t Prtsrt r r qt - PowerPoint PPT Presentation

❖♥ t❤❡ P❛rt❤❛s❛r❛t❤② ❢♦r♠✉❧❛ ❢♦r q✉❛♥t✐③❡❞ ✐rr❡❞✉❝✐❜❧❡ ✢❛❣ ♠❛♥✐❢♦❧❞s ▼❛r❝♦ ▼❛t❛ss❛ ❯♥✐✈❡rs✐té ❈❧❡r♠♦♥t ❆✉✈❡r❣♥❡ ✧■♥❞❡① ❚❤❡♦r② ❛♥❞ ❙✐♥❣✉❧❛r ❙tr✉❝t✉r❡s✧✱ ❚♦✉❧♦✉s❡✱ ❏✉♥❡ ✷✵✶✼ ✶ ✴ ✶✽

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❤❛✈❡ ❛♥ ❛♥❛❧♦❣✉❡ ❢♦r q✉❛♥t✉♠ s②♠♠❡tr✐❝ s♣❛❝❡s✳ ●♦❛❧✿ s♣❡❝tr❛❧ tr✐♣❧❡s ❢♦r q✉❛♥t✐③❡❞ ✐rr❡❞✉❝✐❜❧❡ ✢❛❣ ♠❛♥✐❢♦❧❞s✳ Pr♦✈❡ ❝♦♠♣❛❝t r❡s♦❧✈❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❑rä❤♠❡r✬s ❉✐r❛❝ ♦♣❡r❛t♦rs✳ ❲❤❛t ✐s t❤❡ P❛rt❤❛s❛r❛t❤② ❢♦r♠✉❧❛❄ ●✐✈❡♥ ❛ s②♠♠❡tr✐❝ s♣❛❝❡ M = G / K ✇✐t❤ ❉✐r❛❝ ♦♣❡r❛t♦r D ✱ ✐t ❡①♣r❡ss❡s D ✷ ✐♥ t❡r♠s ♦❢ t❤❡ ❈❛s✐♠✐r ♦❢ G ✱ ♥❛♠❡❧② D ✷ = Ω G + ✶ ✽ ❙❝❛❧ M . ❆❧❧♦✇s t♦ ❞❡t❡r♠✐♥❡ t❤❡ s♣❡❝tr✉♠ ✈✐❛ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ♦❢ G ✳ ✷ ✴ ✶✽

❲❤❛t ✐s t❤❡ P❛rt❤❛s❛r❛t❤② ❢♦r♠✉❧❛❄ ●✐✈❡♥ ❛ s②♠♠❡tr✐❝ s♣❛❝❡ M = G / K ✇✐t❤ ❉✐r❛❝ ♦♣❡r❛t♦r D ✱ ✐t ❡①♣r❡ss❡s D ✷ ✐♥ t❡r♠s ♦❢ t❤❡ ❈❛s✐♠✐r ♦❢ G ✱ ♥❛♠❡❧② D ✷ = Ω G + ✶ ✽ ❙❝❛❧ M . ❆❧❧♦✇s t♦ ❞❡t❡r♠✐♥❡ t❤❡ s♣❡❝tr✉♠ ✈✐❛ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ♦❢ G ✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❤❛✈❡ ❛♥ ❛♥❛❧♦❣✉❡ ❢♦r q✉❛♥t✉♠ s②♠♠❡tr✐❝ s♣❛❝❡s✳ ●♦❛❧✿ s♣❡❝tr❛❧ tr✐♣❧❡s ❢♦r q✉❛♥t✐③❡❞ ✐rr❡❞✉❝✐❜❧❡ ✢❛❣ ♠❛♥✐❢♦❧❞s✳ Pr♦✈❡ ❝♦♠♣❛❝t r❡s♦❧✈❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❑rä❤♠❡r✬s ❉✐r❛❝ ♦♣❡r❛t♦rs✳ ✷ ✴ ✶✽

❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r ♣❛r❛❜♦❧✐❝s ♦❢ ❝♦♠✐♥✉s❝✉❧❡ t②♣❡✳ ❖♥❡ ❞❡✜♥✐t✐♦♥ ✐s t❤❛t ✐s ❛❜❡❧✐❛♥✳ ■♥ t❤✐s ❝❛s❡ ✐s ❛ s②♠♠❡tr✐❝ s♣❛❝❡✳ ❲❡ ❛❧s♦ ❤❛✈❡ t❤❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s ✳ ❆ ❣❡♥❡r❛❧✐③❡❞ ✢❛❣ ♠❛♥✐❢♦❧❞ ✐s ❛ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡ G / P ✱ ✇❤❡r❡ P ✐s ❛ ♣❛r❛❜♦❧✐❝ s✉❜❣r♦✉♣ ✭❝♦♥t❛✐♥s ❛ ❇♦r❡❧ s✉❜❣r♦✉♣✮✳ ❚❤❡② ❛r❡ ❝❧❛ss✐✜❡❞ ❜② s✉❜s❡ts S ⊆ Π ♦❢ s✐♠♣❧❡ r♦♦ts✳ ❲❡ ❤❛✈❡ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ g = u − ⊕ l ⊕ u + ❛s l ✲♠♦❞✉❧❡s✱ ✇❤❡r❡ � � l = h ⊕ g α , u ± = g ± α . α ∈ ∆( l ) α ∈ ∆( u + ) ❚❤❡s❡ ❛r❡ s✉❜❛❧❣❡❜r❛s ♦❢ g ✳ ❙♦♠❡ ♥❛♠❡s✿ p = l ⊕ u + ✐s t❤❡ st❛♥❞❛r❞ ♣❛r❛❜♦❧✐❝ s✉❜❛❧❣❡❜r❛✱ l ✐s t❤❡ ▲❡✈✐ ❢❛❝t♦r ❛♥❞ u + ✐s t❤❡ ♥✐❧r❛❞✐❝❛❧✳ ✸ ✴ ✶✽

❆ ❣❡♥❡r❛❧✐③❡❞ ✢❛❣ ♠❛♥✐❢♦❧❞ ✐s ❛ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡ G / P ✱ ✇❤❡r❡ P ✐s ❛ ♣❛r❛❜♦❧✐❝ s✉❜❣r♦✉♣ ✭❝♦♥t❛✐♥s ❛ ❇♦r❡❧ s✉❜❣r♦✉♣✮✳ ❚❤❡② ❛r❡ ❝❧❛ss✐✜❡❞ ❜② s✉❜s❡ts S ⊆ Π ♦❢ s✐♠♣❧❡ r♦♦ts✳ ❲❡ ❤❛✈❡ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ g = u − ⊕ l ⊕ u + ❛s l ✲♠♦❞✉❧❡s✱ ✇❤❡r❡ � � l = h ⊕ g α , u ± = g ± α . α ∈ ∆( l ) α ∈ ∆( u + ) ❚❤❡s❡ ❛r❡ s✉❜❛❧❣❡❜r❛s ♦❢ g ✳ ❙♦♠❡ ♥❛♠❡s✿ p = l ⊕ u + ✐s t❤❡ st❛♥❞❛r❞ ♣❛r❛❜♦❧✐❝ s✉❜❛❧❣❡❜r❛✱ l ✐s t❤❡ ▲❡✈✐ ❢❛❝t♦r ❛♥❞ u + ✐s t❤❡ ♥✐❧r❛❞✐❝❛❧✳ ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r ♣❛r❛❜♦❧✐❝s ♦❢ ❝♦♠✐♥✉s❝✉❧❡ t②♣❡✳ ❖♥❡ ❞❡✜♥✐t✐♦♥ ✐s t❤❛t u + ✐s ❛❜❡❧✐❛♥✳ ■♥ t❤✐s ❝❛s❡ G / P ✐s ❛ s②♠♠❡tr✐❝ s♣❛❝❡✳ ❲❡ ❛❧s♦ ❤❛✈❡ t❤❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s [ u + , u − ] ⊂ l ✳ ✸ ✴ ✶✽

