Towards an analogue of the Baum-Connesconjecture for quantum groups

Christian Voigt

Westfalische Wilhelms-Universitat Munstercvoigt@math.uni-muenster.de

http://wwwmath.uni-muenster.de/reine/u/cvoigt/

VanderbiltMay 14, 2007

Christian Voigt

The Baum-Connes conjecture

Let G be a second countable locally compact group. TheBaum-Connes conjecture asserts that the assembly map

µ : K top∗ (G ) = KG

∗ (EG ) → K∗(C∗red(G ))

is an isomorphism.Here EG is the universal proper G -space.

More generally, the Baum-Connes conjecture with coefficientsstates that

µ : K top∗ (G ;A) → K∗(G nred A)

is an isomorphism for every G -C ∗-algebra A.Here G nred A is the reduced crossed product of A by G .

What happens if G is a locally compact quantum group?

Christian Voigt

The Baum-Connes conjecture

Let G be a second countable locally compact group. TheBaum-Connes conjecture asserts that the assembly map

µ : K top∗ (G ) = KG

∗ (EG ) → K∗(C∗red(G ))

is an isomorphism.Here EG is the universal proper G -space.

More generally, the Baum-Connes conjecture with coefficientsstates that

µ : K top∗ (G ;A) → K∗(G nred A)

is an isomorphism for every G -C ∗-algebra A.Here G nred A is the reduced crossed product of A by G .

What happens if G is a locally compact quantum group?

Christian Voigt

The Baum-Connes conjecture

Let G be a second countable locally compact group. TheBaum-Connes conjecture asserts that the assembly map

µ : K top∗ (G ) = KG

∗ (EG ) → K∗(C∗red(G ))

is an isomorphism.Here EG is the universal proper G -space.

More generally, the Baum-Connes conjecture with coefficientsstates that

µ : K top∗ (G ;A) → K∗(G nred A)

is an isomorphism for every G -C ∗-algebra A.Here G nred A is the reduced crossed product of A by G .

What happens if G is a locally compact quantum group?

Christian Voigt

The Baum-Connes conjecture

Let G be a second countable locally compact group. TheBaum-Connes conjecture asserts that the assembly map

µ : K top∗ (G ) = KG

∗ (EG ) → K∗(C∗red(G ))

is an isomorphism.Here EG is the universal proper G -space.

More generally, the Baum-Connes conjecture with coefficientsstates that

µ : K top∗ (G ;A) → K∗(G nred A)

is an isomorphism for every G -C ∗-algebra A.Here G nred A is the reduced crossed product of A by G .

What happens if G is a locally compact quantum group?

Christian Voigt

Basic definitions

DefinitionLet G be a locally compact group. A G-C ∗-algebra is a C ∗-algebraA with a strongly continuous action of G by ∗-automorphisms.A ∗-homomorphism f : A → B is equivariant if f (t · a) = t · f (a)for all t ∈ G , a ∈ A.

Let H ⊂ G be a closed subgroup.

I If A is a G -C ∗-algebra then A = resGH(A) becomes anH-C ∗-algebra by restriction of the action.

I If A is an H-C ∗-algebra then the induced (G-C ∗-) algebra is

indGH(B) = {f ∈ Cb(G ,B)|f (xh) = h−1 · f (x),

xH 7→ ||f (x)|| ∈ C0(G/H)}

Christian Voigt

Basic definitions

DefinitionLet G be a locally compact group. A G-C ∗-algebra is a C ∗-algebraA with a strongly continuous action of G by ∗-automorphisms.A ∗-homomorphism f : A → B is equivariant if f (t · a) = t · f (a)for all t ∈ G , a ∈ A.

Let H ⊂ G be a closed subgroup.

I If A is a G -C ∗-algebra then A = resGH(A) becomes anH-C ∗-algebra by restriction of the action.

I If A is an H-C ∗-algebra then the induced (G-C ∗-) algebra is

indGH(B) = {f ∈ Cb(G ,B)|f (xh) = h−1 · f (x),

xH 7→ ||f (x)|| ∈ C0(G/H)}

Christian Voigt

Basic definitions

DefinitionLet G be a locally compact group. A G-C ∗-algebra is a C ∗-algebraA with a strongly continuous action of G by ∗-automorphisms.A ∗-homomorphism f : A → B is equivariant if f (t · a) = t · f (a)for all t ∈ G , a ∈ A.

Let H ⊂ G be a closed subgroup.

I If A is a G -C ∗-algebra then A = resGH(A) becomes anH-C ∗-algebra by restriction of the action.

I If A is an H-C ∗-algebra then the induced (G-C ∗-) algebra is

indGH(B) = {f ∈ Cb(G ,B)|f (xh) = h−1 · f (x),

xH 7→ ||f (x)|| ∈ C0(G/H)}

Christian Voigt

Basic definitions

DefinitionLet G be a locally compact group. A G-C ∗-algebra is a C ∗-algebraA with a strongly continuous action of G by ∗-automorphisms.A ∗-homomorphism f : A → B is equivariant if f (t · a) = t · f (a)for all t ∈ G , a ∈ A.

Let H ⊂ G be a closed subgroup.

I If A is a G -C ∗-algebra then A = resGH(A) becomes anH-C ∗-algebra by restriction of the action.

I If A is an H-C ∗-algebra then the induced (G-C ∗-) algebra is

indGH(B) = {f ∈ Cb(G ,B)|f (xh) = h−1 · f (x),

xH 7→ ||f (x)|| ∈ C0(G/H)}

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

Equivariant Kasparov theory yields a category KKG

I objects in KKG are all separable G -C ∗-algebras.

I morphism sets are the bivariant Kasparov K -groupsKKG (A,B), and composition of morphisms is given byKasparov product.

In fact, the category KKG is triangulated - this allows to dohomological algebra.

A basic example of a triangulated category to have in mind is thehomotopy category of chain complexes CH(R) of R-modules over aring R.

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

Equivariant Kasparov theory yields a category KKG

I objects in KKG are all separable G -C ∗-algebras.

I morphism sets are the bivariant Kasparov K -groupsKKG (A,B), and composition of morphisms is given byKasparov product.

In fact, the category KKG is triangulated - this allows to dohomological algebra.

A basic example of a triangulated category to have in mind is thehomotopy category of chain complexes CH(R) of R-modules over aring R.

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

Equivariant Kasparov theory yields a category KKG

I objects in KKG are all separable G -C ∗-algebras.

I morphism sets are the bivariant Kasparov K -groupsKKG (A,B), and composition of morphisms is given byKasparov product.

In fact, the category KKG is triangulated - this allows to dohomological algebra.

A basic example of a triangulated category to have in mind is thehomotopy category of chain complexes CH(R) of R-modules over aring R.

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

Equivariant Kasparov theory yields a category KKG

I objects in KKG are all separable G -C ∗-algebras.

I morphism sets are the bivariant Kasparov K -groupsKKG (A,B), and composition of morphisms is given byKasparov product.

