Actions of Compact Quantum Groups VI Free and homogeneous actions II - - PowerPoint PPT Presentation

Actions of Compact Quantum Groups VI

Free and homogeneous actions II

Kenny De Commer (VUB, Brussels, Belgium)

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Outline

Homogeneous and free: Galois objects From homogeneous to free and back Homogeneous actions of SUq(2)

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Galois objects

Definition

X

α G Galois object (or quantum torsor) if

• 1. α free,
• 2. α homogeneous,
• 3. C(X) = {0}.

Lemma

If X

α G Galois object, then X ∼

= G equivariantly. No longer true in quantum case!

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Quantum torus

Example (Quantum torus)

Let θ ∈ [0, 2π]. Put C(T2

θ) = C∗(U, V | U, V unitary, UV = eiθV U.}.

Then free and homogeneous T2

θ T2 by

α(w,z)(U) = wU, α(w,z)V = zV.

Remarks: ◮ Check that C(T2

q) not trivial.

◮ Instance of general construction: 2-cocycles on discrete quantum groups.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Twisting procedure

Theorem (Bichon-De Rijdt-Vaes)

• 1. There is a one-to-one-correspondence between (classes of)

◮ Galois objects X for G, ◮ Fiber functors F on Rep(G) (into Hilbert spaces).

• 2. Let X

α G Galois object. Then ∃!H such that

◮ H

β X is (left) Galois object,

◮ α and β commute.

Remark:

◮ Abstractly: H from Tannaka-Krein on F. ◮ Concretely: C(Hred) ⊆ C(Xred) ⊗ C(Xred)op. ◮ One says G and H monoidally equivalent.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Another look at quantum SU(2) and O+(n)

Definition

Take F ∈ GLn(C) with F ¯ F ∈ R. Then C(O+

u (F)) = C∗(uij | 1 ≤ i, j ≤ n, U unitary, FUF −1 = U)

becomes compact quantum group for ∆(uij) =

• k

uik ⊗ ukj.

Example

• 1. For F =
• 1

−q−1

• , C(O+

u (F)) = C(SUq(2)).

• 2. For F = In, C(O+

u (In)) = C(O+ n ).

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Classification of all Galois objects of SUq(2)

Notation

For F ∈ GLn(C) with F ¯ F ∈ R, write cF = −sign(F ¯ F)Tr(F ∗F).

Remark: Always |cF | ≥ 2.

Theorem (Bichon-De Rijdt-Vaes)

◮ {O+(F)} is complete w.r.t. monoidal equivalence. ◮ O+(F1)

∼ =

• mon. eq. O+(F2) iff cF1 = cF2.

◮ O+(F)

∼ =

• mon. eq. SUq(2) for q + q−1 = cF .

In fact, Galois object between O+(F1) and O+(F2) C(O+

u (F1, F2)) = C∗

uij | 1 ≤ i ≤ dim(F1), 1 ≤ j ≤ dim(F2) U unitary, F1UF −1

2

= U

• .

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Morita base change

Lemma

Let X G free, Y = X/G. Assume p ∈ M(C0(Y)) full projection: [C0(Y)pC0(Y)] = C0(Y). Then, with C0(Xp) = pC0(X)p, free action Xp G by αp : C0(Xp) → C0(Xp) ⊗ C(G), a → α(a). Moreover, with C0(Yp) = pC0(Y)p, Xp/G = Yp.

Remarks: ◮ C0(Yp) and C0(Y) (strongly) Morita equivalent (‘non-commutative isomorphism Yp ∼ = Y’). ◮ Then also C0(Xp) and C0(X) Morita equivalent.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Proof

◮ Well-defined coaction: clear. ◮ Free: using C0(X) = [C0(Y)C0(X)C0(Y)],

[αp(C0(Xp))(C0(Xp) ⊗ 1)] = [(p ⊗ 1)α(C0(X))(pC0(X)p ⊗ 1)] = [(p ⊗ 1)α(C0(X))(C0(Y)pC0(Y)C0(X)p ⊗ 1)] = [(p ⊗ 1)α(C0(X))(C0(Y)C0(X)p ⊗ 1)] = [(p ⊗ 1)α(C0(X))(C0(X)p ⊗ 1)] = [pC0(X)p ⊗ C(G)] = C0(Xp) ⊗ C(G).

◮ Xp/G = Yp as exercise.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Reduction to free actions

Recall: X G homogeneous, then C0(X ⋊ G) ∼ = ⊕i∈IB0(Hi).

Corollary (Wassermann construction)

Consider e(i)

00 fixed matrix unit in B0(Hi), and put

p = ⊕e(i)

00 ∈ M(⊕i∈IB0(Hi)). ◮ p full projection for ⊕i∈IB0(Hi). ◮ Xfree G free with C0(Xfree) = pC0(X ⋊ G ⋊

G)p.

◮ C0(Xfree/G) = c0(I).

Proof.

Use (X ⋊ G ⋊ G)/G = X ⋊ G.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

From free to homogeneous and back

Lemma

X G with C0(X/G) = c0(I), and C(Xi) = δiC0(X)δi. Then Xi G homogeneous.

Lemma

Let X G homogeneous. ◮ C0(X ⋊ G) ։ B(L2

Y(X)) ⇒ distinguished block B0(Hi0) ⊆ C0(X ⋊ G).

