đặc trưng chern không giao hoán của c∗− đại số của nhóm lie compact và nhóm lượng tử tương ứng

∗
−
∗
−
∗
−
∗
−
∗
−
∗
−
∗
−
n
∗
−
n
∗
−
C
∗
−
KK−
K−
X H
∗
(X)

X
K
∗
(X) K−
X X
ch : K
∗
(X) −→ H
∗
(X).
G H
∗
D R
(G; Q)
Z/(2)− K
∗
(G) K− G.
G
ch : K
∗
(G) ⊗ Q −→ H
∗
D R
(G; Q).
SU(2n), Sp(n),
SO(2n+1) SU(n)/Sp(n), SU(2n)/SO(2n),
SU(2n + 1)/SO(2n + 1).
KK
A, HE
∗

(A)
{I
α
}
α∈Γ
A τ
α
: I
α
−→ C ad
A
−
HE
∗
(A)
A
ch : K
∗
(A) −→ HE
∗
(A)
K−
K
∗
(A) × HE
∗
(A) −→ C.
A A
A
A

K
∗
(A) × HE
∗
(A) −→ C
A.
K
∗
(A) −→ lim
−→
HE
∗
(I
α
).
∗
−
ch : K
∗
(A) −→ HE
∗
(A)
C
∗
−
A C
∗
−
A C
∗

−
A C
∗
−
C
∗
−
C
∗
− C
∗
−
C
∗
−
∗
−
G C
∗
(G) C
∗
−
G.
C
∗
(G).
C
∗
−
A

HP
∗
(A) HE
∗
(A).
C
∗
− G
X−
C
∗
(G)
C
∗
ϵ
(G) C
∗
− G.
1
C
∗
ϵ
(G).
C
∗
− G
C
∗
ϵ
(G)

∼
=
C(T ) ⊕

e̸=ω∈W

⊕
T
K(H
ω,t
)dt .
K−
HE
∗
C
∗
ϵ
(G). K− HE
∗
C
∗
ϵ
(G)
HE
∗
(A)
C
∗
−
K

∗
(C
∗
ϵ
(G)) HE
∗
(C
∗
ϵ
(G))
C
∗
ϵ
(G)
C
∗
ϵ
(G). HP
∗
(C
∗
ϵ
(G))
C
∗
ϵ
(G)
C
∗
ϵ

(G)
∗
−
n
C
∗
−
S
n
ch
C
∗
: K
∗
(C
∗
(S
n
)) −→ HE
∗
(C
∗
(S
n
))
C
∗
− S
n
.

ch
C
∗
ϵ
: K
∗
(C
∗
ϵ
(S
2n+1
)) −→ HE
∗
(C
∗
ϵ
(S
2n+1
))
C
∗
−
C
∗
−
S
n
. O(n) O(n + 1),
S
n

= O(n + 1)/O(n) C
∗
−
S
n
C
∗
−
K
∗
HE
∗
C
∗
− S
n
HE
∗
K
∗
C
∗
−
1.
2.
3.
4.
5.
∗
−

G C
∗
(G) C
∗
− G.
HP
∗
HP
∗
C
∗
(G).
ch
C
∗
: K
∗
(C
∗
(G)) −→ HE
∗
(C
∗
(G))
C
∗
(G)
ch
alg
: K

alg
∗
(C
∗
(G)) −→ HP
∗
(C
∗
(G))
C
∗
(G)
C
∗
− K−
∗
−
G G
dg. L
1
(G)
L
1
(G) =

f : G −→ C |

G
|f(x)|dx < ∞


,
(f ∗ g)(x) :=

G
f(y)g(y
−1
x)dy,
f
∗
(x) := f(x
−1
)
L
1
−
∥f∥ :=

G
|f(x)|dx.
L
1
(G)
∥f ∗ g∥ ̸= ∥f∥.∥g∥.
L
1
(G).
∥f∥
C
∗
:= sup

π∈

G
∥π(f)∥, (1)

G
G. L
1
(G) (1)
C
C
∗
−
G

G
G.