❆ ♣❛r❛❜♦❧✐❝ s✉❜❛❧❣❡❜r❛ p ✐s ❝♦♠✐♥✉s❝✉❧❡ ✐✛ S = Π \{ α t } ❛♥❞ α t ❤❛s ♠✉❧t✐♣❧✐❝✐t② ✶ ✐♥ t❤❡ ❤✐❣❤❡st r♦♦t ♦❢ g ✳ ❆❧❣✳ ❉②♥❦✐♥ ❞✐❛❣r❛♠ ◆♦♠❡♥❝❧❛t✉r❡ ●r❛ss♠❛♥♥✐❛♥ A r ✶ ✷ k r − ✶ r B r ❖❞❞ ❞✐♠❡♥s✐♦♥❛❧ q✉❛❞r✐❝ r − ✷ r − ✶ r ✶ ✷ ▲❛❣r❛♥❣✐❛♥ ●r❛ss♠❛♥♥✐❛♥ C r ✶ ✷ r − ✷ r − ✶ r r − ✶ D r r ❊✈❡♥ ❞✐♠❡♥s✐♦♥❛❧ q✉❛❞r✐❝ r − ✷ ✶ ✷ r − ✶ ❖rt❤♦❣♦♥❛❧ ●r❛ss♠❛♥♥✐❛♥ D r r ✶ ✷ r − ✷ ✻ ❈❛②❧❡② ♣❧❛♥❡ E ✻ ✶ ✷ ✸ ✹ ✺ ✼ ❯♥♥❛♠❡❞ E ✼ ✶ ✷ ✸ ✹ ✺ ✻ ✹ ✴ ✶✽

✷ ✳ ❙✐♥❝❡ ✷ ◆♦✇ ✇❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ✵ ✇❡ ❤❛✈❡ ✷ ❚❤❡s❡ s♣❛❝❡s ❛r❡ ❝♦♠♣❛❝t ❑ä❤❧❡r ♠❛♥✐❢♦❧❞s✳ P✉t ❛♥ ❍❡r♠✐t✐❛♥ ♠❡tr✐❝ ♦♥ Ω ( ✵ , • ) ✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t❤❡ ❉♦❧❜❡❛✉❧t✲❉✐r❛❝ ♦♣❡r❛t♦rs D = ¯ ∂ + ¯ ∂ ∗ ✳ ❈♦♥❝r❡t❡❧② ♣✐❝❦ ❞✉❛❧ ❜❛s❡s { v i } i ∈ u + ❛♥❞ { w i } i ∈ u − ✳ ❲❡ ✐❞❡♥t✐❢② t❤❡ ❜❛s✐s ♦❢ u + ✇✐t❤ s♦♠❡ r♦♦t ✈❡❝t♦rs { E ξ i } i ∈ g ✳ ❲❡ ❞❡✜♥❡ � ð = E ξ i ⊗ γ − ( w i ) ∈ U ( g ) ⊗ ❈❧ . ξ i ∈ ∆( u + ) ■t t✉r♥s ♦✉t t❤❛t ð = ¯ ∂ ∗ ✳ ❍❡♥❝❡ ✇❡ ❝❛♥ ✇r✐t❡ D = ð + ð ∗ ∈ U ( g ) ⊗ ❈❧ . ✺ ✴ ✶✽

