Croissance des groupes quantiques discrets

Γ
ˆ
A=C(Γ)
C
ˆ
δ:ˆ A→M(ˆ A⊗ ˆ A)ˆ δ(f)(r, s) = f(rs)ˆ A
A=CΓ
ΓAδ:A → A⊗A δ(r) =
r⊗rAσ◦δ=δ σ(a⊗b) = b⊗a
ˆ
A A
H=`2(Γ) ˆ A A
ˆ
A A
C∗c0(Γ) C∗
red(Γ)
Aˆ A
(δ⊗id) ◦δ= (id⊗δ)◦δ
ˆ
δ
C∗A
H
ˆ
A
C∗C∗
A∗δ:A→A⊗A
(δ⊗id)δ= (id⊗δ)δ
δ(A)(1⊗A)δ(A)(A⊗1) A⊗A
δ(A)(1⊗A) = {δ(a)(1⊗b)|a, b ∈A}
C∗
C∗
Γ (A, δ)

A=C∗
red(Γ) A=C∗(Γ) Γ
A=C(G)G δ(f)(r, s) = f(rs)
A=Cq(G)G
q∈]0,1]
q
A=Au(In)Ao(In)
q
Au(Q) = uij,1≤i, j ≤n
(uij)ij Q(u∗
ij)ijQ−1,
Ao(Q) = uij,1≤i, j ≤n
(uij)ij =Q(u∗
ij)ijQ−1,
Q∈GL(n, C)
δ(uij) = Puik⊗ukl
Q¯ Q
As(n)As
h(n)
(A, δ)
α(A, δ)
Hαα∈L(Hα)⊗A(id⊗δ)(α) = α12α13
Mor(α, β)⊂L(Hα, Hβ)
Hα⊕β=Hα⊕HβHα⊗β=Hα⊗HβH¯α=
¯
Hαω L(Hα) (ω⊗id)(α)
α
A ⊂ A
(A, δ)C∗
h∈A0(id⊗h)δ= (h⊗id)δ= 1h
(A, δ)
A(δ, , S)
A=C(G)L(Hα)⊗A'C(G, L(Hα)) (A, δ)

G
h:C(G)→C
A=C∗
red(Γ)
id⊗r∈L(C)⊗A
r∈Γ
Γ
diag(r1, . . . , rn)∈Mn(A)'L(Cn)⊗A
r1, . . . , rn∈Γh
C∗
red(Γ) h(r) = δe,r
Γ
S⊂ΓS=S−1
e /∈S n S S
u= diag(s1, . . . , sn)∈L(Cn)⊗CΓk
uCkn
k S
ΓS
2kΓS
l(r) = min{k|r⊂u⊗k},
pk=n−2kdim Mor(1, u⊗2k) = n−2kh(χ2k
u),
χu= (Tr⊗id)(u) = Puii u
h(χα)=0
α6= 1 h(χ1)=1
C∗(A, δ)
u
n¯u=u16⊂ u
α u
l(α) = min{k|α⊂u⊗k}.
2k
u
pk=n−2kdim Mor(1, u⊗2k) = n−2kh(χ2k
u).

k
sk= #{α|l(α) = k}
Sk=X
l(α)=k
(dim α)2.
Bk=Pk
l=0 Sl
(A, δ)
(Sn)u
k Sk
1sk=Sk
Sk
(A, δ)
(A, δ)
G C(G)
q
A=Ao(In)n≥2
Nα0= 1 α1=u
αk⊗αl=α|k−l|⊕α|k−l|+2 ⊕ · · · ⊕ αk+l.
ndim αk= dim αk+1 +
dim αk−1sk= 1 Sk=
dim αkn≥3
Ao(In)
A=Au(In)n≥2
u¯u
l(α)
α(sk) (Sk)
G
C(G)G

X=ˆ T G R+
X X++ ⊂X
G
X++ $i1≤i≤r
X++ `1
u=L$i
skkr−1
dim α=Y
λ∈R+1 + (α|λ)
(ρ|λ),2ρ=X$i.
α
Sk≤Ckd−1d=r+ 2#R+
G
Bk≈kdd G
SU(3) : R, X++,($1, $2)
G pk
G
pk=n−2kZG χu(g)2kdg=n−2k
w(G)ZT
χu(t)2kδG(t)dt,
δG(t) = Qλ∈R(hλ, ti − 1) w(g) = #W
P(u)⊂X
u χu(t) = Pλ∈P(u)hλ, ti h · ,·i
T X T 'TrX'Zr
hλ, ti=tλ=tλ1
1· · · tλr
r