In fact, the category KKG is triangulated - this allows to dohomological algebra.

A basic example of a triangulated category to have in mind is thehomotopy category of chain complexes CH(R) of R-modules over aring R.

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

Equivariant Kasparov theory yields a category KKG

I objects in KKG are all separable G -C ∗-algebras.

I morphism sets are the bivariant Kasparov K -groupsKKG (A,B), and composition of morphisms is given byKasparov product.

In fact, the category KKG is triangulated - this allows to dohomological algebra.

A basic example of a triangulated category to have in mind is thehomotopy category of chain complexes CH(R) of R-modules over aring R.

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

Equivariant Kasparov theory yields a category KKG

I objects in KKG are all separable G -C ∗-algebras.

I morphism sets are the bivariant Kasparov K -groupsKKG (A,B), and composition of morphisms is given byKasparov product.

In fact, the category KKG is triangulated - this allows to dohomological algebra.

A basic example of a triangulated category to have in mind is thehomotopy category of chain complexes CH(R) of R-modules over aring R.

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

The triangulated structure on KKG is given by the following data.

I The (inverse of the) suspension Σ(A) = C0(R)⊗ A yields thetranslation functor.

I Distinguished triangles are all triangles isomorphic to mappingcone triangles

Σ(B) → Cf → A → B

for equivariant ∗-homomorphisms f : A → B.

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

The triangulated structure on KKG is given by the following data.

I The (inverse of the) suspension Σ(A) = C0(R)⊗ A yields thetranslation functor.

I Distinguished triangles are all triangles isomorphic to mappingcone triangles

Σ(B) → Cf → A → B

for equivariant ∗-homomorphisms f : A → B.

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

The triangulated structure on KKG is given by the following data.

I The (inverse of the) suspension Σ(A) = C0(R)⊗ A yields thetranslation functor.

I Distinguished triangles are all triangles isomorphic to mappingcone triangles

Σ(B) → Cf → A → B

for equivariant ∗-homomorphisms f : A → B.

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

A G -C ∗-algebra is called compactly induced if it is of the formindG

H(B) for a compact subgroup H ⊂ G .

(corresponds to a projective chain complex in CH(R))

A G -C ∗-algebra A is called weakly contractible ifresGH(A) ∼= 0 ∈ KKH for every compact subgroup H ⊂ G .

(corresponds to an exact chain complex in CH(R))

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

A G -C ∗-algebra is called compactly induced if it is of the formindG

H(B) for a compact subgroup H ⊂ G .

(corresponds to a projective chain complex in CH(R))

A G -C ∗-algebra A is called weakly contractible ifresGH(A) ∼= 0 ∈ KKH for every compact subgroup H ⊂ G .

(corresponds to an exact chain complex in CH(R))

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

A G -C ∗-algebra is called compactly induced if it is of the formindG

H(B) for a compact subgroup H ⊂ G .

(corresponds to a projective chain complex in CH(R))

A G -C ∗-algebra A is called weakly contractible ifresGH(A) ∼= 0 ∈ KKH for every compact subgroup H ⊂ G .

(corresponds to an exact chain complex in CH(R))

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

A G -C ∗-algebra is called compactly induced if it is of the formindG

H(B) for a compact subgroup H ⊂ G .

(corresponds to a projective chain complex in CH(R))

A G -C ∗-algebra A is called weakly contractible ifresGH(A) ∼= 0 ∈ KKH for every compact subgroup H ⊂ G .

(corresponds to an exact chain complex in CH(R))

Christian Voigt

Reformulation of the conjecture by Meyer-Nest

A G -C ∗-algebra is called compactly induced if it is of the formindG

H(B) for a compact subgroup H ⊂ G .

(corresponds to a projective chain complex in CH(R))

A G -C ∗-algebra A is called weakly contractible ifresGH(A) ∼= 0 ∈ KKH for every compact subgroup H ⊂ G .

(corresponds to an exact chain complex in CH(R))

Christian Voigt

Reformulation by Meyer-Nest

Let 〈CI〉 be the localising subcategory of KKG generated by allcompactly induced algebras.

A 〈CI〉-simplicial approximation of a G -C ∗-algebra A is a weakequivalence A → A with A ∈ 〈CI〉.

TheoremFor every A ∈ KKG there exists a 〈CI〉-simplicial approximation Awhich is unique up to isomorphism.

TheoremFor the functor F (A) = K∗(G nred A) the transformation

LF (A) = K∗(G nred A) → K∗(G nred A) = F (A)

is isomorphic to the Baum-Connes assembly map.

Christian Voigt

Reformulation by Meyer-Nest

Let 〈CI〉 be the localising subcategory of KKG generated by allcompactly induced algebras.

A 〈CI〉-simplicial approximation of a G -C ∗-algebra A is a weakequivalence A → A with A ∈ 〈CI〉.

TheoremFor every A ∈ KKG there exists a 〈CI〉-simplicial approximation Awhich is unique up to isomorphism.

TheoremFor the functor F (A) = K∗(G nred A) the transformation

LF (A) = K∗(G nred A) → K∗(G nred A) = F (A)

is isomorphic to the Baum-Connes assembly map.

Christian Voigt

Reformulation by Meyer-Nest

Let 〈CI〉 be the localising subcategory of KKG generated by allcompactly induced algebras.

A 〈CI〉-simplicial approximation of a G -C ∗-algebra A is a weakequivalence A → A with A ∈ 〈CI〉.

TheoremFor every A ∈ KKG there exists a 〈CI〉-simplicial approximation Awhich is unique up to isomorphism.

TheoremFor the functor F (A) = K∗(G nred A) the transformation

LF (A) = K∗(G nred A) → K∗(G nred A) = F (A)

is isomorphic to the Baum-Connes assembly map.

Christian Voigt

Reformulation by Meyer-Nest

Let 〈CI〉 be the localising subcategory of KKG generated by allcompactly induced algebras.

A 〈CI〉-simplicial approximation of a G -C ∗-algebra A is a weakequivalence A → A with A ∈ 〈CI〉.

TheoremFor every A ∈ KKG there exists a 〈CI〉-simplicial approximation Awhich is unique up to isomorphism.

TheoremFor the functor F (A) = K∗(G nred A) the transformation

LF (A) = K∗(G nred A) → K∗(G nred A) = F (A)

is isomorphic to the Baum-Connes assembly map.

Christian Voigt

Locally compact quantum groups

DefinitionA Hopf-C ∗-algebra is a C ∗-algebra H together with anondegenerate injective ∗-homomorphism ∆ : H → M(H ⊗ H)such that

H∆ //

��

M(H ⊗ H)

id⊗∆��

M(H ⊗ H)∆⊗id

// M(H ⊗ H ⊗ H)

is commutative and ∆(H)(1⊗ H) and (H ⊗ 1)∆(H) are densesubspaces of H ⊗ H.

