Categorical actions of quantum groups Christian Voigt University of - - PowerPoint PPT Presentation

Categorical actions of quantum groups

Christian Voigt

University of Glasgow christian.voigt@glasgow.ac.uk http://www.maths.gla.ac.uk/~cvoigt/index.xhtml

Oslo 6 August 2019

Actions of quantum groups

Actions of quantum groups

(Quantum) groups and Hopf algebras are often defined and studied via their actions. For instance, ◮ actions on combinatorial or geometric objects, e.g. graphs or manifolds. ◮ action on algebras, e.g. via quantum automorphisms of graphs or (noncommutative) manifolds/varieties. ◮ actions on vector spaces (representations). There are more general types of actions appearing in the theory, for instance actions of tensor categories (module categories). This is usually studied in an algebraic framework. How to include analytical considerations/study continuity for such categorical actions? In this talk we shall define and study (continuous) actions of locally compact (quantum) groups on C ∗-categories.

C ∗-categories

C ∗-categories

Definition

A C ∗-category is a (semi-) category C enriched in Banach spaces together with an antilinear involutive contravariant functor ∗ : C → C which is the identity on objects, satisfying f ∗ ◦ f = f 2 and f ∗ ◦ f ≥ 0 for all morphisms f ∈ C(X , Y ). ◮ semi-category means: all structure and axioms as for a category, but without requiring identity morphisms. ◮ enriched in Banach spaces means: all morphism spaces C(X , Y ) are Banach spaces, composition C(Y , Z) × C(X , Y ) → C(X , Z) is bilinear and satisfies g ◦ f ≤ gf . ◮ an antilinear involutive contravariant functor ∗ : C → C which is the identity on objects means: we have complex antilinear maps C(X , Y ) → C(Y , X ), f → f ∗ such that f ∗∗ = f , (g ◦ f )∗ = f ∗ ◦ g∗.

Examples of C ∗-categories

Examples of C ∗-categories

We have the following basic examples of C ∗-categories: ◮ The category Hilb of all separable Hilbert spaces and compact

• perators as morphisms.

◮ The category HilbA of all countably generated Hilbert A-modules and compact operators as morphisms for a separable C ∗-algebra A. ◮ The full subcategory hilbA ⊂ HilbA of all Hilbert A-modules isomorphic to direct summands of A⊕n for some n ∈ N. For later purposes we shall always require that our categories C are separable, in the sense that all morphism spaces C(X , Y ) are separable. That is, each C(X , Y ) is required to contain a countable dense subset. Difference to purely algebraic setting: C ∗-categories are typically not (finitely) cocomplete – cokernels rarely exist!

Multiplier categories

Multiplier categories

Let C be a C ∗-category. Recall that we do not require the morphism spaces of C to contain identity morphisms.

Multiplier categories

Let C be a C ∗-category. Recall that we do not require the morphism spaces of C to contain identity morphisms. A left multiplier morphism L : X → Y for X , Y ∈ C is a family of uniformly bounded linear maps L(Z) : C(Z, X ) → C(Z, Y ) such that L(W )(h ◦ g) = L(Z)(h) ◦ g for all h ∈ C(Z, X ) and g ∈ C(W , Z). In a similar way one defines right multipliers. A multiplier morphism M : X → Y is a pair M = (L, R) of left and right multiplier morphisms from X to Y such that g ◦ L(W )(f ) = R(Z)(g) ◦ f for all f ∈ C(W , X ) and g ∈ C(Y , Z).

Lemma

Let C be a C ∗-category. Then the category M C with the same objects as C and morphisms given by all multiplier morphisms is naturally a (unital) C ∗-category.

Functors and natural transformations

Functors and natural transformations

Definition

Let C, D be C ∗-categories. A ∗-functor F : C → D (or F : C → M D) is a (semi-) functor satisfying F(f ∗) = F(f )∗ for all morphisms f . A ∗-functor F : C → D is called nondegenerate if [F(C(Y , Y )) ◦ D(F(X ), F(Y ))] = D(F(X ), F(Y )) = [D(F(X ), F(Y )) ◦ F(C(X , X ))] for all X , Y ∈ C.

Lemma

If F : C → M D is a nondegenerate ∗-functor then F extends uniquely to a (unital) ∗-functor M C → M D.

Definition

A natural transformation φ : F ⇒ G between two ∗-functors F, G : C → M D is called unitary if φ(X ) is unitary for all X ∈ C.

Direct sums

Direct sums

Let C be a C ∗-category.

