The Baum-Connes conjecture for Quantum Groupsruben.martos/doc/thesispresentation.pdf · The Baum-Connes conjecture for Quantum Groups Stability properties and K-theory computations

The Baum-Connes conjecture forQuantum Groups

Stability properties and K -theory computations

Ruben MartosAdvisor: Pierre Fima

Institut de Mathematiques de Jussieu-Paris Rive Gauche (IMJ-PRG)Universite Paris Diderot (Paris VII)

Table of contents

1. Introduction

Woronowicz’s approach of quantum groups

The Baum-Connes conjecture2. Meyer-Nest’s approach for the Baum-Connes conjecture

Categorical context and Reformulation

“Quantification” of the conjecture3. Current situation of the quantum Baum-Connes conjecture

Work of C. Voigt and collaborators

Stability properties4. An application: some K -theory computations

General strategy

The Lemeux-Tarrago’s Hq

The Baum-Connes conjecture for Quantum Groups Ruben Martos

Table of contents

1. Introduction







General strategy



Table of contents

1. Introduction







General strategy



Table of contents

1. Introduction







General strategy



Introduction


Definition

Definition of a C.Q.G.A compact quantum group is the data G :“ pA,∆q where A is a unitalC˚-algebra and ∆ : A ÝÑ AbA is a unital ˚-homomorphism such that:

i) ∆ is co-associative, that is,

A

ö∆��

∆ // Ab A

id b∆��

Ab A∆b id

// Ab Ab A

ii) The spaces pAb 1Aq∆pAq et p1A b Aq∆pAq are dense in Ab A(cancellation property)



Motivation

Let G be a compact groupThe pair G :“ pCpGq,∆q with

∆pf qpx , yq :“ f pxyq, @f P CpGq, x , y P G

is a compact quantum group.For this reason we write G “ pCpGq,∆q



Examples I: the basic

Quantum permutation group S`NFree unitary quantum group U`pF qFree orthogonal quantum group O`pF qq-deformations of compact connected semi-simple Lie groups

§ SUqp2q



Examples II: constructions

Quantum direct product GˆH§ Introduced by S. Wang in 1995§ C

`

GˆH˘

“ CmpGq bmax

CmpHq and co-multiplication Θ such that

Θpa b bq “ ∆Gpaqp1q b∆Hpbqp1q b∆Gpaqp2q b∆Hpbqp2q,

for all a P CmpGq and all b P CmpHq.




Quantum semi-direct product Γ˙αG

§ Introduced by S. Wang in 1995§ α : Γ ÝÑ AutpGq, group homomorphism§ C

`

Γ˙αG˘

“ Γ ˙α,m

CmpGq and co-multiplication Θ such that

Θpπpaqq “ pπ b πq`

∆Gpaq˘

and Θpuγq “ uγ b uγ ,

for all a P CmpGq and all γ P Γ




Quantum free product G ˚H§ Introduced by S. Wang in 1995§ C

`

G ˚H˘

“ CmpGq ˚ CmpHq and co-multiplication Θ such that

ΘpνGpaqq “ pνG b νGq`

∆Gpaq˘

and ΘpνHpbqq “ pνH b νHq`

∆Hpbq˘

,

for all a P CmpGq and all b P CmpHq, whereνG : CpGq ÝÑ CmpGq ˚ CmpHq and νH : CpHq ÝÑ CmpGq ˚ CmpHqdenote the canonical inclusions




Free wreath product G o˚ S`N§ Introduced by J. Bichon in 2004§ C

`

G o˚ S`N˘

“ CpGq˚N˚ CpS`N q{I, where

I :“ xtνkpaquki ´ ukiνkpaquNi,k“1y and co-multiplication Θ such that

Θpνipaqq “Nÿ

k“1

νipap1qquik b νkpap2qq and Θpuijq “

Nÿ

k“1

uik b ukj ,

for all a P CpGq, i , j “ 1, . . . ,N.



Importance of Woronowicz’s theory

Main theorem (S. L. Woronowicz, 1998)For every compact quantum group G “ pCpGq,∆q there exists a uniquestate hG P CpGq˚ such that

phG b idq∆paq “ hGpaq1 “ pid b hGq∆paq,

for all a P CpGq. The state hG is called the Haar state of G

Representation theory and harmonic analysis analogue to those of acompact group.

