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▼❛❝❦❡② ❢✉♥❝t♦rs ❛♥❞ ●r❡❡♥ ❢✉♥❝t♦rs

❊❧❛♥❣♦ P❛♥❝❤❛❞❝❤❛r❛♠

❏♦✐♥t ✇♦r❦ ✇✐t❤ Pr♦❢❡ss♦r ❘♦ss ❙tr❡❡t

❈❡♥tr❡ ♦❢ ❆✉str❛❧✐❛♥ ❈❛t❡❣♦r② ❚❤❡♦r② ▼❛❝q✉❛r✐❡ ❯♥✐✈❡rs✐t② ❙②❞♥❡② ❆✉str❛❧✐❛

✶

▼❛✐♥ ❘❡❢❡r❡♥❝❡s

✶✳ ❏✳ ❆✳ ●r❡❡♥✱ ❆①✐♦♠❛t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ❢♦r ✜♥✐t❡ ❣r♦✉♣s✱ ❏✳ P✉r❡ ❛♥❞ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✶ ✭✶✾✼✶✮✱ ✹✶✕✼✼✳ ✷✳ ❆✳ ❲✳ ▼✳ ❉r❡ss✱ ❈♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ t❤❡♦r② ♦❢ ✐♥❞✉❝❡❞ r❡♣r❡s❡♥t❛t✐♦♥s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳ ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✮ ✸✹✷ ✭❆❧❣❡❜r❛✐❝ ❑✲❚❤❡♦r② ■■✮ ✭✶✾✼✸✮✱ ✶✽✸ ✕✷✹✵✳ ✸✳ ❍✳ ▲✐♥❞♥❡r✱ ❆ r❡♠❛r❦ ♦♥ ▼❛❝❦❡② ❢✉♥❝t♦rs✱ ▼❛♥✉s❝r✐♣t❛ ▼❛t❤✳ ✶✽ ✭✶✾✼✻✮✱ ✷✼✸✕✷✼✽✳ ✹✳ ❙✳ ❇♦✉❝✱ ●r❡❡♥ ❢✉♥❝t♦rs ❛♥❞ ●✲s❡ts✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳ ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✮ ✶✻✼✶ ✭✶✾✾✼✮✳

✷

❚❤❡ ❈♦♠♣❛❝t ❝❧♦s❡❞ ❝❛t❡❣♦r② Spn(E )

• ▲❡t E ❜❡ ❛ ✜♥✐t❡❧② ❝♦♠♣❧❡t❡ ❝❛t❡❣♦r②✳
• ❖❜❥❡❝ts ♦❢ Spn(E ) ❛r❡ t❤❡ ♦❜❥❡❝ts ♦❢ E✳
• ▼♦r♣❤✐s♠s U

V ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠s ❝❧❛ss ♦❢

s♣❛♥s ❢r♦♠ U t♦ V ✳

• ❆ s♣❛♥ ❢r♦♠ U t♦ V ✐s ❛ ❞✐❛❣r❛♠✱

(s1,S,s2) : S V

s2

• U

s1

• ❆♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ t✇♦ s♣❛♥s (s1,S,s2) :U

V ❛♥❞

(s′

1,S′,s′ 2) :U

V ✐s ❛♥ ✐♥✈❡rt✐❜❧❡ ❛rr♦✇ h : S S′ s✉❝❤

t❤❛t ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠ ❝♦♠♠✉t❡s✳

S V

s2

• U

s1

• S′

h ∼ =

• s′

1

• s′

2

• ✸

• ❚❤❡ ❝♦♠♣♦s✐t❡ ♦❢ t✇♦ s♣❛♥s (s1,S,s2) :U

V ❛♥❞

(t1,T,t2) : V

W ✐s (s1 ◦ p1,S ×V T,t2 ◦ p2)

S ×V T T

p2

• W

t2

• S

p1

• U

s1

• V

s2

• t1
• ❚❤❡ ✐❞❡♥t✐t② s♣❛♥ (1,U,1) :U

U ✐s

U U

1

• U

1

• ❚❤✐s ❞❡✜♥❡s t❤❡ ❝❛t❡❣♦r② Spn(E )✳
• ❲❡ ✇r✐t❡ Spn(E )(U,V ) ∼

= [E /(U ×V )]✳

✹

• ❚❤❡ ❝❛t❡❣♦r② Spn(E ) ✐s ♠♦♥♦✐❞❛❧✳ ❚❡♥s♦r ♣r♦❞✉❝t

Spn(E )×Spn(E ) × Spn(E )

✐s ❞❡✜♥❡❞ ❜②

(U,V )

U ×V

[U S U′,V T V ′]

[U ×VS×TU′ ×V ′].