A locally compact quantum group is given by a Hopf-C ∗-algebra Htogether with left and right Haar integrals.

Christian Voigt

Locally compact quantum groups

DefinitionA Hopf-C ∗-algebra is a C ∗-algebra H together with anondegenerate injective ∗-homomorphism ∆ : H → M(H ⊗ H)such that

H∆ //

��

M(H ⊗ H)

id⊗∆��

M(H ⊗ H)∆⊗id

// M(H ⊗ H ⊗ H)

is commutative and ∆(H)(1⊗ H) and (H ⊗ 1)∆(H) are densesubspaces of H ⊗ H.

A locally compact quantum group is given by a Hopf-C ∗-algebra Htogether with left and right Haar integrals.

Christian Voigt

Locally compact quantum groups

DefinitionA Hopf-C ∗-algebra is a C ∗-algebra H together with anondegenerate injective ∗-homomorphism ∆ : H → M(H ⊗ H)such that

H∆ //

��

M(H ⊗ H)

id⊗∆��

M(H ⊗ H)∆⊗id

// M(H ⊗ H ⊗ H)

is commutative and ∆(H)(1⊗ H) and (H ⊗ 1)∆(H) are densesubspaces of H ⊗ H.

A locally compact quantum group is given by a Hopf-C ∗-algebra Htogether with left and right Haar integrals.

Christian Voigt

Locally compact quantum groups

Examples

If G is a locally compact group then H = C0(G ) defines a locallycompact quantum group.The comultiplication ∆ : C0(G ) → Cb(G × G ) is given by

∆(f )(s, t) = f (st),

and the integrals are given by left/right Haar measure.

Christian Voigt

Locally compact quantum groups

Examples

If G is a locally compact group then H = C0(G ) defines a locallycompact quantum group.The comultiplication ∆ : C0(G ) → Cb(G × G ) is given by

∆(f )(s, t) = f (st),

and the integrals are given by left/right Haar measure.

Christian Voigt

Locally compact quantum groups

In general we will write H = C red0 (G ) for a locally compact

quantum group.We think of C red

0 (G ) as ” the algebra of functions” on the(imaginary) quantum group G .

For every locally compact quantum group G there exists a duallocally compact quantum group G given by

H = C ∗red(G ) = C red0 (G )

and the Pontrjagin duality theorem holds.

Every locally compact quantum group comes equipped with aHilbert space HG (the GNS-space of the left Haar weight) and amultiplicative unitary W ∈ M(C red

0 (G )⊗ C ∗red(G )).

In the sequel all locally compact quantum groups are assumed tobe strongly regular.

Christian Voigt

Locally compact quantum groups

In general we will write H = C red0 (G ) for a locally compact

quantum group.We think of C red

0 (G ) as ” the algebra of functions” on the(imaginary) quantum group G .

For every locally compact quantum group G there exists a duallocally compact quantum group G given by

H = C ∗red(G ) = C red0 (G )

and the Pontrjagin duality theorem holds.

Every locally compact quantum group comes equipped with aHilbert space HG (the GNS-space of the left Haar weight) and amultiplicative unitary W ∈ M(C red

0 (G )⊗ C ∗red(G )).

In the sequel all locally compact quantum groups are assumed tobe strongly regular.

Christian Voigt

Locally compact quantum groups

In general we will write H = C red0 (G ) for a locally compact

quantum group.We think of C red

0 (G ) as ” the algebra of functions” on the(imaginary) quantum group G .

For every locally compact quantum group G there exists a duallocally compact quantum group G given by

H = C ∗red(G ) = C red0 (G )

and the Pontrjagin duality theorem holds.

Every locally compact quantum group comes equipped with aHilbert space HG (the GNS-space of the left Haar weight) and amultiplicative unitary W ∈ M(C red

0 (G )⊗ C ∗red(G )).

In the sequel all locally compact quantum groups are assumed tobe strongly regular.

Christian Voigt

Locally compact quantum groups

In general we will write H = C red0 (G ) for a locally compact

quantum group.We think of C red

0 (G ) as ” the algebra of functions” on the(imaginary) quantum group G .

For every locally compact quantum group G there exists a duallocally compact quantum group G given by

H = C ∗red(G ) = C red0 (G )

and the Pontrjagin duality theorem holds.

Every locally compact quantum group comes equipped with aHilbert space HG (the GNS-space of the left Haar weight) and amultiplicative unitary W ∈ M(C red

0 (G )⊗ C ∗red(G )).

In the sequel all locally compact quantum groups are assumed tobe strongly regular.

Christian Voigt

Locally compact quantum groups

In general we will write H = C red0 (G ) for a locally compact

quantum group.We think of C red

0 (G ) as ” the algebra of functions” on the(imaginary) quantum group G .

For every locally compact quantum group G there exists a duallocally compact quantum group G given by

H = C ∗red(G ) = C red0 (G )

and the Pontrjagin duality theorem holds.

Every locally compact quantum group comes equipped with aHilbert space HG (the GNS-space of the left Haar weight) and amultiplicative unitary W ∈ M(C red

0 (G )⊗ C ∗red(G )).

In the sequel all locally compact quantum groups are assumed tobe strongly regular.

Christian Voigt

Actions

DefinitionA (left) coaction of a Hopf C ∗-algebra H on a C ∗-algebra A is aninjective nondegenerate ∗-homomorphism α : A → M(H ⊗ A) suchthat the diagram

Aα //

α

��

M(H ⊗ A)

∆⊗id��

M(H ⊗ A)id⊗α

// M(H ⊗ H ⊗ A)

is commutative and α(A)(H ⊗ 1) ⊂ H ⊗ A is dense.

Christian Voigt

Actions

DefinitionA (left) coaction of a Hopf C ∗-algebra H on a C ∗-algebra A is aninjective nondegenerate ∗-homomorphism α : A → M(H ⊗ A) suchthat the diagram

Aα //

α

��

M(H ⊗ A)

∆⊗id��

M(H ⊗ A)id⊗α

// M(H ⊗ H ⊗ A)

is commutative and α(A)(H ⊗ 1) ⊂ H ⊗ A is dense.

Christian Voigt

The Kasparov category

DefinitionLet G be a locally compact quantum group. A G -C ∗-algebra is aC ∗-algebra A with a coaction of C red

0 (G ).

For locally compact groups this recovers the usual definition.

Baaj and Skandalis defined KKG for quantum groups. As in thegroup case one obtains a triangulated category with objects theseparable G -C ∗-algebras and morphisms given by equivariantKasparov groups.

Christian Voigt

The Kasparov category

DefinitionLet G be a locally compact quantum group. A G -C ∗-algebra is aC ∗-algebra A with a coaction of C red

0 (G ).

For locally compact groups this recovers the usual definition.

Baaj and Skandalis defined KKG for quantum groups. As in thegroup case one obtains a triangulated category with objects theseparable G -C ∗-algebras and morphisms given by equivariantKasparov groups.

