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❖♥ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ t✇♦ s❡ts ♦❢ ❤✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛❧ ❣❛♠♠❛ ♠❛tr✐❝❡s ❛♥❞ ❛ ♣r✐♠✐t✐✈❡ ✜❡❧❞ ❡q✉❛t✐♦♥

❉♠✐tr② ❙❤✐r♦❦♦✈

◆❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❯♥✐✈❡rs✐t② ❍✐❣❤❡r ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s ❑❤❛r❦❡✈✐❝❤ ■♥st✐t✉t❡ ❢♦r ■♥❢♦r♠❛t✐♦♥ ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠s ♦❢ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s

❚❤❡ ✷♥❞ ❋r❡♥❝❤✲❘✉ss✐❛♥ ❝♦♥❢❡r❡♥❝❡ ✏❘❛♥❞♦♠ ●❡♦♠❡tr② ❛♥❞ P❤②s✐❝s✑ ❖❝t♦❜❡r ✶✼✲✷✶✱ ✷✵✶✻✱ P❛r✐s✱ ❋r❛♥❝❡

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶ ✴ ✶✾

❉✐r❛❝ ❣❛♠♠❛ ♠❛tr✐❝❡s

γ✵ =     ✶ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ −✶ ✵ ✵ ✵ ✵ −✶     , γ✶ =     ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ −✶ ✵ ✵ −✶ ✵ ✵ ✵     , γ✷ =     ✵ ✵ ✵ −i ✵ ✵ i ✵ ✵ i ✵ ✵ −i ✵ ✵ ✵     , γ✸ =     ✵ ✵ ✶ ✵ ✵ ✵ ✵ −✶ −✶ ✵ ✵ ✵ ✵ ✶ ✵ ✵     . γaγb + γbγa = ✷ηab✶, a, b = ✵, ✶, ✷, ✸. η = ηab = ❞✐❛❣(✶, −✶, −✶, −✶). ❉✐r❛❝ P✳❆✳▼✳✱ Pr♦❝✳ ❘♦②✳ ❙♦❝✳ ▲♦♥❞✳ ❆✶✶✼ ✭✶✾✷✽✮✳ ❉✐r❛❝ P✳❆✳▼✳✱ Pr♦❝✳ ❘♦②✳ ❙♦❝✳ ▲♦♥❞✳ ❆✶✶✽ ✭✶✾✷✽✮✳

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✷ ✴ ✶✾

P❛✉❧✐✬s ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠

❚❤❡♦r❡♠ ✭P❛✉❧✐✮ ❈♦♥s✐❞❡r ✷ s❡ts ♦❢ sq✉❛r❡ ❝♦♠♣❧❡① ♠❛tr✐❝❡s γa, βa, a = ✶, ✷, ✸, ✹. ♦❢ s✐③❡ ✹✳ ▲❡t t❤❡s❡ ✷ s❡ts s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s γaγb + γbγa = ✷ηab✶, η = ❞✐❛❣(✶, −✶, −✶, −✶), βaβb + βbβa = ✷ηab✶. ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ❝♦♠♣❧❡① ❝♦♥st❛♥t✮ ❝♦♠♣❧❡① ♠❛tr✐① T s✉❝❤ t❤❛t γa = T −✶βaT, a = ✶, ✷, ✸, ✹. ❲✳P❛✉❧✐✱ ❈♦♥tr✐❜✉t✐♦♥s ♠❛t❤❡♠❛t✐q✉❡s ❛ ❧❛ t❤❡♦r✐❡ ❞❡s ♠❛tr✐❝❡s ❞❡ ❉✐r❛❝✱ ❆♥♥✳ ■♥st✳ ❍❡♥r✐ P♦✐♥❝❛r❡ ✻✱ ✭✶✾✸✻✮✳

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✸ ✴ ✶✾

❈❧✐✛♦r❞ ❛❧❣❡❜r❛s CℓR(p, q) ❛♥❞ CℓC(p, q) = C ⊗ CℓR(p, q)

❧✐♥❡❛r s♣❛❝❡ E ♦✈❡r R, ❞✐♠ E = ✷n, ❜❛s✐s✿ {e, ea✶, ea✶a✷, . . . , e✶...n}, ✶ ≤ a✶ < · · · < ak ≤ n, ♠✉❧t✐♣❧✐❝❛t✐♦♥✿

