Discrete Quantum Group Discrete Group S = c 0 - L

(1)

Discrete Quantum Group Discrete Group S = c ₀ - L

α∈R B(H _α ), H _α fd

α : S → B(H _α ), p _α = id _H

_α

∈ S, p _α S = B(H _α ) S = alg- L

α∈R B(H _α )

S = c ₀ (Γ), R = Γ, H _α = C α = ev _α , p _α = 11 _α

S = C ^(Γ) δ : S → M (S ⊗S) coassociative, κ : S → S

ε : S → C co-unit (trivial repr. : ε ∈ R)

δ (a)(α, β) = a(αβ)

ε(a) = a(e), κ(a)(α) = a(α ⁻¹ ) Haar weights h _L , h _R defined on S

∀ a ∈ p _α S h _L (a) = m _α Tr (F _α ⁻¹ a) and h _R (a) = m _α Tr (F _α a)

with F _α ∈ B(H _α ) ₊ st Tr F _α = Tr F _α ⁻¹ =: m _α

h _L = h _R counting measure on Γ (S, δ) unimodular : h _L = h _R ⇔ ∀α F _α = id _H

_α

⇔ κ ² = 1 ⇔ ˆ h tracial Λ : S → H GNS construction for h _R

V (Λ⊗Λ)(x⊗y) = (Λ⊗Λ)(δ (x)(1⊗y))

V ∈ M ( ˆ S _r ⊗S) with S ˆ _r = (id⊗B(H ) _∗ )(V ) ⁻

H = ` ² (Γ), H ⊗H ' ` ² (Γ × Γ) V (ξ)(α, β) = ξ (αβ, β)

V = P

ρ _α ⊗11 _α ∈ M (C _r ^∗ (Γ)⊗c ₀ (Γ)) F (a) = (id⊗h _R )(V ^∗ (1⊗a)) ∈ S ˆ _r for a ∈ S

S ˆ = F (S) ⊂ S ˆ _r , ˆ h(F (a) ^∗ F (a)) = h _R (a ^∗ a) F (a) = P

a(α)ρ _α

⁻¹

, S ˆ = CΓ ⊂ C _r ^∗ (Γ)

In the case of an abelian discrete group, C _r ^∗ (Γ) ' C (ˆ Γ) via α 7→ h·, αi. Then V = [(ˆ α, α) 7→ h α, αi] ˆ ∈ C _b (ˆ Γ × Γ) and F (a) = [ˆ α 7→ P

α a(α)h α, α ˆ ⁻¹ i].

(2)

Discrete Quantum Group Compact Group S = c ₀ - L

α∈R B(H _α ), H _α fd

α : S → B(H _α ), p _α = id _H

_α

∈ S, p _α S = B(H _α ) S = alg- L

α∈R B(H _α )

S = C ^∗ (G), R = Irrep(G) H _α space of the repr. α δ : S → M (S ⊗S) coassociative, κ : S → S

ε : S → C co-unit (trivial repr. : ε ∈ R)

δ (U _g ) = U _g ⊗U _g

ε(U _g ) = 1, κ(U _g ) = U _g ⁻¹ Haar weights h _L , h _R defined on S

∀ a ∈ p _α S h _L (a) = m _α Tr (F _α ⁻¹ a) and h _R (a) = m _α Tr (F _α a)

with F _α ∈ B(H _α ) ₊ st Tr F _α = Tr F _α ⁻¹ =: m _α

h _L = h _R “canonical trace” of C ^∗ (G) F _α = id _H

_α

, m _α = dim H _α

Λ : S → H GNS construction for h _R V (Λ⊗Λ)(x⊗y) = (Λ⊗Λ)(δ (x)(1⊗y))

V ∈ M ( ˆ S _r ⊗S) with S ˆ _r = (id⊗B(H ) _∗ )(V ) ⁻

S ˆ _r = C (G), δ ˆ (ˆ a)(g, h) = ˆ a(gh) ˆ h Haar measure of G (trace) F (a) = (id⊗h _R )(V ^∗ (1⊗a)) ∈ S ˆ _r for a ∈ S

