Path cocycles in quantum Cayley trees and L2-cohomology

Roland Vergnioux

Universit´e de Caen Basse-Normandie

Wien, april 12th, 2011

1 Introduction

Universal discrete quantum groups The main result

The strategy

2 Quantum Cayley trees The classical picture Quantum Cayley graphs Quantum path cocycles

Consider the unital∗-algebras defined by generators and relations:

Au(n) =hu_ij |(u_ij) and (u_ij^∗) unitaryi, Ao(n) =hu_ij |uij =u^∗_ij, (uij) unitaryi, with 1≤i,j ≤n. They become Hopf ∗-algebras with

∆(u_ij) =P

u_ik⊗u_kj, S(u_ij) =u^∗_ji, (u_ij) =δ_ij. Moreover there exists a unique positive Haar integral h:A →C. We can consider the GNS construction:

H=L²(A,h), λ:A →B(H), M =λ(A)⁰⁰⊂B(H).

Classical counterpart: A =CG,H =`²(G), withG a discrete group.

Heuristically : Au(n) =CFU_n,Ao(n) =CFO_n, whereFO_n,FU_n are

there are natural mapsAu(n)CF_n,Ao(n)C(Z/2Z)^∗n ;

we have Au(n)B for anyB associated with a unimodular discrete quantum group and some n ;

FU_n,FO_n have the Property of Rapid Decay ;

theC^∗-algebras Au(n)_red,Ao(n)_red are simple, non-nuclear, exact ; M =λ(Ao(n))⁰⁰, λ(Au(n))⁰⁰ are solid II1 factors ;

FO_n is K-amenable ;

FO_n,FU_n satisfy Haagerup’s Approximation Property.

[Banica, V., Vaes, Vander Vennet, Voigt, Brannan]

The case n= 2 behaves differently, e.g. Ao(I2) =C(SU−1(2)) has polynomial growth, and will be excluded in this talk.

For an ICC group G, we can take A = CG and consider the Hochschild cohomology groupsH¹(A,_λH) andH¹(A,_λM).

These groups are moreover right M-modules and we have β₁⁽²⁾(G) = dimMH¹(A,H) = dimMH¹(A,M).

Recall thatβ₁⁽²⁾(Fn) =n−1. In the case of the orthogonal universal discrete quantum groups we have the strongly contrasting result:

Theorem

For n≥3we have H¹(Ao(n),H) =H¹(Ao(n),M) = 0.

In particular β₁⁽²⁾(Ao(n)) = 0. On the other handβ₁⁽²⁾(Au(n))6= 0.

Consider a representation π :A →L(X) on a vector space X. A π-cocycle is a mapc :A →X such that

∀x,y ∈A c(xy) =π(x)c(y) +c(x)(y).

It is trivial ifc(x) =π(x)ξ−ξ(x) for some ξ∈X and all x∈A. H¹(A,X) is the space of π-cocycles modulo trivial cocycles.

Theorem

For n≥3we have H¹(Ao(n),H) =H¹(Ao(n),M) = 0.

In particular β₁⁽²⁾(Ao(n)) = 0. On the other handβ₁⁽²⁾(Au(n))6= 0.

Theorem

For n≥3we have H¹(Ao(n),H) =H¹(Ao(n),M) = 0.

In particular β₁⁽²⁾(Ao(n)) = 0. On the other handβ₁⁽²⁾(Au(n))6= 0.

Remarks:

Collins-H¨artel-Thom: β_k⁽²⁾(Ao(n)) = 0 fork ≥4,

β_k⁽²⁾(Ao(n)) =β_4−k⁽²⁾ (Ao(n)), and Kyed: β₀⁽²⁾(Ao(n)) = 0.

Corollary : δ^∗(Ao)≤1, by [Connes-Shlyakhtenko].

δ(Ao) =δ^∗(Ao) = 1 ifM is embeddable into R^ω.

History : Leuven 11/2008, ArXiv v1 05/2009, ArXiv v2 03/2010

Theorem

For n≥3we have H¹(Ao(n),H) =H¹(Ao(n),M) = 0.

In particular β₁⁽²⁾(Ao(n)) = 0. On the other handβ₁⁽²⁾(Au(n))6= 0.

Strategy (for Ao):

Show that one particular cocycle vanishes: the path cocycle cg :A →Kg with values in the quantum Cayley tree

Prove that this cocycle is “sufficiently universal” and vanishes

“sufficiently strongly” (and use Property RD)

Algebraic version

X

A

cO 66

c

::t

tt tt tt tt t

A⊗A

T

ddIIIII IIIII T =m−(id⊗) is the usual boundary map, c₀=id−1is “the” trivial cocycle on A. Fix a cocycle c :A →X.

