Discrete Quantum Groups

[Woronowicz, van Daele]

Setting Two dual Hopf C^{∗}-algebras (S, δ), (Ŝ_{red}, δ̂) Densely defined co-units and antipodes

[figure: [c0(Γ)]; S; F; GNS; h_{R}; Ŝ_{red}; GNS; ĥ; [C{∗ \atop red} (Γ)]; H; ≃; H; [ℓ2(Γ)]]

Structure of (S, δ) S = ⨁_{α} B(H_{α}) =: ⨁_{α} p_{α}S, S = alg- ⨁ p_{α}S Index set I := {α}, fd representations of S

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}

The Property of Rapid Decay

[cf Haagerup, Jolissaint]

Length on (S, δ) It is an unbounded multiplier L ∈ S^{η} st

L ≥ 0, ε(L) = 0, κ(L) = L

δ(L) ≤ 1⊗L + L⊗1.

p_{n} ∈ M(S) : spectral proj of L for [n, n + 1[.

Sobolev norms For a ∈ S we put ||a||_{2} := h_{R}(a^{∗}a)^{1/2} and

||a||_{2,s} := ||(1 + L)^{s}a||_{2}.

Let H_{L}^{s} ⊂ H be the associated completions.

Definition / Proposition Let L be a central length on (S, δ). We say that (S, δ, L) has Property RD if

∃ C, s ∈ ℝ_{+} ∀a ∈ S ||F(a)|| ≤ C||a||_{2,s} ⟺ H_{L}^{∞} := ⋂_{s≥0} H_{L}^{s} ⊂ Ŝ_{red} inside H ⟺ ∃ P ∈ ℝ[X] ∀k, l, n ∀a ∈ p_{n}S

||p_{l}F(a)p_{k}|| ≤ P (n)||a||_{2}.

The amenable case

Growth We say that (S, δ, L) has polynomial growth if

∃ P ∈ ℝ[X] ∀n ∈ ℕ h_{R}(p_{n}) ≤ P (n)

Proposition (S, δ, L) amenable + RD ⇒ polynomial growth (S, δ, L) polynomial growth ⇒ Prop RD

Example Duals of connected compact Lie groups have Property RD. In fact in this case

H_{L}^{∞} ⊂ Ŝ_{red} ⟺ C^{∞}(G) ⊂ C(G).

Proposition (S, δ) not unimodular ⇒ not polynomial growth The dual of SU_{q}(N) does not have RD.

A necessary condition

If (S, δ, L) has RD there exists P ∈ ℝ[X] st for any inclusion γ ⊂ β ⊗α without multiplicity

∀a ∈ p_{α}S ||p_{γ}F(a)p_{β}|| ≤ P (|α|)||a||_{2}.

Proposition This condition is equivalent to requiring, for any a ∈ L(H_{α}), b ∈ L(H_{β}) :

||δ(p_{γ})(b⊗a)δ(p_{γ})||_{2} ≤ √\frac{m_{γ}}{m_{β}m_{α}} P (|α|) ||b⊗a||_{2}

NB : δ(p_{γ}) is the projection onto H_{γ} ⊂_{∼} H_{β} ⊗H_{α}.

Corollary Non-unimodular DQG cannot have RD. (Consider ε ⊂ ᾱ⊗α.)

Free quantum groups

[Wang, van Daele, Banica]

Recall that

C^{∗}(F_{N}) = C^{∗}(1, u_{i} | ∀i u_{i}u_{i}^{∗} = u_{i}^{∗}u_{i} = 1) One puts for Q ∈ GL(N, ℂ)

A_{u}(Q) = C^{∗}(1, u_{ij} | U and QŪQ^{−1}unitary )

A_{o}(Q) = C^{∗}(1, u_{ij} | U = QŪQ^{−1}unitary )

These are compact quantum groups whose duals are called the free quantum groups.

In the orthogonal case (with Q̄Q ∈ ℂid) I ≃ ℕ with U ≃ α_{1}, ᾱ_{k} ≃ α_{k} and

α_{k} ⊗α_{l} ≃ α_{|k−l|} ⊕ α_{|k−l|+2} ⊕ ⋯ ⊕ α_{k+l} .

In the unitary case I ≃ \{words on u, ū \} with U ≃ u, \overline{wu} ≃ ūw̄, wu⊗uw′ ≃ wuuw′, wu⊗ūw′ ≃ wuūw′⊕w⊗w′.

(S, δ) unimodular ⟺ Q ∈ ℂU(N). If N ≥ 3, m_{α} grows exponentially with |α|.

Free quantum groups

Haagerup proved that F_{N} has Property RD.

Proposition For the duals of A_{o}(Q) and A_{u}(Q), the necessary condition of Slide 5 is sufficient.

Theorem If Q ∈ ℂU(N), the duals of A_{o}(Q) and A_{u}(Q) have Property RD.

(requires a finer description of the representation theory than just the semi-ring structure)

Application to K-theory

[cf Jolissaint, Lafforgue]

L word length on finitely generated (S, δ). D : Dom D ⊂ B(H) → B(H) derivation by L.

Proposition We have Ŝ_{red} ∩ Dom D^{k} ⊂ H_{L}^{k}. If (S, δ, L) has RD we have H_{L}^{k+s} ⊂ Ŝ_{red}∩Dom D^{k}.

Corollary H_{L}^{∞} is a dense subalgebra of Ŝ_{red}, stable under holomorphic functional calculus. The inclusion induces an isomorphism

K_{∗}(H_{L}^{∞}) ⟶^{∼} K_{∗}(Ŝ_{red}).

Similarly, for s big enough, H_{L}^{s} is a dense subalgebra of Ŝ_{red} which is stable under holomorphic functional calculus.

Another Application

U ∈ M_{N} ⊗Ŝ_{red} fundamental corepr. of A_{o}(I_{N}). Consider the operator

Ψ : Ŝ_{red} → Ŝ_{red}
x ↦ (Tr ⊗id)(U^{∗}(1⊗x)U)/N

Proposition There exists λ < 1 such that

∀x ∈ Ŝ_{red} ĥ(x) = 0 ⇒ ||Ψ(x)||_{2} ≤ λ||x||_{2}.

Corollary For A_{o}(I_{N}),

1. Ŝ′′_{red} is a full factor.

2. Ŝ_{red} is simple with a unique tracial state.