Functoriality of Quantum Principal Bundles and Quantum Connections
aa r X i v : . [ m a t h . QA ] F e b FUNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES ANDQUANTUM CONNECTIONS
GUSTAVO AMILCAR SALDA ˜NA MONCADA
Abstract.
The purpose of this work is to present a complet categorical point of view ofthe association between finite dimmensional representations of a compact quantum groupand quantum vector bundles with quantum linear connections using M. Durdevich’s theory[D1], [D2]. This paper is a noncommutative version of the principal result of [SW], whichextends the work presented in [D3] by considering connections as well.
MSC 2010:
Keywords:
Quantum principal bunldes and quantum principal con-nections, quantum association functor. Introduction
In differential geometry the study of principal bundles and principal connections is one ofmost important subjects [KN], for mathematics and also for physics since Yang-Mills theorieshave been develped in this context [B]. An special result in the theory is the fact that takinga smooth principal G –bundle GM over a manifold M one can associate a fiber bundle over M for every single smooth manifold with a smooth G –action. This turns out into a covariantfunctor Ass GM between the category of manifolds with G –actions, MF G , and the category offiber bundles over M , FB M . One of the most important contributions to the study of thesekinds of functors was made by M. Nori in [N] but this paper was developed in the frameworkof algebraic geometry. Another contribution was developed in [SW], which is a generalizationof [N] in the framework of differential geometry considering connections as well, that means,it presents a complete characterization of the (covariant) association functorAss ωGM : MF G −→ FB ∇ M (this last one is the category of fiber bundles over M with non–linear connections) for a givenprincipal G –bundle GM over M with a principal connection ω . In particular, it shows thatevery covariant functor between MF G and FB ∇ M that satisfies certain properties is naturallyisomorphic to Ass ωGM , for an unique (except by isomorphisms) principal bundle GM over M with a principal connection ω . This implies a categorical equivalence between principalbundles over M with principal connections and the category of gauge theory sectors on M with connections, whose objects are essencialy these assocation functors [SW].In noncommutative differential geometry is common to find several definitions of the sameconcept, such is the case of quantum vector bundles with quantum linear connections andquantum principal bundles with quantum principal connections. M. Durdevich in [D1], [D2]developed a complete theory of quantum principal bundles and quantum principal connec-tions considering the notion of quantum groups qG publised by S. L. Woronovicz in [W1] Date : February 11, 2020. and [W3], which will play the role of structure groups. The theory developed in [D1] is semiclassical in the sense that the base space is the algebra of smooth functions of a compactsmooth manifold; while the theory presented in [D2] is completly quantum . This theory wasextended later in order to embrace others clasical notions of princpial bundles, for examplecharacteristic classes [D5]. We have to remark that even when Woronovicz’s quantum groupsare used, Durdevich’s theory presents a different differential calculus that the shown in [W2].On the other hand, the Serre–Swan theorem ([Sw]) gave us a natural way to generalize intononcommutative differential geometry the concept of vector bundle: a finitely generated pro-jective module; although it is posible to use left modules, right modules o even bimodules.A. Connes, Dubois–Violette and others have studied in a deep way the concept of quantumvector bundles and quantum linear connections [C], [DV].The aim of this work is to show a categorical result; more specifically the noncommutativeversion of the categorical equivalence between principal bundles with principal connectionsand gauge theory sectors with connections shown on [SW] using Durdevich’s theory andDubois–Violette’s theory (with certain changes), which corresponds with an extention of thepaper [D3] when one can find a study of these quantum association functors. The approachpresented is important not only because it provides a better support to the general theory,but because talking about categories and functors always involves natural constructions andit promotes a common language. For example the fact that