Direct sums

Let C be a C ∗-category. Let X1, X2 ∈ C. An object X = X1 ⊕ X2 ∈ C together with multiplier morphisms ik : Xk → X is called a direct sum of X1, X2 if idXk = i∗

k ik,

idX = i1i∗

1 + i2i∗ 2

where k = 1, 2. We always assume that our C ∗-categories have all (finite) direct sums.

Direct sums

Let C be a C ∗-category. Let X1, X2 ∈ C. An object X = X1 ⊕ X2 ∈ C together with multiplier morphisms ik : Xk → X is called a direct sum of X1, X2 if idXk = i∗

k ik,

idX = i1i∗

1 + i2i∗ 2

where k = 1, 2. We always assume that our C ∗-categories have all (finite) direct sums. Let (Xn)n∈N be a countable family of objects in C. An object X =

n∈N Xn together with multiplier morphisms ik : Xk → X is called

a direct sum of the Xk if idXk = i∗

k ik,

idX =

∞

• k=1

iki∗

k

for all k ∈ N. We will usually only consider C ∗-categories which admit countable direct sums, and only ∗-functors which preserve them.

Subobjects

Subobjects

Let C be a C ∗-category.

Subobjects

Let C be a C ∗-category. We say that C is subobject complete, if for every X ∈ C and every projection p ∈ C(X , X ) there exists an object Y ∈ C and a morphism i : Y → X such that i∗i = idY and ii∗ = p. Any C ∗-category C can be embedded into a subobject complete C ∗-category Split(C), called the subobject completion (or Karoubi envelope) of C. There is a canonical ∗-functor C → Split(C) with the following property. Every ∗-functor F : C → D whose target D is subobject complete can be extended in an essentially unique way to a ∗-functor Split(C) → D.

Tensor products

Tensor products

Let C, D be C ∗-categories.

Tensor products

Let C, D be C ∗-categories. We define a new category C ⊙ D as the category with objects all pairs (X , Y ) for X ∈ C, Y ∈ D and morphism spaces (C ⊙ D)((X , Y ), (X ′, Y ′)) = C(X , X ′) ⊙ D(Y , Y ′), where ⊙ denotes the algebraic tensor product. Composition of such morphisms is defined in the obvious way, and the ∗-structure is defined by (f ⊙ g)∗ = f ∗ ⊙ g∗. Morphism spaces in the“naive tensor product”category C ⊙ D are typically not complete, and C ⊙ D does not admit finite direct sums in general.

Tensor products

Tensor products

If C, D are C ∗-categories there exists a C ∗-category C ⊠ D, which is

• btained by taking the subobject completion of the category with the

same objects as C ⊙ D and morphisms (C ⊠ D)((X , Y ), (X ′, Y ′)) = C(X , X ′) ⊗ D(Y , Y ′), the minimal tensor product of C(X , X ′) and D(Y , Y ′).

Tensor products

If C, D are C ∗-categories there exists a C ∗-category C ⊠ D, which is

• btained by taking the subobject completion of the category with the

same objects as C ⊙ D and morphisms (C ⊠ D)((X , Y ), (X ′, Y ′)) = C(X , X ′) ⊗ D(Y , Y ′), the minimal tensor product of C(X , X ′) and D(Y , Y ′).

Example

If A and B are separable C ∗-algebras then HilbA ⊠ HilbB ∼ = HilbA⊗B in a natural way, where A ⊗ B denotes the minimal tensor product. There is also a maximal version of the categorical tensor product.

Actions of groups on C ∗-categories

Actions of groups on C ∗-categories

Definition

Let G be a (discrete) group. A representation of G on a C ∗-category C consists of ◮ ∗-functors πt : C → C for every t ∈ G, ◮ unitary natural isomorphisms µr,s : πrπs → πrs for all r, s ∈ G, ◮ a unitary natural isomorphism ǫ : id → πe, such that the diagram

πr πs πt (V ) πrs πt (V ) πr πst (V ) πrst (V ) πr (µs,t (V )) µr,s (πt (V )) µr,st (V ) µrs,t (V )

is commutative for all V ∈ C, and two further constraints for the unit isomorphism ǫ are satisfied.

Examples of actions on C ∗-categories

Examples of actions on C ∗-categories

Example

Let C be a C ∗-category and G arbitrary. Then C becomes a G-C ∗-category with the trivial action of G. That is, we define πt = id for all t ∈ G, and all natural isomorphisms appearing in the definition to be identities. This is called the trivial action of G on C.

Examples of actions on C ∗-categories

Example

Let C be a C ∗-category and G arbitrary. Then C becomes a G-C ∗-category with the trivial action of G. That is, we define πt = id for all t ∈ G, and all natural isomorphisms appearing in the definition to be identities. This is called the trivial action of G on C.