• ■t ✐s ❛❧s♦ ❝♦♠♣❛❝t ❝❧♦s❡❞✳

■♥ ❢❛❝t✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s♦♠♦r♣❤✐s♠s✿

⋆ Spn(E )(U,V ) ∼ = Spn(E )(V,U) ⋆ Spn(E )(U ×V,W ) ∼ = Spn(E )(U,V ×W )

❚❤❡ s❡❝♦♥❞ ✐s♦♠♦r♣❤✐s♠ ❝❛♥ ❜❡ s❤♦✇♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠

S W

• U ×V
• ←→

S W

• U
• V
• ←→

S V ×W

• U
• ✺

❉✐r❡❝t s✉♠s ✐♥ Spn(E )

• ▲❡t E ❜❡ ❛ ❧❡①t❡♥s✐✈❡ ❝❛t❡❣♦r②✳
• ❆ ❝❛t❡❣♦r② E ✐s ❝❛❧❧❡❞ ❧❡①t❡♥s✐✈❡ ✇❤❡♥ ✐t ❤❛s ✜♥✐t❡

❧✐♠✐ts ❛♥❞ ✜♥✐t❡ ❝♦♣r♦❞✉❝ts s✉❝❤ t❤❛t t❤❡ ❢✉♥❝t♦r

E /A ×E /B

E /A +B ;

X

f

• A

, Y

g

• B
• X +Y

f +g

• A +B

✐s ❛♥ ❡q✉✐✈❛❧❛♥❝❡ ♦❢ ❝❛t❡❣♦r✐❡s ❢♦r ❛❧❧ ♦❜❥❡❝ts A ❛♥❞ B✳

• ■♥ ❛ ❧❡①t❡♥s✐✈❡ ❝❛t❡❣♦r②✱ ❝♦♣r♦❞✉❝ts ❛r❡ ❞✐s❥♦✐♥t

❛♥❞ ✉♥✐✈❡rs❛❧ ❛♥❞ 0 ✐s str✐❝t❧② ✐♥✐t✐❛❧✳ ❆❧s♦ ✇❡ ❤❛✈❡ t❤❛t t❤❡ ❝❛♥♦♥✐❝❛❧ ♠♦r♣❤✐s♠

(A ×B)+(A ×C)

A ×(B +C)

✐s ✐♥✈❡rt✐❜❧❡✳

✻

• ■♥ Spn(E ) t❤❡ ♦❜❥❡❝t U +V ✐s t❤❡ ❞✐r❡❝t s✉♠ ♦❢ U

❛♥❞ V. ❚❤✐s ❝❛♥ ❜❡ s❤♦✇♥ ❛s ❢♦❧❧♦✇s✿

Spn(E )(U +V,W ) ∼ = [E /((U +V )×W )] ∼ = [E /((U ×W )+(V ×W ))] ≃ [E /(U ×W )]×[E /(V ×W )] ∼ = Spn(E )(U,W )×Spn(E )(V,W );

❛♥❞ s♦ Spn(E )(W,U +V ) ∼

= Spn(E )(W,U)×Spn(E )(W,V )✳

• ❚❤❡ ❛❞❞✐t✐♦♥ ♦❢ t✇♦ s♣❛♥s (s1,S,s2) :U

V ❛♥❞

(t1,T,t2) :U

V ✐s ❣✐✈❡♥ ❜②

S V

s2

• U

s1

• +

T V

t2

• U

t1

• =

S +T V +V

s2+t2

• V .