we are able to recreate theclassical categorical equivalence could tell us that we are presence of a correct definition ofquantum vector bundles with quantum linear connections and principal qG –bundles withprincipal connections, among others concepts.We going to do some little changes to the notation presented originally in the theories thatwe will use because we want to highlight similarities and constras with the classical theoryshown in [SW]. One of these changes consist in denoting compact quantum spaces as qX andformally represent them as associative unital ∗ –algebras over C , ( X , · , , ∗ ) (interpreted likethe ∗ –algebra of smooth C –valued functions on qX ). We will identify the quantum space withits algebra, so in general, we going to omit the words compact, associative and unital . Alsoall our ∗ –algebra morphisms will be unital. In some cases we going to work with noncompactquantum spaces, in which case we will point how we going to denote them.The paper is organized in four sections. The second one is about the notation and basicconcepts that we will use and it is split into three parts: the category of quantum represen-tations (or corepresentations) of a compact quantum group Rep qG in which we will presentWorononicz’s theory but we going to change the traditional definition of morphisms in orderto add antilinear maps as well; the category of quantum vector bundles with quantum linearconnections qVB q ∇ (and over a fixed quantum space qM , qVB q ∇ qM ) in which we will showour definition of these quantum structures using bimodules and our definition of morphismsbetween them that will include antilinear maps too; and finally the category of quantumprincipal qG –bundles with quantum principal connections, qPB qω (and over a fixed quan-tum space qM , qPB qωqM ). In this part we going to present the general theory and after thatwe will impose several condition on the quantum bundles and on the quantum connections(as the regularity condition [D2]) in order to be able to define quantum association functorsqAss qωqζ . We have to remark that we shall use the theory presented on [D4] to define quan-tum principal connections or to be more precisely, covariant derivatives. The third section UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 3 is about the quantum association functorqAss qωqζ : Rep qG −→ qVB q ∇ qM for a principal qG –bundle over a fixed quantum space qM qζ with a quantum principalconnection qω : its construction; its general properties (its relation with some functors definedfor Rep qG and qVB q ∇ qM , its behaviour in monomorphisms and epimorphisms, etc); and finallywe going to present the cateogrical equivalence between qPB qωqM and the category of quantumgauge theory sectors on qM with quantum connections qGTS q ∇ qM , so it will be necessary away to rebiult the bundle and the connection for a given contravariant functor between Rep qG and qVB q ∇ qM that satisfies certain properties. To do this we will based on the resultshown in [D3]. The last section is about some concluding comments.One of the most important theorems in gauge theory is the Gauge Principle [KMS], [SW],which establishes that given a principal G –bundle GM over M with a principal connectionand its associated vector bundle with the induced linear connection for a linear representation α , there must exist a linear isomorphism between vector bundle–valued m –forms on M andthe space of all basic m –forms of type α on GM for any m ∈ N . This isomorphismcommutes with the twisted exterior covariant derivative on vector bundle–valued m –formsand the exterior covariant derivative associated to the principal connection on basic m –formsof type α . For m = 0, the Gauge Principle will give us a natural way to define the associatedquantum vector bundle for a given quantum principal bundle and a quantum representation qα . Also we going to present the noncommutative version of this theorem for m > Notation and Basic Concepts
In this section we will present some basic notation and concepts that we going to use inthe whole paper; particularly, we will define the necessary categories to fulfill our purpose.We going to use Sweedler notation and given an arbitrary category C , we will denote by Obj ( C ) the class of objects of C and by Mor ( C ) the class of morphisms in C . Furthermore,given c , c ∈ Obj ( C ), we going to denote by Mor C ( c , c ) the class of all morphisms in C between c and c .2.1. Representations of Matrix Compact Quantum Groups.