G G
G
π G ∗− C
∗
(G)

f(π) = π(f) :=

G
π(x)f(x)dx.
1 − 1 π G ∗−
C

∗
(G).
π
n
∈

G n ∈ N. ∀f ∈ C
∗
(G) c = c
f
∥

f(π
n
) − c
f
.Id∥ −→ 0 n −→ ∞,
Id
′
∞

i=1
Mat
n
i
(C) =


f | ∥


f(π
n
) − c
f
.Id∥ −→ 0 khi n −→ ∞

.
′
∞

i=1
Mat
n
i
(C) Mat
n
i
(C)
n
i
.
C
∗
(G)
∼
=
′
∞

i=1

Mat
n
i
(C).
G C
∗
(G) C
∗
− G. I
N
:=
N

i=1
Mat
n
i
(C).
I
N
C
∗
(G) C
∗
−
G I
N
,
C
∗

(G) = lim
−→
I
N
.
C
∗
− A (π
1
, π
2
, F ),
π
1
, π
2
: A −→ £(H
B
) ∗− F ∈ F(H
B
)
F C
∗
− H
B
= l
2
B
C
∗

− B,
π
1
(a)F − F π
2
(a) ∈ K(B),
K(B) H
B
).
K− KK
∗
(A, B)).
G A = C
∗
(G) B = C
K
∗
(C
∗
(G))
∼
=
KK
∗
(C
∗
(G), C).
KK
0
(C

∗
(G), C)
∼
=
K
0
(C
∗
(G)) KK
1
(C
∗
(G), C)
∼
=
K
1
(C
∗
(G)),
K
∗
(C
∗
(G)) K− C
∗
(G).
A
HE
∗

(A) K
∗
(A) × HE
∗
(A) −→ C.
HE
∗
(A).
HE
∗
(A) A {I
α
}
α∈Γ
A
τ
α
: I
α
−→ C ad
A
−
A {I
α
}
α∈Γ
A
τ
α
: I

α
−→ C,
τ
α
∥τ
α
∥ = 1,
τ
α
(aa
∗
) ≥ 0 ∀a ∈ I
α
,
τ
α
(aa
∗
) = 0 a = 0 α ∈ Γ,
τ
α
ad
A
− τ
α
(xa) = τ
α
(ax) ∀x ∈ A, a ∈ I
α
.

α ∈ Γ, τ
α
I
α
⟨a, b⟩
τ
α
:= τ
α
(ab
∗
) ∀a, b ∈ I
α
.
I
α
I
α
Γ
α  β  γ ⇐⇒ I
α
⊆ I
β
⊆ I
γ
, ∀α, β, γ ∈ Γ.
{I
β
, j
β

α
} Γ : ∀α, β, γ ∈ Γ, α  β  γ
j
β
α
: I
α
−→ I
β
j
γ
β
j
β
α
= j
γ
α
: I
α
−→ I
γ
j
α
α
= id.
I
α
⊗(n+1)
(n + 1)− I

α
. I
α
I
α
⊗(n+1)

I
α
= I
α
⊕ C, ∀α ∈ Γ I
α
C
n
(

I
α
) =

φ : (

I
α
)
⊗(n+1)
−→ C | φ (n + 1) −

(n + 1)−


I
α
C
n
(

I
α
)
α, β, γ ∈ Γ α  β  γ,
D
β
α
: C
n
(

I
α
) −→ C
n
(

I
β
),
D
β
α

j
β
α
C
n
(

I
α
)
D
γ
β
D
β
α
= D
γ
α
, D
α
α
= id.
{C
n
(

I
α
), D

β
α
}
α∈Γ
Q =
lim
−→
C
n
(

I
α
).
Q = lim
−→
C
n
(

I
α
)
α ∈ Γ
b
′
: C
n
(


I
α
) −→ C
n+1
(

I
α
)
(b
′
φ)(a
0
, a
1
, , a
n+1
) =
n

j=0
(−1)
j
φ(a
0
, , a
j
a
j+1
, , a

n+1
),
b : C
n
(

I
α
) −→ C
n+1
(

I
α
)
(bφ)(a
0
, a
1
, , a
n+1
) =
n

j=0
(−1)
j
φ(a
0
, , a

j
a
j+1
, , a
n+1
)
+(−1)
n+1
φ(a
n+1
a
0
, , a
n−1
, a
n+1
),
(a
0
, , a
j
a
j+1
, , a
n+1
) ∈

I
α
⊗(n+1)

