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▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

➪❣♦t❛ ❋✐❣✉❧❛ ❛♥❞ Pét❡r ❚✳ ◆❛❣②

❯♥✐✈❡rs✐t② ♦❢ ❉❡❜r❡❝❡♥✱ ❯♥✐✈❡rs✐t② ♦❢ Ó❜✉❞❛

✶✻✲✶✼✳✵✻✳✷✵✶✼✱ ●❚●✱ ❯♥✐✈❡rs✐t② ♦❢ ❚r❡♥t♦

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

Pr❡❧✐♠✐♥❛r✐❡s

❉❡✜♥✐t✐♦♥ ▲❡t g ❜❡ ❛ ▲✐❡ ❛❧❣❡❜r❛ ❛♥❞ G ❜❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥♥❡❝t❡❞ ❛♥❞ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ▲✐❡ ❣r♦✉♣✳ ❆ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (g, ., .) ✐s ❛ ▲✐❡ ❛❧❣❡❜r❛ g t♦❣❡t❤❡r ✇✐t❤ ❛ ❊✉❝❧✐❞❡❛♥ ✐♥♥❡r ♣r♦❞✉❝t ., . ♦♥ g✳ ❚❤✐s ✐♥♥❡r ♣r♦❞✉❝t ., . ♦♥ g ✐♥❞✉❝❡s ❛ ❧❡❢t ✐♥✈❛r✐❛♥t ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ t❤❡ ▲✐❡ ❣r♦✉♣ G ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳ ▲❡t (n, ., .) ❜❡ ❛ ♥✐❧♣♦t❡♥t ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♥✐❧♣♦t❡♥t ▲✐❡ ❣r♦✉♣ N ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ❧❡❢t✲✐♥✈❛r✐❛♥t ♠❡tr✐❝ ❛r✐s✐♥❣ ❢r♦♠ ., . ✐s ❛ ❘✐❡♠❛♥♥✐❛♥ ♥✐❧♠❛♥✐❢♦❧❞✳ ❉❡♥♦t❡ OA(n) t❤❡ ❣r♦✉♣ ♦❢ ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ (n, ., .)✱ ✇❤✐❝❤ ♣r❡s❡r✈❡ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t✳ ❊✳ ❲✐❧s♦♥ ✐♥ ■s♦♠❡tr② ❣r♦✉♣s ♦♥ ❤♦♠♦❣❡♥❡♦✉s ♥✐❧♠❛♥✐❢♦❧❞s✱ ●❡♦♠✳ ❉❡❞✐❝❛t❛ ✶✷ ✭✶✾✽✷✮✱ ✸✸✼✲✸✹✻✱ ❤❛s ❜❡❡♥ ♣r♦✈❡❞ t❤❛t t❤❡ ❣r♦✉♣ I(N) ♦❢ ✐s♦♠❡tr✐❡s ♦❢ (N, ., .) ✭❞✐st❛♥❝❡ ♣r❡s❡r✈✐♥❣ ❜✐❥❡❝t✐♦♥s✮ ✐s t❤❡ s❡♠✐✲❞✐r❡❝t ♣r♦❞✉❝t OA(n) ⋉ N ♦❢ t❤❡ ❣r♦✉♣ OA(n) ❛♥❞ t❤❡ ❣r♦✉♣ N ✐ts❡❧❢✱ ✇❤✐❝❤ ✐s ♥♦r♠❛❧ ✐♥ t❤❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❊✳ ❲✐❧s♦♥ ❞❡s❝r✐❜❡❞ ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ ✐s♦♠❡tr② ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ♦❢ ❘✐❡♠❛♥♥✐❛♥ ♥✐❧♠❛♥✐❢♦❧❞s✳ ❚❤✐s ✐s ❛♣♣❧✐❡❞ ❜② ❏✳ ▲❛✉r❡t ✐♥ ❍♦♠♦❣❡♥❡♦✉s ◆✐❧♠❛♥✐❢♦❧❞s ♦❢ ❉✐♠❡♥s✐♦♥s ✸ ❛♥❞ ✹✱

• ❡♦♠❡tr✐❛❡ ❉❡❞✐❝❛t❛ ✻✽✱ ✭✶✾✾✼✮✱ ✶✹✺✲✶✺✺✱ ❢♦r t❤❡ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢

t❤❡ ✸✲ ❛♥❞ ✹✲❞✐♠❡♥s✐♦♥❛❧ ❘✐❡♠❛♥♥✐❛♥ ♥✐❧♠❛♥✐❢♦❧❞s ✉♣ t♦ ✐s♦♠❡tr② ❛♥❞ t❤❡✐r ✐s♦♠❡tr② ❣r♦✉♣s✳ ❙③✳ ❍♦♠♦❧②❛ ❛♥❞ ❖✳ ❑♦✇❛❧s❦✐ ❤❛✈❡ ❝❧❛ss✐✜❡❞ ✐♥ ❙✐♠♣❧② ❝♦♥♥❡❝t❡❞ t✇♦✲st❡♣ ❤♦♠♦❣❡♥❡♦✉s ♥✐❧♠❛♥✐❢♦❧❞s ♦❢ ❞✐♠❡♥s✐♦♥ ✺✱ ◆♦t❡ ▼❛t❤✳ ✷✻ ✭✷✵✵✻✮✱ ✻✾✲✼✼✱ ❛❧❧ ✺✲❞✐♠❡♥s✐♦♥❛❧ ✷✲st❡♣ ♥✐❧♣♦t❡♥t ❘✐❡♠❛♥♥✐❛♥ ♥✐❧♠❛♥✐❢♦❧❞s ❛♥❞ t❤❡✐r ✐s♦♠❡tr② ❣r♦✉♣s✳ ❚♦❣❡t❤❡r ✇✐t❤ P✳❚✳ ◆❛❣② ✇❡ ✇❛♥t t♦ ❞❡t❡r♠✐♥❡ ❡①♣❧✐❝✐t❧② ❛❧❧ ✺✲❞✐♠❡♥s✐♦♥❛❧ ♥♦♥ ✷✲st❡♣ ❘✐❡♠❛♥♥✐❛♥ ♥✐❧♠❛♥✐❢♦❧❞s ❛♥❞ t❤❡ ❣r♦✉♣s ♦❢ t❤❡✐r ✐s♦♠❡tr✐❡s✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❆ s✉❜❛❧❣❡❜r❛ h ♦❢ ❛ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ g ✐s t♦t❛❧❧② ❣❡♦❞❡s✐❝ ✐❢ ❢♦r ❛❧❧ Y , Z ∈ h✱ X ∈ h⊥ ♦♥❡ ❤❛s [X, Y ], Z + [X, Z], Y = ✵. ❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s ❝❤♦s❡♥ s♦ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ▲✐❡ s✉❜❣r♦✉♣ H ♦❢ h ✐s ❛ t♦t❛❧❧② ❣❡♦❞❡s✐❝ s✉❜♠❛♥✐❢♦❧❞ r❡❧❛t✐✈❡ t♦ t❤❡ ❧❡❢t ✐♥✈❛r✐❛♥t ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ❞❡✜♥❡❞ ❜② t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ t❤❡ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ▲✐❡ ❣r♦✉♣ G ♦❢ g✳ ❚❤✐s ♥♦t✐♦♥ ✐s ♠♦t✐✈❛t❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤❡ ❧❡❢t ❝♦s❡ts ♦❢ H ❞❡✜♥❡ ❛ t♦t❛❧❧② ❣❡♦❞❡s✐❝ ❢♦❧✐❛t✐♦♥ ♦♥ G✳ P✳❚✳ ◆❛❣② ❛♥❞ ❙③✳ ❍♦♠♦❧②❛ ✐♥ ●❡♦❞❡s✐❝ ✈❡❝t♦rs ❛♥❞ s✉❜❛❧❣❡❜r❛s ✐♥ t✇♦✲st❡♣ ♥✐❧♣♦t❡♥t ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s✱ ❆❞✈✳ ●❡♦♠❡tr② ✶✺ ✭✷✵✶✺✮ ❤❛✈❡ ❜❡❡♥ ♣r♦✈❡❞ t❤❛t ❢♦r ✷✲st❡♣ ♥✐❧♣♦t❡♥t ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s t❤❡ ❧✐♥❡❛r str✉❝t✉r❡ ♦❢ ✢❛t t♦t❛❧❧② ❣❡♦❞❡s✐❝ s✉❜❛❧❣❡❜r❛s ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥❧② ♦♥ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss ♦❢ t❤❡ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛ n✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