We going to use thetheory developed in [W3] by S. L. Woronovicz but with a little change of notation. One canalso check [MVD].A compact quantum group (cqg) [W3] will be denote by qG ; while its dense ∗ –Hopf(sub)algebra will be denote by qG ∞ := ( G , · , , φ, ǫ, κ, ∗ ) , where φ is the comultiplication, ǫ is the counity and κ is the coinverse. It shall treat as the algebra of all smooth C –valued functions definend on qG . In other words, qG ∞ defines asmooth structure on qG or a Lie group structure [W2]. GUSTAVO AMILCAR SALDA ˜NA MONCADA
Definition 2.1 (Quantum representation) . For a given cqg qG , a (smooth right) qG –representationon a C –vector space V is a linear map qα : V −→ V ⊗ G such that (1) V qα −−−−−−−−−−→ V ⊗ G id V y (cid:9) y id V ⊗ ǫ V −−−−−−−−−−→ ∼ = V ⊗ C (where the horizontal arrow at the bottom of the diagram is the canonical isomorphism v v ⊗ ) and (2) V qα −−−−−−−−−−→ V ⊗ G qα y (cid:9) y id V ⊗ φ V ⊗ G −−−−−−−−−−→ qα ⊗ id G V ⊗ G ⊗ G . We say that the representation is finite dimensional if dim C ( V ) < | N | . qα usually receivesthe name of (right) coaction or (right) corepresentation of qG on V . We have to remark that in the general theory presented in [W1], [W3], Diagram (1) is notnecessary.
Example 2.2.
Given a cqp qG and a C –vector space, one can always take qα triv V : V −→ V ⊗ G v v ⊗ .qα triv V turns out to be a qG –representation on V which is called the trivial quantum represen-tation on V . It is easy to see that a linear map qα that satisfies Diagrams (1), (2) can be thought as qα = X i f i ⊗ g i ∈ B ( V ) ⊗ G (where B ( V ) = { f : V −→ V | f is linear } ) such that X i,j f i ◦ f j ⊗ g i ⊗ g j = ( φ ⊗ id V )( qα ) and id V ∼ = X i ǫ ( g i ) f i . Definition 2.3 (Corepresentation morphism) . Let qG be a cqg and qα i be a qG –representationon V i . A corepresentation morphism of degree between them is a linear map f : V −→ V such that (3) V qα −−−−−−−−−−→ V ⊗ G f y (cid:9) y f ⊗ id G V −−−−−−−−−−→ qα V ⊗ G . UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 5
A correpresentation morphism of degree between these qG –representations is an antilinearmap f : V −→ V such that (4) V qα −−−−−−−−−−→ V ⊗ G f y (cid:9) y f ⊗ ∗ V −−−−−−−−−−→ qα V ⊗ G . We will denote by
Mor Rep qG ( qα , qα ) the set of all degree corepresentation morphismsbetween two finite dimensional corepresentations qα , qα ; and by Mor Rep qG ( qα , qα ) theset of all degree corepresentation morphisms between these qG –representations. Finally wedefine the set of corepresentation morphisms between qα and qα as Mor
Rep qG ( qα , qα ) := Mor Rep qG ( qα , qα ) ∪ Mor Rep qG ( qα , qα ) . It is easy to show that finite dimensional qG –representations turn into a category wherecomposition of morphisms is composition of maps and the identity morphism is just theidentity map. With this composition, morphisms have a natural Z –grading. Definition 2.4 (The category of qG –representations) . For a fixed cqg qG , let us define Rep qG as the category whose objects are finite dimensional qG –representations and whosemorphisms are corepresentation morphisms. The category of qG –representations of any di-mension will be denote by Rep ∞ qG . A qG –representation is unitary if qα is an unitary element of B ( V ) ⊗G and it can be proventhat every finite dimmensional qG –representation is isomorphic with a degree 0 morphismto an unitary one [MVD]. In this way we will considerate that every object in Rep qG isunitary.Now, for qα , qα , qβ , qβ ∈ Obj ( Rep qG ) we define the set of cross corepresentationmorphisms between ( qα , qα ) and ( qβ , qβ ) as Mor