Example

Let C = HilbA for a G-C ∗-algebra A. Then C becomes a G-C ∗-category with the action πt(E) = Et = E ⊗A At, where At = A is the standard Hilbert A-module AA with left action a · ξ = πt(a)ξ.

Examples of actions on C ∗-categories

Example

Let C be a C ∗-category and G arbitrary. Then C becomes a G-C ∗-category with the trivial action of G. That is, we define πt = id for all t ∈ G, and all natural isomorphisms appearing in the definition to be identities. This is called the trivial action of G on C.

Example

Let C = HilbA for a G-C ∗-algebra A. Then C becomes a G-C ∗-category with the action πt(E) = Et = E ⊗A At, where At = A is the standard Hilbert A-module AA with left action a · ξ = πt(a)ξ.

Example

Let C = HilbA and let M be an invertible Hilbert A-bimodule. Then πn(F) = F ⊗A M⊗An defines an action of Z on HilbA.

Equivariant functors

Equivariant functors

Definition

Let C, D be G-categories. A G-equivariant functor from C to D is a ∗-functor F : C → D together with unitary natural isomorphisms γs : πD

s F → FπC s for all s ∈ G such that the diagrams

πD s πD t F(V ) πD s FπC t (V ) FπC st (V ) πD st F(V ) FπC s πC t (V ) πs (γt (V )) µs,t (F(V )) γst (V ) F(µs,t (V )) γs (πt (V ))

are commutative for all s, t ∈ G.

Equivariant natural transformations

Equivariant natural transformations

A G-equivariant natural transformation φ : F → G between G-equivariant functors is a natural transformation between the underlying functors such that γG

s (V )πD s (φ(V )) = φ(πC s (V ))γF s (V )

for all s ∈ G and V ∈ C, or equivalently, the diagram

πD s F(V ) FπC s (V ) πD s G(V ) GπC s (V ) πD s (φ(V )) γF s (V ) γG s (V ) φ(πC s (V ))

is commutative for s ∈ G and V ∈ C.

Fixed points

Fixed points

Recall that any C ∗-category becomes a G-category with the trivial action

• f G, that is, πt = id for all t ∈ G, and all natural isomorphisms

appearing in the definition being identities.

Fixed points

Recall that any C ∗-category becomes a G-category with the trivial action

• f G, that is, πt = id for all t ∈ G, and all natural isomorphisms

appearing in the definition being identities.

Definition

Let C be a G-category. A fixed point in C is a G-equivariant functor F : Hilb → C, where Hilb is equipped with the trivial action.

Fixed points

Recall that any C ∗-category becomes a G-category with the trivial action

• f G, that is, πt = id for all t ∈ G, and all natural isomorphisms

appearing in the definition being identities.

Definition

Let C be a G-category. A fixed point in C is a G-equivariant functor F : Hilb → C, where Hilb is equipped with the trivial action.

Lemma

Let C be a G-category. A fixed point in C is the same thing as an object V ∈ C together with unitary isomorphisms γt : πt(V ) → V such that γrπr(γt) = γrtµr,t for all r, t ∈ G.

Fixed points

Recall that any C ∗-category becomes a G-category with the trivial action

• f G, that is, πt = id for all t ∈ G, and all natural isomorphisms

appearing in the definition being identities.

Definition

Let C be a G-category. A fixed point in C is a G-equivariant functor F : Hilb → C, where Hilb is equipped with the trivial action.

Lemma

Let C be a G-category. A fixed point in C is the same thing as an object V ∈ C together with unitary isomorphisms γt : πt(V ) → V such that γrπr(γt) = γrtµr,t for all r, t ∈ G.

Example

Let C = Hilb with the trivial action. Then a fixed point in C is the same thing as a Hilbert space V together with a unitary representation of G

• n V .

Fixed points

Example

More generally, if C = HilbA for a G-C ∗-algebra A then a fixed point structure on E ∈ C is equivalent to a G-Hilbert A-module structure on E.

Fixed points

Example

More generally, if C = HilbA for a G-C ∗-algebra A then a fixed point structure on E ∈ C is equivalent to a G-Hilbert A-module structure on E.

Definition

A morphism of fixed points V , W ∈ C is a morphism f : V → W in C such that γW

r πr(f ) = f γV r

for all r ∈ G. The fixed points for C together with morphisms of fixed points form a C ∗-category CG. This is also known as the equivariantization of C. In the above example C = HilbA, morphisms of fixed points are precisely the G-equivariant morphisms of G-Hilbert A-modules.