▼♦r❡♦✈❡r✱ ✐♥ ●✳ ❈❛✐r♥s✱ ❆✳ ❍✐♥✐➣ ●❛❧✐➣✱ ❨✉✳ ◆✐❦♦❧❛②❡✈s❦②✱ ❚♦t❛❧❧② ❣❡♦❞❡s✐❝ s✉❜❛❧❣❡❜r❛s ♦❢ ✜❧✐❢♦r♠ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s✱ ❏✳ ▲✐❡ ❚❤❡♦r②✱ ✷✸ ✭✷✵✶✸✮ t❤❡ ❛✉t❤♦rs ❞❡t❡r♠✐♥❡ t❤❡ ♠❛①✐♠❛❧ ❞✐♠❡♥s✐♦♥ ♦❢ t♦t❛❧❧② ❣❡♦❞❡s✐❝ s✉❜❛❧❣❡❜r❛s ♦❢ ✜❧✐❢♦r♠ ♥✐❧♣♦t❡♥t ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s ❛♥❞ s❤♦✇ t❤❛t t❤✐s ❜♦✉♥❞ ✐s ❛tt❛✐♥❡❞✳ ❚♦❣❡t❤❡r ✇✐t❤ P✳❚✳ ◆❛❣② ✇❡ ✇✐s❤ t♦ ❞❡t❡r♠✐♥❡ t❤❡ st❛♥❞❛r❞ ✜❧✐❢♦r♠ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❈❛♥♦♥✐❝❛❧ ❜❛s✐s ♦❢ ♥♦♥ t✇♦✲st❡♣ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♥♦♥ t✇♦✲st❡♣ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺ ✇❤✐❝❤ ❛r❡ ♥♦t ❞✐r❡❝t ♣r♦❞✉❝ts ♦❢ ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❧♦✇❡r ❞✐♠❡♥s✐♦♥✳ ❆❝❝♦r❞✐♥❣ t♦ ❲✳ ❆✳ ❞❡ ●r❛❛❢✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✻✲❞✐♠❡♥s✐♦♥❛❧ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ♥♦t ✷✱ ❏✳ ❆❧❣❡❜r❛ ✸✵✾ ✭✷✵✵✼✮✱ ✻✹✵ ✕ ✻✺✸✱ t❤❡ ❧✐st ♦❢ t❤❡s❡ ▲✐❡ ❛❧❣❡❜r❛s ✐s ❣✐✈❡♥ ✉♣ t♦ ✐s♦♠♦r♣❤✐s♠ ❜② t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ❝♦♠♠✉t❛t♦rs ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❞✐st✐♥❣✉✐s❤❡❞ ❜❛s✐s {E✶, E✷, . . . }✿ l✺,✺ : [E✶, E✷] = E✹✱ [E✶, E✹] = E✺✱ [E✷, E✸] = E✺❀ l✺,✻ : [E✶, E✷] = E✸✱ [E✶, E✸] = E✹✱ [E✶, E✹] = E✺✱ [E✷, E✸] = E✺❀ l✺,✼ : [E✶, E✷] = E✸✱ [E✶, E✸] = E✹✱ [E✶, E✹] = E✺❀ l✺,✾ : [E✶, E✷] = E✸✱ [E✶, E✸] = E✹✱ [E✷, E✸] = E✺✳ ❚❤✐s ❜❛s✐s ✇❡ ❝❛❧❧ t❤❡ ❝❛♥♦♥✐❝❛❧ ❜❛s✐s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ▲✐❡ ❛❧❣❡❜r❛✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❍❡✉r✐st✐❝ ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤❡✉r✐st✐❝ ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s ✉♣ t♦ ✐s♦♠❡tr✐❝ ✐s♦♠♦r♣❤✐s♠s✿

✶ ❙❡❧❡❝t ❛ ❜❛s✐s E = {E✶, E✷, . . . , En} ♦❢ t❤❡ ▲✐❡ ❛❧❣❡❜r❛ g✱ s✉❝❤

t❤❛t t❤❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s ❤❛✈❡ ❛ s✐♠♣❧❡ ❢♦r♠✳

✷ ❋♦r ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ., . ♦♥ g ❧❡t F = {F✶, F✷, . . . , Fn} ❜❡

t❤❡ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ t❤❡ ❢♦r♠ Fi = n

k=i aikEk ✇✐t❤

aik ∈ R ❛♥❞ aii > ✵ ♦❜t❛✐♥❡❞ ❢r♦♠ E = {E✶, E✷, . . . , En} ❜② t❤❡

• r❛♠✲❙❝❤♠✐❞t ♣r♦❝❡ss✳ ❚❤❡s❡ ❜❛s❡s ♣❛r❛♠❡tr✐③❡ t❤❡ ✐♥♥❡r

♣r♦❞✉❝ts ♦♥ g✳

✸ ❈♦♠♣✉t❡ t❤❡ ▲✐❡ ❜r❛❝❦❡t ❡①♣r❡ss✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s

F ❛♥❞ ✜♥❞ t❤❡✐r ❣❡♥❡r❛❧ s❤❛♣❡s ❞❡♣❡♥❞✐♥❣ ♦♥ r❡❛❧ ♣❛r❛♠❡t❡rs✳ ❋✐♥❞ ❢♦r ❡❛❝❤ ▲✐❡ ❜r❛❝❦❡t ♦♣❡r❛t✐♦♥ ❣✐✈❡♥ ❜② t❤❡s❡ r❡❛❧ ♣❛r❛♠❡t❡rs ❛❧❧ ♣♦ss✐❜❧❡ ✐♥♥❡r ♣r♦❞✉❝ts✳ ❚❤❡ ❝❧❛ss ♦❢ t❤❡s❡ ✐♥♥❡r ♣r♦❞✉❝ts ❞❡t❡r♠✐♥❡s t❤❡ ❝❧❛ss ♦❢ ✐s♦♠❡tr✐❝ ✐s♦♠♦r♣❤✐❝ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

✶ ❈❤♦♦s❡ ❢♦r ❛♥② ❝❧❛ss ♦❢ ✐s♦♠❡tr✐❝ ✐s♦♠♦r♣❤✐❝ ♠❡tr✐❝ ▲✐❡

❛❧❣❡❜r❛s ❛ r❡♣r❡s❡♥t✐♥❣ ✐♥♥❡r ♣r♦❞✉❝t ❡①♣r❡ss❡❞ ❜② ✐ts ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s F✳ ❚❤❡s❡ ✐♥♥❡r ♣r♦❞✉❝ts t♦❣❡t❤❡r ✇✐t❤ t❤❡ ▲✐❡ ❜r❛❝❦❡t ♦♣❡r❛t✐♦♥ ❣✐✈❡ ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s✳ ❲❡ ✐❧❧✉str❛t❡ t❤✐s ❝❧❛ss✐✜❝❛t✐♦♥ ♠❡t❤♦❞ ♦♥ t❤❡ ▲✐❡ ❛❧❣❡❜r❛ l✺,✺✿ [E✶, E✷] = E✹✱ [E✶, E✹] = E✺✱ [E✷, E✸] = E✺✳ ❲❡ ✜♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡r✐❡s i✺ < i✹ < · · · < i✶ < l✺,✺ = i✵ ♦❢ ✐❞❡❛❧s ♦❢ l✺,✺ ✇✐t❤ ❞✐♠(ik/ik+✶) = ✶✿ t❤❡ ❝❡♥t❡r Z(l✺,✺) ♦❢ l✺,✺ ✐s R E✺✱ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ l′

✺,✺ ✐s R E✹ + R E✺✳ ❚❤❡ ♣r❡✐♠❛❣❡

π−✶(Z(l✺,✺/Z(l✺,✺))) ♦❢ t❤❡ ❝❡♥t❡r Z(l✺,✺/Z(l✺,✺)) ♦❢ t❤❡ ❢❛❝t♦r ❛❧❣❡❜r❛ l✺,✺/Z(l✺,✺) ✐♥ l✺,✺ ✐s R E✸ + R E✹ + R E✺ ❛♥❞ t❤❡ ❝❡♥tr❛❧✐③❡r Cl✺,✺(l′