Algebraic version

X

A

cO 66

c

::t

tt tt tt tt t

A⊗A

T

ddIIIII IIIII

mc

OO

Fix a cocycle c :A →X.

Observe that the cocycle relation for c :A →X reads

π(x)c(y) =c((m−id⊗)(x⊗y)) = (c◦T)(x⊗y) Definemc :A⊗A →X by puttingmc(x⊗y) =π(x)c(y).

Algebraic version

X

A

cO 66

c

::t

tt tt tt tt t

cg

$$I

II II II II I

A⊗A

T

ddIIIII IIIII

mc

OO

Fix a cocycle c :A → X.

Definem_c :A⊗A →X by puttingm_c(x⊗y) =π(x)c(y).

Assume we can “lift” c0 to a cocyclecg :A → A⊗A, ie T ◦cg =c0. We obtainc =mc◦cg.

Algebraic version

X

A

cO 66

c

::t

tt tt tt tt t

cg

$$I

II II II II I

A⊗A

T

ddIIIII IIIII

mc

OO

Fix a cocycle c :A → X.

Definem_c :A⊗A →X by puttingm_c(x⊗y) =π(x)c(y).

Assume we can “lift” c0 to a cocyclecg :A → A⊗A, ie T ◦cg =c0. We obtainc =mc ◦cg. Hence ifcg is trivial with fixed vectorξg ∈A⊗A, all cocyclesc are trivial with fixed vectorξ=m_c(ξ_g).

Hilbertian version

M

A

cO 66

c

::u

uu uu uu uu u

cg

$$I

II II II II

H⊗H₁

T

ddIIIIIIIII

mc

OO

Fix a cocycle c :A → M.

Definemc :H⊗H₁ →X by puttingmc(x⊗y) =π(x)c(y).

Assume we can “lift” c₀ to a cocyclec_g :A → H⊗H₁, ie T ◦c_g =c₀. We obtainc =m_c◦c_g. Hence ifc_g is trivial with fixed vectorξ_g ∈H⊗H₁, all cocyclesc are trivial with fixed vectorξ=mc(ξg).

With H₁ ⊂H finite-dimensional...

The correct version

M

A

cO 66

c

::u

uu uu uu uu u

cg

$$I

II II II II

H⊗H₁

T

ddIIIIIIIII

mc

OO

Fix a cocycle c :A → M.

Definemc :H⊗H₁ →X by puttingmc(x⊗y) =π(x)c(y).

Assume we can “lift” c₀ to a cocyclec_g :A → Kg⁰, ie T ◦c_g =c₀. We obtainc =m_c ◦c_g. Hence ifc_g is trivial with fixed vectorξ_g ∈M⊗H₁, all cocyclesc are trivial with fixed vectorξ=mc(ξg).

With Kg⁰⊂(A⊗A1)⊕KerT...

1 Introduction

Universal discrete quantum groups The main result

The strategy

2 Quantum Cayley trees The classical picture Quantum Cayley graphs Quantum path cocycles

In the case of the free group Fn=hSi, consider:

the Cayley graph of (F_n,S),

the`²-spaceK_g of antisymmetric edges, for w ∈F_n, the sumc_g(w)∈K_g of edges along the path from e to w: “path cocycle”

e

w

In the case of the free group Fn=hSi, consider:

the Cayley graph of (F_n,S),

the`²-spaceK_g of antisymmetric edges, for w ∈F_n, the sumc_g(w)∈K_g of edges along the path from e to w: “path cocycle”

e

w

This cocycle is non-trivial, even proper =⇒ a-T-menability.

A

Fix the following data:

a discrete groupG,

a finite subsetS ⊂G such that S⁻¹ =S,e∈/ S.

The Cayley graph associated with (G,S) is given by:

the set of verticesG, the set of edges G×S,

the target map t: (α, γ)→αγ, the reversing map θ(α, γ) = (αγ, γ⁻¹).

Fix the following data:

a discrete groupG,

a finite subsetS ⊂G such that S⁻¹ =S,e∈/ S.

The HilbertianCayley graph associated with (G,S) is given by:

the set of verticesG, the set of edges G×S,

the target map t: (α, γ)→αγ, the reversing map θ(α, γ) = (αγ, γ⁻¹).

Fix the following data:

a discrete groupG,

a finite subsetS ⊂G such that S⁻¹ =S,e∈/ S.