Rep Z qG (( qα , qα ) , ( qβ , qβ )) := Mor Rep Z qG (( qα , qα ) , ( qβ , qβ )) ∪ Mor Rep Z qG (( qα , qα ) , ( qβ , qβ )) , where elements of Mor Rep Z qG (( qα , qα ) , ( qβ , qβ ))are ordered pairs ( f , f ) with f ∈ Mor Rep qG ( qα , qβ ) and f ∈ Mor Rep qG ( qα , qβ ); andelements of Mor Rep Z qG (( qα , qα ) , ( qβ , qβ ))are ordered pairs ( f , f ) with f ∈ Mor Rep qG ( qα , qβ ) and f ∈ Mor Rep qG ( qα , qβ ). GUSTAVO AMILCAR SALDA ˜NA MONCADA
Definition 2.5 (The cross category of qG –representations) . For a fixed cqg qG , we de-fine Rep Z qG as the category whose objects are ordered pairs ( qα , qα ) where qα , qα ∈ Obj ( Rep qG ) and whose morphisms are cross corepresentation morphisms . In [W1] there are some functors for qG –representations and correpresentation morphismsof degree 0 that we will adapt to Rep qG and Rep Z qG . First of all, for every C –vector space V , we can consider the complex conjugate vector space of V (this space and V are equal asadditive groups but multiplication by scalars is given by λ · v = λ ∗ v ). This vector space willbe denote by V and its elements by v . There is a canonical antilinear mapid V : V −→ Vv ¯ v Definition 2.6 (Conjugate functor) . Let us define the conjugate functor on
Rep qG as thegraded–preserving covariant enofunctor − : Rep qG −→ Rep qG such that on objects is given by − qα := qα, where qα := X i (id V ◦ f i ◦ id − V ) ⊗ g ∗ i if qα = X i f i ⊗ g i ∈ B ( V ) ⊗ G ; and on morphisms isgiven by − ( f ) := id V ◦ f ◦ id − V , if qα i coacts on V i for i = 1 , and f : V −→ V . qα recives the name of complex conjugaterepresentation of qα . Given two linear f : V −→ V , f : W −→ W , let us consider the twisted direct sum of f with f f ⊕ T f : V ⊕ V −→ W ⊕ W ( v , v ) ( f ( v ) , f ( v )) . Definition 2.7 (Direct sum functor) . Let us define the direct sum functor on qG –representationsas the graded–preserving covariant functor M : Rep Z qG −→ Rep qG such that on objects is given by M ( qα , qα ) := qα ⊕ qα , where qα ⊕ qα := ( i ⊗ id G ) ◦ qα ◦ ( π ⊗ id G ) + ( i ⊗ id G ) ◦ qα ◦ ( π ⊗ id G ) Composition of morphisms is given by ( h ◦ f , h ◦ f ) when ( f , f ) has degree 0 and ( h ◦ f , h ◦ f ) when ( f , f ) has degree 1, where ( f , f ) ∈ Mor
Rep Z qG (( qα , qα ) , ( qβ , qβ )) and ( h , h ) ∈ Mor
Rep Z qG (( qβ , qβ ) , ( qγ , qγ )). We must notice that morphisms have a natural Z –grading with respectto composition. Finally the identity morphism of any object ( qα , qα ) is (id V , id V ), if qα i coacts on V i , for i = 1, 2. UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 7 with ι i : V i −→ V ⊕ V and π i : V ⊕ V −→ V i the canonical inclusion and projections(assuming that qα i coacts on V i ); and on morphisms is given by M ( f , f ) := f ⊕ f if ( f , f ) has degree , and M ( f , f ) := f ⊕ T f if ( f , f ) has degree . qα ⊕ qα receives the name of direct sum of qα and qα . Given two linear f : V −→ V , f : W −→ W , let us consider the twisted tensor productof f with f f ⊗ T f : V ⊗ V −→ W ⊗ W such that f ⊗ T f ( v ⊗ v ) = f ( v ) ⊗ f ( v ) . Definition 2.8 (Tensor product functor) . We define the tensor product functor on
Rep qG as the graded–preserving covariant functor O : Rep Z qG −→ Rep qG such that on objects is defined by O ( qα , qα ) := qα ⊗ qα , where qα ⊗ qα := X i,j f i ⊗ f j ⊗ g i g j considering qα = X i f i ⊗ g i , qα = X j f j ⊗ g j (viewed as elements of B ( V i ) ⊗ G ); andon morphisms is defined by O ( f , f ) := f ⊗ f if ( f , f ) has degree , and O ( f , f ) := f ⊗ T f if ( f , f ) has degree . qα ⊗ qα is usually called the tensor product of qα and qα . Quantum Vector Bundles and Quantum Linear Connections.