✶ ❞✐str✐❜✉t✐✈✐t②✱

❛ss♦❝✐❛t✐✈✐t②✱ e ✲ ✐❞❡♥t✐t② ❡❧❡♠❡♥t✱

✷ ea✶ . . . eak = ea✶...ak,

✶ ≤ a✶ < · · · < ak ≤ n,

✸ eaeb +ebea = ✷ηabe,

η = ||ηab|| = ❞✐❛❣(✶, . . . , ✶

p

, −✶, . . . , −✶

• q

), p +q = n. Cℓ(p, q) ∋ U = ue +

• a

uaea +

• a<b

uabeab + · · · + u✶...ne✶...n = uAeA. Cℓ(p, q) =

n

• k=✵

Cℓk(p, q), Cℓk(p, q) = {

• |A|=k

uAeA}. Cℓ(p, q) = Cℓ❊✈❡♥(p, q) ⊕ Cℓ❖❞❞(p, q), Cℓ❊✈❡♥(p, q) =

• k−even

Cℓk(p, q), Cℓ❖❞❞(p, q) =

• k−odd

Cℓk(p, q).

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✹ ✴ ✶✾

❚❤❡♦r❡♠ cenCℓ(p, q) = Cℓ✵(p, q), ✐❢ n ✲ ❡✈❡♥❀ Cℓ✵(p, q) ⊕ Cℓn(p, q), ✐❢ n ✲ ♦❞❞✳ ❚❤❡♦r❡♠ ✭❈❛rt❛♥ ✶✾✵✽✱ ❇♦tt ✶✾✻✵✮ CℓR(p, q) ≃              ▼❛t(✷

n ✷ , R),

✐❢ p − q ≡ ✵; ✷ ♠♦❞ ✽❀ ▼❛t(✷

n−✶ ✷ , R) ⊕ ▼❛t(✷ n−✶ ✷ , R),

✐❢ p − q ≡ ✶ ♠♦❞ ✽❀ ▼❛t(✷

n−✶ ✷ , C),

✐❢ p − q ≡ ✸; ✼ ♠♦❞ ✽❀ ▼❛t(✷

n−✷ ✷ , H),

✐❢ p − q ≡ ✹; ✻ ♠♦❞ ✽❀ ▼❛t(✷

n−✸ ✷ , H) ⊕ ▼❛t(✷ n−✸ ✷ , H),

✐❢ p − q ≡ ✺ ♠♦❞ ✽✳ ❚❤❡♦r❡♠ CℓC(p, q) ≃ ▼❛t(✷

n ✷ , C),

✐❢ n ✲ ❡✈❡♥❀ ▼❛t(✷

n−✶ ✷ , C) ⊕ ▼❛t(✷ n−✶ ✷ , C),

✐❢ n ✲ ♦❞❞✳

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✺ ✴ ✶✾

▲❡t t❤❡ s❡t ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ❡❧❡♠❡♥ts s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s βa ∈ Cℓ(p, q), βaβb + βbβa = ✷ηabe. ❚❤❡♥ t❤❡ s❡t γa = T −✶βaT, ∀ ✐♥✈❡rt✐❜❧❡ T ∈ Cℓ(p, q) s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s γaγb + γbγa = ✷ηabe. ❘❡❛❧❧②✱ γaγb + γbγa = T −✶βaTT −✶βbT + T −✶βbTT −✶βaT = = T −✶(βaβb + βbβa)T = T −✶✷ηabeT = ✷ηabe.