Christian Voigt

The Kasparov category

DefinitionLet G be a locally compact quantum group. A G -C ∗-algebra is aC ∗-algebra A with a coaction of C red

0 (G ).

For locally compact groups this recovers the usual definition.

Baaj and Skandalis defined KKG for quantum groups. As in thegroup case one obtains a triangulated category with objects theseparable G -C ∗-algebras and morphisms given by equivariantKasparov groups.

Christian Voigt

Braided tensor products

DefinitionLet G be a locally compact quantum group and H = C red

0 (G ) andlet H = C ∗red(G ). A G-Yetter-Drinfeld algebra is a C ∗-algebra A

equipped with a coaction α of H and a coaction λ of H such thatthe diagram

Aλ //

α

��

M(H ⊗ A)id⊗α

// M(H ⊗ H ⊗ A)

σ⊗id��

M(H ⊗ A)id⊗λ

// M(H ⊗ H ⊗ A)ad(W )

// M(H ⊗ H ⊗ A)

is commutative.

Here σ : H ⊗ H → H ⊗ H is the flip map.

Christian Voigt

Braided tensor products

DefinitionLet G be a locally compact quantum group and H = C red

0 (G ) andlet H = C ∗red(G ). A G-Yetter-Drinfeld algebra is a C ∗-algebra A

equipped with a coaction α of H and a coaction λ of H such thatthe diagram

Aλ //

α

��

M(H ⊗ A)id⊗α

// M(H ⊗ H ⊗ A)

σ⊗id��

M(H ⊗ A)id⊗λ

// M(H ⊗ H ⊗ A)ad(W )

// M(H ⊗ H ⊗ A)

is commutative.

Here σ : H ⊗ H → H ⊗ H is the flip map.

Christian Voigt

Braided tensor products

Examples

I G ordinary locally compact group - then every G -C ∗-algebraA is a G -YD-algebra with the trivial coactionλ : A → M(C ∗red(G )⊗ A) given by

λ(a) = 1⊗ a.

I G discrete group - then coactions of C ∗red(G ) correspond toFell bundles. A YD-structure is equivalent to having aG -equivariant Fell bundle.

I in general - if H ⊂ G is a quantum subgroup then the inducedalgebra indG

H(A) of a H-YD-algebra A is a G -YD-algebra.

Christian Voigt

Braided tensor products

Examples

I G ordinary locally compact group - then every G -C ∗-algebraA is a G -YD-algebra with the trivial coactionλ : A → M(C ∗red(G )⊗ A) given by

λ(a) = 1⊗ a.

I G discrete group - then coactions of C ∗red(G ) correspond toFell bundles. A YD-structure is equivalent to having aG -equivariant Fell bundle.

I in general - if H ⊂ G is a quantum subgroup then the inducedalgebra indG

H(A) of a H-YD-algebra A is a G -YD-algebra.

Christian Voigt

Braided tensor products

Examples

I G ordinary locally compact group - then every G -C ∗-algebraA is a G -YD-algebra with the trivial coactionλ : A → M(C ∗red(G )⊗ A) given by

λ(a) = 1⊗ a.

I G discrete group - then coactions of C ∗red(G ) correspond toFell bundles. A YD-structure is equivalent to having aG -equivariant Fell bundle.

I in general - if H ⊂ G is a quantum subgroup then the inducedalgebra indG

H(A) of a H-YD-algebra A is a G -YD-algebra.

Christian Voigt

Braided tensor products

Examples

I G ordinary locally compact group - then every G -C ∗-algebraA is a G -YD-algebra with the trivial coactionλ : A → M(C ∗red(G )⊗ A) given by

λ(a) = 1⊗ a.

I G discrete group - then coactions of C ∗red(G ) correspond toFell bundles. A YD-structure is equivalent to having aG -equivariant Fell bundle.

I in general - if H ⊂ G is a quantum subgroup then the inducedalgebra indG

H(A) of a H-YD-algebra A is a G -YD-algebra.

Christian Voigt

Braided tensor products

Examples

I G ordinary locally compact group - then every G -C ∗-algebraA is a G -YD-algebra with the trivial coactionλ : A → M(C ∗red(G )⊗ A) given by

λ(a) = 1⊗ a.

I G discrete group - then coactions of C ∗red(G ) correspond toFell bundles. A YD-structure is equivalent to having aG -equivariant Fell bundle.

I in general - if H ⊂ G is a quantum subgroup then the inducedalgebra indG

H(A) of a H-YD-algebra A is a G -YD-algebra.

Christian Voigt

Braided tensor products and the Drinfeld double

DefinitionLet G be a locally compact quantum group. The Drinfeld doubleD(G ) is the locally compact quantum group given byC red

0 (D(G )) = C red0 (G )⊗ C ∗red(G ) with the comultiplication

∆D(G) = (id⊗σ ⊗ id)(id⊗ad(W )⊗ id)(∆⊗ ∆)

where ad(W )(x) = WxW ∗ for x ∈ C red0 (G )⊗ C ∗red(G ).

Proposition

A G-Yetter-Drinfeld algebra is the same thing as aD(G )-C ∗-algebra.

Christian Voigt

Braided tensor products and the Drinfeld double

DefinitionLet G be a locally compact quantum group. The Drinfeld doubleD(G ) is the locally compact quantum group given byC red

0 (D(G )) = C red0 (G )⊗ C ∗red(G ) with the comultiplication

∆D(G) = (id⊗σ ⊗ id)(id⊗ad(W )⊗ id)(∆⊗ ∆)

where ad(W )(x) = WxW ∗ for x ∈ C red0 (G )⊗ C ∗red(G ).

Proposition

A G-Yetter-Drinfeld algebra is the same thing as aD(G )-C ∗-algebra.

Christian Voigt

Braided tensor products and the Drinfeld double

DefinitionLet G be a locally compact quantum group. The Drinfeld doubleD(G ) is the locally compact quantum group given byC red

0 (D(G )) = C red0 (G )⊗ C ∗red(G ) with the comultiplication

∆D(G) = (id⊗σ ⊗ id)(id⊗ad(W )⊗ id)(∆⊗ ∆)

where ad(W )(x) = WxW ∗ for x ∈ C red0 (G )⊗ C ∗red(G ).

Proposition

A G-Yetter-Drinfeld algebra is the same thing as aD(G )-C ∗-algebra.

Christian Voigt

Braided tensor products

DefinitionLet A be a G -YD-algebra and let B be a G -algebra. The braidedtensor product is

A � B = [λ(A)12β(B)13] ⊂ L(HG ⊗ A⊗ B).

I A � B is a C ∗-algebra, and λ (resp. β) define injective∗-homomorphisms ιA : A → M(A � B) (resp.ιB : B → M(A � B)).

I There is a coaction A � B → M(C red0 (G )⊗ (A � B)) such

that ιA and ιB are equivariant.

I If B is a YD-algebra then A � B is a YD-algebra and

(A � B) � C ∼= A � (B � C )

for all G -algebras C .