∇

• U +U

s1+t1

• U

∇

• [s1,t1]
• [s2,t2]
• Spn(E ) ✐s ❛ ♠♦♥♦✐❞❛❧ ❝♦♠♠✉t❛t✐✈❡✲♠♦♥♦✐❞✲❡♥r✐❝❤❡❞

❝❛t❡❣♦r②✳

✼

▼❛❝❦❡② ❢✉♥❝t♦rs ♦♥ E

• ❆ ▼❛❝❦❡② ❢✉♥❝t♦r

M : E

Modk

❝♦♥s✐sts ♦❢ t✇♦ ❢✉♥❝t♦rs M∗ : (E )♦♣

Modk,

M∗ : E

Modk s✉❝❤ t❤❛t

⋆ M∗(U) = M∗(U) (= M(U)) ❢♦r ❛❧❧ U ✐♥ E✳ ⋆ ❋♦r ❛❧❧ ♣✉❧❧❜❛❝❦s P V

q

• W ,

s

• U

p

• r
• ✐♥ E✱ t❤❡ sq✉❛r❡✭▼❛❝❦❡② sq✉❛r❡✮

M(U) M(W )

M∗(r)

• M(V )

M∗(s)

• M(P)

M∗(p)

• M∗(q)

❝♦♠♠✉t❡s✳

✽

⋆ ❋♦r ❛❧❧ ❝♦♣r♦❞✉❝t ❞✐❛❣r❛♠s U

i

U +V

V

j

• ✐♥ E✱ t❤❡ ❞✐❛❣r❛♠

M(U)

M∗i

M(U +V )

M∗i

• M∗j
• M(V )

M∗j

• ✐s ❛ ❞✐r❡❝t s✉♠ s✐t✉❛t✐♦♥ ✐♥ Modk✳

✭❚❤✐s ✐♠♣❧✐❡s M(U +V ) ∼

= M(U)⊕ M(V )✳✮

• ❆ ♠♦r♣❤✐s♠ θ : M

N ♦❢ ▼❛❝❦❡② ❢✉♥❝t♦rs ✐s ❛

❢❛♠✐❧② θU : M(U)

N(U) ♦❢ ♠♦r♣❤✐s♠s ❢♦r U ✐♥ E✳

❚❤✐s ❣✐✈❡s ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥s θ∗ : M∗

N∗ ❛♥❞

θ∗ : M∗

N∗✳

• Pr♦♣♦s✐t✐♦♥✿ ✭❉✉❡ t♦ ▲✐♥❞♥❡r✮

❚❤❡ ❝❛t❡❣♦r② Mky(E ,Modk) ♦❢ ▼❛❝❦❡② ❢✉♥❝t♦rs ✐s ❡q✉✐✈❛❧❡♥t t♦ [Spn(E ),Modk]+ ♦❢ t❤❡ ❝❛t❡❣♦r② ♦❢ ❝♦♣r♦❞✉❝t✲♣r❡s❡r✈✐♥❣ ❢✉♥❝t♦rs✳ ❚❤❛t ✐s✿

Mky(E ,Modk) ≃ [Spn(E ),Modk]+

✾

• Pr♦♦❢

▲❡t M : E

Modk ❜❡ ❛ ▼❛❝❦❡② ❢✉♥❝t♦r✳

❲❡ ❉❡✜♥❡ ❛ ♠♦r♣❤✐s♠ M : Spn(E )

Modk ❜②

M(U) = M∗(U) = M∗(U) ❛♥❞ M

• S

V

s2

• U

s1

• =
• M(U)

M(S)

M∗(s1)

M(V )

M∗(s2)

• .

❈♦♥✈❡rs❡❧②✱ ❧❡t M : Spn(E )

Modk ❜❡ ❛ ❢✉♥❝t♦r✳

❚❤❡♥ ✇❡ ❝❛♥ ❞❡✜♥❡ t✇♦ ❢✉♥❝t♦rs M∗ ❛♥❞ M∗✱

E Spn(E )

(−)∗

• Modk ,

M

• E ♦♣

(−)∗

• ❜② ♣✉tt✐♥❣ M∗ = M ◦(−)∗ ❛♥❞ M∗ = M ◦(−)∗✳
• ❉❡♥♦t❡ Mky = Mky(E ,Modk) ≃ [Spn(E ),Modk]+