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✻ ✴ ✶✾

❚❤❡♦r❡♠ ✭❈❛s❡ ♦❢ ❡✈❡♥ n✮ ❈♦♥s✐❞❡r r❡❛❧ ✭♦r ❝♦♠♣❧❡①✐✜❡❞✮ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ Cℓ(p, q) ♦❢ ❡✈❡♥ ❞✐♠❡♥s✐♦♥ n = p + q✳ ▲❡t t❤❡ ❢♦❧❧♦✇✐♥❣ ✷ s❡ts ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ❡❧❡♠❡♥ts γa, βa, a = ✶, ✷, . . . , n s❛t✐s❢② ❝♦♥❞✐t✐♦♥s γaγb + γbγa = ✷ηabe, βaβb + βbβa = ✷ηabe. ❚❤❡♥ ❜♦t❤ s❡ts ♦❢ ❡❧❡♠❡♥ts ❣❡♥❡r❛t❡ ❜❛s❡s ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ r❡❛❧ ✭❝♦♠♣❧❡①✮ ❝♦♥st❛♥t✮ ❡❧❡♠❡♥t T ∈ Cℓ(p, q) s✉❝❤ t❤❛t γa = T −✶βaT, ∀a = ✶, . . . , n. ▼♦r❡♦✈❡r✱ ✇❡ ❝❛♥ ❛❧✇❛②s ✜♥❞ t❤✐s ❡❧❡♠❡♥t T ✐♥ t❤❡ ❢♦r♠ T =

• A

βAFγA, γA = (γA)−✶ ✇❤❡r❡ F ✐s ❛♥② ❡❧❡♠❡♥t ♦❢ ❛ s❡t ✶) {γA, A ∈ I❊✈❡♥} ✐❢ β✶...n = −γ✶...n; ✷) {γA, A ∈ I❖❞❞} ✐❢ β✶...n = γ✶...n s✉❝❤ t❤❛t ❝♦rr❡s♣♦♥❞✐♥❣ T ✐s ♥♦♥③❡r♦ T = ✵✳

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✼ ✴ ✶✾

❈❛s❡ ♦❢ ❡✈❡♥ n ✐♥ ♠❛tr✐① ❢♦r♠❛❧✐s♠

❚❤❡♦r❡♠ ▲❡t n ✲ ❡✈❡♥ ❛♥❞ ✷ s❡ts ♦❢ sq✉❛r❡ ♠❛tr✐❝❡s γa, βa, a = ✶, ✷, . . . , n s❛t✐s❢② ❝♦♥❞✐t✐♦♥s γaγb + γbγa = ✷ηab✶, βaβb + βbβa = ✷ηab✶. ■❢ ♠❛tr✐❝❡s ❛r❡ ❝♦♠♣❧❡① ♦❢ t❤❡ ♦r❞❡r ✷

n ✷ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ❛

❝♦♠♣❧❡① ❝♦♥st❛♥t✮ ♠❛tr✐① T s✉❝❤ t❤❛t ■❢ s✐❣♥❛t✉r❡ ✐s p − q ≡ ✵, ✷ ♠♦❞ ✽ ❛♥❞ ♠❛tr✐❝❡s ❛r❡ r❡❛❧ ♦❢ t❤❡ ♦r❞❡r ✷

n ✷ ✱ t❤❡♥

t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ❛ r❡❛❧ ❝♦♥st❛♥t✮ ♠❛tr✐① T s✉❝❤ t❤❛t ■❢ s✐❣♥❛t✉r❡ ✐s p − q ≡ ✹, ✻ ♠♦❞ ✽ ❛♥❞ ♠❛tr✐❝❡s ❛r❡ ♦✈❡r t❤❡ q✉❛t❡r♥✐♦♥s ♦❢ t❤❡ ♦r❞❡r ✷

n−✷ ✷ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ❛ r❡❛❧ ❝♦♥st❛♥t✮ ♠❛tr✐① T

s✉❝❤ t❤❛t γa = T −✶βaT, a = ✶, . . . n.

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✽ ✴ ✶✾

❚❤❡ ❝❛s❡ ♦❢ ♦❞❞ n

❊①❛♠♣❧❡ ✶✿ CℓR(✷, ✶) ≃ ▼❛t(✷, R) ⊕ ▼❛t(✷, R) ✇✐t❤ ❣❡♥❡r❛t♦rs e✶, e✷, e✸✳ ❲❡ ❝❛♥ t❛❦❡ γ✶ = e✶, γ✷ = e✷, γ✸ = e✶e✷. ❚❤❡♥ γaγb + γbγa = ✷ηab✶✳ ❊❧❡♠❡♥ts γ✶, γ✷, γ✸ ❣❡♥❡r❛t❡ ♥♦t CℓR(✷, ✶)✱ ❜✉t ❣❡♥❡r❛t❡ CℓR(✷, ✵) ≃ ▼❛t(✷, R)✳ ❊①❛♠♣❧❡ ✷✿ CℓR(✸, ✵) ≃ ▼❛t(✷, C) ✇✐t❤ ❣❡♥❡r❛t♦rs e✶, e✷, e✸✳ β✶ = σ✶ =