The 2-category Rep(G)

The 2-category Rep(G)

The collection of all G-categories, G-equivariant functors, and G-natural transformations forms a (strict) 2-category, which we will denote by Rep(G). This is a categorical analogue of the category Rep(G) of representations

• f G. Note that Rep(G) is contained in Rep(G) as the fixed point

category of the trivial representation on Hilb. So far we have considered the group G to be discrete. How to generalize to locally compact (quantum) groups?

Actions of quantum groups on C ∗-categories

Actions of quantum groups on C ∗-categories

Let G be a (second countable, regular) locally compact (quantum) group with (reduced) C ∗-algebra of functions C0(G). We write ∆ : HilbC0(G) → M (HilbC0(G) ⊠ HilbC0(G)) for the nondegenerate ∗-functor induced by the ∗-homomorphism ∆ : C0(G) → M (C0(G) ⊗ C0(G)).

Actions of quantum groups on C ∗-categories

Let G be a (second countable, regular) locally compact (quantum) group with (reduced) C ∗-algebra of functions C0(G). We write ∆ : HilbC0(G) → M (HilbC0(G) ⊠ HilbC0(G)) for the nondegenerate ∗-functor induced by the ∗-homomorphism ∆ : C0(G) → M (C0(G) ⊗ C0(G)).

Definition

Let C be a C ∗-category. A (left) action of G on C is a nondegenerate faithful ∗-functor γ : C → M (HilbC0(G) ⊠ C) together with a unitary natural isomorphism α : (∆ ⊠ id)γ → (id ⊠γ)γ such that the diagram

(∆ ⊠ γ)γ (∆ ⊠ id ⊠ id)(∆ ⊠ id)γ (id ⊠ id ⊠γ)(id ⊠γ)γ (id ⊠∆ ⊠ id)(∆ ⊠ id)γ (id ⊠∆ ⊠ id)(id ⊠γ)γ

is commutative.

Fixed points

Write 1 for the Hilbert module C0(G) ∈ HilbC0(G).

Fixed points

Write 1 for the Hilbert module C0(G) ∈ HilbC0(G).

Definition

Let C be a C ∗-category equipped with an action γ : C → M (HilbC0(G) ⊠ C) of G. We say that an object V ∈ C is a fixed point under γ if there exists a unitary isomorphism τ : γ(V ) → 1 ⊠ V such that

∆(1) ⊠ V (∆ ⊠ id)γ(V ) 1 ⊠ 1 ⊠ V (id ⊠γ)γ(V ) (id ⊠γ)(1 ⊠ V )

∼ = (∆⊠id)(τ) α (id ⊠γ)(τ) id ⊠τ

is commutative.

G-C ∗-categories

Definition

A G-C ∗-category is a C ∗-category C together with an action γ : C → M (HilbC0(G) ⊠ C) such that for every fixed point V ∈ C we have [(C0(G) ⊗ 1)γ(C(V , V ))] = C0(G) ⊗ C(V , V ) inside M (HilbC0(G) ⊠ C)(γ(V ), γ(V )) ∼ = M (C0(G) ⊗ C(V , V )).

G-C ∗-categories

Definition

A G-C ∗-category is a C ∗-category C together with an action γ : C → M (HilbC0(G) ⊠ C) such that for every fixed point V ∈ C we have [(C0(G) ⊗ 1)γ(C(V , V ))] = C0(G) ⊗ C(V , V ) inside M (HilbC0(G) ⊠ C)(γ(V ), γ(V )) ∼ = M (C0(G) ⊗ C(V , V )).

Example

Let C = HilbA for a G-C ∗-algebra A. Then C is a G-C ∗-category with coaction C → M (C0(G) ⊠ C) given by γ(E) = E ⊗A (C0(G) ⊗ A).

Fixed points in G-C ∗-categories

Write φ : HilbC0(G) → Hilb for the functor given by tensoring with the regular representation L2(G), viewed as a Hilbert C0(G)-C-bimodule.

Fixed points in G-C ∗-categories

Write φ : HilbC0(G) → Hilb for the functor given by tensoring with the regular representation L2(G), viewed as a Hilbert C0(G)-C-bimodule. Let C be a G-C ∗-category and consider V = (φ ⊠ id)γ(U ) ∈ C for U ∈ C. Applying the coaction gives an isomorphism γ(V ) ∼ = (φ ⊠ id ⊠ id)(id ⊠γ)γ(U ) ∼ = (φ ⊠ id ⊠ id)(∆ ⊠ id)γ(U ) ∼ = 1 ⊠ (φ ⊠ id)γ(U ) = 1 ⊠ V . Write τ : γ(V ) → 1 ⊠ V for this morphism.

