Discrete Quantum Group  Discrete Group
S = c_{0}- ⨁_{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- ⨁_{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) \atop 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 ⇔ ĥ tracial

Λ : S → H GNS construction for h_{R} V (Λ⊗Λ)(x⊗y) = (Λ⊗Λ)(δ(x)(1⊗y)) V ∈ M(Ŝ_{r}⊗S) with Ŝ_{r} = (id⊗B(H)_{∗})(V)^{−}

H = ℓ^{2}(Γ), H⊗H ≃ ℓ^{2}(Γ × Γ) ; V (ξ)(α, β) = ξ(αβ, β) ; V = ∑ ρ_{α}⊗11_{α} ∈ M(C_{r}^{∗}(Γ)⊗c_{0}(Γ))

{F(a) = (id⊗h_{R})(V^{∗}(1⊗a)) ∈ Ŝ_{r} for a ∈ S \atop Ŝ = F(S) ⊂ Ŝ_{r}, ĥ(F(a)^{∗}F(a)) = h_{R}(a^{∗}a)}  F(a) = ∑ a(α)ρ_{α^{−1}}, Ŝ = ℂΓ ⊂ C_{r}^{∗}(Γ)

In the case of an abelian discrete group, C_{r}^{∗}(Γ) ≃ C(Γ̂) via α ↦ ⟨·, α⟩. Then V = [(α̂, α) ↦ ⟨α̂, α⟩] ∈ C_{b}(Γ̂ × Γ) and F (a) = [α̂ ↦ ∑_{α} a(α)⟨α̂, α^{−1}⟩].

Discrete Quantum Group  Compact Group

S = c_{0}- ⨁_{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- ⨁_{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} \atop 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(Ŝ_{r}⊗S) with Ŝ_{r} = (id⊗B(H)_{∗})(V)^{−} Ŝ_{r} = C(G), δ̂(â)(g, h) = â(gh) ĥ Haar measure of G (trace)

{F(a) = (id⊗h_{R})(V^{∗}(1⊗a)) ∈ Ŝ_{r} for a ∈ S \atop Ŝ = F(S) ⊂ Ŝ_{r}, ĥ(F(a)^{∗}F(a)) = h_{R}(a^{∗}a)} for a ∈ p_{α}S, F(a) is a coefficient of α