Christian Voigt

Braided tensor products

DefinitionLet A be a G -YD-algebra and let B be a G -algebra. The braidedtensor product is

A � B = [λ(A)12β(B)13] ⊂ L(HG ⊗ A⊗ B).

I A � B is a C ∗-algebra, and λ (resp. β) define injective∗-homomorphisms ιA : A → M(A � B) (resp.ιB : B → M(A � B)).

I There is a coaction A � B → M(C red0 (G )⊗ (A � B)) such

that ιA and ιB are equivariant.

I If B is a YD-algebra then A � B is a YD-algebra and

(A � B) � C ∼= A � (B � C )

for all G -algebras C .

Christian Voigt

Braided tensor products

DefinitionLet A be a G -YD-algebra and let B be a G -algebra. The braidedtensor product is

A � B = [λ(A)12β(B)13] ⊂ L(HG ⊗ A⊗ B).

I A � B is a C ∗-algebra, and λ (resp. β) define injective∗-homomorphisms ιA : A → M(A � B) (resp.ιB : B → M(A � B)).

I There is a coaction A � B → M(C red0 (G )⊗ (A � B)) such

that ιA and ιB are equivariant.

I If B is a YD-algebra then A � B is a YD-algebra and

(A � B) � C ∼= A � (B � C )

for all G -algebras C .

Christian Voigt

Braided tensor products

DefinitionLet A be a G -YD-algebra and let B be a G -algebra. The braidedtensor product is

A � B = [λ(A)12β(B)13] ⊂ L(HG ⊗ A⊗ B).

I A � B is a C ∗-algebra, and λ (resp. β) define injective∗-homomorphisms ιA : A → M(A � B) (resp.ιB : B → M(A � B)).

I There is a coaction A � B → M(C red0 (G )⊗ (A � B)) such

that ιA and ιB are equivariant.

I If B is a YD-algebra then A � B is a YD-algebra and

(A � B) � C ∼= A � (B � C )

for all G -algebras C .

Christian Voigt

Braided tensor products

DefinitionLet A be a G -YD-algebra and let B be a G -algebra. The braidedtensor product is

A � B = [λ(A)12β(B)13] ⊂ L(HG ⊗ A⊗ B).

I A � B is a C ∗-algebra, and λ (resp. β) define injective∗-homomorphisms ιA : A → M(A � B) (resp.ιB : B → M(A � B)).

I There is a coaction A � B → M(C red0 (G )⊗ (A � B)) such

that ιA and ιB are equivariant.

I If B is a YD-algebra then A � B is a YD-algebra and

(A � B) � C ∼= A � (B � C )

for all G -algebras C .

Christian Voigt

Braided tensor products and Kasparov theory

TheoremLet A1,B1 and D be G-YD algebras and let A2,B2 be G-algebras.There is an exterior Kasparov product

KKD(G)∗ (A1,B1�D)×KKG

∗ (D �A2,B2) → KKG∗ (A1�A2,B1�B2)

which is functorial and associative.

Christian Voigt

Braided tensor products and Kasparov theory

TheoremLet A1,B1 and D be G-YD algebras and let A2,B2 be G-algebras.There is an exterior Kasparov product

KKD(G)∗ (A1,B1�D)×KKG

∗ (D �A2,B2) → KKG∗ (A1�A2,B1�B2)

which is functorial and associative.

Christian Voigt

The quantum group SUq(2)

DefinitionFix q ∈ (0, 1]. The unital ∗-algebra O(SUq(2)) (over C) isgenerated by elements α and γ satisfying the relations

αγ = qγα, αγ∗ = qγ∗α, γγ∗ = γ∗γ,

α∗α + γ∗γ = 1, αα∗ + q2γγ∗ = 1.

These relations are equivalent to saying that the fundamentalmatrix (

α −qγ∗

γ α∗

)is unitary.

Christian Voigt

The quantum group SUq(2)

DefinitionFix q ∈ (0, 1]. The unital ∗-algebra O(SUq(2)) (over C) isgenerated by elements α and γ satisfying the relations

αγ = qγα, αγ∗ = qγ∗α, γγ∗ = γ∗γ,

α∗α + γ∗γ = 1, αα∗ + q2γγ∗ = 1.

These relations are equivalent to saying that the fundamentalmatrix (

α −qγ∗

γ α∗

)is unitary.

Christian Voigt

The quantum group SUq(2)

DefinitionFix q ∈ (0, 1]. The unital ∗-algebra O(SUq(2)) (over C) isgenerated by elements α and γ satisfying the relations

αγ = qγα, αγ∗ = qγ∗α, γγ∗ = γ∗γ,

α∗α + γ∗γ = 1, αα∗ + q2γγ∗ = 1.

These relations are equivalent to saying that the fundamentalmatrix (

α −qγ∗

γ α∗

)is unitary.

Christian Voigt

The quantum group SUq(2)

The comultiplication ∆ : O(SUq(2)) → O(SUq(2))⊗O(SUq(2)) isdefined by

∆

(α −qγ∗

γ α∗

)=

(α −qγ∗

γ α∗

)⊗

(α −qγ∗

γ α∗

)

In fact, O(SUq(2)) is a Hopf-∗-algebra.

Christian Voigt

The quantum group SUq(2)

The comultiplication ∆ : O(SUq(2)) → O(SUq(2))⊗O(SUq(2)) isdefined by

∆

(α −qγ∗

γ α∗

)=

(α −qγ∗

γ α∗

)⊗

(α −qγ∗

γ α∗

)In fact, O(SUq(2)) is a Hopf-∗-algebra.

Christian Voigt

The quantum group SUq(2)

The ∗-algebra O(SUq(2)) can be completed uniquely to aC ∗-algebra C (SUq(2)). This yields a (locally) compact quantumgroup.

For q = 1 one obtains in this way the algebras O(SU(2)) andC (SU(2)) of polynomial and continuous functions on SU(2),respectively.

Christian Voigt

The quantum group SUq(2)

The ∗-algebra O(SUq(2)) can be completed uniquely to aC ∗-algebra C (SUq(2)). This yields a (locally) compact quantumgroup.

For q = 1 one obtains in this way the algebras O(SU(2)) andC (SU(2)) of polynomial and continuous functions on SU(2),respectively.

Christian Voigt

The Podles sphere

The maximal torus T = S1 ⊂ SUq(2) is given by the projectionπ : C (SUq(2)) → C (T ) ⊃ C[z , z−1] given by

π

(α −qγ∗

γ α∗

)=

(z 00 z−1

)

The (standard) Podles sphere is the homogenous space SUq(2)/Tgiven by the algebra of coinvariants

C (SUq(2)/T ) = {x ∈ C (SUq(2))|(id⊗π)∆(x) = x ⊗ 1}

under right translations.We remark that for q ∈ (0, 1) one has C (SUq(2)/T ) ∼= K+. Thereis an algebraic version O(SUq(2)/T ) as well.