• ✵

✶ ✶ ✵

• ,

β✷ = σ✷ =

• ✵

−i i ✵

• ,

β✸ = σ✸ =

• ✶

✵ ✵ −✶

• .

γa = −σa, a = ✶, ✷, ✸. ❚❤❡♥ γ✶✷✸ = −β✶✷✸✳ ❙✉♣♣♦s❡✱ t❤❛t ✇❡ ❤❛✈❡ s✉❝❤ T ∈ ●▲(✷, C) t❤❛t γa = T −✶βaT✳ ❚❤❡♥ γ✶✷✸ = T −✶β✶TT −✶β✷TT −✶β✸T = T −✶β✶β✸β✸T = β✶✷✸ ❛♥❞ ✇❡ ♦❜t❛✐♥ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ✭✇❡ ✉s❡ t❤❛t β✶✷✸ = σ✶✷✸ = i

• ✶

✵ ✵ ✶

• = i✶✮✳

❇✉t ✇❡ ❤❛✈❡ s✉❝❤ ❡❧❡♠❡♥t T = ✶ t❤❛t γa = −T −✶βaT✳

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✾ ✴ ✶✾

❚❤❡♦r❡♠ ❈♦♥s✐❞❡r r❡❛❧ ✭♦r ❝♦♠♣❧❡①✐✜❡❞✮ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ Cℓ(p, q) ♦❢ ♦❞❞ ❞✐♠❡♥s✐♦♥ n = p + q✳ ▲❡t t❤❡ ❢♦❧❧♦✇✐♥❣ ✷ s❡ts ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ❡❧❡♠❡♥ts γa, βa, a = ✶, ✷, . . . , n s❛t✐s❢② ❝♦♥❞✐t✐♦♥s γaγb + γbγa = ✷ηabe, βaβb + βbβa = ✷ηabe. ❚❤❡♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ Cℓ(p, q) ♦❢ s✐❣♥❛t✉r❡ p − q ≡ ✶ ♠♦❞ ✹ ❡❧❡♠❡♥ts γ✶...n ❛♥❞ β✶...n ❡q✉❛❧s ±e✶...n ❛♥❞ t❤❡♥ ❝♦rr❡s♣♦♥❞✐♥❣ s❡ts ❣❡♥❡r❛t❡ ❜❛s❡s ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ♦r ❡q✉❛❧s ±e ❛♥❞ t❤❡♥ ❝♦rr❡s♣♦♥❞✐♥❣ s❡ts ❞♦♥✬t ❣❡♥❡r❛t❡ ❜❛s❡s✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ Cℓ(p, q) ♦❢ s✐❣♥❛t✉r❡ p − q ≡ ✸ ♠♦❞ ✹ ❡❧❡♠❡♥ts γ✶...n ❛♥❞ β✶...n ❡q✉❛❧s ±e✶...n✱ ❛♥❞ t❤❡♥ ❝♦rr❡s♣♦♥❞✐♥❣ s❡ts ❣❡♥❡r❛t❡ ❜❛s❡s ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ♦r ✭♦♥❧② ❢♦r CℓC(p, q)✮ ❡q✉❛❧s ±ie ❛♥❞ t❤❡♥ ❝♦rr❡s♣♦♥❞✐♥❣ s❡ts ❞♦♥✬t ❣❡♥❡r❛t❡ ❜❛s❡s✳

