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❙②♠♣❧❡❝t✐❝ ♥✐❧♠❛♥✐❢♦❧❞s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❱✐❝t♦r ▼✳ ❇✉❝❤st❛❜❡r � ❜✉❝❤st❛❜❅♠✐✳r❛s✳r✉ � ❙t❡❦❧♦✈ ▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡✱ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s✱ ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❖s❛❦❛ ❉❡❝❡♠❜❡r ✷✱ ✷✵✶✶✳

❆❜str❛❝t✳ ❚❤❡ t❛❧❦ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ r❡♠❛r❦❛❜❧❡ t♦✇❡rs ♦❢ ❜✉♥❞❧❡s M n → M n − 1 → · · · → S 1 , n � 2 , ✇✐t❤ ✜❜❡r t❤❡ ❝✐r❝❧❡ S 1 ✳ ❚❤✐s t♦✇❡rs ❛r❡ ❞❡✜♥❡❞ ❜② t❤❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣s ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ r❡❛❧ ❧✐♥❡✳ ❊❛❝❤ M n ✱ n � 2 ✱ ✐s ❛ s♠♦♦t❤ ♥✐❧♠❛♥✐❢♦❧❞ ✇✐t❤ ❛ ✷✲❢♦r♠ ✇❤✐❝❤ ❣✐✈❡s ❛ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡ ♦♥ ❛♥② M 2 k ✳ ❙✉❝❤ ♠❛♥✐❢♦❧❞s ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ❞✐✛❡r❡♥t ❛r❡❛s ♦❢ ♠❛t❤❡♠❛t✐❝s✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss t❤❡ ❞✐✛❡r❡♥t✐❛❧✲❣❡♦♠❡tr✐❝ ❛♥❞ ❛❧❣❡❜r♦✲t♦♣♦❧♦❣✐❝ r❡s✉❧ts ❛♥❞ ✉♥s♦❧✈❡❞ ♣r♦❜❧❡♠s✱ ❝♦♥❝❡r♥✐♥❣ t❤✐s ♠❛♥✐❢♦❧❞s✳ ✶

●r♦✉♣s ♦❢ ♣♦❧②♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥s✳ P✉t L n = { p x ( t ) = t + � n k =1 x k t k +1 , x k ∈ R } ✳ ❲❡ ❤❛✈❡ L n ∼ = R n : p x ( t ) ⇒ x = ( x 1 , . . . , x n ) ✳ ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r L n ❛s t❤❡ n ✲❞✐♠ ❣r♦✉♣ ♦❢ ♣♦❧②♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ R → R : t �→ p x ( t ) , ✇✐t❤ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ x ∗ y = z ✱ ✇❤❡r❡ mod t n +2 . ( p x ∗ p y )( t ) = p z ( t ) = p y ( p x ( t )) ✷

❊①❛♠♣❧❡✳ ❋♦r n = 4 ✿ 4 mod t 6 : y k p x ( t ) k +1 � p z ( t ) = ( p x ∗ p y )( t ) = p x ( t ) + k =1 z 1 = x 1 + y 1 , z 2 = x 2 + 2 x 1 y 1 + y 2 , z 3 = x 3 + (2 x 2 + x 2 1 ) y 1 + 3 x 1 y 2 + y 3 z 4 = x 4 + 2( x 3 + x 1 x 2 ) y 1 + 3( x 2 + x 2 1 ) y 2 + 4 x 1 y 3 + y 4 . ✸

◆✐❧♣♦t❡♥t ❣r♦✉♣ str✉❝t✉r❡ ♦♥ R n ✳ ❚❤❡ ❣r♦✉♣ L n ∼ = R n ❤❛s t❤❡ str✉❝t✉r❡ ♦❢ ♥✐❧♣♦t❡♥t ❣r♦✉♣ ✇✐t❤ t❤❡ ✉♣♣❡r ❝❡♥tr❛❧ s❡r✐❡s L n n ⊂ · · · ⊂ L n q ⊂ · · · ⊂ L n 0 = L n , ✇❤❡r❡ L n n = { 0 ∈ R } ✱ n R n − q ∼ = L n x k t k +1 } . � q = { p x ( t ) = t + k = q +1 ❲❡ ❤❛✈❡ q = { x ∈ L n | ∀ y ∈ L n : L n [ x, y ] ∈ L n q +1 } q − 1 ∼ ❛♥❞ L n q /L n = R ✐s t❤❡ ❝❡♥t❡r ♦❢ L n /L n q ✱ q = 0 , . . . , n − 1 ✳ ✹