§ IrrpGq:= set of all unitary equivalence classes of irreducible unitaryfinite dimensional representations of G

Two versions of the same object: CmpGq and Cr pGq




Existence of a dual discrete quantum group pG such that

i) c0ppGq :“c0À

xPIrrpGqBpHx q ñ Non unital!

ii) V :“À

xPIrrpGqw xP Mpc0ppGq b CpGqq is a unitary representation

iii) the co-multiplication p∆ : c0ppGq ÝÑ Mpc0ppGq b c0ppGqq is such thatgiven x P IrrpGq and T P BpHx q, we have

p∆pT q ˝ Φ “ Φ ˝ T ,

for every Φ P Morpx , y j zq and y , z P IrrpGq

Quantum Pontryagin duality: p

pG – G

Notion of pG-C˚-algebra A ñ crossed product constructions pG˙r

A

and pG˙m

A




Existence of a dual discrete quantum group pG such that

i) c0ppGq :“c0À

xPIrrpGqBpHx q ñ Non unital!

ii) V :“À

xPIrrpGqw xP Mpc0ppGq b CpGqq is a unitary representation

iii) the co-multiplication p∆ : c0ppGq ÝÑ Mpc0ppGq b c0ppGqq is such thatgiven x P IrrpGq and T P BpHx q, we have

p∆pT q ˝ Φ “ Φ ˝ T ,

for every Φ P Morpx , y j zq and y , z P IrrpGq

Quantum Pontryagin duality: p

pG – G

Notion of pG-C˚-algebra A ñ crossed product constructions pG˙r

A

and pG˙m

A


The Baum-Connes conjecture

Definition, motivation and relevance

The Baum-Connes conjecture (with coefficients), 1982The assembly map

µA : K top˚ pG ; Aq ÝÑ K˚pG ˙

rAq

is an isomorphism, for every G-C˚-algebra A.

Abstraction of the Atiyah-Singer’s index theoremLot of consequences

§ Kadison-Kaplansky conjecture for torsion-free discrete groups§ Novikov conjecture

Satisfied by a large class of groups: those with the Haagerupproperty (Higson-Kasparov, 2001), hyperbolic groups (V. Lafforgue,2002)


Meyer-Nest’s approach for theBaum-Connes conjecture


Main ingredients

Triangulated categories: data pT ,Σ,∆Σ, 5 axiomesq where T is anadditive category, Σ is an auto-equivalence of T and ∆Σ is a class ofdistinguished triangles, that is, diagrams of the form

X uÝÑ Y v

ÝÑ Z wÝÑ ΣpX q

Theory of localization of functors by means of complementary pairof subcategories pL ,N q

§ L Ă N $ :“ tX P ObjpT q : HomT pX ,N q “ p0qu§ For every object X P T there exists a distinguished triangle of the

formL uÝÑ X v

ÝÑ N wÝÑ ΣpLq

§ Theorem: Such triangles are unique (up to isomorphism) and thisassociation is functorial



Main ingredients

Triangulated categories: data pT ,Σ,∆Σ, 5 axiomesq where T is anadditive category, Σ is an auto-equivalence of T and ∆Σ is a class ofdistinguished triangles, that is, diagrams of the form

X uÝÑ Y v

ÝÑ Z wÝÑ ΣpX q

Theory of localization of functors by means of complementary pairof subcategories pL ,N q

§ L Ă N $ :“ tX P ObjpT q : HomT pX ,N q “ p0qu§ For every object X P T there exists a distinguished triangle of the

formL uÝÑ X v

ÝÑ N wÝÑ ΣpLq

§ Theorem: Such triangles are unique (up to isomorphism) and thisassociation is functorial



Main ingredients

If F : T ÝÑ A is a functor, its localization with respect to pL ,N q

isLF :“ F ˝ L : T ÝÑ A

§ There exists a natural transformation

η : LF ÝÑ F

In order to construct such complementary pairs, R.Meyer and R.Nest develop an adapted homological algebra for triangulatedcategories