✶✵

❚❡♥s♦r ♣r♦❞✉❝ts ✐♥ Mky

• ▲❡t T ❜❡ ❣❡♥❡r❛❧ ❝♦♠♣❛❝t ❝❧♦s❡❞✱ ❝♦♠♠✉t❛t✐✈❡✲

♠♦♥♦✐❞✲❡♥r✐❝❤❡❞ ❝❛t❡❣♦r②✳ ✭❚❤❡ ♠❛✐♥ ❡①❛♠♣❧❡ ✐s

Spn(E )✮✳

• ❚❤❡ t❡♥s♦r ♣r♦❞✉❝t ♦❢ ▼❛❝❦❡② ❢✉♥❝t♦rs ❝❛♥ ❜❡

❞❡✜♥❡❞ ❜② ❝♦♥✈♦❧✉t✐♦♥ ✐♥ [T ,Modk]+ s✐♥❝❡ T ✐s ❛ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r②✳

• ❚❤❡ t❡♥s♦r ♣r♦❞✉❝t ✐s✿

(M ∗ N)(Z) = X ,Y T (X ⊗Y,Z)⊗ M(X )⊗k N(Y ) ∼ = X ,Y T (Y,X ∗ ⊗ Z)⊗ M(X )⊗k N(Y ) ∼ = X M(X )⊗k N(X ∗ ⊗ Z) ∼ = Y M(Z ⊗Y ∗)⊗k N(Y ).

✶✶

❍♦♠ ❢✉♥❝t♦r ❛♥❞ ❇✉r♥s✐❞❡ ❢✉♥❝t♦r

• ▲❡t T = Spn(E ) ✇❤❡r❡ E t❤❡ ❝❛t❡❣♦r② ♦❢ ✜♥✐t❡

G✲s❡ts ❢♦r t❤❡ ✜♥✐t❡ ❣r♦✉♣ G✳

• ❚❤❡ ❍♦♠ ▼❛❝❦❡② ❢✉♥❝t♦r ✐s

Hom(M,N)(V ) = Mky(M(V ×−),N),

❢✉♥❝t♦r✐❛❧❧② ✐♥ V ✳

(L ∗ M)(U)

N(U)

L(V )⊗k M(V ×U)

N(U)

L(V )

Homk(M(V ×U),N(U))

L(V )

• U

❍♦♠k(M(V ×U),N(U))

L(V )

Mky(M(V ×−),N)

• ❚❤❡ ❇✉r♥s✐❞❡ ❢✉♥❝t♦r J : E

Modk ❤❛s ✈❛❧✉❡ ❛t U

❡q✉❛❧ t♦ t❤❡ ❢r❡❡ ❦✲♠♦❞✉❧❡ ♦♥ Spn(E )(1,U) = [E /U]✳

✶✷

• r❡❡♥ ❢✉♥❝t♦rs ♦♥ E
• ❆ ●r❡❡♥ ❢✉♥❝t♦r A : E

Modk ✐s

⋆ ❆ ▼❛❝❦❡② ❢✉♥❝t♦r ✭t❤❛t ✐s✱ ❛ ❝♦♣r♦❞✉❝t

♣r❡s❡r✈✐♥❣ ❢✉♥❝t♦r A : Spn(E )

Modk✮ ✇✐t❤

⋆ ❆ ♠♦♥♦✐❞❛❧ str✉❝t✉r❡ ♠❛❞❡ ✉♣ ♦❢ ❛ ♥❛t✉r❛❧

tr❛♥s❢♦r♠❛t✐♦♥

µ : A(U)⊗k A(V )

A(U ×V ),

❢♦r ✇❤✐❝❤ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ µ(a⊗b) = a.b ❢♦r

a ∈ A(U)✱ b ∈ A(V )✱ ❛♥❞ ⋆ ❛ ♠♦r♣❤✐s♠ η : k

A(1) s✉❝❤ t❤❛t η(1) = 1✳

• ●r❡❡♥ ❢✉♥❝t♦rs ❛r❡ t❤❡ ♠♦♥♦✐❞s ✐♥ Mky✳
• ❚❤❡ ❇✉r♥s✐❞❡ ❢✉♥❝t♦r J ❛♥❞ ❍♦♠(A, A) ❛r❡ ♠♦♥♦✐❞s