❚❤❡ ❝❛♥♦♥✐❝❛❧ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥✳ ❚❤❡ ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ∗ ❣✐✈❡s t❤❡ ❝❛♥♦♥✐❝❛❧ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ � � � � 1 1 ( x : v → x ∗ v ) : ρ : L n → GT ( n + 1) : ρ ( p x ( t )) = v x ∗ v ✐♥t♦ t❤❡ ❣r♦✉♣ ♦❢ ❧♦✇❡r tr✐❛♥❣✉❧❛r ( n + 1) × ( n + 1) ✲♠❛tr✐❝❡s ✇✐t❤ ♦♥❡s ♦♥ t❤❡ ❞✐❛❣♦♥❛❧✿ ρ ( p x ( t )) = X = ( x ik ) , i, k = 0 , . . . , n, ✇❤❡r❡ x i,k = [ p x ( t ) k +1 ] i +1 ✐s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t i +1 ✐♥ p x ( t ) k ✳ ✺

❊①❛♠♣❧❡✳ ❋♦r n = 4 ✿     1 1 1 x 1 v 1     � � 1      2 x 1 1    ρ ( p x ( t )) = x 2 v 2 .     v  2 x 2 + x 2    x 3 3 x 1 1 v 3     1     3( x 2 + x 2 v 4 x 4 2( x 3 + x 1 x 2 ) 1 ) 4 x 1 1 ✻

❉❡❢♦r♠❛t✐♦♥ t♦ t❤❡ st❛♥❞❛r❞ ❣r♦✉♣ str✉❝t✉r❡✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ∗ ♦♥ R n ❝❛♥ ❜❡ ✇r✐tt❡♥ ❞♦✇♥ ❛s x ∗ y = x + y + A ( x ) y, ✇❤❡r❡ A ( x ) = ( a ik ( x )) ✐s t❤❡ ❧♦✇❡r tr✐❛♥❣✉❧❛r ( n × n ) ✲♠❛tr✐① ✇✐t❤ ③❡r♦s ♦♥ t❤❡ ❞✐❛❣♦♥❛❧ ❛♥❞ a ik ( x ) = x i,k = [ p x ( t ) k +1 ] i +1 , i � = k. ✼

❆♥② ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ B : R n → R n ♦❢ ❝♦♦r❞✐♥❛t❡s ✐♥ R n ❜② B ∈ GL ( n, R ) ❣✐✈❡s ❛ tr❛♥s❢♦r♠❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ R n ✿ def = B − 1 (( Bx ) ∗ ( By )) = x ∗ B y = B − 1 ( Bx + By + A ( Bx ) By ) = = x + y + ( B − 1 A ( Bx ) B ) y. ■♥ t❤❡ ❝❛s❡ ♦❢ ❛ s❝❛❧❛r ♠❛tr✐① τE ✱ ✇❡ ♦❜t❛✐♥ x ∗ τ y = x + y + A ( τx ) y. ❚❤✐s ❣✐✈❡s ❛ ❞❡❢♦r♠❛t✐♦♥ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✯ ✭ τ = 1 ✮ t♦ t❤❡ st❛♥❞❛r❞ ❛❞❞✐t✐♦♥ ✭ τ = 0 ✮ ♦♥ R n ✳ ✽

❊①❛♠♣❧❡✳ ❋♦r n = 4 ✿ x ∗ y = x + y + τA 1 ( x ) y + τ 2 A 2 ( x ) y, ✇❤❡r❡    0  0 0 0 2 x 1 0     A 1 ( x ) =  , A 2 ( x ) =  .     x 2 2 x 2 3 x 1 0 0 0     1   3 x 2 2 x 3 3 x 2 4 x 1 0 2 x 1 x 2 0 0 1 ✾

❈♦❝♦♠♣❛❝t ❧❛tt✐❝❡s✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ∗ ❣✐✈❡s t❤❡ ❢r❡❡ ❛❝t✐♦♥s ♦❢ L n ♦♥ R n ✿ ❚❤❡ ❧❡❢t s❤✐❢t v → x ∗ v ❣✐✈❡s ❛ ❧✐♥❡❛r ❛❝t✐♦♥ ρ, ❚❤❡ r✐❣❤t s❤✐❢t v → v ∗ x ❣✐✈❡s ❛ ♥♦♥✲❧✐♥❡❛r ❛❝t✐♦♥ . ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❝❛♥♦♥✐❝❛❧ ❧❛tt✐❝❡✿ Γ n = { p x ( t ) ∈ L n : x i ∈ Z } ✇✐t❤ t❤❡ ✉♣♣❡r ❝❡♥tr❛❧ s❡r✐❡s✿ Γ n n ⊂ · · · ⊂ Γ n q ⊂ · · · ⊂ Γ n 0 = Γ n . ❚❤✐s ❧❛tt✐❝❡ Γ n ∼ = Z n ✐s ❝♦❝♦♠♣❛❝t ✭✉♥✐❢♦r♠✮✳ ✶✵