Main ingredients

If F : T ÝÑ A is a functor, its localization with respect to pL ,N q

isLF :“ F ˝ L : T ÝÑ A

§ There exists a natural transformation

η : LF ÝÑ F

In order to construct such complementary pairs, R.Meyer and R.Nest develop an adapted homological algebra for triangulatedcategories



Kasparov category

If G is a locally compact group (second-countable), K K G denotesthe G-equivariant Kasparov category:

§ ObjpK K Gq :“ separable G-C˚-algebras

§ HomK K G pA,Bq :“ KK GpA,Bq

Suspension of C˚-algebras, Σ, is an auto-equivalence (by Bottperiodicity)Given an equivariant ˚-homomorphism f : A ÝÑ B, a mapping conetriangle is the following diagram

ΣpBq ÝÑ Cf ÝÑ A fÝÑ B

Theorem (Meyer-Nest, 2006): pK K G ,Σ,∆Σq is a triangulatedcategory.



Choice of the complementary pair in K K G

F :“ family of all compact subgroups of GCompactly induced objects

L :“ xtA P ObjpK K Gq | A – IndGH pBq, H P F , B P K K Huy

Compactly contractible objects

N :“ tA P ObjpK K Gq | ResGH pAq – p0q @H P Fu

Theorem (Meyer-Nest, 2006): pL ,N q is a complementary pair inK K G .



Categorification of the Baum-Connes conjecture

Consider the functor

F : K K G ÝÑ A bZ{2

A ÞÝÑ F pAq :“ K˚pG ˙r

Aq

The categorical Baum-Connes assembly map is the naturaltransformation

ηG : LF ÝÑ F

Definition: Categorical Baum-Connes conjecture§ We say that G satisfies the (categorical) Baum-Connes conjecture if ηG is

a natural equivalence.§ We say that G satisfies the strong (categorical) Baum-Connes conjecture

if L “ K K G



Categorification of the Baum-Connes conjecture

Consider the functor

F : K K G ÝÑ A bZ{2

A ÞÝÑ F pAq :“ K˚pG ˙r

Aq

The categorical Baum-Connes assembly map is the naturaltransformation

ηG : LF ÝÑ F

Definition: Categorical Baum-Connes conjecture§ We say that G satisfies the (categorical) Baum-Connes conjecture if ηG is

a natural equivalence.§ We say that G satisfies the strong (categorical) Baum-Connes conjecture

if L “ K K G



Reformulation

Reformulation of the Baum-Connes conjecture (R. Meyer and R.Nest, 2006)The following assertions are equivalent:

i) G satisfies the Baum-Connes conjecture (with coefficients): µA is anisomorphism, for every G-C˚-algebra A.

ii) The natural transformation ηG : LF ÝÑ F is a natural equivalence.

Strong Baum-Connes conjecture ú Dirac-dual Dirac method



Reformulation

Reformulation of the Baum-Connes conjecture (R. Meyer and R.Nest, 2006)The following assertions are equivalent:

i) G satisfies the Baum-Connes conjecture (with coefficients): µA is anisomorphism, for every G-C˚-algebra A.

ii) The natural transformation ηG : LF ÝÑ F is a natural equivalence.

Strong Baum-Connes conjecture ú Dirac-dual Dirac method


“Quantification” of the conjecture

Observations

No classifying space for proper actions ñ No geometricconstructions ñ Replacement: G ú G (L.C. quantum group)Problem: the pair pL ,N q is defined in terms of the compactsubgroups of G

§ What is such a subgroup for G?§ Technical problems with the functors IndG

p¨q and ResGp¨qPartial solution: regard the discrete case ñ G is a compactquantum group ñ Replacement: Γ ú pG

§ If G :“ Γ is a discrete group, then pL ,N q is defined in terms of thetorsion of Γ

§ Notion of torsion for discrete quantum groups (Meyer-Nest in 2007and Arano-De Commer in 2015)



Observations

No classifying space for proper actions ñ No geometricconstructions ñ Replacement: G ú G (L.C. quantum group)Problem: the pair pL ,N q is defined in terms of the compactsubgroups of G

§ What is such a subgroup for G?§ Technical problems with the functors IndG

p¨q and ResGp¨qPartial solution: regard the discrete case ñ G is a compactquantum group ñ Replacement: Γ ú pG