S ˆ = F (S) ⊂ S ˆ _r , ˆ h(F (a) ^∗ F (a)) = h _R (a ^∗ a) for a ∈ p _α S , F (a) is a coefficient of α

Discrete Quantum Group Discrete Group S = c 0 - L

Discrete Quantum Group Discrete Group S = c 0 - L

α∈R B(H α ), H α fd

α : S → B(H α ), p α = id H

∈ S, p α S = B(H α ) S = alg- L

α∈R B(H α )

S = c 0 (Γ), R = Γ, H α = C α = ev α , p α = 11 α

S = C (Γ) δ : S → M (S ⊗S) coassociative, κ : S → S

ε : S → C co-unit (trivial repr. : ε ∈ R)

δ (a)(α, β) = a(αβ)

ε(a) = a(e), κ(a)(α) = a(α −1 ) Haar weights h L , h R defined on S

∀ a ∈ p α S h L (a) = m α Tr (F α −1 a) and h R (a) = m α Tr (F α a)

with F α ∈ B(H α ) + st Tr F α = Tr F α −1 =: m α

h L = h R counting measure on Γ (S, δ) unimodular : h L = h R ⇔ ∀α F α = id H

⇔ κ 2 = 1 ⇔ ˆ h tracial Λ : S → H GNS construction for h R

V (Λ⊗Λ)(x⊗y) = (Λ⊗Λ)(δ (x)(1⊗y))

V ∈ M ( ˆ S r ⊗S) with S ˆ r = (id⊗B(H ) ∗ )(V ) −

H = ` 2 (Γ), H ⊗H ' ` 2 (Γ × Γ) V (ξ)(α, β) = ξ (αβ, β)

V = P

ρ α ⊗11 α ∈ M (C r ∗ (Γ)⊗c 0 (Γ)) F (a) = (id⊗h R )(V ∗ (1⊗a)) ∈ S ˆ r for a ∈ S

S ˆ = F (S) ⊂ S ˆ r , ˆ h(F (a) ∗ F (a)) = h R (a ∗ a) F (a) = P

a(α)ρ α

, S ˆ = CΓ ⊂ C r ∗ (Γ)

In the case of an abelian discrete group, C r ∗ (Γ) ' C (ˆ Γ) via α 7→ h·, αi. Then V = [(ˆ α, α) 7→ h α, αi] ˆ ∈ C b (ˆ Γ × Γ) and F (a) = [ˆ α 7→ P

α a(α)h α, α ˆ −1 i].

Discrete Quantum Group Compact Group S = c 0 - L

α∈R B(H α ), H α fd

α : S → B(H α ), p α = id H

∈ S, p α S = B(H α ) S = alg- L

α∈R B(H α )

S = C ∗ (G), R = Irrep(G) H α space of the repr. α δ : S → M (S ⊗S) coassociative, κ : S → S

ε : S → C co-unit (trivial repr. : ε ∈ R)

δ (U g ) = U g ⊗U g

ε(U g ) = 1, κ(U g ) = U g −1 Haar weights h L , h R defined on S

∀ a ∈ p α S h L (a) = m α Tr (F α −1 a) and h R (a) = m α Tr (F α a)

with F α ∈ B(H α ) + st Tr F α = Tr F α −1 =: m α

h L = h R “canonical trace” of C ∗ (G) F α = id H

, m α = dim H α

Λ : S → H GNS construction for h R V (Λ⊗Λ)(x⊗y) = (Λ⊗Λ)(δ (x)(1⊗y))

V ∈ M ( ˆ S r ⊗S) with S ˆ r = (id⊗B(H ) ∗ )(V ) −

S ˆ r = C (G), δ ˆ (ˆ a)(g, h) = ˆ a(gh) ˆ h Haar measure of G (trace) F (a) = (id⊗h R )(V ∗ (1⊗a)) ∈ S ˆ r for a ∈ S