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✵ ✴ ✶✾

❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ❛ ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥t ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ❝❡♥t❡r✮ ❡❧❡♠❡♥t T s✉❝❤ t❤❛t ✶) γa = T −✶βaT, ∀a = ✶, . . . , n ⇔ β✶...n = γ✶...n, ✷) γa = −T −✶βaT, ∀a = ✶, . . . , n ⇔ β✶...n = −γ✶...n, ✸) γa = e✶...nT −✶βaT, ∀a = ✶, . . . , n ⇔ β✶...n = e✶...nγ✶...n, ✹) γa = −e✶...nT −✶βaT, ∀a = ✶, . . . , n ⇔ β✶...n = −e✶...nγ✶...n, ✺) γa = ie✶...nT −✶βaT, ∀a = ✶, . . . , n ⇔ β✶...n = ie✶...nγ✶...n, ✻) γa = −ie✶...nT −✶βaT, ∀a = ✶, . . . , n ⇔ β✶...n = −ie✶...nγ✶...n. ◆♦t❡✱ t❤❛t ❛❧❧ ✻ ❝❛s❡s ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠ γa = (β✶...nγ✶...n)T −✶βaT.

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✶ ✴ ✶✾

▼♦r❡♦✈❡r✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ r❡❛❧ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ CℓR(p, q) ♦❢ s✐❣♥❛t✉r❡ p − q ≡ ✸ ♠♦❞ ✹ ✇❡ ❝❛♥ ❛❧✇❛②s ✜♥❞ t❤✐s ❡❧❡♠❡♥t T ✐♥ t❤❡ ❢♦r♠

• A∈I❊✈❡♥

βAFγA, ✇❤❡r❡ F ✐s ❛♥② ❡❧❡♠❡♥t ♦❢ t❤❡ s❡t γA, A ∈ I❊✈❡♥, s✉❝❤ t❤❛t ❝♦rr❡s♣♦♥❞✐♥❣ T ✐s ♥♦♥③❡r♦ T = ✵✳ ■♥ t❤❡ ❝❛s❡ ♦❢ r❡❛❧ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ CℓR(p, q) ♦❢ s✐❣♥❛t✉r❡ p − q ≡ ✶ ♠♦❞ ✹ ❛♥❞ ❝♦♠♣❧❡①✐✜❡❞ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ CℓC(p, q) ✇❡ ❝❛♥ ❛❧✇❛②s ✜♥❞ t❤✐s ❡❧❡♠❡♥t T ✐♥ t❤❡ ❢♦r♠

• A∈I❊✈❡♥

βAFγA, ✇❤❡r❡ F ✐s ♦♥❡ ♦❢ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ s❡t γA + γB, A, B ∈ I❊✈❡♥.

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✷ ✴ ✶✾

❈❧✐✛♦r❞ ✜❡❧❞ ✈❡❝t♦rs

Rp,q, p + q = n, η = ηµν, µ, ν = ✶, . . . , n xµ → ´ xµ = pµ

ν xν,

O(p, q) = {P = pµ

ν ∈ ▼❛t(n, R) : PTηP = η}.

❚❡♥s♦r ✜❡❧❞s ✇✐t❤ ✈❛❧✉❡s ✐♥ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛✿ Uµ✶...µr

ν✶...νs ∈ Cℓ(p, q)❚r s ✇❤❡r❡

❝♦♠♣♦♥❡♥ts Uµ✶...µr

ν✶...νs ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s ❢✉♥❝t✐♦♥s Rp,q → Cℓ(p, q)✳

■❢ hµ = hµ(x) ❛r❡ ❝♦♠♣♦♥❡♥ts ♦❢ ✈❡❝t♦r ✜❡❧❞ ✇✐t❤ ✈❛❧✉❡s ✐♥ Cℓ(p, q) hµ(x) = uµ(x)e + uµ

a (x)ea +

• a✶<a✷

uµ

a✶a✷(x)ea✶a✷ + . . . + uµ ✶...n(x)e✶...n,

t❤❛t s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s✿ hµ(x)hν(x) + hν(x)hµ(x) = ✷ηµνe, µ, ν = ✶, . . . , n ✭✶✮ ❢♦r ❛♥② ∀x ∈ Rp,q ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥ π✵(h✶ . . . hn) = ✵, ✭✷✮ t❤❡♥ t❤❡ ✈❡❝t♦r hµ ∈ Cℓ(p, q)❚✶ ✐s ❝❛❧❧❡❞ ❛ ❈❧✐✛♦r❞ ✜❡❧❞ ✈❡❝t♦r✳ ❲❡ ❝❛❧❧ ❛♥ ❛❧❣❡❜r❛ ✇✐t❤ t❤❡ ❜❛s✐s {e, hµ, hµν, . . . , h✶...n} ❛♥ ❛❧❣❡❜r❛ ♦❢ h✲❢♦r♠s Cℓ[h](p, q)✳ ■t ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❆t✐②❛❤✲❑☎ ❛❤❧❡r ❛❧❣❡❜r❛✳