§ If G :“ Γ is a discrete group, then pL ,N q is defined in terms of thetorsion of Γ

§ Notion of torsion for discrete quantum groups (Meyer-Nest in 2007and Arano-De Commer in 2015)



Observations

Current quantum formulation of theBaum-Connes conjecture:

torsion-free discrete quantum groups pG

LpG :“ xtc0ppGq b B, B P ObjpK K quy

NpG :“ tA P ObjpK K

pGq | A – 0 in K K u

F : K KpG ÝÑ A bZ{2

A ÞÝÑ F pAq :“ K˚ppG˙r

Aq


Current situation of the quantumBaum-Connes conjecture


Main results

Discrete quantum groups satisfying the strong Baum-Connesconjecture:

§ 2011 ù {SUqp2q and any free orthogonal quantum group, {O`pF q§ 2013 (with R. Vergnioux) ù Free unitary quantum group, {U`pF q§ 2015 ù Quantum permutation group, xS`N

Applications:§ K -amenability for {O`pF q, {U`pF q and xS`N§ Explicit K -theory computations for O`pF q, U`pF q, SUqp2q and S`N

§ e.g. K0pCpS`N qq “ ZN2´2N`2 and K1pCpS`N qq “ Z§ Classification of the C˚-algebras defining quantum permutation

groupsCr pS`N q – Cr pS`M q ô N “ M



Main results








Main results








What can we do?

No results (beyond the free quantum group case)for general constructions of quantum groups

ó

Stability properties



Relevant constructions

Duals of classical compact groups

§ already studied by R. Meyer and R. Nest for connected groups§ duals of q-deformations of compact connected semi-simple Lie

groups? ù recent work of A. Monk and C. Voigt

Discrete quantum subgroups

§ true for divisible discrete quantum subgroups

Quantum direct products

§ connection with the Kunneth formula

Quantum semi-direct products

§ compact bicrossed product ÐÑ torsion-freeness assumption

Quantum free products

§ already studied by R. Vergnioux and C. Voigt

Free wreath products

§ specially interesting because of its torsion phenomena




Duals of classical compact groups§ already studied by R. Meyer and R. Nest for connected groups§ duals of q-deformations of compact connected semi-simple Lie








Quantum free products

§ already studied by R. Vergnioux and C. Voigt














Quantum free products§ already studied by R. Vergnioux and C. Voigt








Discrete quantum subgroups§ true for divisible discrete quantum subgroups














Quantum direct products§ connection with the Kunneth formula













Quantum semi-direct products§ compact bicrossed product ÐÑ torsion-freeness assumption











Quantum semi-direct products§ compact bicrossed product ÐÑ torsion-freeness assumption


Free wreath products§ specially interesting because of its torsion phenomena



Assumption

From now on, all discrete quantum groupsare supposed to be torsion-free



Semi-direct products: the classical case

The Baum-Connes conjecture for semi-direct products (J. Chabertand S. Echterhoff, 2001)

Let F :“ Γ˙ G be a semi-direct product of locally compact groups.Assume that for every compact subgroup Λ ă Γ, FΛ :“ Λ˙ Gsatisfies the Baum-Connes conjecture.

F satisfies the Baum-Connes conjecture with coefficients in A if and onlyif Γ satisfies the Baum-Connes conjecture with coefficients in G ˙

rA.



Semi-direct products: the classical case

The Baum-Connes conjecture for semi-direct products (H.Oyono-Oyono, 2001)

Let F :“ Γ˙ G be a semi-direct product of discrete groups.Assume that for every finite subgroup Λ ă Γ, FΛ :“ Λ˙ G satisfiesthe Baum-Connes conjecture.

F satisfies the Baum-Connes conjecture with coefficients in A if and onlyif Γ satisfies the Baum-Connes conjecture with coefficients in G ˙

rA.