✺,✺) ♦❢ l′ ✺,✺ ✐s R E✷ + R E✸ + R E✹ + R E✺✳ ❚❤❡ ●r❛♠✲❙❝❤♠✐❞t

♣r♦❝❡ss ❛♣♣❧✐❡❞ t♦ t❤❡ ♦r❞❡r❡❞ ❝❛♥♦♥✐❝❛❧ ❜❛s✐s (E✺, E✹, E✸, E✷, E✶) ②✐❡❧❞s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s {F✶, F✷, F✸, F✹, F✺} ♦❢ l✺,✺✱ ✇❤❡r❡ t❤❡ ✈❡❝t♦r Fi ✐s ❛ ♣♦s✐t✐✈❡ ♠✉❧t✐♣❧❡ ♦❢ Ei ♠♦❞✉❧♦ t❤❡ s✉❜s♣❛❝❡ Ej; j > i ❣❡♥❡r❛t❡❞ ❜② {Ej; j > i} ❛♥❞ ♦rt❤♦❣♦♥❛❧ t♦ Ej; j > i✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❲❡ ❤❛✈❡ F✺ ∈ Z(l✺,✺)✱ F✹ ∈ l′

✺,✺✱ F✸ ∈ π−✶(Z(l✺,✺/Z(l✺,✺))) ❛♥❞

F✷ ∈ Cl✺,✺(l′

✺,✺)✳ ❍❡♥❝❡ l✺,✺ ❤❛s t❤❡ ♦rt❤♦❣♦♥❛❧ ❞✐r❡❝t s✉♠

❞❡❝♦♠♣♦s✐t✐♦♥ l✺,✺ = RF✶ ⊕ · · · ⊕ RF✺ ✐♥t♦ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s RF✶✱ ✳ ✳ ✳ ✱ RF✺ ✇❤✐❝❤ ✐s ❞❡t❡r♠✐♥❡❞ ✉♥✐q✉❡❧② ❜② t❤❡ ❛❧❣❡❜r❛✐❝ ❛♥❞ ♠❡tr✐❝ str✉❝t✉r❡ ♦❢ (l✺,✺, ., .)✳ ❙✐♥❝❡ Fi = ✺

k=i aikEk ✇✐t❤ aii > ✵ ✇❡ ❤❛✈❡

✭✶✮ [F✶, F✷] = aF✹+bF✺, [F✶, F✸] = cF✺, [F✶, F✹] = dF✺, [F✷, F✸] = fF✺, a, d, f > ✵✱ a, b, c, d, f ∈ R✱ ✇❤❡r❡ a = a✶✶a✷✷ a✹✹ , b = a✹✹ (a✶✶a✷✹ + a✶✷a✷✸ − a✶✸a✷✷) − a✶✶a✷✷a✹✺ a✹✹a✺✺ , c = a✶✶a✸✹ + a✶✷a✸✸ a✺✺ , d = a✶✶a✹✹ a✺✺ , f = a✷✷a✸✸ a✺✺ .

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❉❡✜♥✐t✐♦♥ ▲❡t {G✶, G✷, G✸, G✹, G✺} ❜❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ✈❡❝t♦r s♣❛❝❡ E✺ ❛♥❞ a, b, c, d, f r❡❛❧ ♥✉♠❜❡rs ✇✐t❤ a, d, f = ✵✳ ▲❡t n✺,✺(a, b, c, d, f ) ❞❡♥♦t❡ t❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ ❞❡✜♥❡❞ ♦♥ E✺ ❜② t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ❝♦♠♠✉t❛t♦rs [G✶, G✷] = aG✹+bG✺, [G✶, G✸] = cG✺, [G✶, G✹] = dG✺, [G✷, G✸] = fG✺. ❚❤❡ ♠❛♣ E✶ → E✶✱ E✷ → adE✷ + bE✹✱ E✸ →

f ad E✸ + cE✹✱

E✹ → dE✹✱ E✺ → E✺ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ l✺,✺ → n✺,✺(a, b, c, d, f )✳ ❈❤❛♥❣✐♥❣ t❤❡ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s✿ ˜ F✶ = −F✶✱ ˜ F✷ = −F✷✱ ˜ F✸ = F✸✱ ˜ F✹ = F✹✱ ˜ F✺ = −F✺ ✇❡ ♦❜t❛✐♥ [ ˜ F✶, ˜ F✷] = a ˜ F✹−b ˜ F✺, [ ˜ F✶, ˜ F✸] = c ˜ F✺, [ ˜ F✶, ˜ F✹] = d ˜ F✺, [ ˜ F✷, ˜ F✸] = f ˜ F✺.

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❙✐♠✐❧❛r❧②✱ ✇✐t❤ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❜❛s✐s✿ ˜ F✶ = F✶✱ ˜ F✷ = −F✷✱ ˜ F✸ = F✸✱ ˜ F✹ = −F✹✱ ˜ F✺ = −F✺ ♦♥❡ ❤❛s [ ˜ F✶, ˜ F✷] = a ˜ F✹+b ˜ F✺, [ ˜ F✶, ˜ F✸] = −c ˜ F✺, [ ˜ F✶, ˜ F✹] = d ˜ F✺, [ ˜ F✷, ˜ F✸] = f ˜ F✺. ❍❡♥❝❡ t❤❡r❡ ✐s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s s✉❝❤ t❤❛t ✐♥ t❤❡ ❝♦♠♠✉t❛t♦rs ✭✶✮ t❤❡ ❝♦❡✣❝✐❡♥ts b ❛♥❞ c ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡✳ ❚❤❡♦r❡♠ ▲❡t ., . ❜❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ t❤❡ ✺✲❞✐♠❡♥s✐♦♥❛❧ t❤r❡❡✲st❡♣ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛ l✺,✺✳ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ n✺,✺(a, b, c, d, f ) ✇✐t❤ a, d, f > ✵✱ b, c ≥ ✵✱ ✇❤✐❝❤ ✐s ✐s♦♠❡tr✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (l✺,✺, ., .)✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

■t t✉r♥s ♦✉t t❤❛t ✉♣ t♦ ♦♥❡ ❡①❝❡♣t✐♦♥❛❧ ❝❧❛ss ❛❧❧ ❤✐❣❤❡r✲st❡♣ ♥✐❧♣♦t❡♥t ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺ ❤❛✈❡ s✉❝❤ ♦rt❤♦❣♦♥❛❧ ❞✐r❡❝t s✉♠ ❞❡❝♦♠♣♦s✐t✐♦♥✿ ❉❡✜♥✐t✐♦♥ ▲❡t (g, ., .) ❜❡ ❛ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ ❞✐♠❡♥s✐♦♥ n✳ ❆♥ ♦rt❤♦❣♦♥❛❧ ❞✐r❡❝t s✉♠ ❞❡❝♦♠♣♦s✐t✐♦♥ g = V✶ ⊕ · · · ⊕ Vn ♦♥ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s V✶✱ ✳ ✳ ✳ ✱ Vn ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ❛ ❢r❛♠✐♥❣ ♦❢ (g, ., .)✱ ✐❢ ✐t ✐s ❞❡t❡r♠✐♥❡❞ ✉♥✐q✉❡❧② ❜② t❤❡ ❛❧❣❡❜r❛✐❝ ❛♥❞ ♠❡tr✐❝ str✉❝t✉r❡ ♦❢ (g, ., .)✳ ❚❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (g, ., .) ✐s ❝❛❧❧❡❞ ❢r❛♠❡❞✱ ✐❢ ✐t ❤❛s ❛ ❢r❛♠✐♥❣✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

▲❡t (g, ., .) ❛♥❞ (g∗, ., .∗) ❜❡ ❢r❛♠❡❞ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ n ✇✐t❤ ❢r❛♠✐♥❣s g = R e✶ ⊕ · · · ⊕ R en ❛♥❞ g∗ = R e∗

✶ ⊕ · · · ⊕ R e∗ n✱ ✇❤❡r❡ (e✶, . . . , en)✱ r❡s♣❡❝t✐✈❡❧② (e∗ ✶, . . . , e∗ n)