n

• ✶
• ✶

✶

ν✶ ν ∈ C

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✸ ✴ ✶✾

❚❤❡♦r❡♠ ✭▲♦❝❛❧ ❣❡♥❡r❛❧✐③❡❞ P❛✉❧✐✬s t❤❡♦r❡♠ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❡✈❡♥ n✮ ▲❡t n ❜❡ ❡✈❡♥ ♥✉♠❜❡r ❛♥❞ ha = ha(x)✱ a = ✶, . . . , n ❛r❡ ❢✉♥❝t✐♦♥s Ω → Cℓ(p, q) ♦❢ ❝❧❛ss C k(Ω) s✉❝❤ t❤❛t ha(x)hb(x) + hb(x)ha(x) = ✷ηabe, a, b = ✶, . . . , n, ∀x ∈ Ω. ❚❤❡♥ ❢♦r ❛♥② x✵ ∈ Ω t❤❡r❡ ❡①✐sts ε > ✵ ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥ T = T(x) : Oε(x✵) → Cℓ(p, q)✱ s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s

✶ T(x) ✕ ❢✉♥❝t✐♦♥ ♦❢ ❝❧❛ss C k(Oε(x✵))❀ ✷ T(x) ✕ ❛♥ ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥t ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ Cℓ(p, q) ❢♦r ❛♥② x ∈ Oε(x✵)❀ ✸ ea = T −✶(x)ha(x)T(x)✱ a = ✶, . . . , n✱ ∀x ∈ Oε(x✵)❀ ✹ ❚❤❡ ❢✉♥❝t✐♦♥ T(x) ✐s ❞❡✜♥❡❞ ✉♣ t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ✭r❡❛❧ ✐♥ t❤❡ ❝❛s❡

CℓR(p, q) ♦r ❝♦♠♣❧❡① ✐♥ t❤❡ ❝❛s❡ CℓC(p, q)✮ ❢✉♥❝t✐♦♥ ♦❢ ❝❧❛ss C k(Oε(x✵)) t❤❛t ✐s ♥♦t ❡q✉❛❧ t♦ ③❡r♦ ❢♦r ❛♥② ♣♦✐♥t ♦❢ Oε(x✵)✳

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✹ ✴ ✶✾

Pr✐♠✐t✐✈❡ ✜❡❧❞ ❡q✉❛t✐♦♥✱ ❣❛✉❣❡ ✐♥✈❛r✐❛♥❝❡

▲✐❡ ❛❧❣❡❜r❛✿ Cℓ(p, q) = Cℓ(p, q) \ ❝❡♥Cℓ(p, q). ❚❤❡♦r❡♠ ▲❡t hν ∈ Cℓ(p, q)❚✶ ❜❡ ❛ ❈❧✐✛♦r❞ ✜❡❧❞ ✈❡❝t♦r ❛♥❞ Cµ ∈ Cℓ(p, q)❚✶ s❛t✐s❢② t❤❡ ♣r✐♠✐t✐✈❡ ✜❡❧❞ ❡q✉❛t✐♦♥ ∂µhρ − [Cµ, hρ] = ✵, ∀µ, ρ = ✶, . . . , n. ▲❡t S : Rp,q → Cℓ×(p, q) ❜❡ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ✈❛❧✉❡s ✐♥ Cℓ×(p, q) s✉❝❤ t❤❛t S−✶∂µS ∈ Cℓ(p, q)❚✶. ❚❤❡♥✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣♦♥❡♥ts ♦❢ ❝♦✈❡❝t♦rs ´ hρ = S−✶hρS ∈ Cℓ(p, q)❚✶, ´ Cµ = S−✶CµS − S−✶∂µS ∈ Cℓ(p, q)❚✶ ❛❧s♦ s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ ∂µ´ hρ − [ ´ Cµ, ´ hρ] = ✵, ∀µ, ρ = ✶, . . . , n.