Semi-direct products: the quantum case

Goal: generalization for a quantum semi-direct product, F :“ Γ˙αG

Strategy: translate the Chabert-Echterhoff’s arguments using theMeyer-Nest machinery




Starting point: we consider the functor

Z : K KpF ÝÑ K K Γ

A ÞÝÑ ZpAq :“ pG˙r

A

§ Z triangulated§ Z is such that ZpL

pFq Ă LΓ




Main Lemma

For every pF-C˚-algebra A there exists an element

ψ P KK ΓppG˙r

LpAq, L1ppG˙r

Aqq

such that the following diagram is commutative

LF pAq

ηpFA

��

Ψ // LF 1ppG˙r

Aq

ηΓpG˙

rA

��F pAq F 1ppG˙

rAq

where Ψ :“ F 1pψq




Put T :“ pG˙r

LpAq

A P ObjpK KpFq ñ ΣpNpAqq ÝÑ LpAq u

ÝÑ A ÝÑ NpAqpG˙

rA P ObjpK K Γq ñ

ΣpN 1pG˙r

Aqq ÝÑ L1ppG˙r

Aq u1ÝÑ pG˙

rA ÝÑ N 1ppG˙

rAq

Since ZpLpFq Ă LΓ, then T P LΓ

pLΓ,NΓq is a complementary pair ñ LΓ Ă N $Γ ñ

pu1q˚ : KK ΓpT , L1pG˙r Aqq ÝÑ KK ΓpT , G˙r Aq

is an isomorphismTake ψ :“ pu1q´1

˚ pZpuqqThe Baum-Connes conjecture for Quantum Groups Ruben Martos



Put T :“ pG˙r

LpAq



rA P ObjpK K Γq ñ

ΣpN 1pG˙r

Aqq ÝÑ L1ppG˙r

Aq u1ÝÑ pG˙

rA ÝÑ N 1ppG˙

rAq








Put T :“ pG˙r

LpAq



rA P ObjpK K Γq ñ

ΣpN 1pG˙r

Aqq ÝÑ L1ppG˙r

Aq u1ÝÑ pG˙

rA ÝÑ N 1ppG˙

rAq








Theorem (R.M., 2016)Let F :“ Γ˙

αG be a quantum semi-direct product

Assume that pF is torsion-freeLet A be any pF-C˚-algebra

The following assertions are equivalenti) pF satisfies the Baum-Connes conjecture with coefficients in A.ii) pG satisfies the Baum-Connes conjecture with coefficients in A and Γ

satisfies the Baum-Connes conjecture with coefficients in pG˙r

A.



Wreath products: the classical case

The Haagerup property for wreath products (Y. Cornulier, Y.Stalder, A. Valette, 2012)

Let F :“ G o H be a wreath product of countable groups

If G and H have the Haagerup property, then F has the Haagerupproperty.

Applications:§ F satisfies the strong Baum-Connes conjecture (by Higson-Kasparov)§ Explicit K -theory computations in concrete examples

§ lamplighter group (R. Flores, S. Pooya, A. Valette, 2016)§ G o Fn with G finite (S. Pooya, 2017)



Wreath products: the quantum case

Goal: generalization for a free wreath product, F :“ G o˚ S`NStrategy:

i) Study the torsion phenomena of G o˚ S`N , which suggests a suitabledefinition of L

{Go˚S`Nii) Re-define the strong Baum-Connes property by requiring the abstract

condition L{Go˚S`N

“ K K{Go˚S`N




Alternative picture of free wreath products (F. Lemeux and P.Tarrago, 2014)

Let F :“ G o˚ S`N be a free wreath productDenote by pHq the discrete quantum subgroup of {G ˚ SUqp2qgenerated by uxu with x P IrrpGq

There exists 0 ă |q| ă 1 with q ` q´1 “?

N such that Hq is monoidallyequivalent to G o˚ S`N

Accordingly, we can replace G o˚ S`N by Hq for all results concerningthe torsion phenomena or the strong Baum-Connes conjecture



How to handle the torsion phenomena?

Definition of torsion (a la Meyer-Nest)

A discrete quantum group pG is called torsion-free if any torsion action ofG is G-equivariantly Morita equivalent to the trivial G-C˚-algebra C.TorppGq :“ set of all torsion actions of G

Terminology and notation:§ Action of G on a C˚-algebra A: ˚-homomorphismδ : A ÝÑ MpAb CpGqq compatible with ∆

§ Torsion action of G: A is finite dimensional and δ is ergodic

Exemples:§ Consistent with torsion for classical discrete groups§ pH ă pG finite discrete quantum subgroup§ G classical compact group, TorppGq ú topology of G



How to handle the torsion phenomena?