✐♥ Mky ❛♥❞ t❤❡r❡❢♦r❡ ❛r❡ ●r❡❡♥ ❢✉♥❝t♦rs✳

✶✸

❋✐♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ▼❛❝❦❡② ❢✉♥❝t♦rs

• ▲❡t Mky✜♥ ❜❡ t❤❡ ❝❛t❡❣♦r② ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✲

✈❛❧✉❡❞ ▼❛❝❦❡② ❢✉♥❝t♦rs✳ ❉❡✜♥❡ Mky✜♥ = [T ,Vect✜♥]+✳

• ▲❡t C ❜❡ t❤❡ ❢✉❧❧ s✉❜✲❝❛t❡❣♦r② ♦❢ T ❝♦♥s✐st✐♥❣ ♦❢

t❤❡ ❝♦♥♥❡❝t❡❞ G✲s❡ts✳ ❚❤❡ ❢✉♥❝t♦r F : C → T ✐s ❛ ❢✉❧❧② ❢❛✐t❤❢✉❧ ❢✉♥❝t♦r✳ ❚❤❡ ❝❛t❡❣♦r② C ❤❛s ✜♥✐t❡❧② ♠❛♥② ♦❜❥❡❝ts✳ ❊❛❝❤ X ∈ T ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s

X ∼ =

n

• i=1

F(Ui).

• ❲❡ ❝❛♥ s❤♦✇ t❤❛t

M(X ) ∼ = C T (C,X )⊗ M(C).

• ▲❡♠♠❛ ■❢ S ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞

❜② ❛ ✜♥✐t❡ s❡t ♦❢ ❡❧❡♠❡♥ts s1,...,sm ❛♥❞ V ✐s ❛ ✈❡❝✲ t♦r s♣❛❝❡ ✇✐t❤ ❜❛s✐s v1,...,vn t❤❡♥ S ⊗V ✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r s♣❛❝❡✳

✶✹

• ❚❤❡ t❡♥s♦r ♣r♦❞✉❝t M,N ∈ Mky✜♥ ✐s ✜♥✐t❡ ❞✐♠❡♥✲

s✐♦♥❛❧✳

(M ∗ N)(Z) = X ,Y T (X ×Y,Z)⊗ M(X )⊗k N(Y ) ∼ = X ,Y,C,D T (X ×Y, Z)⊗T (C,X )⊗T (D,Y )⊗ M(C)⊗k N(D) ∼ = C,D T (C ×D,Z)⊗ M(C)⊗k N(D).

❍❡r❡ T (C ×D,Z) ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❝♦♠♠✉✲ t❛t✐✈❡ ♠♦♥♦✐❞ ❛♥❞ M(C) ❛♥❞ N(D) ❛r❡ ✜♥✐t❡ ❞✐♠❡♥✲ s✐♦♥❛❧✳

• ❚❤❡ ♣r♦♠♦♥♦✐❞❛❧ str✉❝t✉r❡ ♦♥ Mky✜♥ ❢♦r t❤❡ ▼❛❝❦❡②

❢✉♥❝t♦rs M,N, ❛♥❞ L ✐s

P(M,N;L) = ◆❛tX ,Y,Z(T (X ×Y,Z)⊗ M(X )⊗k N(Y ),L(Z)) ∼ = ◆❛tX ,Y (M(X )⊗k N(Y ),L(X ×Y )) ∼ = ◆❛tX ,Z(M(X )⊗k N(X ∗ ×W ),L(Z)) ∼ = ◆❛tY,Z(M(Z ×Y ∗)⊗k N(Y ),L(Z)).

❚❤❡r❡❢♦r❡ t❤❡ ❝❛t❡❣♦r② Mky✜♥ ✐s ♠♦♥♦✐❞❛❧ ❢♦r t❤❡ ♣r♦♠♦♥♦✐❞❛❧ str✉❝t✉r❡❀ t❤❛t ✐s✱

P(M,N;L) ∼ = Mky✜♥(M ∗ N,L).