This subsectionwill be based on the general theory [DV] with some changes to adapt it to our purposes.Based on the Serre–Swan theorem [Sw] we have
Definition 2.9 (Quantum vector bundle) . Let qM = ( M , · , , ∗ ) be a quantum space. Aquantum vector bundle (qvb) over qM , is a quantum structure qζ formally represented by a M –bimodule (Γ( qM, qV M ) , + , · ) which is finitely generated and projective as left and as right M –module. It represents thespace of smooth sections on qζ and we will identify qζ with (Γ( qM, qV M ) , + , · ) . GUSTAVO AMILCAR SALDA ˜NA MONCADA
According to the Serre–Swan theorem, trivial vector bundles are free projective C ∞ ( M )–modules [Sw], so in this way we say that a qvb over qM is trivial if there exist a left andright M –basis { x i } ni =1 of Γ( qM, qV M ).For a given qζ over a quantum space qM , a graded differential ∗ –algebra(Ω • ( M ) , d, ∗ ) , Ω • ( M ) := M k ≥ Ω k ( M )is an admissible differential ∗ –calculus if Ω ( M ) = M and there exists a graded–preserving M –bimodule isomorphism(5) σ : Ω • ( M ) ⊗ M Γ( qM, qV M ) −→ Γ( qM, qV M ) ⊗ M Ω • ( M ) . Whenever we are using an admissible differential ∗ –calculus for a qvb we will think that themorphism σ is fixed. It is important to notice that we can endow toΩ • ( M ) ⊗ M Γ( qM, qV M )with a graded Ω • ( M )–bimodule structure where the left multiplication · : Ω • ( M ) ⊗ (Ω • ( M ) ⊗ M Γ( qM, qV M )) −→ Ω • ( M ) ⊗ M Γ( qM, qV M )is just(6) m Ω • ( M ) ⊗ M id Γ( qM,qV M ) and the right multiplication · : (Ω • ( M ) ⊗ M Γ( qM, qV M )) ⊗ Ω • ( M ) −→ Ω • ( M ) ⊗ M Γ( qM, qV M )is given by(7) σ − ◦ (id Γ( qM,qV M ) ⊗ M m Ω • ( M ) ) ◦ ( σ ⊗ id Ω • ( M ) ) , with m Ω • ( M ) the product on Ω • ( M ). Clearly there is a similar graded Ω • ( M )–bimodulestructure for Γ( qM, qV M ) ⊗ M Ω • ( M ) . With this new structure, σ becomes into a graded Ω • ( M )–bimodule isomorphism.The following definition is clearly a noncommutative version of the classical concept oflinear connection [DV]. Definition 2.10 (Quantum linear connection) . Let us consider a qvb qζ = (Γ( qM, qV M ) , + , · ) and an admissible differential ∗ –calculus (Ω • ( M ) , d, ∗ ) on it. A quantum linear connection(qlc) on qζ is a linear map q ∇ : Γ( qM, qV M ) −→ Ω ( M ) ⊗ M Γ( qM, qV M ) that satisfies left and right Leibniz rule: for all p ∈ M and all x ∈ Γ( qM, qV M ) q ∇ ( px ) = p q ∇ ( x ) + dp ⊗ M x,q ∇ ( xp ) = q ∇ ( x ) p + σ − ( x ⊗ M dp ) . A qvb with a qlc will be denote as ( qζ , q ∇ ) . One has to notice that qlcs depend on the choice of the admissible differential ∗ –calculus(Ω • ( M ) , d, ∗ ) ( quantum differential forms on qM ). UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 9
Example 2.11.