Definition of torsion (a la Meyer-Nest)

A discrete quantum group pG is called torsion-free if any torsion action ofG is G-equivariantly Morita equivalent to the trivial G-C˚-algebra C.TorppGq :“ set of all torsion actions of G

Terminology and notation:§ Action of G on a C˚-algebra A: ˚-homomorphismδ : A ÝÑ MpAb CpGqq compatible with ∆

§ Torsion action of G: A is finite dimensional and δ is ergodic

Exemples:§ Consistent with torsion for classical discrete groups§ pH ă pG finite discrete quantum subgroup§ G classical compact group, TorppGq ú topology of G




Theorem (A. Freslon and R. M., 2017)

Let F :“ G o˚ S`N be a free wreath productAssume that pG is torsion-freeAssume that pG satisfies the strong Baum-Connes property

The following properties holdi) pF is never torsion-free and pCN , αNq is the only (up to Morita

equivalence) non-trivial torsion action of G o˚ S`N .ii) If we put

L{Go˚S`N

:“ xtpG o˚ S`N ˙r T q b B, T P Tor , B P ObjpK K quy

then {G o˚ S`N satisfies the strong Baum-Connes conjecture.




i) Sketch of the non-torsion-freeness.§ Replacement G o˚ S`N ú Hq§ Classification of torsion actions for a free product {G ˚H§

#

Non-trivial torsionactions of G o˚ S`N

+

ÐÑ

#

All torsionactions of G

+

ii) Sketch of the strong Baum-Connes property.§ Replacement G o˚ S`N ú Hq

§ {G ˚ SUqp2q is torsion-free and K K{G˚SUqp2qq “ L

{G˚SUqp2q

§ B P ObjpK KpHq q ñ Ind

{G˚SUqp2qpHq

pBq P L{G˚SUqp2q

§ Explicit description of T P TorppHqq ` definition of LpHqñ

Res{G˚SUqp2q

pHqpL

{G˚SUqp2qq Ă L

pHq

§ Res{G˚SUqp2q

pHq

´

Ind{G˚SUqp2q

pHqpBq

¯

P LpHqñ B P L

pHq




i) Sketch of the non-torsion-freeness.§ Replacement G o˚ S`N ú Hq§ Classification of torsion actions for a free product {G ˚H§

#

Non-trivial torsionactions of G o˚ S`N

+

ÐÑ

#

All torsionactions of G

+

ii) Sketch of the strong Baum-Connes property.§ Replacement G o˚ S`N ú Hq

§ {G ˚ SUqp2q is torsion-free and K K{G˚SUqp2qq “ L

{G˚SUqp2q

§ B P ObjpK KpHq q ñ Ind

{G˚SUqp2qpHq

pBq P L{G˚SUqp2q

§ Explicit description of T P TorppHqq ` definition of LpHqñ

Res{G˚SUqp2q

pHqpL

{G˚SUqp2qq Ă L

pHq

§ Res{G˚SUqp2q

pHq

´

Ind{G˚SUqp2q

pHqpBq

¯

P LpHqñ B P L

pHq


An application: some K -theorycomputations

General strategy

Torsion-free case

“Algorithm”:1. pG satisfies the strong Baum-Connes conjecture2. pG torsion-free ` strong BC ñ pG is K -amenable3. Construction of a 1-length projective resolution for C in K K

pG

0 ÝÑ P1δ1ÝÑ P0

δ0ÝÑ C ÝÑ 0

4. Meyer-Nest’s homological algebra ` Bott periodicity ñ six-termexact sequence

K0ppG˙ P1qK0ppG˙ δ1q // K0ppG˙ P0q

K0ppG˙ δ0q // K0pCpGqq

��K1pCpGqq

OO

K1ppG˙ P0qK1ppG˙ δ0q

oo K1ppG˙ P1qK1ppG˙ δ1q

oo


General strategy

Torsion-case

“Algorithm”:1. Classification of TorppGq2. Re-definition of the subcategory L

pG,

LpG :“ xtG˙ T b B with T P TorppGq and B P ObjpK K quy

3. pG satisfies the strong Baum-Connes conjecture: LpG “ K K

pG

4. Definition of LpG ` strong BC ñ pG is K -amenable

5. Construction of a 1-length projective resolution for C in K KpG

taking into account TorppGq6. Classification of TorppGq ` Meyer-Nest’s homological algebra ` Bott

