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 � ≤ � g �� f � . ◮ 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 operators as morphisms. ◮ The category Hilb A of all countably generated Hilbert A -modules and compact operators as morphisms for a separable C ∗ -algebra A . ◮ The full subcategory hilb A ⊂ Hilb A 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 X 1 , X 2 ∈ C . An object X = X 1 ⊕ X 2 ∈ C together with multiplier morphisms i k : X k → X is called a direct sum of X 1 , X 2 if id X k = i ∗ id X = i 1 i ∗ 1 + i 2 i ∗ k i k , 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 X 1 , X 2 ∈ C . An object X = X 1 ⊕ X 2 ∈ C together with multiplier morphisms i k : X k → X is called a direct sum of X 1 , X 2 if id X k = i ∗ id X = i 1 i ∗ 1 + i 2 i ∗ k i k , 2 where k = 1 , 2 . We always assume that our C ∗ -categories have all (finite) direct sums . Let ( X n ) n ∈ N be a countable family of objects in C . An object X = � n ∈ N X n together with multiplier morphisms i k : X k → X is called a direct sum of the X k if ∞ � id X k = i ∗ i k i ∗ id X = k i k , k k =1 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 = id Y 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 obtained 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 obtained 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 Hilb A ⊠ Hilb B ∼ = Hilb A ⊗ 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 π t : C → C for every t ∈ G , ◮ unitary natural isomorphisms µ r , s : π r π s → π rs for all r , s ∈ G , ◮ a unitary natural isomorphism ǫ : id → π e , such that the diagram µ r , s ( π t ( V )) π r π s π t ( V ) π rs π t ( V ) π r ( µ s , t ( V )) µ rs , t ( V ) π r π st ( V ) π rst ( V ) µ r , st ( 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 π t = 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 π t = 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 = Hilb A for a G - C ∗ -algebra A . Then C becomes a G - C ∗ -category with the action π t ( E ) = E t = E ⊗ A A t , where A t = A is the standard Hilbert A -module A A with left action a · ξ = π t ( a ) ξ .