S ˆ = F (S) ⊂ S ˆ r , ˆ h(F (a) ∗ F (a)) = h R (a ∗ a) for a ∈ p α S , F (a) is a coefficient of α

Discrete Quantum Group Discrete Group S = c ₀ - L

α∈R B(H _α ), H _α fd

α : S → B(H _α ), p _α = id _H

∈ S, p _α S = B(H _α ) S = alg- L

α∈R B(H _α )

S = c ₀ (Γ), R = Γ, H _α = C α = ev _α , p _α = 11 _α

S = C ^(Γ) δ : S → M (S ⊗S) coassociative, κ : S → S

ε(a) = a(e), κ(a)(α) = a(α ⁻¹ ) Haar weights h _L , h _R defined on S

∀ a ∈ p _α S h _L (a) = m _α Tr (F _α ⁻¹ a) and h _R (a) = m _α Tr (F _α a)

with F _α ∈ B(H _α ) ₊ st Tr F _α = Tr F _α ⁻¹ =: m _α

h _L = h _R counting measure on Γ (S, δ) unimodular : h _L = h _R ⇔ ∀α F _α = id _H

⇔ κ ² = 1 ⇔ ˆ h tracial Λ : S → H GNS construction for h _R

V ∈ M ( ˆ S _r ⊗S) with S ˆ _r = (id⊗B(H ) _∗ )(V ) ⁻

H = ` ² (Γ), H ⊗H ' ` ² (Γ × Γ) V (ξ)(α, β) = ξ (αβ, β)

ρ _α ⊗11 _α ∈ M (C _r ^∗ (Γ)⊗c ₀ (Γ)) F (a) = (id⊗h _R )(V ^∗ (1⊗a)) ∈ S ˆ _r for a ∈ S

S ˆ = F (S) ⊂ S ˆ _r , ˆ h(F (a) ^∗ F (a)) = h _R (a ^∗ a) F (a) = P

a(α)ρ _α

, S ˆ = CΓ ⊂ C _r ^∗ (Γ)

In the case of an abelian discrete group, C _r ^∗ (Γ) ' C (ˆ Γ) via α 7→ h·, αi. Then V = [(ˆ α, α) 7→ h α, αi] ˆ ∈ C _b (ˆ Γ × Γ) and F (a) = [ˆ α 7→ P

α a(α)h α, α ˆ ⁻¹ i].

Discrete Quantum Group Compact Group S = c ₀ - L

α∈R B(H _α ), H _α fd

α : S → B(H _α ), p _α = id _H

∈ S, p _α S = B(H _α ) S = alg- L

α∈R B(H _α )

S = C ^∗ (G), R = Irrep(G) H _α space of the repr. α δ : S → M (S ⊗S) coassociative, κ : S → S

δ (U _g ) = U _g ⊗U _g

ε(U _g ) = 1, κ(U _g ) = U _g ⁻¹ Haar weights h _L , h _R defined on S

∀ a ∈ p _α S h _L (a) = m _α Tr (F _α ⁻¹ a) and h _R (a) = m _α Tr (F _α a)

with F _α ∈ B(H _α ) ₊ st Tr F _α = Tr F _α ⁻¹ =: m _α

h _L = h _R “canonical trace” of C ^∗ (G) F _α = id _H

, m _α = dim H _α

Λ : S → H GNS construction for h _R V (Λ⊗Λ)(x⊗y) = (Λ⊗Λ)(δ (x)(1⊗y))

V ∈ M ( ˆ S _r ⊗S) with S ˆ _r = (id⊗B(H ) _∗ )(V ) ⁻

S ˆ _r = C (G), δ ˆ (ˆ a)(g, h) = ˆ a(gh) ˆ h Haar measure of G (trace) F (a) = (id⊗h _R )(V ^∗ (1⊗a)) ∈ S ˆ _r for a ∈ S

S ˆ = F (S) ⊂ S ˆ _r , ˆ h(F (a) ^∗ F (a)) = h _R (a ^∗ a) for a ∈ p _α S , F (a) is a coefficient of α