Algèbre de quasi-Hopf, associateur de Drinfel`d

GT
GT
Gal(Q/Q)
ΦKZ
C
⊗:C ×C → C a: (U⊗V)⊗W→U⊗(V⊗W)
c:U⊗V→V⊗U
((U⊗V)⊗W)⊗X
aU⊗V,W,X
ttiiiiiiiiiiiiiiiiaU,V,W ⊗IdX **
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
(U⊗V)⊗(W⊗X)
aU,V W ⊗X
(U⊗(V⊗W)) ⊗X
aU,V ⊗W,X
U⊗(V⊗(W⊗X)) U⊗((V⊗W)⊗X)
IdU⊗aV,W,X
oo

U⊗(V⊗W)cU,V ⊗W
//(V⊗W)⊗U
aV,W,U
((
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(U⊗V)⊗W
aU,V,W
66
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m
cU,V ⊗IdW
((
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Q
V⊗(W⊗U)
(V⊗U)⊗WaV,U,W//V⊗(U⊗W)
IdV⊗cU,W
66
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m
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(U⊗V)⊗WcU⊗V,W
//W⊗(U⊗V)
a−1
W,U,V
((
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Q
Q
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U⊗(V⊗W)
a−1
U,V,W
66
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IdU⊗cV,W
((
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Q
(W⊗U)⊗V
U⊗(W⊗V)a−1
U,W,V//(U⊗W)⊗V
cU,W ⊗IdV
66
m
m
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m
(A, ∆, ε, Φ) A
k∆ : A→A⊗A
εa →k∆(1) = 1 ε(1) = 1 Φ
A⊗A⊗A
(Id ⊗∆)(∆(a)) = Φ.(∆ ⊗Id(∆(a)).Φ−1
(Id ⊗Id ⊗∆)(Φ).(∆ ⊗Id ⊗Id)(Φ) = (1 ⊗Φ).(Id ⊗∆⊗Id)(Φ).(Φ ⊗1)
(ε⊗Id) ◦∆ = Id = (Id ⊗ε)◦∆
(Id ⊗ε⊗Id)(Φ) = 1
(A, ∆, ε, Φ, R) (A, ∆, ε, Φ)
R A ⊗A
∆0(a) = R∆(a)R−1
(∆ ⊗Id)(R)=Φ312R13(Φ132)−1R23Φ
(Id ⊗∆)(R) = (Φ231)−1R13Φ213R12Φ−1
∆0=σ◦∆σ R =Pai⊗bi
R12 =Pai⊗bi⊗1R13 =Pai⊗1biR23 =P1⊗ai⊗biΦ = Pxi⊗yi⊗zi
Φ312 =Pyi⊗zi⊗xi
A A

R12 6=R−1
F A ⊗A(A, ∆, ε, Φ, R)
(Id ⊗ε)(F)=1=(ε⊗Id)
˜
∆(a) = F∆(a)F−1
˜
Φ = F23(Id ⊗∆)(F)Φ(∆ ⊗Id)(F−1)(F12)−1
˜
R=R21RF −1
(A, ˜ ∆, ε, ˜ Φ,˜ R)F(A, δ, ε, Φ, R)
k
k[[h]] (A, ∆, ε, Φ, R)A/hA
A
k[[h]] V[[h]] V k
Φ≡1 mod h R ≡1 mod h F ≡1 mod h
gk t ∈g⊗g g
A=Ughg
R=eht/2Φ∈A⊗A⊗A
(A, ∆, ε, Φ, R)
QΦτ=ht
PQΦ
exp(P(ht12, ht23))
ΦKZ
GT
C
(V1⊗V2)⊗V3
∼
→V1⊗(V2⊗V3)
(V1⊗V2)⊗V3(V⊗V)⊗V V

CB3σ1
c⊗Id σ2a−1(Id ⊗c)a α B3
(V1⊗V2)⊗V3)∼ →(Vi1⊗Vi2)⊗Vi3(i1, i2, i3)
α(V1⊗V2)⊗V3K3= ker(B3→S3)ϕ∈K3
ψ∈K2
ψ∈K2σ2m
λ= 2m+ 1 ϕ∈K3f(σ2
1, σ2 2)(σ1σ2)3nn∈Z
f(X, Y ) (σ1σ2)3= (σ2σ1)3
B3
(λ, f, n)n= 0
(λ, f)
f(Y, X) = f(X, Y )−1
f(X3, X1)Xm
3f(X2, X3)Xm 2f(X1, X2)Xm 1= 1 X1X2X3= 1, m = (λ−1)/2
f(x12, x23x24)f(x13x23, x34) = f(x23x34)f(x12x13, x24x34)fx12, x23),
16i < j 6n xij = (σj−1· · · σi)σ2
i(σj−1· · · σi)−1Kn
(λ, f)λ∈1 + 2Z
C
C0F:C1→ C2F
C0
1C0 2
(λ1, f1).(λ2, f2) =
(λ, f)
λ=λ1λ2,
f(X, Y ) = f1(f2(X, Y )Xλ2f2(X, Y )−1, , Y λ2).f2(X, Y )
(A, ∆, ε, Φ, R)
A
(λ, f) Φ R
R=R(R21R)m= (RR21)mR m = (λ−1)/2
Φ = Φf(R21R12,Φ−1R32R23Φ)
=f(ΦR21R12Φ−1, R32R23)Φ
(λ, f)

f(X, Y )
X Y λ =±1f(X, Y ) = YrXr
k
λ∈k f(X, Y )k
f(X, Y ) exp(F(ln X, ln Y))
F k
GT(k) (λ, f) GT(k)
(λ, f)∈GT(k) (A, ∆, ε, Φ, R)
k[[h]]
GT(k)
Ug g k[[h]]
R=eht/2Φ = exp(P(ht12, ht23)) p
RΦR=eλht/2Φ = exp(P(ht12, ht23))
P k
Gal(Q/Q)
ΦKZ
(A, ∆, ε, Φ, R) Φ
A
A
g g k =C
ΦA R =eαt t
g g ⊗g alpha h
iπ
G0(z) = 1
2iπ u0 z+u11−zG(z)
GC C << u0, u1>>
C
D=C\(] − ∞; 0] ∪[1; ∞[)
G0D G0(z)∼ z→0zu0
G1D G1(z)∼ z→1 (1 −z−u1
zu0= exp(uoln(z)) f∼
ag fg−1
a1a
ΦKZ =G0G−1
1∈C<< u0;u1>>
z