Taking a trivial qvb qζ triv = (Γ( qM, qV M ) , + , · ) , we say that a qlc q ∇ istrivial if q ∇ ( x ) = n X i dp i ⊗ M x i , where x = n X i p i x i , p i ∈ M and { x i } ni =1 is a left and right M –basis of Γ( qM, qV M ) . Everytrivial qlc will be denote by q ∇ triv . Inspiring in the classical case, one can think Γ( qM, qV M ) and Ω ( M ) ⊗ M Γ( qM, qV M )as qvb–valued 0–forms on qM and qvb–valued 1–forms on qM , respectively (and by σ ,Γ( qM, qV M ) ⊗ M Ω ( M ) is also the space of qvb–valued 1–forms on qM ). In this way a qlc q ∇ can be extended for qvb–valued forms on qMd q ∇ L : Ω • ( M ) ⊗ M Γ( qM, qV M ) −→ Ω • ( M ) ⊗ M Γ( qM, qV M ) d q ∇ R : Γ( qM, qV M ) ⊗ M Ω • ( M ) −→ Γ( qM, qV M ) ⊗ M Ω • ( M )(8)by means of d q ∇ L ( µ ⊗ M x ) = dµ ⊗ M x + ( − k µ q ∇ x if µ ∈ Ω k ( M ); and d q ∇ R ( x ⊗ M µ ) = (( σ ◦ q ∇ )( x )) µ + x ⊗ M dµ. These maps satisfy d q ∇ L ( µ ψ ) = ( − k µ ( d q ∇ L ψ ) + ( dµ ) ψ for all ψ ∈ Ω • ( M ) ⊗ M Γ( qM, qV M ), µ ∈ Ω k ( M ) and; d q ∇ R ( ˆ ψ η ) = ( d q ∇ R ˆ ψ ) η + ( − k ˆ ψ ( dη )for all ˆ ψ ∈ Γ( qM, qV M ) ⊗ M Ω k ( M ), η ∈ Ω • ( M ) . Definition 2.12 (Curvature) . Given ( qζ , q ∇ ) a qvb with qlc, we define the curvature of q ∇ as R q ∇ := d q ∇ L ◦ q ∇ : Γ( qM, qV M ) −→ Ω ( M ) ⊗ M Γ( qM, qV M ) . Now it should be clear how one can define morphisms between these structures.
Definition 2.13 (Parallel quantum vector bundle morphism) . Let ( qζ i , q ∇ i ) be a qvb over qM i = ( M i , · , , ∗ ) with a qlc and admissible differential ∗ –calculus (Ω • ( M i ) , d, ∗ ) , for i = 1 , . A parallel qvb morphism of type II and degree (pqvb morphism II − ) or a morphismof qvbs with qlcs of type II and degree is a pair ( F, A ) , where F : Ω • ( M ) −→ Ω • ( M ) is a graded differential ∗ –algebra morphism; A : Γ( qM , qV M ) −→ Γ( qM , qV M ) is a linear map such that A ( pxp ′ ) = F ( p ) A ( x ) F ( p ′ ) for all p , p ′ ∈ M , x ∈ Γ( qM , qV M ) ; (9) Γ( qM , qV M ) q ∇ −−−−−−−−−−→ Ω ( M ) ⊗ M Γ( qM , qV M ) A y (cid:9) y F ⊗ M A Γ( qM , qV M ) −−−−−−−−−−→ q ∇ Ω ( M ) ⊗ M Γ( qM , qV M ) and (10) Ω • ( M ) ⊗ M Γ( qM , qV M ) σ −−−−−−−−−−→ Γ( qM , qV M ) ⊗ M Ω • ( M ) F ⊗ M A y (cid:9) y A ⊗ M F Ω • ( M ) ⊗ M Γ( qM , qV M ) −−−−−−−−−−→ σ Γ( qM , qV M ) ⊗ M Ω • ( M ) . If (Ω • ( M ) , d, ∗ ) = (Ω • ( M ) , d, ∗ ) , a pqvb morphism of type I and degree or a pqvb mor-phism of degree is a pqvb morphism II − with F = id Ω • ( M ) . These kinds of morphismswill be denote just by A .A parallel qvb morphism of type II and degree (pqvb