◮ Associated projection δi0 ∈ c0(I) ⊆ C0(Xfree) is full. ⇒ C0(Xfree) ∼ =

Morita C0(X).

Theorem

G CQG. The above gives one-to-one correspondence between ◮ (Irreducible) free actions X′ G with X′/G classical discrete set (up to iso) ◮ Homogeneous actions X G (up to ‘equivariant Morita equivalence’). ⇒ classifying homogeneous actions = classifying certain free actions.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Free actions and fiber functors

Definition

I a set. Monoidal category (IHilbI, ⊠): ◮ Objects: I-bigraded Hilbert spaces, H = ⊕

k,l kHl,

◮ Tensor product:

k(H ⊠G)l = ⊕ m kHm ⊗mGl.

Theorem (DC-Yamashita)

There is a one-to-one correspondence between

• 1. Free actions X G with X/G classical discrete set I (up to isomorphism)
• 2. Tensor C∗-functors Repfd(G) → IHilbI (up to ‘equivalence’).

Concrete Tannaka-Krein reconstruction process.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Reduction scheme

Classifying homogeneous actions of SUq(2).

• Classifying free actions of SUq(2) with discrete quotient space
• Classifying Monoidal C∗-functors Repfd(SUq(2)) → IHilbI.

But... Repfd(SUq(2)) easy generators and relations... Classifying Monoidal C∗-functors Repfd(SUq(2)) → IHilbI.

• Combinatorial data.

Remark: ∃q, Rep(O+(F)) = Rep(SUq(2)), so: classification homogeneous X O+(F).

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Representation category of SUq(2)

Lemma

Repfd(SUq(2)) is ‘completion’ of tensor C∗-category with

◮ Objects: finite ordinals ◮ Basis for morphisms: non-crossing 2-partitions

• and

◮ Tensor product: horizontal juxtaposition ◮ Composition: vertical stacking with rule ★ = −q − q−1. ◮ ∗-structure: ∩∗ = −sgn(q)∪.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Reciprocal random walks

Definition (DC-Yamashita)

Let δ ∈ R0. A δ-reciprocal random walk consists of a quadruple (Γ, w, sgn, i) where ◮ Γ = (Γ(0), Γ(1), s, t) is a graph with source and target maps s and t, ◮ w is a weight function w: Γ(1) → R+

0 ,

◮ sgn a sign function sgn: Γ(1) → {±1}, ◮ i is an involution e → e on Γ(1) interchanging source and target, s.t. ◮ for all e, w(e)w(¯ e) = 1, ◮ for all e, sgn(e)sgn(¯ e) = sgn(δ), ◮ for all v,

s(e)=v 1 |δ| w(e) = 1.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Examples

◮ Action SUq(2) on non-standard Podle´ s sphere S2

q,x

· · · •

qx+q−x qx−1+q−x+1

qx−1+q−x+1 qx+q−x

• qx+1+q−x−1

qx+q−x

• · · ·

qx+q−x qx+1+q−x−1

• Figure: δ = −(q + q−1) (q > 0, x ∈ R)

◮ Action O+

n on SN−1 +

• 1
• N−1

1 N−1

• 1
• N(N−2)

N−1

• · · ·

N−1 N(N−2)

• Figure: δ = N

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Abundance of reciprocal random walks

Lemma (DC-Yamashita)

Let (Γ, w, sgn, i) δ-reciprocal random walk. Then Γ bounded degree: sup

v∈Γ(0) #{e ∈ Γ(1) | s(e) = v} < ∞.

Theorem (DC-Yamashita)

Γ bounded degree ⇒ ∃δ and δ-reciprocal random walk on Γ.

‘Proof’.

By Frobenius-Perron theory.

Theorem (Kronecker)

ADE-classification for 2-reciprocal random walks.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

A one-to-one correspondence

Theorem (DC-Yamashita)

Fix q = 0, put δq = −q − q−1. There is (up to appropriate equivalence) a one-to-one correspondence between ◮ Tensor C∗-functors F : Rep → IHilbI, and ◮ δq-reciprocal random walks Γ = (Γ(0), Γ(1), s, t) with Γ(0) = I. Construction of F from Γ: ◮ F( ) = l2(Γ(0)), ◮ F(•) = l2(Γ(1)), ◮ F(∩)(δv) =

• s(e)=v

sgn(e)w(e)1/2δe ⊗ δ¯

e.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Wassermann’s theorem

Lemma

Let H ⊆ G compact Hausdorff groups. Let π irreducible H-representation. Write C(G, B(Hπ))H = {f ∈ C(G, B(Hπ)) | f(hg) = π(h)f(g)π(h)∗}. Then homogeneous action G

α

C(G, B(Hπ))H, αg(f)(g′) = f(g′g).

Theorem (Wassermann)

X SU(2) homogeneous, then ∃H ⊆ SU(2) and irreducible H-representation s.t. C(X) ∼ =

G-equivariantly C(G, B(Hπ))H.

Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SUq(2)

Further questions

Project

X SUq(2) homogeneous.

• 1. Analyse C(X) and L∞(X) in terms of their weighted graph.
• 2. Classify all X with C(X) or L∞(X) type I/simplicial/factorial.