❛r❡ ♦rt❤♦♥♦r♠❛❧ ❜❛s❡s✳ ■❢ Φ : g → g∗ ✐s ❛♥ ✐s♦♠❡tr✐❝ ✐s♦♠♦r♣❤✐s♠✱ ✐t ♠❛♣s R ei → R e∗

i ✱

i = ✶, . . . , n✱ ✐✳❡✳ Φ(ei) = εie∗

i ✇✐t❤ εi = ±✶✳ ❍❡♥❝❡

Φ[ei, ej] = Σn

k=✶ck i,jεke∗ k = [Φ(ei), Φ(ej)]∗ = εiεjΣn k=✶c∗k i,je∗ k,

❝♦♥s❡q✉❡♥t❧② |ck

i,j| = |c∗k i,j|✳

▲❡♠♠❛ ❆ss✉♠❡ t❤❛t t❤❡ ❝♦♠♠✉t❛t♦rs [., .] ✐♥ g ❛♥❞ [., .]∗ ✐♥ g∗ ❛r❡ ❣✐✈❡♥ ✐♥ t❤❡ ❢♦r♠ [ei, ej] = Σn

k=✶ck i,jek✱ [e∗ i , e∗ j ]∗ = Σn k=✶c∗k i,je∗ k✱

i, j, k = ✶, . . . , n✳ ❚❤❡♥ ♦♥❡ ❤❛s ck

i,j = ±c∗k i,j ❢♦r ❛❧❧

i, j, k = ✶, . . . , n✳ P❛rt✐❝✉❧❛r❧②✱ ✐❢ ck

i,j ≥ ✵ ❛♥❞ c∗k i,j ≥ ✵ t❤❡♥

ck

i,j = c∗k i,j✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❚❤❡♦r❡♠ ❚❤❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✐s♦♠❡tr② ❣r♦✉♣ I(N) ♦❢ ❛ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❘✐❡♠❛♥♥✐❛♥ ♥✐❧♠❛♥✐❢♦❧❞ (N, ., .) ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❛♠❡❞ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (n, ., .) ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ▲✐❡ ❣r♦✉♣ N✳ Pr♦♦❢✳ ▲❡♠♠❛ ✶ ②✐❡❧❞s t❤❛t ❛♥② ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠ Φ : n → n ✐s ❣✐✈❡♥ ❜② Φ(ei) = εiei ✇✐t❤ εi = ±✶✳ ❍❡♥❝❡ t❤❡ ❣r♦✉♣ OA(n) ♦❢ ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ (n, ., .) ✐s ❛ s✉❜❣r♦✉♣ ♦❢ Z✷ × · · · × Z✷✱ ✇❤❡r❡ t❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝t♦rs ≤ ❞✐♠ n✳ ❙✐♥❝❡ I(N) ∼ = N ⋊ OA(n) t❤❡ ❛ss❡rt✐♦♥ ❢♦❧❧♦✇s✳ ❍❡♥❝❡ t❤❡ ▲✐❡ ❛❧❣❡❜r❛ n✺,✺(a, b, c, d, f )✱ a > ✵✱ b ≥ ✵✱ c ≥ ✵✱ d > ✵✱ f > ✵ ✐s ✐s♦♠❡tr✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ n✺,✺(a∗, b∗, c∗, d∗, f ∗)✱ a∗ > ✵✱ b∗ ≥ ✵✱ c∗ ≥ ✵✱ d∗ > ✵✱ f ∗ > ✵ ♣r❡❝✐s❡❧② ✐❢ a = a∗✱ b = b∗✱ c = c∗✱ d = d∗✱ f = f ∗✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

n✺,✺(a, b, c, d, f )✿ [G✶, G✷] = aG✹ + bG✺✱ [G✶, G✸] = cG✺✱ [G✶, G✹] = dG✺✱ [G✷, G✸] = fG✺✳ ■❢ T : n✺,✺(a, b, c, d, f ) → n✺,✺(a, b, c, d, f ) ✐s ❛♥ ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ n✺,✺(a, b, c, d, f ) t❤❡♥ TGi = εiGi ❛♥❞ [TGi, TGj] = [εiGi, εjGj] = T[Gi, Gj] ❢♦r i, j = ✶, . . . , ✺✱ ✇❤❡r❡ εi, εj = ±✶✳ ▲❡t b = c = ✵✳ ❋r♦♠ t❤❡ ▲✐❡ ❜r❛❝❦❡ts [ε✶G✶, ε✷G✷] = aε✹G✹✱ [ε✶G✶, ε✹G✹] = dε✺G✺✱ [ε✷G✷, ε✸G✸] = f ε✺G✺ ✇❡ ♦❜t❛✐♥ ε✹ = ε✶ε✷✱ ε✺ = ε✶ε✹ = ε✷ε✸✳ ❍❡♥❝❡ ε✷ = ε✺ = ε✷ε✸✱ ❛♥❞ ε✸ = ✶✱ ε✶ε✹ = ε✷ = ε✺✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❣r♦✉♣ ♦❢ ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ n(a,✵,✵,d,f ) ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❣r♦✉♣ Z✷ × Z✷✳ ■❢ b = ✵✱ c > ✵✱ t❤❡♥ ✇❡ ❤❛✈❡ ✐♥ ❛❞❞✐t✐♦♥ [ε✶G✶, ε✸G✸] = cε✺G✺✱ ✇❤✐❝❤ ②✐❡❧❞s t❤❛t ε✶ε✸ = ε✺✳ ❍❡♥❝❡ ✇❡ ❣❡t ε✶ = ε✷ = ε✺✱ ε✸ = ✶ = ε✹✳ ❚❤❡ ❣r♦✉♣ ♦❢ ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ n(a,✵,c,d,f ) ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❣r♦✉♣ Z✷✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❚❤❡♦r❡♠ ▲❡t ., . ❜❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ t❤❡ ✺✲❞✐♠❡♥s✐♦♥❛❧ t❤r❡❡✲st❡♣ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛ l✺,✺✳ ✭✶✮ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ n✺,✺(a, b, c, d, f ) ✇✐t❤ a, d, f > ✵✱ b, c ≥ ✵✱ ✇❤✐❝❤ ✐s ✐s♦♠❡tr✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (l✺,✺, ., .)✳ ✭✷✮ ❚❤❡ ❣r♦✉♣ ♦❢ ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ n✺,✺(a, b, c, d, f ) ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ♠❛tr✐① ❣r♦✉♣✿

✶ ❢♦r b = c = ✵✿

✭✷✮                  ε✶ ✵ ✵ ✵ ✵ ✵ ε✶ε✹ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ε✹ ✵ ✵ ✵ ✵ ✵ ε✶ε✹       , ε✶, ε✹ = ±✶            ∼ = Z✷ × Z✷,

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

✶ ❢♦r b = ✵, c > ✵✿

✭✸✮                  ε✶ ✵ ✵ ✵ ✵ ✵ ε✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ε✶       , ε✶ = ±✶            ∼ = Z✷,

✷ ❢♦r b > ✵, c = ✵✿

✭✹✮                  ✶ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ε✷       , ε✷ = ±✶            ∼ = Z✷,

✸ ✐❢ b > ✵, c > ✵✱ t❤❡♥ ✐t ✐s tr✐✈✐❛❧

✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s {G✶, G✷, G✸, G✹, G✺}✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❚❤❡ ▲✐❡ ❛❧❣❡❜r❛ l✺,✾ ❤❛✈✐♥❣ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❡♥t❡r

❉❡✜♥✐t✐♦♥ ▲❡t {G✶, G✷, G✸, G✹, G✺} ❜❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ✈❡❝t♦r s♣❛❝❡ E✺✳ ▲❡t n✺,✾(l, m, n, p, q)✱ l, m, n, p, q ∈ R ✇✐t❤ l, p, q = ✵✱ ❞❡♥♦t❡ t❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ ❞❡✜♥❡❞ ♦♥ E✺ ❜② t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ❝♦♠♠✉t❛t♦rs ✭✺✮ [G✶, G✷] = lG✸ + mG✹ + nG✺, [G✶, G✸] = pG✹, [G✷, G✸] = qG✺. ■❢ n = ✵✱ p = q ✇❡ ❞❡♥♦t❡ ˜ n✺,✾(l, m, p) = n✺,✾(l, m, ✵, p, p)✳ ❚❤❡ ♠❛♣ G✶ → G✶ + n

qG✸✱ G✷ → G✷ − m p G✸✱ G✸ → lG✸✱

G✹ → plG✹, G✺ → qlG✺ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ n✺,✾(l, m, n, p, q) t♦ l✺,✾✳ ❚❤❡♦r❡♠ ❚❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (l✺,✾, ., .) ✐s ✐s♦♠❡tr✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ ❛ ✉♥✐q✉❡ n✺,✾(l, m, n, p, q) ✇✐t❤ l, m, n, p, q ∈ R s✉❝❤ t❤❛t l > ✵✱ q > p > ✵ ❛♥❞ m, n ≥ ✵✱ ♦r t♦ ❛ ✉♥✐q✉❡ ˜ n✺,✾(l, m, p) ✇✐t❤ l, m, p ∈ R s✉❝❤ t❤❛t l, p > ✵ ❛♥❞ m ≥ ✵✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❚❤❡ ▲✐❡ ❛❧❣❡❜r❛ l✺,✾ ✐s ❞❡✜♥❡❞ ❜②✿ [E✶, E✷] = E✸✱ [E✶, E✸] = E✹✱ [E✷, E✸] = E✺✳ ❚❤❡ ❝❡♥t❡r Z(l✺,✾) ♦❢ l✺,✾ ✐s R E✹ + R E✺ ❛♥❞ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ l′