✶✺

• ❆ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r② V ✐s ∗✲❛✉t♦♥♦♠♦✉s ✇❤❡♥ ✐t

✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛♥ ❡q✉✐✈❛❧❡♥❝❡ S : V ♦♣

V ♦❢

❝❛t❡❣♦r✐❡s ❛♥❞

V (A ⊗B,SC) ∼ = V (B ⊗C,S−1A).

■♥ t❤❡ ❝❛t❡❣♦r② Mky✜♥ ✇❡ ❝❛♥ ✇r✐t❡ (SA)X = A(X ∗)∗.

• ❚❤❡♦r❡♠ ❚❤❡ ❝❛t❡❣♦r② Mky✜♥ ✐s ∗✲❛✉t♦♥♦♠♦✉s✳
• Pr♦♦❢ ❚❤❡ ♣r♦♠♦♥♦✐❞❛❧ str✉❝t✉r❡ P(M,N;SL) ❢♦r

t❤❡ ❝❛t❡❣♦r② Mky✜♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s✿

P(M,N;SL) = ◆❛tX ,Y (M(X )⊗k N(Y ),L(X ∗ ×Y ∗)∗) ∼ = ◆❛tX ,Y (N(Y )⊗L(X ∗ ×Y ∗),(MX )∗) ∼ = ◆❛tX ,Y (N(Y )⊗L(X ×Y ∗),M∗(X )) ∼ = P(N,L;M∗).

• ❚❤❡r❡ ✐s ❛ ♣♦ss✐❜✐❧✐t② t❤❛t ❢♦r ❛ ❝❧❛ss ♦❢ ✜♥✐t❡ G

✭✐♥❝❧✉❞✐♥❣ t❤❡ ❝②❝❧✐❝ ♦♥❡s✮ t❤❛t Mky✜♥ ❝♦✉❧❞ ❜❡ ❝♦♠♣❛❝t ✭❛✉t♦♥♦♠♦✉s✮✳

✶✻

▼♦❞✉❧❡s ♦✈❡r ❛ ●r❡❡♥ ❢✉♥❝t♦r

• ❆ ♠♦❞✉❧❡ M ♦✈❡r A✱ ♦r A✲♠♦❞✉❧❡ ♠❡❛♥s A ❛❝ts

♦♥ M ✈✐❛ t❤❡ ❝♦♥✈♦❧✉t✐♦♥✳

• ❚❤❡ ♠♦♥♦✐❞❛❧ ❛❝t✐♦♥ αM : A ∗ M

M ✐s ❞❡✜♥❡❞ ❜②

❛ ❢❛♠✐❧② ♦❢ ♠♦r♣❤✐s♠s

¯ αM

U,V : A(U)⊗k M(V )

M(U ×V ),

✇❤❡r❡ ✇❡ ♣✉t ¯

αM

U,V (a⊗m) = a.m ❢♦r a ∈ A(U)✱ m ∈ M(V )✳

• ■❢ M ✐s ❛♥ A✲♠♦❞✉❧❡✱ t❤❡♥ M ✐s ♦❢ ❝♦✉rs❡ ❛ ▼❛❝❦❡②

❢✉♥❝t♦r✳

• ▲❡t Mod(A) ❞❡♥♦t❡ t❤❡ ❝❛t❡❣♦r② ♦❢ ❧❡❢t A✲♠♦❞✉❧❡s✳

❖❜❥❡❝ts ❛r❡ A✲♠♦❞✉❧❡s ❛♥❞ ♠♦r♣❤✐s♠s ❛r❡ A✲♠♦❞✉❧❡ ♠♦r♣❤✐s♠s✳

✶✼

▼♦r✐t❛ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ●r❡❡♥ ❢✉♥❝t♦rs

• ❋♦r ❛♥② ❣♦♦❞ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r② W ✇❡ ❤❛✈❡ t❤❡