The Hilbertian Cayley graph associated with (G,S) is given by:

the space of vertices H=`²(G), the set of edges G×S,

the target map t: (α, γ)→αγ, the reversing map θ(α, γ) = (αγ, γ⁻¹).

Fix the following data:

a discrete groupG,

a finite subsetS ⊂G such that S⁻¹ =S,e∈/ S.

The Hilbertian Cayley graph associated with (G,S) is given by:

the space of vertices H=`²(G),

the space of edgesK =H⊗H₁, whereH1=`²(S), the target map t: (α, γ)→αγ,

the reversing map θ(α, γ) = (αγ, γ⁻¹).

Fix the following data:

a discrete groupG,

a finite subsetS ⊂G such that S⁻¹ =S,e∈/ S.

The Hilbertian Cayley graph associated with (G,S) is given by:

the space of vertices H=`²(G),

the space of edgesK =H⊗H₁, whereH1=`²(S), the target operatorT :δ_α⊗δ_β 7→δ_αβ,

the reversing map θ(α, γ) = (αγ, γ⁻¹).

Fix the following data:

a discrete groupG,

a finite subsetS ⊂G such that S⁻¹ =S,e∈/ S.

The Hilbertian Cayley graph associated with (G,S) is given by:

the space of vertices H=`²(G),

the space of edgesK =H⊗H₁, whereH1=`²(S), the target operatorT :δ_α⊗δ_β 7→δ_αβ,

the reversing operator Θ :δ_α⊗δ_γ7→δ_αγ⊗δ_γ−1.

Fix the following data:

a discrete groupG,

a finite subsetS ⊂G such that S⁻¹ =S,e∈/ S.

The Hilbertian Cayley graph associated with (G,S) is given by:

the space of vertices H=`²(G),

the space of edgesK =H⊗H₁, whereH1=`²(S), the target operatorT :δ_α⊗δ_β 7→δ_αβ,

the reversing operator Θ :δ_α⊗δ_γ7→δ_αγ⊗δ_γ−1. C_red^∗ (G) acts on H and on the first factor ofH⊗p₁H.

T and Θ are intertwining operators.

Fix the following data:

a discrete quantum group,

a finite subsetS ⊂G such that S⁻¹ =S,e∈/ S.

The Hilbertian Cayley graph associated with (G,S) is given by:

the space of vertices H=`²(G),

the space of edgesK =H⊗H₁, whereH1=`²(S), the target operatorT :δ_α⊗δ_β 7→δ_αβ,

the reversing operator Θ :δ_α⊗δ_γ7→δ_αγ⊗δ_γ−1. C_red^∗ (G) acts on H and on the first factor ofH⊗p₁H.

T and Θ are intertwining operators.

Fix the following data:

a discrete quantum group,

a finite subsetS ⊂IrrC such that ¯S =S and ˆ /∈S. The Hilbertian Cayley graph associated with (G,S) is given by:

the space of vertices H=`²(G),

the space of edgesK =H⊗H₁, whereH1=`²(S), the target operatorT :δ_α⊗δ_β 7→δ_αβ,

the reversing operator Θ :δ_α⊗δ_γ7→δ_αγ⊗δ_γ−1. C_red^∗ (G) acts on H and on the first factor ofH⊗p₁H.

T and Θ are intertwining operators.

Fix the following data:

a discrete quantum group,

a finite subsetS ⊂IrrC such that ¯S =S and ˆ /∈S. The quantumCayley graph associated with (C,S) is given by:

the space of vertices H,

the space of edgesK =H⊗H₁, whereH1=pSH, the target operatorT :δ_α⊗δ_β 7→δ_αβ,

the reversing operator Θ :δ_α⊗δ_γ7→δ_αγ⊗δ_γ−1. C_red^∗ (G) acts on H and on the first factor ofH⊗p₁H.

T and Θ are intertwining operators.

Fix the following data:

a discrete quantum group,

a finite subsetS ⊂IrrC such that ¯S =S and ˆ /∈S. The quantum Cayley graph associated with (C,S) is given by:

the space of vertices H,

the space of edgesK =H⊗H₁, whereH1=pSH, the target operatorT =m:K →H,

the reversing operator Θ =· · ·, such thatTΘ =id⊗.

C_red^∗ (G) acts on H and on the first factor ofH⊗p₁H.

T and Θ are intertwining operators.

Fix the following data:

a discrete quantum group,

a finite subsetS ⊂IrrC such that ¯S =S and ˆ /∈S. The quantum Cayley graph associated with (C,S) is given by:

the space of vertices H,

the space of edgesK =H⊗H₁, whereH1=pSH, the target operatorT =m:K →H,

the reversing operator Θ =· · ·, such thatTΘ =id⊗.