✺,✾ ✐s R E✸ + R E✹ + R E✺✳ ▲❡t ❜❡ ❣✐✈❡♥ ❛♥ ✐♥♥❡r

♣r♦❞✉❝t ., . ♦♥ l✺,✾ ❛♥❞ ❛♣♣❧② t❤❡ ●r❛♠✲❙❝❤♠✐❞t ♣r♦❝❡ss t♦ t❤❡ ♦r❞❡r❡❞ ❝❛♥♦♥✐❝❛❧ ❜❛s✐s (E✺, E✹, E✸, E✷, E✶) ♦❢ l✺,✾✳ ❲❡ ♦❜t❛✐♥ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s E ∗

i = ✺ k=i aikEk✱ i = ✺, . . . , ✶ ♦❢ l✺,✾ ✇✐t❤

aii > ✵ s✉❝❤ t❤❛t [E ∗

✶ , E ∗ ✷ ] = lE ∗ ✸ +mE ∗ ✹ +nE ∗ ✺ , [E ∗ ✶ , E ∗ ✸ ] = pE ∗ ✹ +rE ∗ ✺ , [E ∗ ✷ , E ∗ ✸ ] = qE ∗ ✺ ,

✇❤❡r❡ l = a✶✶a✷✷

a✸✸

> ✵✱ p = a✶✶a✸✸

a✹✹

> ✵✱ q = a✷✷a✸✸

a✺✺

> ✵✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❚♦ ❞✐st✐♥❣✉✐s❤ ❛ ✶✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ t❤❡ ❝❡♥t❡r Z(l✺,✾) = E ∗

✹ , E ∗ ✺ ❛♥❞ ❛ ✶✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ t❤❡

(l′

✺,✾)⊥ = E ∗ ✶ , E ∗ ✷ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♦rt❤♦❣♦♥❛❧ ✉♥✐t ✈❡❝t♦rs F(t)✱

F(t + π

✷ ) ✐♥ (l′ ✺,✾)⊥✿

F(t) = ❝♦s tE ∗

✶ + s✐♥ tE ∗ ✷ ,

F(t + π ✷ ) = − s✐♥ tE ∗

✶ + ❝♦s tE ∗ ✷ .

❲❡ ❤❛✈❡ [F(t), F(t + π

✷ )] = [E ∗ ✶ , E ∗ ✷ ] ❛♥❞

Φ(t) = [F(t), E ∗

✸ ] = a✶✶a✸✸ a✹✹

❝♦s tE ∗

✹ +

a✸✸ a✺✺ a✶✷a✹✹ − a✶✶a✹✺ a✹✹ ❝♦s t + a✷✷ s✐♥ t

• E ∗

✺ ,

❛♥❞ Φ(t + π

✷ ) = [F(t + π ✷ ), E ∗ ✸ ] = − a✶✶a✸✸ a✹✹

s✐♥ tE ∗

✹ +

a✸✸ a✺✺ −a✶✷a✹✹ + a✶✶a✹✺ a✹✹ s✐♥ t + a✷✷ ❝♦s t

• E ∗

✺ .

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

Φ(t✵) ❛♥❞ Φ(t✵ + π

✷ ) ❛r❡ ♦rt❤♦❣♦♥❛❧ ✐❢

✶ ✷

• a✷

✷✷a✷ ✹✹ − a✷ ✶✶a✷ ✺✺ − (a✶✷a✹✹ − a✶✶a✹✺)✷

s✐♥ ✷t✵+ a✷✷a✹✹ (a✶✷a✹✹ − a✶✶a✹✺) ❝♦s ✷t✵ = ✵. ▲❡♠♠❛ ❚❤❡ ✈❡❝t♦rs Φ(t) ❛♥❞ Φ(t + π

✷ ) ❛r❡ ♦rt❤♦❣♦♥❛❧ ❢♦r ❛♥② t ∈ R ✐❢ ❛♥❞

♦♥❧② ✐❢ ✭✻✮ a✶✷a✹✹ = a✶✶a✹✺ ❛♥❞ a✶✶a✺✺ = a✷✷a✹✹. ❖t❤❡r✇✐s❡✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✵ ≤ t✵ < π

✷ s✉❝❤ t❤❛t Φ(t) ❛♥❞

Φ(t + π

✷ ) ❛r❡ ♦rt❤♦❣♦♥❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t = t✵ + k π ✷ ✱ k ∈ Z✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

■♥ t❤❡ s❡❝♦♥❞ ❝❛s❡ ✇❡ ❤❛✈❡ ❡✐t❤❡r Φ(t✵) < Φ(t✵ + π

✷ ) ❛♥❞

❤❡♥❝❡ ✇❡ ❞❡✜♥❡ F✶ = F(t✵), F✷ = F(t✵ + π ✷ ), F✸ = E ∗

✸ , F✹ =

Φ(t✵) Φ(t✵) , ✭✼✮ F✺ = Φ(t✵ + π

✷ )

Φ(t✵ + π

✷ ) ,

♦r ✐❢ Φ(t✵) = Φ(t✵ + π) > Φ(t✵ + π

✷ ) ✱ t❤❡♥ ✇❡ ❞❡✜♥❡

F✶ = F(t✵ + π ✷ ), F✷ = F(t✵ + π), F✸ = E ∗

✸ , F✹ =

Φ(t✵ + π

✷ )

Φ(t✵ + π

✷ ) ,

✭✽✮ F✺ = Φ(t✵ + π) Φ(t✵ + π) .

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❋r♦♠ t❤✐s ❝♦♥str✉❝t✐♦♥ ✇❡ ♦❜t❛✐♥ t❤❛t ✭✾✮ [F✶, F✷] = lF✸ + mF✹ + nF✺, [F✶, F✸] = pF✹, [F✷, F✸] = qF✺ ✇✐t❤ l > ✵✱ q > p > ✵✳ ❯s✐♥❣ t❤❡ ❜❛s✐s ❝❤❛♥❣❡ ˜ F✶ = −F✶✱ ˜ F✷ = F✷✱ ˜ F✸ = −F✸✱ ˜ F✹ = F✹✱ ˜ F✺ = −F✺ ♦♥❡ ❤❛s [ ˜ F✶, ˜ F✷] = l ˜ F✸ − m ˜ F✹ + n ˜ F✺✱ [ ˜ F✶, ˜ F✸] = p ˜ F✹✱ [ ˜ F✷, ˜ F✸] = q ˜ F✺✳ ❲✐t❤ t❤❡ ❜❛s✐s ❝❤❛♥❣❡ ˜ F✶ = F✶✱ ˜ F✷ = −F✷✱ ˜ F✸ = −F✸✱ ˜ F✹ = −F✹✱ ˜ F✺ = F✺ ♦♥❡ ❤❛s [ ˜ F✶, ˜ F✷] = l ˜ F✸ + m ˜ F✹ − n ˜ F✺✱ [ ˜ F✶, ˜ F✸] = p ˜ F✹✱ [ ˜ F✷, ˜ F✸] = q ˜ F✺✳ ❍❡♥❝❡ ✇❡ ❝❛♥ ❝❤♦♦s❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s s✉❝❤ t❤❛t ✐♥ t❤❡ ❝♦♠♠✉t❛t♦rs ✭✾✮ t❤❡ ❝♦❡✣❝✐❡♥ts s❛t✐s❢② m, n ≥ ✵✳ ❚❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s R F✶✱ R F✷✱ R F✸✱ R F✹✱ R F✺ ❢♦r♠ ❛ ❢r❛♠✐♥❣ ♦❢ (l✺,✾, ., .) s✐♥❝❡ t❤❡ s✉❜s♣❛❝❡ RF✺ ⊂ Z(l✺,✾) ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ✈❡❝t♦r ✐♥ {Φ(t✵), Φ(t✵ + π