♠♦♥♦✐❞❛❧ ❜✐❝❛t❡❣♦r② Mod(W )✳ ❲❡ s♣❡❧❧ t❤✐s ♦✉t ✐♥ t❤❡ ❝❛s❡ W ❂ Mky✿

⋆ ❖❜❥❡❝ts ❛r❡ ♠♦♥♦✐❞s A ✐♥ W ✭✐✳❡✳ A : E

Modk

❛r❡ ●r❡❡♥ ❢✉♥❝t♦rs✮

⋆ ♠♦r♣❤✐s♠s ❛r❡ ♠♦❞✉❧❡s M : A

• B ✇✐t❤ ❛ t✇♦✲

s✐❞❡❞ ❛❝t✐♦♥ αM : A ∗ M ∗B

M✱ t❤❛t ✐s

αM

U,V,W : A(U)⊗k M(V )⊗k B(W )

M(U ×V ×W )

⋆ ❈♦♠♣♦s✐t✐♦♥ ♦❢ ♠♦r♣❤✐s♠s M : A

• B ❛♥❞

N : B

• C ✐s M ∗B N ❛♥❞ ✐t ✐s ❞❡✜♥❡❞ ✈✐❛ t❤❡

❝♦❡q✉❛❧✐③❡r

M ∗B ∗ N

αM∗1N

• 1M∗αN

M ∗ N M ∗B N = N ◦ M

t❤❛t ✐s✱

(M∗BN)(U) =

• X ,Y

Spn(E )(X ×Y,U)⊗M(X )⊗kN(Y )/ ∼B . ⋆ ❚❤❡ ✐❞❡♥t✐t② ♠♦r♣❤✐s♠ ✐s ❣✐✈❡♥ ❜② A : A

• A.

✶✽

⋆ ❚❤❡ ✷✲❝❡❧❧s ❛r❡ ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥s θ : M

M′

✇❤✐❝❤ r❡s♣❡❝t t❤❡ ❛❝t✐♦♥s

A(U)⊗k M(V )⊗k B(W ) M(U ×V ×W )

¯ αM

U,V,W

• M′(U ×V ×W ) .

θU×V ×W

• A(U)⊗k M′(V )⊗k B(W )

1⊗kθV ⊗k1

• ¯

αM′

U,V,W

• ⋆ ❚❤❡ t❡♥s♦r ♣r♦❞✉❝t ♦♥ Mod(W ) ✐s t❤❡ ❝♦♥✈♦❧✉t✐♦♥

∗✳ ❚❤❡ t❡♥s♦r ♣r♦❞✉❝t ♦❢ t❤❡ ♠♦❞✉❧❡s M : A

• B

❛♥❞ N : C

• D ✐s M ∗ N : A ∗C
• B ∗D✳
• ❉❡✜♥✐t✐♦♥✿ ●r❡❡♥ ❢✉♥❝t♦rs A ❛♥❞ B ❛r❡ s❛✐❞ t♦

❜❡ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t ✇❤❡♥ t❤❡② ❛r❡ ❡q✉✐✈❛❧❡♥t ✐♥

Mod(W )✳

• Pr♦♣♦s✐t✐♦♥✿ ■❢ A ❛♥❞ B ❛r❡ ❡q✉✐✈❛❧❡♥t ✐♥ Mod(W )

t❤❡♥ Mod(A) ≃ Mod(B) ❛s ❝❛t❡❣♦r✐❡s✳

• Pr♦♦❢ Mod(W )(−, J) : Mod(W )♦♣

CAT ✐s ❛ ♣s❡✉❞♦ ❢✉♥❝✲

t♦r ❛♥❞ s♦ t❛❦❡s ❡q✉✐✈❛❧❡♥❝❡s t♦ ❡q✉✐✈❛❧❡♥❝❡s✳

✶✾

• ◆♦✇ ✇❡ ❡♥r✐❝❤❡❞ Mod(A) t♦ ❛ W ✲❝❛t❡❣♦r② P A✳
• ❚❤❡ W ✲❝❛t❡❣♦r② P A ❤❛s ✉♥❞❡r❧②✐♥❣ ❝❛t❡❣♦r②

Mod(W )(J, A)✳

❚❤❡ ♦❜❥❡❝ts ❛r❡ ♠♦❞✉❧❡s M : J

• A

❛♥❞ ❤♦♠s ❛r❡ ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛❧✐③❡r✳

Mod(A)(M,N) Hom(M,N)