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✺ ✴ ✶✾

• ❡♥❡r❛❧ s♦❧✉t✐♦♥ ♦❢ ♣r✐♠✐t✐✈❡ ✜❡❧❞ ❡q✉❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t hν ∈ Cℓ(p, q)❚✶ ❜❡ ❛ ❈❧✐✛♦r❞ ✜❡❧❞ ✈❡❝t♦r ❛♥❞ Cµ ∈ Cℓ(p, q)❚✶✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ s②st❡♠s ♦❢ ❡q✉❛t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ∂µhρ − [Cµ, hρ] = ✵, µ, ρ = ✶, . . . , n ⇔ Cµ =

´ n

• k=✶

µkπ[h]k((∂µhρ)hρ), ✭✸✮ ✇❤❡r❡ ´ n = n ❢♦r ❡✈❡♥ n✱ ´ n = n − ✶ ❢♦r ♦❞❞ n ❛♥❞ µk = ✶ n − (−✶)k(n − ✷k). ■♥ t❤❡ ❝❛s❡ n = ✷ Cµ =

✷

• k=✶

µkπ[h]k((∂µhρ)hρ) = ✶ ✷π[h]✶((∂µhρ)hρ) + ✶ ✹π[h]✷((∂µhρ)hρ) = ✶ ✷(∂µhρ)hρ − ✶ ✶✻hα(∂µhρ)hρhα − ✸ ✸✷hβhα(∂µhρ)hρhαhβ.

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✻ ✴ ✶✾

■♥ t❤❡ ❝❛s❡ n = ✸ Cµ =

✷

• k=✶

µkπ[h]k((∂µhρ)hρ) = ✶ ✹π[h]✶((∂µhρ)hρ) + ✶ ✹π[h]✷((∂µhρ)hρ) = ✶ ✹π[h]✶✷((∂µhρ)hρ) = ✸ ✶✻(∂µhρ)hρ − ✶ ✶✻hα(∂µhρ)hρhα. ■♥ t❤❡ ❝❛s❡ n = ✹ Cµ =

✹

• k=✶

µkπ[h]k((∂µhρ)hρ) = ✶ ✻π[h]✶((∂µhρ)hρ) + ✶ ✹π[h]✷((∂µhρ)hρ) +✶ ✷π[h]✸((∂µhρ)hρ) + ✶ ✽π[h]✹((∂µhρ)hρ) = ✶ ✹(∂µhρ)hρ + ✻✼ ✺✼✻hα(∂µhρ)hρhα + ✼✸ ✷✸✵✹hβhα(∂µhρ)hρhαhβ − ✶✾ ✷✸✵✹hγhβhα(∂µhρ)hρhαhβhγ − ✷✺ ✾✷✶✻hδhγhβhα(∂µhρ)hρhαhβhγhδ.

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✼ ✴ ✶✾

• ❧♦❜❛❧ P❛✉❧✐✬s t❤❡♦r❡♠ ❛♥❞ ♣r✐♠✐t✐✈❡ ✜❡❧❞ ❡q✉❛t✐♦♥

❊①❛♠♣❧❡✿ n = ✷✱ (p, q) = (✷, ✵)✱ γa = ea✱ βa = ha(x)✱ a = ✶, ✷✳ ❋r♦♠ ∂µeρ − [Cµ, eρ] = ✵ ✇❡ ♦❜t❛✐♥ Cµ = ✵✳ ❋r♦♠ ∂µhρ − [ ´ Cµ, hρ] = ✵ ✐♥ ♣❛rt✐❝✉❧❛r ❝❛s❡ hµ = uµ

a (x)ea ∈ Cℓ✶(p, q) ✇❡ ♦❜t❛✐♥ ´

Cµ = ✶

✹(∂µhρ)hρ✳

❯s✐♥❣ ❣❛✉❣❡ ✐♥✈❛r✐❛♥❝❡ ´ hρ = S−✶eρS, ´ Cµ = S−✶CµS − S−✶∂µS ✇❡ ♦❜t❛✐♥ ❡q✉❛t✐♦♥ ´ Cµ = −S−✶∂µS ♦r ∂µ(S−✶) = ´ CµS−✶✳ ■❢ h✶(x) = ❝♦sϕ(x)e✶ + s✐♥ϕ(x)e✷, h✷(x) = −s✐♥ϕ(x)e✶ + ❝♦sϕ(x)e✷, t❤❡♥ ´ Cµ = ✶ ✹(∂µhρ)hρ = −∂µϕ ✷ e✶✷. ❲❡ ♦❜t❛✐♥ ∂µ(S−✶) = − ∂µϕ