✷ )} ❤❛✈✐♥❣ ❣r❡❛t❡r ♥♦r♠✳

❚❤❡ s✉❜s♣❛❝❡ RF✹ ⊂ Z(l✺,✾) ✐s ♦rt❤♦❣♦♥❛❧ t♦ RF✺✳ ❚❤❡ s✉❜s♣❛❝❡ R F✸ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ ❛♥❞ ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ❝❡♥t❡r✳ ❚❤❡ ♦rt❤♦❣♦♥❛❧ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s R F✶✱ R F✷ ❛r❡ ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ ❛♥❞ t❤❡ s✉❜s♣❛❝❡ R[F✷, F✸] ✐s ❝♦♥t❛✐♥❡❞ ✐♥ RF✺✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

■❢ Φ(t) ❛♥❞ Φ(t + π

✷ ) ❛r❡ ♦rt❤♦❣♦♥❛❧ ❢♦r ❛♥② t ∈ R✱ t❤❡♥

Φ(t) = Φ(t + π

✷ ) = a✶✶a✸✸ a✹✹

= ❝♦♥st✳ ❋♦r E ∗

✹ ✱ E ∗ ✺ ✇❡ ♦❜t❛✐♥

E ∗

✹ = a✹✹ a✶✶a✸✸

• ❝♦s tΦ(t) − s✐♥ tΦ(t + π

✷ )

• ✱

E ∗

✺ = a✹✹ a✶✶a✸✸

• s✐♥ tΦ(t) + ❝♦s tΦ(t + π

✷ )

• ✳ ❚❤❡ ▲✐❡ ❜r❛❝❦❡t

[F(t), F(t + π

✷ )] = [E ∗ ✶ , E ∗ ✷ ] = lE ∗ ✸ + mE ∗ ✹ + nE ∗ ✺ ✱ l > ✵✱ ❝❛♥ ❜❡

✇r✐tt❡♥ ✐♥t♦ t❤❡ ❢♦r♠ l E ∗

✸ +

+ a✹✹ a✶✶a✸✸

• (m ❝♦s t + n s✐♥ t)Φ(t) + (−m s✐♥ t + n ❝♦s t)Φ(t + π

✷ )

• .

■❢ m = n = ✵ ✇❡ ♣✉t Fi = E ∗

i ✳ ❋♦r (m, n) = (✵, ✵) t❤❡r❡ ✐s ✉♥✐q✉❡

t✵ ∈ (− π

✷ , π ✷ ] s✉❝❤ t❤❛t t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❡q✉❛t✐♦♥

−m s✐♥ t + n ❝♦s t = ✵ ❛r❡ t✵ + k π

✷ ✱ k ∈ Z✳ ❚❤❡♥ ✇❡ ❞❡✜♥❡

F✶ = F(t✵)✱ F✷ = F(t✵ + π

✷ )✱

F✸ = E ∗

✸ , F✹ =

a✹✹ a✶✶a✸✸ Φ(t✵), F✺ = a✹✹ a✶✶a✸✸ Φ(t✵ + π ✷ ).

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❲❡ ♦❜t❛✐♥ t❤❛t t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ❜r❛❝❦❡ts [Fi, Fj]✱ i, j = ✶, . . . , ✺✱ ❤❛✈❡ t❤❡ s❤❛♣❡ ✭✶✵✮ [F✶, F✷] = lF✸ + mF✹, [F✶, F✸] = pF✹, [F✷, F✸] = pF✺, ✇✐t❤ s♦♠❡ m ∈ R✱ l, p > ✵✳ ❯s✐♥❣ t❤❡ ✐s♦♠❡tr✐❝ ✐s♦♠♦r♣❤✐s♠ ˜ F✶ = −F✶✱ ˜ F✷ = F✷✱ ˜ F✸ = −F✸✱ ˜ F✹ = F✹✱ ˜ F✺ = −F✺ ✇❡ ♦❜t❛✐♥ [ ˜ F✶, ˜ F✷] = l ˜ F✸ − m ˜ F✹, [ ˜ F✶, ˜ F✸] = p ˜ F✹, [ ˜ F✷, ˜ F✸] = p ˜ F✺✳ ❍❡♥❝❡ ✇❡ ❝❛♥ ❛ss✉♠❡ m ≥ ✵✳ ■❢ m = ✵✱ t❤❡♥ t❤❡ s✉❜s♣❛❝❡ R F✺ ⊂ Z(˜ n✺,✾(l, m, p) ✐s ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ▲✐❡ ❜r❛❝❦❡t ♦❢ ❛♥② t✇♦ ✈❡❝t♦rs ❝♦♥t❛✐♥❡❞ ✐♥ R F✶ ⊕ R F✷✳ ❚❤❡ s✉❜s♣❛❝❡ RF✹ ⊂ Z(˜ n✺,✾(l, m, p)) ✐s ♦rt❤♦❣♦♥❛❧ t♦ RF✺✳ ❚❤❡ s✉❜s♣❛❝❡ R F✸ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ ❛♥❞ ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ❝❡♥t❡r✳ ❚❤❡ ♦rt❤♦❣♦♥❛❧ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s R F✶✱ R F✷ ❛r❡ ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ ❛♥❞ t❤❡ s✉❜s♣❛❝❡ R[F✷, F✸] ✐s ❝♦♥t❛✐♥❡❞ ✐♥ RF✺✳ ❍❡♥❝❡ t❤❡ s✉❜s♣❛❝❡s R F✶✱ R F✷✱ R F✸✱ R F✹✱ R F✺ ❢♦r♠ ❛ ❢r❛♠✐♥❣ ♦❢ t❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (˜ n✺,✾(l, m, p), ., .)✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

■❢ m = ✵ t❤❡♥ ✇❡ ❤❛✈❡ ✭✶✶✮ [F✶, F✷] = lF✸, [F✶, F✸] = pF✹, [F✷, F✸] = pF✺. ❋r♦♠ t❤✐s ✇❡ ❝❛♥ s❡❡ t❤❛t Z(˜ n✺,✾(l, ✵, p)) ✐s ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ▲✐❡ ❜r❛❝❦❡t ♦❢ ❛♥② t✇♦ ✈❡❝t♦rs ❝♦♥t❛✐♥❡❞ ✐♥ R F✶ ⊕ R F✷✳ ❍❡♥❝❡ t❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (˜ n✺,✾(l, ✵, p), ., .) ✐s ♥♦t ❢r❛♠❡❞✳ ■♥ t❤✐s ❝❛s❡ ❢♦r ❛♥② ✐s♦♠❡tr✐❝ ✐s♦♠♦r♣❤✐s♠ Φ : ˜ n✺,✾(l, ✵, p) → ˜ n✺,✾(l∗, m∗, p∗) ♦♥❡ ❤❛s Φ(F✶) = ❝♦s tF✶ ± s✐♥ tF✷✱ Φ(F✷) = ∓ s✐♥ tF✶ + ❝♦s tF✷✱ Φ(F✸) = εF✸✱ Φ(F✹) = ❝♦s tF✹ ± s✐♥ tF✺✱ Φ(F✺) = ∓ s✐♥ tF✹ + ❝♦s tF✺ ❢r♦♠ ✇❤✐❝❤ ❡❛s✐❧② ❢♦❧❧♦✇s m∗ = ✵✱ l = l∗✱ p = p∗✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❚❤❡♦r❡♠ ▲❡t ., . ❜❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ t❤❡ ✺✲❞✐♠❡♥s✐♦♥❛❧ t❤r❡❡✲st❡♣ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛ l✺,✾✳ ✭✶✮ ❚❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (l✺,✾, ., .) ✐s ✐s♦♠❡tr✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ ❛ ✉♥✐q✉❡ n✺,✾(l, m, n, p, q) ✇✐t❤ l, m, n, p, q ∈ R s✉❝❤ t❤❛t l > ✵✱ q > p > ✵ ❛♥❞ m, n ≥ ✵✱ ♦r t♦ ❛ ✉♥✐q✉❡ ˜ n✺,✾(l, m, p) ✇✐t❤ l, m, p ∈ R s✉❝❤ t❤❛t l, p > ✵ ❛♥❞ m ≥ ✵✳ ✭✷✮ ❚❤❡ ❣r♦✉♣s ♦❢ ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐① ❣r♦✉♣s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s {G✶, G✷, G✸, G✹, G✺}✿