Proposition

With the notation as above, the morphism τ : γ(V ) → 1 ⊠ V turns V into a fixed point.

Dualization

Dualization

Let C be a G-C ∗-category, and consider the corresponding fixed point category CG.

Dualization

Let C be a G-C ∗-category, and consider the corresponding fixed point category CG. Let X ∈ CG. Then C(X , X ) becomes a G-C ∗-algebra via the ∗-homomorphism γU : C(X , X ) → M (C0(G) ⊗ C(X , X )) given by γU (T) = U γ(T)U ∗, where U : γ(X ) → C0(G) ⊠ X is the trivializer. We define a new category G ⋉red CG as follows. Take the subobject completion of the C ∗-category with the same objects as CG, but with the morphism spaces (G ⋉red CG)(X , Y ) = G ⋉red C(X , Y ). We obtain indeed a well-defined subobject complete C ∗-category this way.

The dual action

The dual action

The category G ⋉red CG admits a coaction of C0( ˆ G) defined as follows. On the full subcategory consisting of objects X ∈ CG we define ˆ γ(X ) = C0( ˆ G) ⊠ X and ˆ γ(T) = (ad(U ) ⊗ id ⊗ id)(1 ⊗ T) for T ∈ G ⋉red C(X , Y ). Here U = ΣV Σ ∈ M (C0( ˆ G) ⊗ K(L2(G))) is the multiplicative unitary implementing the dual action. For a general object (X , p) with p ∈ M (G ⋉red C(X , X )) we define ˆ γ(X , p) ∈ M (HilbC0( ˆ

G) ⊠ (G ⋉red CG)) to be the subobject of ˆ

γ(X ) given by the projection ˆ γ(p) with the map ˆ γ defined as above.

Theorem

This defines the structure of a ˆ G-C ∗-category on G ⋉red CG.

Disintegration

Disintegration

Let C be a G-C ∗-category and let V ∈ CG. If K(L2(G)) ⊂ M (C(V , V )) is embedded as a nondegenerate G-C ∗-subalgebra we say that V can be disintegrated if there exists W ∈ CG such that (φ ⊠ id)γ(W ) ∼ = V , in a way compatible with the

• trivializers. We say that W is a disintegration of V in this case.

Definition

A G-C ∗-category admits disintegration if for every object of CG and any nondegenerate embedding K(L2(G)) → M (C(V , V )) there exists a disintegration. This is automatically the case if G is finite.

Disintegration

Let C be a G-C ∗-category and let V ∈ CG. If K(L2(G)) ⊂ M (C(V , V )) is embedded as a nondegenerate G-C ∗-subalgebra we say that V can be disintegrated if there exists W ∈ CG such that (φ ⊠ id)γ(W ) ∼ = V , in a way compatible with the

• trivializers. We say that W is a disintegration of V in this case.

Definition

A G-C ∗-category admits disintegration if for every object of CG and any nondegenerate embedding K(L2(G)) → M (C(V , V )) there exists a disintegration. This is automatically the case if G is finite.

Example

Let C = HilbA for a G-C ∗-algebra A, and let V = L2(G) ⊗ E for a G-Hilbert module E. Then W = E is a disintegration of V .

Categorical Fourier duality

Theorem

Let G be a (regular) locally compact quantum group and let C be a G-C ∗-category which admits disintegration. Then we have a natural equivalence ˆ FF(C) ≃ C

• f G-C ∗-categories.

Categorical Fourier duality

Theorem

Let G be a (regular) locally compact quantum group and let C be a G-C ∗-category which admits disintegration. Then we have a natural equivalence ˆ FF(C) ≃ C

• f G-C ∗-categories.

Let Rep(G) be the 2-category of G-C ∗-categories which admit disintegration.

Theorem (Fourier transform)

The assignment F(C) = G ⋉red CG determines a homomorphism F : Rep(G) → Rep( ˆ G).

Categorical Fourier duality

Categorical Fourier duality

Theorem (Fourier duality)

Let G be a (regular) locally compact quantum group. Then the categorical Fourier transform F : Rep(G) → Rep( ˆ G) is a 2-equivalence

• f 2-categories.

Categorical Fourier duality

Theorem (Fourier duality)

Let G be a (regular) locally compact quantum group. Then the categorical Fourier transform F : Rep(G) → Rep( ˆ G) is a 2-equivalence

• f 2-categories.

The following corollary is originally due to Asashiba (in the case of classical finite groups).

Theorem

Let G be a finite quantum group. Then the 2-categories of Rep(G)-module categories and Rep( ˆ G)-module categories are 2-equivalent.