• Hom(A ∗ M,N)

Hom(αM,1)

• Hom(A ∗ M, A ∗ N)

(A∗−)

• Hom(1,αN)
• ❚❤❡ ❈❛✉❝❤② ❝♦♠♣❧❡t✐♦♥ QA ♦❢ ❆ ✐s t❤❡ ❢✉❧❧

s✉❜✲W ✲❝❛t❡❣♦r② ♦❢ P A ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ ♠♦❞✉❧❡s

M : J

• A ✇✐t❤ r✐❣❤t ❛❞❥♦✐♥ts N : A
• J✳
• ❘❡❝❛❧❧ t❤❡ ❝❧❛ss✐❝❛❧ r❡s✉❧t ❢r♦♠ ❡♥r✐❝❤❡❞ ❝❛t❡❣♦r②

t❤❡♦r②✿

• ❚❤❡♦r❡♠✿
• r❡❡♥ ❢✉♥❝t♦rs A ❛♥❞ B ❛r❡ ▼♦r✐t❛

❡q✉✐✈❛❧❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ QA ≃ QB ❛s W ✲❝❛t❡❣♦r✐❡s✳

✷✵

• ■♥ ♦✉r ❝❛s❡ t❤✐s t❤❡♦r❡♠ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ✈✐❛ ♦✉r

❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❈❛✉❝❤② ❝♦♠♣❧❡t✐♦♥✳

• ❚❤❡♦r❡♠✿

❚❤❡ ❈❛✉❝❤② ❝♦♠♣❧❡t✐♦♥ QA ♦❢ t❤❡ ♠♦♥♦✐❞ A ✐♥ Mky ❝♦♥s✐sts ♦❢ ❛❧❧ t❤❡ r❡tr❛❝ts ♦❢ ♠♦❞✉❧❡s ♦❢ t❤❡ ❢♦r♠

k

• i=1

A(Yi ×−)

❢♦r s♦♠❡ Yi ∈ Spn(E )✳

✷✶

❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ▼❛❝❦❡② ❢✉♥❝t♦rs

• ▲❡t Rep(G) ❜❡ t❤❡ ❝❛t❡❣♦r② ♦❢ k✲❧✐♥❡❛r r❡♣r❡s❡♥t❛✲

t✐♦♥s ♦❢ t❤❡ ✜♥✐t❡ ❣r♦✉♣ G✳ ❚❤❡ ❝❛t❡❣♦r② Mky(G) ♣r♦✈✐❞❡s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ♦r❞✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✳ ❋♦r ❡①❛♠♣❧❡✱ Rep(G) ❝❛♥ ❜❡ r❡❣❛r❞❡❞ ❛s ❛ ❢✉❧❧ r❡✢❡❝t✐✈❡ ♠♦♥♦✐❞❛❧ s✉❜✲❝❛t❡❣♦r② ♦❢ Mky(G)✳

• ▼❛❝❦❡② ❢✉♥❝t♦rs ♣r♦✈✐❞❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ λ✲ ❛♥❞

µ✲✐♥✈❛r✐❛♥ts ✐♥ ■✇❛s❛✇❛ t❤❡♦r② ❛♥❞ ❜❡t✇❡❡♥ ▼♦r❞❡❧❧✲

❲❡✐❧ ❣r♦✉♣s✱ ❙❤❛❢❛r❡✈✐❝❤✲❚❛t❡ ❣r♦✉♣s✱ ❙❡❧♠❡r ❣r♦✉♣s ❛♥❞ ③❡t❛ ❢✉♥❝t✐♦♥s ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✭❲✳ ❇❧❡② ❛♥❞ ❘✳ ❇♦❧t❥❡✱ ❈♦❤♦♠♦❧♦❣✐❝❛❧ ▼❛❝❦❡② ❢✉♥❝t♦rs ✐♥ ♥✉♠❜❡r t❤❡♦r②✱ ❏✳ ◆✉♠❜❡r ❚❤❡♦r② ✶✵✺ ✭✷✵✵✹✮✱ ✶✕✸✼✮✳

✷✷