✭❆✮ ❢♦r n✺,✾(l, m, n, p, q)

✭✐✮ ✐❢ m = n = ✵✿ ✭✶✷✮                  ε✶ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ε✶ε✷ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ε✶       , ε✶, ε✷ = ±✶            ∼ = Z✷ × Z✷,

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

✭✐✐✮ ✐❢ m = ✵✱ n > ✵✿ ✭✶✸✮                  ε✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ε✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ε✶       , ε✶ = ±✶            ∼ = Z✷, ✭✐✐✐✮ ✐❢ m > ✵✱ n = ✵✿ ✭✶✹✮                  ✶ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ✶       , ε✷ = ±✶            ∼ = Z✷, ✭✐✈✮ ✐❢ m, n > ✵✱ t❤❡♥ ✐t ✐s tr✐✈✐❛❧❀

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❋♦r t❤❡ ▲✐❡ ❛❧❣❡❜r❛ ˜ n✺,✾(l, m, p)✿ (i) ✐❢ m = ✵✿ ✭✶✺✮                  ❝♦s t ε✷ s✐♥ t ✵ ✵ ✵ − s✐♥ t ε✷ ❝♦s t ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ε✷ ❝♦s t s✐♥ t ✵ ✵ ✵ −ε✷ s✐♥ t ❝♦s t       , ε✷ = ±✶, t ∈ [✵, ✷π)            , (ii) ✐❢ m > ✵✱ t❤❡♥ ✐t ✐s t❤❡ ❣r♦✉♣ ✭✶✻✮                  ✶ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ε✷ ✵ ✵ ✵ ✵ ✵ ✶       , ε✷ = ±✶            ∼ = Z✷.

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❚❤❡♦r❡♠ ❊✈❡r② ✜❧✐❢♦r♠ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ ✐s ❛ ❢r❛♠❡❞ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛✳ Pr♦♦❢✳ ❊✈❡r② ✜❧✐❢♦r♠ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛ g ❤❛s ❛ ❜❛s✐s {E✶, · · · , En}✿ ✭✶✼✮ [E✶, Ei] = Ei+✶, ❢♦r ❛❧❧ i ≥ ✷, ✭❞❡✜♥✐♥❣ r❡❧❛t✐♦♥s ♦❢ t❤❡ st❛♥❞❛r❞ ✜❧✐❢♦r♠ ▲✐❡ ❛❧❣❡❜r❛s✮ ✭✶✽✮ [Ei, Ej] ∈ gi+j = ❙♣❛♥(Ei+j, · · · , En), ❢♦r ❛❧❧ i, j : i+j = n+✶ ❛♥❞ t❤❡r❡ ❡①✐sts α ∈ R ✇✐t❤ ✭✶✾✮ [Ei, En−i+✶] = (−✶)iαEn, ❢♦r ❛❧❧ ✷ ≤ i ≤ n − ✶. ■❢ n ✐s ♦❞❞✱ t❤❡♥ α = ✵✳ ❖♥❡ s❡t Ei = ✵ ❢♦r i > n ✭❝❢✳ ❚❤❡♦r❡♠ ✹✳✶ ✐♥ ●✳ ❈❛✐r♥s✱ ❆✳ ❍✐♥✐➣ ●❛❧✐➣✱ ❨✉✳ ◆✐❦♦❧❛②❡✈s❦②✱ ❚♦t❛❧❧② ❣❡♦❞❡s✐❝ s✉❜❛❧❣❡❜r❛s ♦❢ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s✱ ❏✳ ▲✐❡ ❚❤❡♦r②✱ ✷✸ ✭✷✵✶✸✮✱ ✶✵✷✸ ✕ ✶✵✹✾✮✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

Pr♦♦❢✳ ❚❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s C✵(g) = g, . . . , Ci(g) = [g, Ci−✶(g)] = R Ei+✷ + · · · + R En, . . . , Cn−✷(g) ♦❢ g ❢♦r♠s ❛ s❡r✐❡s ♦❢ ✐♥✈❛r✐❛♥t ✐❞❡❛❧s ✇✐t❤ ❞✐♠(Ci(g)/Ci+✶(g)) = ✶✳ ❚❤❡ ●r❛♠✲❙❝❤♠✐❞t ♣r♦❝❡ss ❛♣♣❧✐❡❞ t♦ t❤❡ ♦r❞❡r❡❞ ❜❛s✐s (En, En−✶, . . . , E✷, E✶) ②✐❡❧❞s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s F = {F✶, F✷, . . . , Fn} s✉❝❤ t❤❛t t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ g′ = C✶(g) ❤❛s ❛ ❢r❛♠✐♥❣ g′ = R F✸ ⊕ · · · ⊕ R Fn✳ ❚❤❡ ♦rt❤♦❣♦♥❛❧ ❝♦♠♣❧❡♠❡♥t❡r ♦❢ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ ❤❛s ❞✐♠❡♥s✐♦♥ ✷✳ ■❢ g ✐s ❛ st❛♥❞❛r❞ ✜❧✐❢♦r♠ ▲✐❡ ❛❧❣❡❜r❛✱ t❤❡♥ t❤❡ ❝❡♥tr❛❧✐③❡r Cg(g′) ♦❢ t❤❡ ❝♦♠♠✉t❛t♦r ❛❧❣❡❜r❛ g′ ✐s t❤❡ ✐❞❡❛❧ R E✷ + · · · + R En✳ ❍❡♥❝❡ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ g = R F✶ ⊕ · · · ⊕ R Fn ✐s ❛ ❢r❛♠✐♥❣ ♦❢ t❤❡ ♠❡tr✐❝ st❛♥❞❛r❞ ✜❧✐❢♦r♠ ❛❧❣❡❜r❛ (g, ., .)✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

Pr♦♦❢✳ ❋r♦♠ t❤❡ ❝♦♠♠✉t❛t♦r r❡❧❛t✐♦♥s ✭✶✼✲✶✾✮ ✇❡ ❤❛✈❡ [E✷, C✶(g)] ⊂ C✸(g) = E✺, · · · , En✳ ❚❤❡ ❢❛❝t♦r ▲✐❡ ❛❧❣❡❜r❛ g/C✸(g) = ¯ E✶, ¯ E✷, ¯ E✸, ¯ E✹ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ▲✐❡ ❜r❛❝❦❡ts [ ¯ E✶, ¯ E✷] = ¯ E✸✱ [ ¯ E✶, ¯ E✸] = ¯ E✹✳ ❍❡♥❝❡ ✐t ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ✹✲❞✐♠❡♥s✐♦♥❛❧ st❛♥❞❛r❞ ✜❧✐❢♦r♠ ▲✐❡ ❛❧❣❡❜r❛ s✹✳ ❚❤❡ ❝❡♥tr❡ Z(g/C✸(g)) ♦❢ t❤❡ ❢❛❝t♦r ▲✐❡ ❛❧❣❡❜r❛ g/C✸(g) ✐s R ¯ E✹✱ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ (g/C✸(g))′ ♦❢ g/C✸(g) ✐s R ¯ E✸ + R ¯ E✹ ❛♥❞ t❤❡ ❝❡♥tr❛❧✐③❡r Cg/C✸(g)(g/C✸(g))′ ♦❢ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❛❧❣❡❜r❛ (g/C✸(g))′ ♦❢ t❤❡ ❢❛❝t♦r ❛❧❣❡❜r❛ g/C✸(g) ✐s R ¯ E✷ + R ¯ E✸ + R ¯ E✹✳ ❚❤❡r❡❢♦r❡ t❤❡ ♣r❡✐♠❛❣❡ ♦❢ t❤❡ ❝❡♥tr❛❧✐③❡r Cg/C✸(g)(g/C✸(g))′ ✐♥ g ✐s R E✷ + R E✸ + R E✹ + R E✺ + · · · + R En✳ ❆♣♣❧②✐♥❣ t❤❡