;
λ : C
n
(

I
α
) −→ C
n
(

I
α
)
(λφ)(a
0
, a
1
, , a
n
) = (−1)
n
φ(a
n
, a
0
, , a
n−1
),
S : C

n+1
(

I
α
) −→ C
n
(

I
α
),
(Sφ)(a
0
α
, a
1
α
, , a
n
α
) = φ(1, a
0
, , a
n
),

I
α
= I

α
⊕ C I
α
b, b
′
b
2
= (b
′
)
2
= 0. b, b
′
b, b
′
, λ S
b, b
′
: lim
−→
C
n
(

I
α
) −→ lim
−→
C
n+1

(

I
α
),
λ : lim
−→
C
n
(

I
α
) −→ lim
−→
C
n
(

I
α
),
S : lim
−→
C
n+1
(

I
α

) −→ lim
−→
C
n
(

I
α
).
N = 1 + λ + λ + + λ
n
, N λ
k
.
N
N : lim
−→
C
n
(

I
α
) −→ lim
−→
C
n
(

I

α
).
A {I
α
}
α∈Γ
A, τ
α
: A
α
−→ C ad
A
−
C
n
(A) := Hom(lim
−→
C
n
(

I
α
), C)
lim
−→
C
n
(


I
α
)
b, b
′
, λ, S
b
∗
, (b
′
)
∗
, λ
∗
, S
∗
b, b
′
, λ, S.
b, b
′
, λ, S b
∗
, (b
′
)
∗
, λ
∗
, S

∗
b
2
= b
′
2
= 0 N(1 − λ) = (1 − λ)N = 0
(b
∗
)
2
= (b
′
∗
)
2
= 0 N
∗
(1 − λ
∗
) = (1 − λ
∗
)N
∗
= 0 .
b
∗
, (b
′
)

∗
, λ
∗
, N
∗
A
C(A))
(−b
′
)
∗



b
∗



(−b
′
)
∗



←−−
1−λ
∗
C

1
(A) ←−−
N
∗
C
1
(A) ←−−
1−λ
∗
C
1
(A) ←−−
N
∗
···
C(A)
(−b
′
)
∗



b
∗



(−b
′

)
∗



←−−
1−λ
∗
C
0
(A) ←−−
N
∗
C
0
(A) ←−−
1−λ
∗
C
0
(A) ←−−
N
∗
···
b
∗
, (−b
′
)
∗

∗
C(A) C(A)
T ot(C(A))
even
= T ot(C(A))
odd
:=

n≥0
C
n
(A).
C(A) 2.
∂ :

n≥0
C
n
(A) 

n≥0
C
n
(A),
∂ = d
v
+ d
h
d
v

d
h
A
C(A)
A, HP
∗
(A).
(f
n
)
n≥0
∈ C(A)

n≥0
n!

n
2

!
∥f
n
∥z
n
z ∈ C.
C(A)
C
e
(A). C
e

(A)
C(A).
C
e
(A) C(A)
T ot(C
e
(A))
even
= T ot(C
e
(A))
odd
:=

n≥0
C
e
n
(A),
C
e
n
(A) n−
∂ 2,
∂ :

n≥0
C
e

n
(A) 

n≥0
C
e
n
(A),
∂ = d
v
+ d
h
d
v
, d
h
A
C
e
(A)
A, HE
∗
(A).
[13], [14],
HE
∗
(A) :
A, B {A
λ
}

λ∈Γ
,
{B
λ
}
λ∈Γ
A, B φ
t
= (φ
λ
t
)
λ
∈ Γ,
φ
λ
t
: A
λ
−→ B
λ
, t ∈ [0, 1]
δ
t
= (δ
λ
t
)
λ∈Γ
,

δ
λ
t
: A
λ
−→ B
λ
φ
t
.
φ
1∗
= φ
0∗
: HE
∗
(A) −→ HE
∗
(B).
A {A
λ
}
∈Γ
A. i = (i
λ
)
λ∈Γ
, i
λ
i

λ
:

A
λ
−→ Mat
q
(

A
λ
)
a
λ
−→



a
λ
0 . . . 0
0 0 . . . 0
0 0 . . . 0



q ≥ 1, ∀λ ∈ Γ. i
λ
HE
∗

(A).
G H
∗
D R
(G, Q)
G
K
∗
(G)
K− G
G
ch : K
∗
(G) ⊗ Q −→ H
∗
DR
(G; Q)
K− G.
C
∗
−
ch : K
∗
(C
∗
(G)) −→ HP
∗
(C
∗
(G))

K−
HP
∗
HE
∗
KK−
C
∗
− G.
e M
k
(A)
k ∈ N) φ = ∂ψ φ ∈ C
n
(

I
α
) ψ ∈ C
n+1
(

I
α
) n
< e, φ >=

n≥0
(−1)
n

n!
φ(e, e, , e) = 0.
A
ch
C
∗
: K
∗
(A) −→ HE
∗
(A).