periodicity ñ six-term exact sequence


General strategy

Torsion-case


pG,



pG






General strategy

Torsion-case


pG,



pG






General strategy

Torsion-case


pG,



pG







Project description

Ideal Goal: compute K0`

CpG o˚ S`N q˘

and K1`

CpG o˚ S`N q˘

Problem: no 1-length resolution found ñ spectral sequences?Effective Goal: compute K0

`

CpHqq˘

and K1`

CpHqq˘

Strategy: pHq ă {G ˚ SUqp2q ñ restriction technique inspired by theworks of R. Vergnioux and C. Voigt

§ G :“ O`pnq§ G :“ U`1 ˚ . . . ˚ U`k ˚ O`1 ˚ . . . ˚ O`l§ G :“ Fn



Project description


CpG o˚ S`N q˘

and K1`

CpG o˚ S`N q˘


`

CpHqq˘

and K1`

CpHqq˘





Project description


CpG o˚ S`N q˘

and K1`

CpG o˚ S`N q˘


`

CpHqq˘

and K1`

CpHqq˘





After Z-linear algebra computations...

Theorem (A. Freslon and R. M., 2017)

Let n ě 2 and N ě 4 be natural numbers. Let O`pnq o˚ S`N be the freewreath product of the free orthogonal quantum group O`pnq by S`N . IfHq denotes the corresponding Lemeux-Tarrago’s compact quantum groupwhich is monoidally equivalent to O`pnq o˚ S`N , then

K0pCpHqqq “ Z‘ Z2 and K1pCpHqqq “ Z2



Preliminaries computations

pHq ă {G ˚ SUqp2q, put F :“ G ˚ SUqp2qDescription of c0ppFq as pHq-C˚-algebra: combinatorics of ReppFq `classification of TorppHqq ñ

RespFpHq

`

c0ppFq˘

“ Hq ˙ T ‘´

à

Nc0ppHqq

¯

,

where T P TorppHqq and T „M

C

Then K0`

pHq ˙ c0ppFq˘

“ Z‘´

À

NZ¯

and K1`

pHq ˙ c0ppFq˘

“ p0q



G :“ O`pnq computations

F :“ O`pnq ˚ SUqp2qChoice of the 1-length projective resolution for C in K K

pF: work ofR. Vergnioux and C. Voigt

P1 :“ c0ppFq ‘ c0ppFq and P0 :“ c0ppFq

δ1 :“ pTu ´ dimpuqidq ‘ pTv ´ dimpvqidq and δ0 :“ pλ

Restriction to K KpHq ñ Six-term exact sequence

´

Z‘´

À

NZ¯¯‘2 du ‘ dv // Z‘

´

À

NZ¯

// K0pCpHqqq

��K1pCpHqqq

OO

0oo 0oo

where du :“ Bu ´ 2 id and dv :“ Bv ´ n idThe Baum-Connes conjecture for Quantum Groups Ruben Martos

Conclusion: summary and openquestions

Conclusion: summary and open questions

(Strong) Baum-Connes conjecture for general constructions of Q. G.§ examples of Q.G. not satisfying the B.C. conjecture?§ improve the study of the quantum direct product§ complete the study for compact bicrossed products



Torsion phenomena for general constructions of Q. G.§ problematic not completely explored yet§ is torsion-freeness preserved under divisible discrete quantum

subgroups?



K -theory computations§ concrete examples of quantum semi-direct products introduced by S.

WangH ˙ G , Opnq ˙ O`pnq, Upnq ˙ U`pnq

§ deep study of the free wreath product



Formulation of a quantum B.C. conjecture§ problematic case: torsion different from finite discrete quantum

subgroups§ is L

pG such that pLpG,L

$

pGq is a complementary pair in K K

pG ?



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The Baum-Connes conjecture for Quantum Groupsruben.martos/doc/thesispresentation.pdf · The Baum-Connes conjecture for Quantum Groups Stability properties and K-theory computations