Christian Voigt

The Podles sphere

The maximal torus T = S1 ⊂ SUq(2) is given by the projectionπ : C (SUq(2)) → C (T ) ⊃ C[z , z−1] given by

π

(α −qγ∗

γ α∗

)=

(z 00 z−1

)

The (standard) Podles sphere is the homogenous space SUq(2)/Tgiven by the algebra of coinvariants

C (SUq(2)/T ) = {x ∈ C (SUq(2))|(id⊗π)∆(x) = x ⊗ 1}

under right translations.We remark that for q ∈ (0, 1) one has C (SUq(2)/T ) ∼= K+. Thereis an algebraic version O(SUq(2)/T ) as well.

Christian Voigt

The Podles sphere

The maximal torus T = S1 ⊂ SUq(2) is given by the projectionπ : C (SUq(2)) → C (T ) ⊃ C[z , z−1] given by

π

(α −qγ∗

γ α∗

)=

(z 00 z−1

)

The (standard) Podles sphere is the homogenous space SUq(2)/Tgiven by the algebra of coinvariants

C (SUq(2)/T ) = {x ∈ C (SUq(2))|(id⊗π)∆(x) = x ⊗ 1}

under right translations.

We remark that for q ∈ (0, 1) one has C (SUq(2)/T ) ∼= K+. Thereis an algebraic version O(SUq(2)/T ) as well.

Christian Voigt

The Podles sphere

The maximal torus T = S1 ⊂ SUq(2) is given by the projectionπ : C (SUq(2)) → C (T ) ⊃ C[z , z−1] given by

π

(α −qγ∗

γ α∗

)=

(z 00 z−1

)

The (standard) Podles sphere is the homogenous space SUq(2)/Tgiven by the algebra of coinvariants

C (SUq(2)/T ) = {x ∈ C (SUq(2))|(id⊗π)∆(x) = x ⊗ 1}

under right translations.We remark that for q ∈ (0, 1) one has C (SUq(2)/T ) ∼= K+. Thereis an algebraic version O(SUq(2)/T ) as well.

Christian Voigt

The Baum-Connes conjecture

In the sequel we let q ∈ (0, 1] and write G = SUq(2) as well as Gfor its dual.

We shall formulate and prove an analogue of the Baum-Connesconjecture for the dual quantum group G of SUq(2).

...what is the Baum-Connes conjecture in this situation?

Christian Voigt

The Baum-Connes conjecture

In the sequel we let q ∈ (0, 1] and write G = SUq(2) as well as Gfor its dual.

We shall formulate and prove an analogue of the Baum-Connesconjecture for the dual quantum group G of SUq(2).

...what is the Baum-Connes conjecture in this situation?

Christian Voigt

The Baum-Connes conjecture

In the sequel we let q ∈ (0, 1] and write G = SUq(2) as well as Gfor its dual.

We shall formulate and prove an analogue of the Baum-Connesconjecture for the dual quantum group G of SUq(2).

...what is the Baum-Connes conjecture in this situation?

Christian Voigt

The Baum-Connes conjecture

In the sequel we let q ∈ (0, 1] and write G = SUq(2) as well as Gfor its dual.

We shall formulate and prove an analogue of the Baum-Connesconjecture for the dual quantum group G of SUq(2).

...what is the Baum-Connes conjecture in this situation?

Christian Voigt

The Baum-Connes conjecture

The discrete quantum group G is torsion-free.The proper homogeneous G -algebra corresponding to the trivialsubgroup is C ∗(G ) = C0(G ).

We write 〈CI〉 for the localizing subcategory of KK G generated byalgebras of the form C ∗(G )⊗ A where A is some C ∗-algebra andthe coaction is inherited from C ∗(G ).

TheoremOne has 〈CI〉 = KK G .

Christian Voigt

The Baum-Connes conjecture

The discrete quantum group G is torsion-free.The proper homogeneous G -algebra corresponding to the trivialsubgroup is C ∗(G ) = C0(G ).

We write 〈CI〉 for the localizing subcategory of KK G generated byalgebras of the form C ∗(G )⊗ A where A is some C ∗-algebra andthe coaction is inherited from C ∗(G ).

TheoremOne has 〈CI〉 = KK G .

Christian Voigt

The Baum-Connes conjecture

The discrete quantum group G is torsion-free.The proper homogeneous G -algebra corresponding to the trivialsubgroup is C ∗(G ) = C0(G ).

We write 〈CI〉 for the localizing subcategory of KK G generated byalgebras of the form C ∗(G )⊗ A where A is some C ∗-algebra andthe coaction is inherited from C ∗(G ).

TheoremOne has 〈CI〉 = KK G .

Christian Voigt

The Baum-Connes conjecture

The discrete quantum group G is torsion-free.The proper homogeneous G -algebra corresponding to the trivialsubgroup is C ∗(G ) = C0(G ).

We write 〈CI〉 for the localizing subcategory of KK G generated byalgebras of the form C ∗(G )⊗ A where A is some C ∗-algebra andthe coaction is inherited from C ∗(G ).

TheoremOne has 〈CI〉 = KK G .

Christian Voigt

Outline of the proof

Let us concentrate on the following part of the argument.

TheoremWe have C ∈ 〈CI〉 ⊂ KK G .

Theorem (Baaj-Skandalis)

The reduced crossed product functor KK G → KKG is anequivalence of categories.

As a consequence, in order to prove C ∈ 〈CI〉 it suffices to showC (G ) ∈ 〈C〉 ∈ KKG .

Christian Voigt

Outline of the proof

Let us concentrate on the following part of the argument.

TheoremWe have C ∈ 〈CI〉 ⊂ KK G .

Theorem (Baaj-Skandalis)

The reduced crossed product functor KK G → KKG is anequivalence of categories.

As a consequence, in order to prove C ∈ 〈CI〉 it suffices to showC (G ) ∈ 〈C〉 ∈ KKG .

Christian Voigt

Outline of the proof

Let us concentrate on the following part of the argument.

TheoremWe have C ∈ 〈CI〉 ⊂ KK G .

Theorem (Baaj-Skandalis)

The reduced crossed product functor KK G → KKG is anequivalence of categories.

As a consequence, in order to prove C ∈ 〈CI〉 it suffices to showC (G ) ∈ 〈C〉 ∈ KKG .

Christian Voigt

Outline of the proof

Let us concentrate on the following part of the argument.

TheoremWe have C ∈ 〈CI〉 ⊂ KK G .

Theorem (Baaj-Skandalis)

The reduced crossed product functor KK G → KKG is anequivalence of categories.

As a consequence, in order to prove C ∈ 〈CI〉 it suffices to showC (G ) ∈ 〈C〉 ∈ KKG .

Christian Voigt

Outline of the proof

We have C (G ) ∈ 〈C (G/T )〉 in KKG - this follows from (thevalidity of) the Baum-Connes conjecture for T and induction.Hence it suffices to show

TheoremWe have C (G/T ) ∼= C⊕ C in KKG .

