Path cocycles in quantum Cayley trees and L2-cohomology

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Path cocycles in quantum Cayley trees and L

²

-cohomology

Roland Vergnioux

Universit´e de Caen Basse-Normandie

Luminy, Sept. 27, 2010

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Introduction

Outline

1 Introduction

Universal discrete quantum groups The Main Result

The Strategy

2 Quantum Cayley trees Quantum Cayley graphs Path cocycles

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Introduction Universal discrete quantum groups

Universal discrete quantum groups

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

Au(In) =hu_ij |(u_ij) and (u^∗_ij) unitaryi, Ao(In) =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.

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Introduction Universal discrete quantum groups

Analogies with free group algebras

there are natural mapsAu(I_n)CF_n,Ao(I_n)C(Z/2Z)^∗n ; we haveAu(I_n)B for anyB associated with a unimodular discrete quantum group and somen ;

there is a natural correspondence between irreducible corepresentations ofAu(In) and words onu, ¯u ;

theC^∗-algebras Au(In)red,Ao(In)red are simple, non-nuclear, exact ; the discrete quantum groups associated withAu(I_n),Ao(I_n) have the Property of Rapid Decay ;

M =λ(Ao(In))⁰⁰ is a full and primeII1 factor.

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

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Introduction The Main Result

The Main result

For an ICC group G, we can takeA =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) = dim_MH¹(A,H) = dim_MH¹(A,M).

Recall that β₁⁽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(In),H) =H¹(Ao(In),M) = 0.

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

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Introduction The Main Result

The Main result

Theorem

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

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

Remarks:

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

β_k⁽²⁾(Ao(In)) =β⁽²⁾_4−k(Ao(In)), and Kyed: β⁽²⁾₀ (Ao(In)) = 0.

Voigt: Baum-Connes and K-amenability for Ao(In), K0(Ao(In)) =K1(Ao(In)) =Z

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

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Introduction The Strategy

More on the strategy

Strategy (for Ao):

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

Prove that this cocycle is “sufficiently universal” and vanishes

“sufficiently strongly” (and use Property RD)

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∈Land all x∈A. H¹(A,X) is the space of π-cocycles modulo trivial cocycles.

We putc0(x) =λ(x)ξ0−ξ0(x), where ξ0 = Λ(1)∈X =H.

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More on the strategy

Algebraic version

Assume we can “lift” c₀ to a cocyclec_g :A → A⊗A, ie (m−(id⊗))(c_g(x)) =x−(x)1 Observe that the cocycle relation for c :A →X reads

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

We obtainc =mc◦cg.

Hence if cg is trivial with fixed vector ξg ∈A⊗A, all cocyclesc are trivial with fixed vector ξ=m_c(ξ_g).

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More on the strategy

Hilbertian version

Assume we can “lift” c₀ to a cocyclec_g :A → H⊗H₁, ie (m−(id⊗))(c_g(x)) =x−(x)1 Observe that the cocycle relation for c :A →M reads

π(x)c(y) =c((m−id⊗)(x⊗y)) Definem_c :H⊗H₁ →X by puttingm_c(x⊗y) =π(x)c(y).

We obtainc =mc◦cg.

Hence if cg is trivial with fixed vectorξg ∈ H⊗H₁, all cocycles c are trivial with fixed vector ξ=m_c(ξ_g).

With H₁ ⊂H finite-dimensional...

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More on the strategy

The correct version

Assume we can “lift” c0 to a cocyclecg :A → Kg⁰, ie (m−(id⊗))(c_g(x)) =x−(x)1 Observe that the cocycle relation for c :A →M reads

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

We obtainc =m_c◦c_g.

Hence ifc_g is trivial with fixed vector ξ_g ∈M⊗H₁, all cocyclesc are trivial with fixed vector ξ=mc(ξg).

With Kg⁰⊂(A⊗A1)∩Ker(m−id⊗)...

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Quantum Cayley trees

Outline

1 Introduction

Universal discrete quantum groups The Main Result

The Strategy

2 Quantum Cayley trees Quantum Cayley graphs Path cocycles

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Quantum Cayley trees Quantum Cayley graphs

The quantum Cayley graph

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).

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The quantum Cayley graph

a discrete groupG,

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

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

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The quantum Cayley graph

a discrete groupG,

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

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

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The quantum Cayley graph

a discrete groupG,

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).

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The quantum Cayley graph

a discrete groupG,

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).

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The quantum Cayley graph

a discrete groupG,

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

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The quantum Cayley graph

a discrete groupG,

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.

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The quantum Cayley graph

a discrete quantum group,

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The quantum Cayley graph

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

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The quantum Cayley graph

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,

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The quantum Cayley graph

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 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.

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The quantum Cayley graph

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 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.

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Classical subgraphs

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

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

WhenA =CG,q0K =Q0K =K. WhenA =Ao(In),q0K =Q0K 6=K. WhenA =Au(I_n), q₀K 6=Q₀K 6=K.

The classical and quasi-classical subgraphs are the hilbertian counterparts of “real” graphs as follows:

vertices are elements of IrrC,

edges depend on the fusions rules inIrrC,

target operator with weights depending on the quantum dimensions, q0H,Q0H are not stable under the action ofA.

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Classical subgraphs

In the case of A_u(I_n) we have a “classical” binary tree and a

“quasiclassical” union of half lines:

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Classical subgraphs

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Classical subgraphs

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Classical subgraphs

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Quantum Cayley trees Path cocycles

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.

Example: in the Cayley tree of F_n, denote byc_g(g)∈K_g the sum of the antisymmetric edges on the path from the origin to g.

T

+1 +1

−1

−1 +1

e g

−1

g

e

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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.

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

T

+1 +1

−1

−1 +1

e g

−1

g

e

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Some general results

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.

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The orthogonal case

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_(α_n_,α_n+1₎−ξ_(α_n+1_,α_n₎ pdim_qα_ndim_qα_n+1dim_qα₁.

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

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The unitary case

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(I_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(I_n) non-amenable =⇒ β⁽²⁾₁ (Au(I_n))6= 0

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The unitary case

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):

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The unitary case

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The unitary case

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