• r❛♠✲❙❝❤♠✐❞t ♣r♦❝❡ss t♦ t❤❡ ♦r❞❡r❡❞ ❜❛s✐s (En, · · · , E✶) ✇❡ ♦❜t❛✐♥

❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s {F✶, · · · , Fn} s✉❝❤ t❤❛t t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ R F✶ ⊕ · · · ⊕ R Fn ✐s ❛ ❢r❛♠✐♥❣ ♦❢ t❤❡ ♠❡tr✐❝ ✜❧✐❢♦r♠ ❛❧❣❡❜r❛ (g, ., .)✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

sn : [E✶, E✷] = E✸, . . . , [E✶, Ei] = Ei+✶, . . . , [E✶, En−✶] = En. [G✶, G✷] = c✷,✷G✸ + c✷,✸G✹ + · · · + c✷,n−✶Gn [G✶, Gi] = ci,iGi+✶ + · · · + ci,n−✶Gn [G✶, Gn−✶] = cn−✶,n−✶Gn✳ ❉❡✜♥✐t✐♦♥ ▲❡t {G✶, . . . , Gn} ❜❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ✐♥ t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥ ✈❡❝t♦r s♣❛❝❡ En ❛♥❞ ❧❡t C = {cj,k ∈ R; ✷ ≤ k ≤ j ≤ n − ✶} ❜❡ ❛ ❧♦✇❡r tr✐❛♥❣✉❧❛r n − ✷ × n − ✷ ♠❛tr✐① ✇✐t❤ ♣♦s✐t✐✈❡ ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts✳ ❲❡ ❞❡♥♦t❡ ❜② n C t❤❡ ▲✐❡ ❛❧❣❡❜r❛ ❛♥❞ ❜② [., .]C ✐ts ▲✐❡ ❜r❛❝❦❡t ❞❡✜♥❡❞ ♦♥ En ❜② t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ❝♦♠♠✉t❛t♦rs ✭✷✵✮ [G✶, Gi]C = −[Gi, G✶]C =

n−✶

• t=i

ct,i Gt+✶, i = ✷, . . . , n − ✶. ❚❤❡ ▲✐❡ ❛❧❣❡❜r❛ n C ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ., .C ✐♥❞✉❝❡❞ ❜② t❤❡ ❊✉❝❧✐❞❡❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦❢ En ✐s ❛ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (n C, ., .C)✳ ▲❡♠♠❛ ❚❤❡ ✲❞✐♠❡♥s✐♦♥❛❧ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ st❛♥❞❛r❞ ✜❧✐❢♦r♠ ▲✐❡ ❛❧❣❡❜r❛ ✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❯s✐♥❣ ❛♥ ✐s♦♠❡tr✐❝ ✐s♦♠♦r♣❤✐s♠ ✇❡ ❛❝❤✐✈❡ t❤❛t t❤❡ s✐❣♥ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ck,i s✉❝❤ t❤❛t k − i ✐s ♦❞❞ s✐♠✉❧t❛♥❡♦✉s❧② ❝❤❛♥❣❡❞✳ ■❢ t❤❡ s❡t P = {(k, i) : ck,i = ✵ ❛♥❞ k − i ✐s ♦❞❞} ✐s ♥♦t ❡♠♣t②✱ t❤❡♥ t❤❡r❡ ✐s ❛ s✉✐t❛❜❧❡ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s s❛t✐s❢②✐♥❣ ck✵,i✵ > ✵ ❢♦r t❤❡ ♠✐♥✐♠❛❧ ❡❧❡♠❡♥t (k✵, i✵) ♦❢ P ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛♥t✐✲❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ♦❢ ♣❛✐rs✳ ❚❤❡♦r❡♠ ▲❡t ., . ❜❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ st❛♥❞❛r❞ ✜❧✐❢♦r♠ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛ sn✳ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (n C, ., .C) s❛t✐s❢②✐♥❣

✶ (n C, ., .C) ✐s ✐s♦♠❡tr✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ (sn, ., .)✱ ✷ ✐❢ t❤❡ s❡t P = {(k, i) : ck,i = ✵ ❛♥❞ k − i ✐s ♦❞❞} ✐s ♥♦t ❡♠♣t②

t❤❡♥ ck✵,i✵ > ✵ ❢♦r t❤❡ ♠✐♥✐♠❛❧ ❡❧❡♠❡♥t (k✵, i✵) ♦❢ P ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛♥t✐✲❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ♦❢ ♣❛✐rs✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❚❤❡♦r❡♠ ❚❤❡ ❣r♦✉♣ ♦❢ ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ n C ✐s t❤❡ ♠❛tr✐① ❣r♦✉♣

✶ ✐❢ {(i, k) : ck,i = ✵ ❛♥❞ k − i ✐s ♦❞❞} = ∅✿

✭✷✶✮                     ε✶ ✵ ✵ . . . ✵ ✵ ε✷

✶ε✷

✵ . . . ✵ ✵ ✵ ε✸

✶ε✷

. . . ✵ ✵ ✵ ✵ ✳✳✳ ✵ ✵ ✵ ✵ ✵ εn

✶ε✷

       , ε✶, ε✷ = ±✶              ∼ = Z✷×Z✷,

✷ ✐❢ {(i, k) : ck,i = ✵ ❛♥❞ k − i ✐s ♦❞❞} = ∅✿

✭✷✷✮               ✶ ✵ . . . ✵ ✵ ε✷ . . . ✵ ✵ ✵ ✳✳✳ ✵ ✵ ✵ ✵ ε✷      , ε✷ = ±✶          ∼ = Z✷, ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s {G✶, . . . , Gn}✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

l✺,✻✿ [E✶, E✷] = E✸✱ [E✶, E✸] = E✹✱ [E✶, E✹] = E✺✱ [E✷, E✸] = E✺ ❉❡✜♥✐t✐♦♥ ▲❡t {G✶, G✷, G✸, G✹, G✺} ❜❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ✈❡❝t♦r s♣❛❝❡ E✺ ❛♥❞ a, b, c, d, f , g, h r❡❛❧ ♥✉♠❜❡rs ✇✐t❤ a, d, g, h = ✵✳ ❚❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ ❞❡✜♥❡❞ ♦♥ E✺ ❜② t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ❝♦♠♠✉t❛t♦rs [G✶, G✷] = aG✸ + bG✹ + cG✺✱ [G✶, G✸] = dG✹ + fG✺✱ [G✶, G✹] = gG✺✱ [G✷, G✸] = hG✺ ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② n(a, b, c, d, f , g, h)✳ ❚❤❡ ♠❛♣ E✶ → E✶✱ E✷ → w(adg E✷ + bg E✸ + (c − bf

d )E✹)✱

E✸ → wdg E✸, E✹ → w(g E✹ − f d E✺), E✺ → wE✺, ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ l✺,✻ → n(a, b, c, d, f , g, h)✱ ✇❤❡r❡ w =

h ad✷g✷ ✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺

❚❤❡♦r❡♠ ▲❡t ., . ❜❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ t❤❡ ✺✲❞✐♠❡♥s✐♦♥❛❧ ✜❧✐❢♦r♠ ❜✉t ♥♦t st❛♥❞❛r❞ ✜❧✐❢♦r♠ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛ l✺,✻✳ ✭✶✮ ❚❤❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ (l✺,✻, ., .) ✐s ✐s♦♠❡tr✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ ❛ ✉♥✐q✉❡ ♠❡tr✐❝ ▲✐❡ ❛❧❣❡❜r❛ n(a, b, c, d, f , g, h) ✇✐t❤ a, b, c, d, f , g, h ∈ R s✉❝❤ t❤❛t ❡✐t❤❡r a, b, d, g, h > ✵✱ ♦r a, d, g, h > ✵✱ b = ✵✱ f ≥ ✵✳ ✭✷✮ ❚❤❡ ❣r♦✉♣s ♦❢ ♦rt❤♦❣♦♥❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ n(a, b, c, d, f , g, h) ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐① ❣r♦✉♣s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s {G✶, G✷, G✸, G✹, G✺}✿

✭✐✮ ❢♦r b = f = ✵✿ ✭✷✸✮                  ε✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ε✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ε✶       , ε✶ = ±✶            ∼ = Z✷, ✭✐✐✮ ❢♦r b✷ + f ✷ = ✵✱ ✐t ✐s tr✐✈✐❛❧✳

➪❣♦t❛ ❋✐❣✉❧❛ ▼❡tr✐❝ ♥✐❧♣♦t❡♥t ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❞✐♠❡♥s✐♦♥ ✺