In the case q = 1 this is a consequence of equivariant Poincareduality for G/T .

We need some information about the equivariant K -theory andK -homology of the Podles sphere G/T .

Christian Voigt

Outline of the proof

We have C (G ) ∈ 〈C (G/T )〉 in KKG - this follows from (thevalidity of) the Baum-Connes conjecture for T and induction.Hence it suffices to show

TheoremWe have C (G/T ) ∼= C⊕ C in KKG .

In the case q = 1 this is a consequence of equivariant Poincareduality for G/T .

We need some information about the equivariant K -theory andK -homology of the Podles sphere G/T .

Christian Voigt

Outline of the proof

We have C (G ) ∈ 〈C (G/T )〉 in KKG - this follows from (thevalidity of) the Baum-Connes conjecture for T and induction.Hence it suffices to show

TheoremWe have C (G/T ) ∼= C⊕ C in KKG .

In the case q = 1 this is a consequence of equivariant Poincareduality for G/T .

We need some information about the equivariant K -theory andK -homology of the Podles sphere G/T .

Christian Voigt

Outline of the proof

We have C (G ) ∈ 〈C (G/T )〉 in KKG - this follows from (thevalidity of) the Baum-Connes conjecture for T and induction.Hence it suffices to show

TheoremWe have C (G/T ) ∼= C⊕ C in KKG .

In the case q = 1 this is a consequence of equivariant Poincareduality for G/T .

We need some information about the equivariant K -theory andK -homology of the Podles sphere G/T .

Christian Voigt

K -theory of the Podles sphere

The quantum group G acts on the homogenous space G/T fromthe left.

Natural elements in KG0 (C (G/T )) are given by the finitely

generated projective O(G/T )-modules

Γ(G ×T Ck) = {x ∈ O(SUq(2))|(id⊗π)∆(x) = x ⊗ z−k}

for k ∈ Z.

Geometrically, Γ(G ×T Ck) corresponds to an induced bundle onG/T .

Christian Voigt

K -theory of the Podles sphere

The quantum group G acts on the homogenous space G/T fromthe left.

Natural elements in KG0 (C (G/T )) are given by the finitely

generated projective O(G/T )-modules

Γ(G ×T Ck) = {x ∈ O(SUq(2))|(id⊗π)∆(x) = x ⊗ z−k}

for k ∈ Z.

Geometrically, Γ(G ×T Ck) corresponds to an induced bundle onG/T .

Christian Voigt

K -theory of the Podles sphere

The quantum group G acts on the homogenous space G/T fromthe left.

Natural elements in KG0 (C (G/T )) are given by the finitely

generated projective O(G/T )-modules

Γ(G ×T Ck) = {x ∈ O(SUq(2))|(id⊗π)∆(x) = x ⊗ z−k}

for k ∈ Z.

Geometrically, Γ(G ×T Ck) corresponds to an induced bundle onG/T .

Christian Voigt

K -theory of the Podles sphere

The induced bundles yield elements

[Γ(G ×T Ck)] ∈ KKG (C,C (G/T )).

Taking into account the left action of C (G/T ) by multiplicationwe obtain in fact elements

[[Γ(G ×T Ck)]] ∈ KKG (C (G/T ),C (G/T ))

such that[Γ(G ×T Ck)] = [1] · [[Γ(G ×T Ck)]]

where [1] ∈ KKG (C,C (G/T )) is the class of the unithomomorphism.

Christian Voigt

K -theory of the Podles sphere

The induced bundles yield elements

[Γ(G ×T Ck)] ∈ KKG (C,C (G/T )).

Taking into account the left action of C (G/T ) by multiplicationwe obtain in fact elements

[[Γ(G ×T Ck)]] ∈ KKG (C (G/T ),C (G/T ))

such that[Γ(G ×T Ck)] = [1] · [[Γ(G ×T Ck)]]

where [1] ∈ KKG (C,C (G/T )) is the class of the unithomomorphism.

Christian Voigt

The Dirac operator for the Podles sphere

Dabrowski and Sitarz have constructed a spectral triple(O(G/T ), L2(G ×T S),D) for the Podles sphere representing theDirac operator on G/T .

I The Hilbert space L2(G ×T S) is the completion of

Γ(G ×T C1)⊕ Γ(G ×T C−1)

for the scalar product induced from L2(G ).

I Representation of O(G/T ) by left multiplication.

I

D =

(0 D−

D+ 0

), D±|l ,m〉± = [l + 1/2]q|l ,m〉∓.

This defines an element [D] ∈ KKG (C (G/T ), C).

Christian Voigt

The Dirac operator for the Podles sphere

Dabrowski and Sitarz have constructed a spectral triple(O(G/T ), L2(G ×T S),D) for the Podles sphere representing theDirac operator on G/T .

I The Hilbert space L2(G ×T S) is the completion of

Γ(G ×T C1)⊕ Γ(G ×T C−1)

for the scalar product induced from L2(G ).

I Representation of O(G/T ) by left multiplication.

I

D =

(0 D−

D+ 0

), D±|l ,m〉± = [l + 1/2]q|l ,m〉∓.

This defines an element [D] ∈ KKG (C (G/T ), C).

Christian Voigt

The Dirac operator for the Podles sphere

Dabrowski and Sitarz have constructed a spectral triple(O(G/T ), L2(G ×T S),D) for the Podles sphere representing theDirac operator on G/T .

I The Hilbert space L2(G ×T S) is the completion of

Γ(G ×T C1)⊕ Γ(G ×T C−1)

for the scalar product induced from L2(G ).

I Representation of O(G/T ) by left multiplication.

I

D =

(0 D−

D+ 0

), D±|l ,m〉± = [l + 1/2]q|l ,m〉∓.

This defines an element [D] ∈ KKG (C (G/T ), C).

Christian Voigt

The Dirac operator for the Podles sphere

Dabrowski and Sitarz have constructed a spectral triple(O(G/T ), L2(G ×T S),D) for the Podles sphere representing theDirac operator on G/T .

I The Hilbert space L2(G ×T S) is the completion of

Γ(G ×T C1)⊕ Γ(G ×T C−1)

for the scalar product induced from L2(G ).

I Representation of O(G/T ) by left multiplication.

I

D =

(0 D−

D+ 0

), D±|l ,m〉± = [l + 1/2]q|l ,m〉∓.

This defines an element [D] ∈ KKG (C (G/T ), C).

Christian Voigt

The Dirac operator for the Podles sphere

Dabrowski and Sitarz have constructed a spectral triple(O(G/T ), L2(G ×T S),D) for the Podles sphere representing theDirac operator on G/T .

I The Hilbert space L2(G ×T S) is the completion of

Γ(G ×T C1)⊕ Γ(G ×T C−1)

for the scalar product induced from L2(G ).

I Representation of O(G/T ) by left multiplication.