◆✐❧♠❛♥✐❢♦❧❞s✳ ❲✐t❤ r❡s♣❡❝t t♦ t❤❡ r✐❣❤t s❤✐❢ts ✇❡ ♦❜t❛✐♥ ❛ s♠♦♦t❤ ❝❧♦s❡❞ ❛♥❞ ❝♦♠♣❛❝t ♥✐❧♠❛♥✐❢♦❧❞ M n = R n / Γ n . ❚❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ ♦❢ M n ✐s T ( M n ) = R n × Γ n R n → M n = R n / Γ n ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❧✐♥❡❛r ❛❝t✐♦♥ ρ ✭❧❡❢t s❤✐❢t✮ ♦♥ ❛ ✜❜❡r R n ✳ ✶✶

❲❡ ❤❛✈❡ t❤❡ t♦✇❡rs ♦❢ ❣r♦✉♣s L n → L n − 1 → · · · → L 1 , Γ n → Γ n − 1 → · · · → Γ 1 ❛♥❞ t❤❡ ✐♥❞✉❝❡❞ t♦✇❡r M n → M n − 1 → · · · → M 1 = S 1 ♦❢ ❜✉♥❞❧❡s M n → M n − 1 ✇✐t❤ t❤❡ ✜❜❡r S 1 ✳ ❋♦r ❡❛❝❤ n t❤❡ ♠♦♥♦♠♦r♣❤✐s♠ ❤♦❧❞s i n : L 1 → L n : i n ( x 1 ) = ( x 1 , . . . , x k 1 , . . . , x n 1 ) . ■ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ♣r♦❥❡❝t✐♦♥ L n → L 1 ✐s t❤❡ ✐❞❡♥t✐t② ♠❛♣✳ ❚❤✉s ❢♦r ❡❛❝❤ n t❤❡ ❜✉♥❞❧❡ M 1 → S 1 ✇✐t❤ t❤❡ ✜❜❡r L n 1 / Γ n 1 ❤❛s ❛ s❡❝t✐♦♥✳ ✶✷

▲❡❢t ✐♥✈❛r✐❛♥t ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✳ ▲❡t ✉s ✜① t❤❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ R [ x 1 , . . . , x n ] ❛s t❤❡ r✐♥❣ ♦❢ ❢✉♥❝t✐♦♥s ♦♥ L n ∼ = R n ✳ P✉t ❢♦r f ( x ) ∈ R [ x 1 , . . . , x n ] def R y D I ( f ( x )) y I � x f ( x ) = f ( x ∗ y ) = | I | � 0 ✇❤❡r❡ R y x ✐s t❤❡ r✐❣❤t s❤✐❢t ♦♣❡r❛t♦r✱ I = ( i 1 , . . . , i n ) ❛♥❞ y I = y i 1 1 . . . y i n n ✳ ❋r♦♠ t❤❡ ❛ss♦❝✐❛t✐✈✐t② ❡q✉❛t✐♦♥ R y y R y x R z x = R z x ✇❡ ❤❛✈❡ D I D J f ( x ) y J z I = D K f ( x )( y ∗ z ) K . � � � | I | � 0 | J | � 0 | K | � 0 ✶✸

❊①❛♠♣❧❡ n = 3 ✳ ❲❡ ❤❛✈❡ D 0 f ( x ) = f ( x ) ✱ ∂ ∂ 1 ) ∂ + (2 x 2 + x 2 D (1 , 0 , 0) = + 2 x 1 , ∂x 1 ∂x 2 ∂x 3 ∂ ∂ D (0 , 1 , 0) = + 3 x 1 , ∂x 2 ∂x 3 ∂ D (0 , 0 , 1) = . ∂x 3 D (1 , 0 , 0) D (0 , 1 , 0) = D (1 , 1 , 0) + 2 D (0 , 0 , 1) , D (0 , 1 , 0) D (1 , 0 , 0) = D (1 , 1 , 0) + 3 D (0 , 0 , 1) . ✶✹

❚❤❡ ❛❧❣❡❜r❛ A n ❣❡♥❡r❛t❡❞ ❜② t❤❡ ♦♣❡r❛t♦rs D I ✐s t❤❡ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❧❡❢t ✐♥✈❛r✐❛♥t ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ R [ x 1 , . . . , x n ] ❢♦r t❤❡ ❧❡❢t s❤✐❢t L z x ✿ L z x f ( x ) = f ( z ∗ x ) , t❤❛t ✐s L z x D I f ( x ) = D I L z x f ( x ) ❢♦r z ❛s ♣❛r❛♠❡t❡r✳ ✶✺