A_red acts on H and on the first factor of H⊗p₁H.

T and Θ are intertwining operators.

Quasi-classicalsubgraph Q₀K ⊂K: maximal subspace on which Θ²=id.

Classical subgraphq₀K ⊂Q₀K: fixed points for the adjoint repr. of ˆA.

q0K,Q0K are not stable under the action ofA ! WhenA =CG,q₀K =Q₀K =K.

WhenA =Ao(n), q0K =Q0K 6=K. WhenA =Au(n), q0K 6=Q0K 6=K.

Quasi-classicalsubgraph Q₀K ⊂K: maximal subspace on which Θ²=id.

Classical subgraphq₀K ⊂Q₀K: fixed points for the adjoint repr. of ˆA.

Case of Au(n) : q0K ⊂ Q0K ⊂ K.

edge α in the “real” graph I normed vector ξα∈q0K vertex v in the “real” graph I normed vectorζv ∈q0H operators T, Θ induced by the classical operations

Quasi-classicalsubgraph Q₀K ⊂K: maximal subspace on which Θ²=id.

Classical subgraphq₀K ⊂Q₀K: fixed points for the adjoint repr. of ˆA.

Case of Au(n) : q0K ⊂ Q0K ⊂ K.

Why this picture ?

vertices v are irreducible corepresentations edges come from fusion rulesv⊗u =L

v⁰

weights depend on dimensions of corepresentations

Quantum Cayley trees Quantum path cocycles

Quantum path cocycles

We look for cocycles with values in the space of geometric, or antisymmetric, edges Kg =Ker(Θ +id).

Recall that T =m= (id⊗)Θ, so thatm−id⊗= 2T on K_g. Definition

A path cocycleis a cocycle c_g :A →K_g such thatT ◦c_g =c₀.

e w

T

+1

−1

e

−1 +1

w

+1

−1

We look for cocycles with values in the space of geometric, or antisymmetric, edges Kg =Ker(Θ +id).

Recall that T =m= (id⊗)Θ, so thatm−id⊗= 2T on K_g. Definition

A path cocycleis a cocycle c_g :A →K_g such thatT ◦c_g =c₀.

Example: in the Cayley tree of Fn, denote bycg(w)∈Kg the sum of the antisymmetric edges on the path from the origin to w.

e w

T

+1

−1

e

−1 +1

w

+1

−1

We consider a free product of Ao’s and Au’s withn ≥3.

We denote Kg⁰ the orthogonal projection ofA⊗A ontoK_g. Proposition

If T is injective on Kg⁰, then there exists a unique path cocycle c_g :A →K_g⁰.

In the case of F_n we haveK_g⁰ =K_g ∩(A⊗A) andT is injective only on this dense subspace. On the “purely quantum part” of our quantum trees we have the much stronger property:

Theorem

T is injective with closed range on (1−Q₀)K_g.

Proposition

In the case of A_o(Q), with Q ∈GL(n,C), QQ¯ ∈CI_n, n ≥3, the target operator T :Kg →H is invertible. As a result there exists a unique path cocycle c_g :A →K_g, and it is trivial.

The main reason is that q₀K_g =Q₀K_g comes from the half-line:

(0,1) (1,2) (2,3) (3,4) (4,5) (5,6)

We can even compute the fixed vector ξg =T⁻¹ξ0 for cg:

ξg =X

n≥0

ξ_(αn,αn+1)−ξ_(αn+1,αn) pdim_qα_ndim_qα_n+1dim_qα₁.

By property RD it lies in M⊗H₁. =⇒ β₁⁽²⁾(Ao(n)) = 0

The quasiclassical subgraph is a union of trees ⇒ T injective onKg⁰. Hence we have a unique path cocycle c_g :A →Kg⁰.

Let γ ∈M_n⊗A be the fundamental corepresentation of Au(n).

We consider αk =γ^k andβk =γ¯γγ· · ·. Proposition

We have k(id⊗c_g)(α_k)k ≥C√

k andk(id⊗c_g)(β_k)k ≤D for all k and constants C , D >0.

As a result c_g is neither trivial (bounded) nor proper.

Au(n) non-amenable =⇒ β₁⁽²⁾(Au(n))6= 0

Heuristically, the Proposition holds because there is no multiplicity above the zigzag path (β_k), and a lot of multiplicity above the straight line (α_k):

IsM =λ(Ao(n))⁰⁰ strongly solid ? IsM =λ(Ao(n))⁰⁰ L²-rigid ?