![{ displaystyle [{ mathfrak {g}}, [{ mathfrak {g}}, [{ mathfrak {g}}, dots, [{ mathfrak {g}}, { mathfrak {g}}] nuqta]]] = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5f16c3a41309e4c9a17b3aaedbf6ee0d391a08)
![N _ {{{{mathfrak {g}}}} (K): = {x in { mathfrak {g}} | | [x, K] subseteq K }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3288e64b108730fceab2bc9f14fe68ca861bfc27)
![{ mathfrak {g _ { alpha}}}: = {x in { mathfrak {g}} | [h, x] = alfa (h) x , forall h in { mathfrak {h }} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/35e2c18b12b973aad2d7649e480c8ae39635f1f1)
![{ displaystyle D { mathfrak {g}} = [{ mathfrak {g}}, { mathfrak {g}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d6fd4c85dff63c5be5eca6ae2c2bc8b6cb34407)
![[{ mathfrak {g '}}, { mathfrak {g'}}] subseteq { mathfrak {g '}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b93362411c63816edde3258fb3b94a7b20a9cd0)