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❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ ▲❛♣❧❛❝❡ ❛♥❞ ❉✐r❛❝ ♦♣❡r❛t♦rs ✕ t♦✇❛r❞s s✉♣❡rs②♠♠❡tr✐❡s ❏❡❛♥✲P❤✐❧✐♣♣❡ ▼✐❝❤❡❧ ✭❯♥✐✈❡rs✐té ❞❡ ▲✐è❣❡✱ P❆■ ❉❨●❊❙❚✮ ❋✉♥❞❛♠❡♥t❛❧ ❋r♦♥t✐❡rs ✐♥ P❤②s✐❝s ✷✵✶✹ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧②

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉ ❉❡✜♥✐t✐♦♥ ♦❢ ❤✐❣❤❡r s②♠♠❡tr✐❡s ▲❡t ❉ ∈ D ( ▼ ) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ♦✈❡r ▼ ✱ ❡✳❣✳ ❉ = ∆ ♦♥ R ♥ ✳ ❚❤r❡❡ t②♣❡s ♦❢ s②♠♠❡tr✐❡s✿ ❳ ∈ ❱❡❝t ( ▼ ) s✳t✳ [ ❉ , ❳ ] = ✵✱ ▲✐❡ ❛❧❣❡❜r❛ ✴ ♣r❡s❡r✈❡ ❡✐❣❡♥s♣❛❝❡s❀ ❆ ∈ D ( ▼ ) s✳t✳ [ ❉ , ❆ ] = ✵✱ ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛ ✴ ♣r❡s❡r✈❡ ❡✐❣❡♥s♣❛❝❡s❀ � ❝♦♠♠✉t✐♥❣ s②♠♠❡tr✐❡s✱ ❆ ∈ D ( ▼ ) s✳t✳ ❉ ◦ ❆ = ❇ ◦ ❉ ✱ ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛ ✴ ♣r❡s❡r✈❡ ❦❡r ❉ ❀ � ❍✐❣❤❡r ❙②♠♠❡tr✐❡s✱ tr✐✈✐❛❧ ❍❙ ✐❢ ❆ = ❆ ✵ ◦ ❉ ✳ ❘❡♠❛r❦✿ ❣❡♥❡r❛❧✐③❡ t♦ ❉ ∈ D ( ▼ , ❊ ) ❛♥❞ ❉ ∈ D ( ▼ ; ❊ , ❋ ) ✳ ❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✷ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉ ❙❤♦rt r❡✈✐❡✇ ❛♥❞ ❛✐♠ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ✿ s②♠♠❡tr✐❡s ♦❢ ♦r❞❡r ✷ ← → s❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ♦❢ ∆ φ = ❊ φ ✱ ❍❙ ♦❢ ♦r❞❡r ✷ ❝❧❛ss✐✜❡❞ ✐♥ t❤❡ ✢❛t ❝❛s❡ ❬❇♦②❡r✱ ❑❛❧♥✐♥s✱ ▼✐❧❧❡r ✬✼✻❪ ✱ ❝♦♠♠✉t✐♥❣ s②♠♠❡tr✐❡s ♦❢ ♦r❞❡r ✷ ✐♥ t❤❡ ❝✉r✈❡❞ ❝❛s❡ ❬❈❛rt❡r ✬✼✼❪ ✱ ❢✉❧❧ ❛❧❣❡❜r❛ ♦❢ ❍❙ ❞❡t❡r♠✐♥❡❞ ✐♥ t❤❡ ✢❛t ❝❛s❡ ❬❊❛st✇♦♦❞ ✬✵✺❪ ✱ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ❤✐❣❤❡r s♣✐♥ ✜❡❧❞ t❤❡♦r② ❬❱❛s✐❧✐❡✈ ❡t ❛❧✳❪ ✳ ❍❙ ♦❢ ♦t❤❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ❤❛✈❡ ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞✿ ∆ ❦ ✱ / ❉ ✱ ✳✳✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ ♣r❡s❡♥t ❊❛st✇♦♦❞✬s r❡s✉❧t ✈✐❛ q✉❛♥t✐③❛t✐♦♥ ♠❡t❤♦❞s✱ ❡①t❡♥s✐♦♥ t♦ t❤❡ ❉✐r❛❝ ♦♣❡r❛t♦r ❛♥❞ t♦ t❤❡ s②st❡♠ ∆ ⊕ / ❉ ✳ ❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✸ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉ ❉❡q✉❛♥t✐③❛t✐♦♥ ❛♥❞ s②♠❜♦❧ s♣❛❝❡ ❖r❞❡r✿ ❧♦❝❛❧❧②✱ ❉ ✐s ♦❢ ♦r❞❡r ❦ ✐❢ ❉ = ❉ ✐ ✶ ··· ✐ ❦ ( ① ) ∂ ✐ ✶ · · · ∂ ✐ ❦ + ❧✳♦✳t✳✳ � ■♥❝r❡❛s✐♥❣ ✜❧tr❛t✐♦♥✿ D ( ▼ ) = � D ❦ ( ▼ ) ✇✐t❤ D ❦ ( ▼ ) · D ❧ ( ▼ ) ⊆ D ❦ + ❧ ( ▼ ) ✳ D ( ▼ ) ✐s ❛♥ ❛ss♦❝✐❛t✐✈❡ ✜❧t❡r❡❞ ❛❧❣❡❜r❛ ❙②♠❜♦❧s✿ S ( ▼ ) = � S ❦ ( ▼ ) ❛♥❞ S ❦ ( ▼ ) := D ❦ ( ▼ ) / D ❦ − ✶ ( ▼ ) ∼ = Γ( S ❦ ❚▼ ) ∼ = P♦❧ ❦ ( ❚ ∗ ▼ ) ✳ σ ❦ : ❉ �→ σ ❦ ( ❉ ) := ❉ ✐ ✶ ··· ✐ ❦ ( ① ) ∂ ✐ ✶ ⊙ · · · ⊙ ∂ ✐ ❦ = ❉ ✐ ✶ ··· ✐ ❦ ( ① ) ♣ ✐ ✶ · · · ♣ ✐ ❦ Pr♦♣❡rt✐❡s✿ ✐❢ {· , ·} = ∂ ♣ ✐ ⊗ ∂ ① ✐ − ∂ ① ✐ ⊗ ∂ ♣ ✐ t❤❡ P♦✐ss♦♥ ❜r❛❝❦❡t ♦♥ ❚ ∗ ▼ ✱ t❤❡♥ σ ❦ + ❧ ( ❆ ◦ ❇ ) = σ ❦ ( ❆ ) σ ❧ ( ❇ ) , σ ❦ + ❧ − ✶ ([ ❆ , ❇ ]) = { σ ❦ ( ❆ ) , σ ❧ ( ❇ ) } . S ( ▼ ) ✐s ❛ ❣r❛❞❡❞ P♦✐ss♦♥ ❛❧❣❡❜r❛ ❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✹ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉ ❚❤❡ ▲❛♣❧❛❝✐❛♥✱ ✐ts s②♠❜♦❧ ❛♥❞ s②♠♠❡tr✐❡s ❉❡✜♥✐t✐♦♥✿ ♦♥ ( ▼ , ❣ ) ✱ ✇❡ s❡t ∆ := ∇ ✐ ě ✐❥ ∇ ❥ + ❝❘ ✱ ✇✐t❤ ❝ ∈ R ❛♥❞ ❘ t❤❡ s❝❛❧❛r ❝✉r✈❛t✉r❡✳ ❙②♠❜♦❧✿ ❍ = σ ✷ (∆) = ě ✐❥ ∂ ✐ ⊙ ∂ ❥ = ě ✐❥ ♣ ✐ ♣ ❥ ✳ ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ ♦❢ ❍ ♦♥ ❚ ∗ ▼ ě ← → ❣❡♦❞❡s✐❝ s♣r❛② ♦♥ ❚▼ ❙②♠♠❡tr✐❡s✿ [∆ , ❳ ] = ✵ ⇒ { ❍ , σ ✶ ( ❳ ) } = ✵ = ▲ ❳ ě , ❳ ❑✐❧❧✐♥❣ ✈❡❝t♦r ✜❡❧❞❀ [∆ , ❆ ] = ✵ ⇒ { ❍ , σ ❦ ( ❆ ) } = ✵ , σ ❦ ( ❆ ) ❑✐❧❧✐♥❣ t❡♥s♦r ✜❡❧❞❀ � ❤✐❞❞❡♥ s②♠♠❡tr②✱ ✐✳❡✳ s②♠♠❡tr② ♦❢ ( ❚ ∗ ▼ , ❍ ) ♥♦t ♦❢ ( ▼ , ě ) ❀ ∆ ◦ ❆ = ❇ ◦ ∆ ⇒ { ❍ , σ ❦ ( ❆ ) } ∈ ( ❍ ) , σ ❦ ( ❆ ) ❝♦♥❢♦r♠❛❧ ❑✐❧❧✐♥❣ t❡♥s♦r ✜❡❧❞❀ � ✐❢ ❆ = ❳ + ❢ ∈ D ✶ ( ▼ ) ✱ t❤❡♥ ▲ ❳ ě = ❋ ě ✳ ❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✺ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉ ❈♦♥❝❡♣t ♦❢ q✉❛♥t✐③❛t✐♦♥ ❉❡✜♥✐t✐♦♥✿ ❛ q✉❛♥t✐③❛t✐♦♥ ✐s ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ Q : S ( ▼ ) → D ( ▼ ) , s✉❝❤ t❤❛t σ ❦ ◦ Q ( P ) = P ❢♦r ❛❧❧ P ∈ S ❦ ( ▼ ) ✳ ❋❛❝ts✿ ♥♦ q✉❛♥t✐③❛t✐♦♥ s✳t✳ [ Q ( P ✶ ) , Q ( P ✷ )] = Q ( { P ✶ , P ✷ } ) ❢♦r ❛❧❧ P ✶ , P ✷ ∈ S ( ▼ ) ✱ ✐❢ [ Q ( ❳ ) , Q ( ❨ )] = Q ( { ❳ , ❨ } ) ❢♦r ❛❧❧ ❳ , ❨ ∈ S ✶ ( ▼ ) ✱ t❤❡♥ Q ( ❳ ) = ▲ ❳ = ❳ + λ ❞✐✈ ❳ ✐s t❤❡ ▲✐❡ ❞❡r✐✈❛t✐✈❡ ♦♥ λ ✲❞❡♥s✐t✐❡s✱ λ ∈ R ✱ ♥♦ q✉❛♥t✐③❛t✐♦♥ s✳t✳ [ Q ( ❳ ) , Q ( P )] = Q ( { ❳ , P } ) ❢♦r ❛❧❧ ❳ ∈ S ✶ ( ▼ ) ✱ P ∈ S ( ▼ ) ✳ ❊①❛♠♣❧❡s ♦♥ ❚ ∗ R ♥ ✿ ♥♦r♠❛❧ ♦r❞❡r✐♥❣ N : P ✐ ✶ ··· ✐ ❦ ( ① ) ♣ ✐ ✶ · · · ♣ ✐ ❦ �→ P ✐ ✶ ··· ✐ ❦ ( ① ) ∂ ✐ ✶ · · · ∂ ✐ ❦ ✱ N ( ❳ ) = ▲ ❳ ♦♥ ✵✲❞❡♥s✐t✐❡s✱ ❡q✉✐✈✳ ✉♥❞❡r gl ( ♥ ) ⋉ heis ( ✷ ♥ ) ∼ = � ✶ , ① ✐ , ♣ ✐ , ① ✐ ♣ ❥ � ❀ � ✶ � ❲❡②❧ q✉❛♥t✐③❛t✐♦♥ Q ❲ = N ◦ ❡①♣ ✷ ❞✐✈ ✱ ✇✐t❤ ❞✐✈ = ∂ ✐ ∂ ♣ ✐ t❤❡ ❞✐✈❡r❣❡♥❝❡✱ Q ❲ ( ❳ ) = ▲ ❳ ♦♥ ✶ ✷ ✲❞❡♥s✐t✐❡s✱ ❡q✉✐✈✳ ✉♥❞❡r sp ( ✷ ♥ ) ⋉ heis ( ✷ ♥ ) ∼ = � ✶ , ① ✐ , ♣ ✐ , ① ✐ ① ❥ , ① ✐ ♣ ❥ , ♣ ✐ ♣ ❥ � ✳ ❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✻ ✴ ✶✺