❚❤❡s❡ s♣❛❝❡s ❛r❡ ❝♦♠♣❛❝t ❑ä❤❧❡r ♠❛♥✐❢♦❧❞s✳ P✉t ❛♥ ❍❡r♠✐t✐❛♥ ♠❡tr✐❝ ♦♥ Ω ( ✵ , • ) ✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t❤❡ ❉♦❧❜❡❛✉❧t✲❉✐r❛❝ ♦♣❡r❛t♦rs D = ¯ ∂ + ¯ ∂ ∗ ✳ ❈♦♥❝r❡t❡❧② ♣✐❝❦ ❞✉❛❧ ❜❛s❡s { v i } i ∈ u + ❛♥❞ { w i } i ∈ u − ✳ ❲❡ ✐❞❡♥t✐❢② t❤❡ ❜❛s✐s ♦❢ u + ✇✐t❤ s♦♠❡ r♦♦t ✈❡❝t♦rs { E ξ i } i ∈ g ✳ ❲❡ ❞❡✜♥❡ � ð = E ξ i ⊗ γ − ( w i ) ∈ U ( g ) ⊗ ❈❧ . ξ i ∈ ∆( u + ) ■t t✉r♥s ♦✉t t❤❛t ð = ¯ ∂ ∗ ✳ ❍❡♥❝❡ ✇❡ ❝❛♥ ✇r✐t❡ D = ð + ð ∗ ∈ U ( g ) ⊗ ❈❧ . ∂ ✷ = ✵ ✇❡ ❤❛✈❡ ◆♦✇ ✇❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ D ✷ ✳ ❙✐♥❝❡ ¯ D ✷ = ðð ∗ + ð ∗ ð . ✺ ✴ ✶✽

❈❛♥ ❢♦r❣❡t ❛❜♦✉t t❡r♠s ✐♥ ❈❧ ✱ s✐♥❝❡ t❤❡s❡ ❛r❡ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs✳ ❲r✐t❡ ✐❢ ✇✐t❤ ❈❧ ✳ ❙✐♥❝❡ ✇❡ ❣❡t ✷ ◆❡①t ✇❡ ❤❛✈❡ t❤❡ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ r❡❧❛t✐♦♥s ✶ ❙✐♥❝❡ ✇❡ ❛r❡ ✉s✐♥❣ ❞✉❛❧ ❜❛s❡s ✇❡ ❤❛✈❡ ✳ ❍❡♥❝❡ ✷ ✶ ❊①♣❧✐❝✐t❡❧② ✇❡ ❝❛♥ ✇r✐t❡ � � D ✷ = E ξ i F ξ j ⊗ γ − ( w i ) γ + ( v j ) + F ξ i E ξ j ⊗ γ + ( v i ) γ − ( w j ) . i , j i , j ✻ ✴ ✶✽

◆❡①t ✇❡ ❤❛✈❡ t❤❡ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ r❡❧❛t✐♦♥s ✶ ❙✐♥❝❡ ✇❡ ❛r❡ ✉s✐♥❣ ❞✉❛❧ ❜❛s❡s ✇❡ ❤❛✈❡ ✳ ❍❡♥❝❡ ✷ ✶ ❊①♣❧✐❝✐t❡❧② ✇❡ ❝❛♥ ✇r✐t❡ � � D ✷ = E ξ i F ξ j ⊗ γ − ( w i ) γ + ( v j ) + F ξ i E ξ j ⊗ γ + ( v i ) γ − ( w j ) . i , j i , j ❈❛♥ ❢♦r❣❡t ❛❜♦✉t t❡r♠s ✐♥ U ( l ) ⊗ ❈❧ ✱ s✐♥❝❡ t❤❡s❡ ❛r❡ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs✳ ❲r✐t❡ A ∼ B ✐❢ A − B = C ✇✐t❤ C ∈ U ( l ) ⊗ ❈❧ ✳ ❙✐♥❝❡ [ u + , u − ] ⊂ l ✇❡ ❣❡t � D ✷ ∼ E ξ i F ξ j ⊗ ( γ − ( w i ) γ + ( v j ) + γ + ( v j ) γ − ( w i )) . i , j ✻ ✴ ✶✽