I

D =

(0 D−

D+ 0

), D±|l ,m〉± = [l + 1/2]q|l ,m〉∓.

This defines an element [D] ∈ KKG (C (G/T ), C).

Christian Voigt

The Dirac operator for the Podles sphere

Dabrowski and Sitarz have constructed a spectral triple(O(G/T ), L2(G ×T S),D) for the Podles sphere representing theDirac operator on G/T .

I The Hilbert space L2(G ×T S) is the completion of

Γ(G ×T C1)⊕ Γ(G ×T C−1)

for the scalar product induced from L2(G ).

I Representation of O(G/T ) by left multiplication.

I

D =

(0 D−

D+ 0

), D±|l ,m〉± = [l + 1/2]q|l ,m〉∓.

This defines an element [D] ∈ KKG (C (G/T ), C).

Christian Voigt

K-homology of the Podles sphere

Define elements α ∈ KKG0 (C (G/T ), C2) by

α = [D]⊕ [[Γ(G ×T C−1)]] · [D]

and β ∈ KKG0 (C2,C (G/T )) by

β = Γ(G ×T C1)⊕−Γ(G ×T C0).

Proposition

We haveβ ◦ α = 1, α ◦ β = 1

and hence C (G/T ) ∼= C2 in KKG .

This finishes the proof of the theorem.

Christian Voigt

K-homology of the Podles sphere

Define elements α ∈ KKG0 (C (G/T ), C2) by

α = [D]⊕ [[Γ(G ×T C−1)]] · [D]

and β ∈ KKG0 (C2,C (G/T )) by

β = Γ(G ×T C1)⊕−Γ(G ×T C0).

Proposition

We haveβ ◦ α = 1, α ◦ β = 1

and hence C (G/T ) ∼= C2 in KKG .

This finishes the proof of the theorem.

Christian Voigt

K-homology of the Podles sphere

Define elements α ∈ KKG0 (C (G/T ), C2) by

α = [D]⊕ [[Γ(G ×T C−1)]] · [D]

and β ∈ KKG0 (C2,C (G/T )) by

β = Γ(G ×T C1)⊕−Γ(G ×T C0).

Proposition

We haveβ ◦ α = 1, α ◦ β = 1

and hence C (G/T ) ∼= C2 in KKG .

This finishes the proof of the theorem.

Christian Voigt

Equivariant Poincare duality

Let us call two G -YD-algebras P and Q equivariantly Poincaredual to each other if there exists a natural isomorphism

KKG (P � A,B) ∼= KKG (A,Q � B)

for all G -algebras A and B.

Given Poincare dual algebras P and Q we have natural elements

α ∈ KKG (P � Q, C)

andβ ∈ KKG (C,Q � P).

Christian Voigt

Equivariant Poincare duality

Let us call two G -YD-algebras P and Q equivariantly Poincaredual to each other if there exists a natural isomorphism

KKG (P � A,B) ∼= KKG (A,Q � B)

for all G -algebras A and B.

Given Poincare dual algebras P and Q we have natural elements

α ∈ KKG (P � Q, C)

andβ ∈ KKG (C,Q � P).

Christian Voigt

Equivariant Poincare duality

Let us call two G -YD-algebras P and Q equivariantly Poincaredual to each other if there exists a natural isomorphism

KKG (P � A,B) ∼= KKG (A,Q � B)

for all G -algebras A and B.

Given Poincare dual algebras P and Q we have natural elements

α ∈ KKG (P � Q, C)

andβ ∈ KKG (C,Q � P).

Christian Voigt

Equivariant Poincare duality for the Podles sphere

TheoremLet G = SUq(2). The Podles sphere C (G/T ) is equivariantlyPoincare dual to itself. In fact,

KKD(G)(C (G/T ) � A,B) ∼= KKD(G)(A,C (G/T ) � B)

for all G-YD-algebras A and B.

The element α ∈ KKD(G)(C (G/T ) � C (G/T ), C) implementingthis duality is given by the Dirac operator D acting on L2(G ×T S).

The representation φ of C (G/T ) � C (G/T ) on L2(G ×T Σ) is

φ(f � g)(h) = fgh.

Christian Voigt

Equivariant Poincare duality for the Podles sphere

TheoremLet G = SUq(2). The Podles sphere C (G/T ) is equivariantlyPoincare dual to itself. In fact,

KKD(G)(C (G/T ) � A,B) ∼= KKD(G)(A,C (G/T ) � B)

for all G-YD-algebras A and B.

The element α ∈ KKD(G)(C (G/T ) � C (G/T ), C) implementingthis duality is given by the Dirac operator D acting on L2(G ×T S).

The representation φ of C (G/T ) � C (G/T ) on L2(G ×T Σ) is

φ(f � g)(h) = fgh.

Christian Voigt

Equivariant Poincare duality for the Podles sphere

TheoremLet G = SUq(2). The Podles sphere C (G/T ) is equivariantlyPoincare dual to itself. In fact,

KKD(G)(C (G/T ) � A,B) ∼= KKD(G)(A,C (G/T ) � B)

for all G-YD-algebras A and B.

The element α ∈ KKD(G)(C (G/T ) � C (G/T ), C) implementingthis duality is given by the Dirac operator D acting on L2(G ×T S).

The representation φ of C (G/T ) � C (G/T ) on L2(G ×T Σ) is

φ(f � g)(h) = fgh.

Christian Voigt

Equivariant Poincare duality for the Podles sphere

TheoremLet G = SUq(2). The Podles sphere C (G/T ) is equivariantlyPoincare dual to itself. In fact,

KKD(G)(C (G/T ) � A,B) ∼= KKD(G)(A,C (G/T ) � B)

for all G-YD-algebras A and B.

The element α ∈ KKD(G)(C (G/T ) � C (G/T ), C) implementingthis duality is given by the Dirac operator D acting on L2(G ×T S).

The representation φ of C (G/T ) � C (G/T ) on L2(G ×T Σ) is

φ(f � g)(h) = fgh.

Christian Voigt

Remarks

I The Baum-Connes conjecture for a torsion-free discrete groupG implies the Kadison-Kaplansky conjecture: There are nonontrivial idempotents in C ∗red(G ).This is not true for discrete quantum groups, C (SUq(2))contains lots of nontrivial idempotents.

I Look at general q-deformations!

Christian Voigt

Remarks

I The Baum-Connes conjecture for a torsion-free discrete groupG implies the Kadison-Kaplansky conjecture: There are nonontrivial idempotents in C ∗red(G ).This is not true for discrete quantum groups, C (SUq(2))contains lots of nontrivial idempotents.

I Look at general q-deformations!

Christian Voigt

Remarks

I The Baum-Connes conjecture for a torsion-free discrete groupG implies the Kadison-Kaplansky conjecture: There are nonontrivial idempotents in C ∗red(G ).This is not true for discrete quantum groups, C (SUq(2))contains lots of nontrivial idempotents.

I Look at general q-deformations!

Christian Voigt