✷ e✶✷S−✶ ❛♥❞ S−✶ = ❡①♣( −ϕ ✷ e✶✷)C,

C ∈ Cℓ(p, q)✳ ❋✐♥❛❧❧②✱ S(x) = ❡①♣(ϕ ✷ e✶✷) = ❝♦sϕ ✷ e + s✐♥ϕ ✷ e✶✷, hρ(x) = S−✶(x)eρS(x).

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✽ ✴ ✶✾

P❛♣❡rs✿ ❙❤✐r♦❦♦✈ ❉✳ ❙✳✱ ❊①t❡♥s✐♦♥ ♦❢ P❛✉❧✐✬s t❤❡♦r❡♠ t♦ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛s✱ ❉♦❦❧✳ ▼❛t❤✳✱ ✽✹✱ ✷✱ ✻✾✾ ✲ ✼✵✶ ✭✷✵✶✶✮✳ ❙❤✐r♦❦♦✈ ❉✳ ❙✳✱P❛✉❧✐ t❤❡♦r❡♠ ✐♥ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ♥✲❞✐♠❡♥s✐♦♥❛❧ s♣✐♥♦rs ✐♥ t❤❡ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ❢♦r♠❛❧✐s♠✱❚❤❡♦r❡t✳ ❛♥❞ ▼❛t❤✳ P❤②s✳✱ ✶✼✺✿✶ ✭✷✵✶✸✮✱ ✹✺✹ ✲ ✹✼✹✳ ❙❤✐r♦❦♦✈ ❉✳ ❙✳✱ ❈❛❧❝✉❧❛t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ♦❢ s♣✐♥ ❣r♦✉♣s ✉s✐♥❣ ❣❡♥❡r❛❧✐③❡❞ P❛✉❧✐✬s t❤❡♦r❡♠✱ ❆❞✈❛♥❝❡s ✐♥ ❆♣♣❧✐❡❞ ❈❧✐✛♦r❞ ❆❧❣❡❜r❛s✱ ❱♦❧✉♠❡ ✷✺✱ ◆✉♠❜❡r ✶✱ ✷✷✼ ✲ ✷✹✹✱ ✭✷✵✶✺✮ ❛r❳✐✈✿✶✹✵✾✳✷✹✹✾ ❬♠❛t❤✲♣❤❪✳ ◆✳●✳▼❛r❝❤✉❦✱ ❉✳❙✳❙❤✐r♦❦♦✈✱ ●❡♥❡r❛❧ s♦❧✉t✐♦♥s ♦❢ ♦♥❡ ❝❧❛ss ♦❢ ✜❡❧❞ ❡q✉❛t✐♦♥s✱ ❘❡♣♦rts ♦♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s ✭✷✵✶✻✱ t♦ ❛♣♣❡❛r✮✳ ❇♦♦❦s✿ ❮✳➹✳❒àð÷óê✱ ➘✳Ñ✳Øèðîêîâ✱ ➶âåäåíèå â òåîðèþ àëãåáð ✃ëèôôîðäà✱ Ôàçèñ✱ ❒îñêâà✱ ✷✵✶✷✱ ✺✾✵ ❝✳ ➘✳Ñ✳Øèðîêîâ✱ ❐åêöèè ïî àëãåáðàì ✃ëèôôîðäà è ñïèíîðàì✱ ❐åêöèîííûå êóðñû ❮❰Ö✱ ✶✾✱ ❒➮➚❮✱ ❒✳✱ ✷✵✶✷✱ ✶✽✵ ñ✳ ❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶✾ ✴ ✶✾