morphism II − ) or a morphismof qvbs with qlcs of type II and degree is a pair ( F, A ) , with F : Ω • ( M ) −→ Ω • ( M ) a graded differential antilinear and ∗ –antimultiplicative map; A : Γ( qM , qV M ) −→ Γ( qM , qV M ) an antilinear map such that A ( pxp ′ ) = F ( p ′ ) A ( x ) F ( p ) for all p , p ′ ∈ M , x ∈ Γ( qM , qV M ) ; (11) Γ( qM , qV M ) q ∇ −−−−−−−−−−→ Ω ( M ) ⊗ M Γ( qM , qV M ) A y (cid:9) y F ⊗ T M A Γ( qM , qV M ) −−−−−−−−−−→ σ ◦ q ∇ Γ( qM , qV M ) ⊗ M Ω ( M ) and (12) Ω • ( M ) ⊗ M Γ( qM , qV M ) σ −−−−−−−−−−→ Γ( qM , qV M ) ⊗ M Ω • ( M ) F ⊗ T M A y (cid:9) y A ⊗ T M F Γ( qM , qV M ) ⊗ M Ω • ( M ) ←−−−−−−−−−− σ Ω • ( M ) ⊗ M Γ( qM , qV M ) , where F ⊗ T M A and A ⊗ T M F are the twisted tensor product of F with A and A with F ,respectively. If (Ω • ( M ) , d, ∗ ) = (Ω • ( M ) , d, ∗ ) , a pqvb morphism of type I and degree or apqvb morphism of degree is a pqvb morphism II − with F = ∗ . These kinds of morphismswill be denote just by A .We going to define Mor qVB q ∇ (( qζ , q ∇ ) , ( qζ , q ∇ )) as the set of all pqvb morphisms II − between ( qζ , q ∇ ) and ( qζ , q ∇ ) ; and Mor qVB q ∇ (( qζ , q ∇ ) , ( qζ , q ∇ )) as the setof all pqvb morphisms II − between ( qζ , q ∇ ) and ( qζ , q ∇ ) . Finally we define the set ofall pqvb morphisms II between ( qζ , q ∇ ) and ( qζ , q ∇ ) as Mor qVB q ∇ (( qζ , q ∇ ) , ( qζ , q ∇ )) := UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 11
Mor qVB q ∇ (( qζ , q ∇ ) , ( qζ , q ∇ )) ∪ Mor qVB q ∇ (( qζ , q ∇ ) , ( qζ , q ∇ )) . In the same way we define
Mor qVB q ∇ qM (( qζ , q ∇ ) , ( qζ , q ∇ )) := Mor qVB q ∇ qM (( qζ , q ∇ ) , ( qζ , q ∇ )) ∪ Mor qVB q ∇ qM (( qζ , q ∇ ) , ( qζ , q ∇ )) . One can show easily that qvbs with qlcs and pqvb morphisms (of any type) form a categorywhere composition of morphisms is ( F , A ) ◦ ( F , A ) := ( F ◦ F , A ◦ A ) and the identitymorphism is (id Ω • ( M ) , id Γ( qM,qV qα M ) ). With this composition, morphisms have a natural Z –grading. Definition 2.14 (The categories of quantum vector bundles with quantum linear connec-tions) . We will denote by qVB q ∇ the category whose objects are qvbs and whose morphismsare pqvb morphisms of type II. Also for a fixed quantum space qM , we define qVB q ∇ qM asthe category whose objects are qvbs over a fixed qM with qlcs and whose morphisms are pqvbmorphisms. Using definition 2.13 it is easy to get relations between the maps d q ∇ L , d q ∇ R , for example(13) Ω • ( M ) ⊗ M Γ( qM , qV M ) d q ∇ L 1 −−−−−−−−−−→ Ω • ( M ) ⊗ M Γ( qM , qV M ) F ⊗ M A y (cid:9) y F ⊗ M A Ω • ( M ) ⊗ M Γ( qM , qV M ) −−−−−−−−−−→ d q ∇ L 2 Ω • ( M ) ⊗ M Γ( qM , qV M )for all ( F, A ) ∈ Mor qVB q ∇ (( qζ , q ∇ ) , ( qζ , q ∇ )).Now in a similar way that we defined cross corepresentation morphisms we can define crosspqvb morphisms of type II Mor qVB q ∇ Z (( qζ , q ∇ ) , ( qζ , q ∇ ); ( qζ , q ∇ ) , ( qζ , q ∇ ))and cross pqvb morphisms Mor qVB q ∇ Z qM (( qζ , q ∇ ) , ( qζ , q ∇ ); ( qζ , q ∇ ) , ( qζ , q ∇ )) . Definition 2.15 (The cross categories of quantum vector bundles with quantum linearconnections) . We define qVB q ∇ Z as the category whose objects are ordered pairs (( qζ , q ∇ ) , ( qζ , q ∇ )) where ( qζ , q ∇ ) , ( qζ , q ∇ ) ∈ Obj ( qVB q ∇ ) and whose morphisms are cross pqvbmorphisms II . In a similar way we define the category qVB q ∇ Z qM . To finish we going to present a version of the funtors defined in last subsection for qVB q ∇ qM and qVB q ∇ Z qM .Let qζ = (Γ( qM, qV M ) , + , · ) be a qvb. Then the following multiplications · : M × Γ( qM, qV M ) −→ Γ( qM, qV M ) , ( p, x ) x p ∗ · : Γ( qM, qV M ) × M −→ Γ( qM, qV M )( x, p ) p ∗ x Composition of morphisms follows the same rules as cross corepresentation morphisms. The identitymorphism of any object (( qζ , q ∇ ) , ( qζ , q ∇ )) is ((id Ω( M ) , id Γ( qM,qV qα M ) ) , (id Ω( M ) , id Γ( qM,qV qα M ) )). endow to (Γ( qM, qV M ) , +) with another M –bimodule structure, which will be denote by(Γ( qM, qV M ) , + , · ) and it turns out to be a finitely generated projective left–right M –module as well. We going to use the notation x for an element of Γ( qM, qV M ). In this way,taking a qvb over qM , qζ = (Γ( qM, qV M ) , + , · ), we define the conjugate qvb of it as the qvbover qM qζ = (Γ( qM, qV M ) , + , · ) . Just like we did for
Rep qG , one can always takeid Γ( qM,qV M ) : Γ( qM, qV M ) −→ Γ( qM, qV M ) x x. With this, if (Ω • ( M ) , d, ∗ ) is an admissible differential ∗ –calculus for qζ , the map σ := ( ∗ ⊗ T M id Γ( qM,qV M ) ) ◦ σ − ◦ ( ∗ ⊗ T M id − qM,qV M ) )tells us that (Ω • ( M ) , d, ∗ ) is an admissible differential ∗ –calculus for qζ as well, where σ isgiven in Equation (5). Even more for every qlc q ∇ on qζ , the linear map q ∇ := (id Γ( qM,qV M ) ⊗ T M ∗ ) ◦ σ ◦ q ∇ ◦ id − qM,qV M ) is a qlc on qζ which is usually known as the conjugate qlc of q ∇ . Definition 2.16 (Conjugate functor) . Let us define the conjugate functor on qVB q ∇ qM as thegraded–preserving covariant endofunctor − : qVB q ∇ qM −→ qVB q ∇ qM such that on objects is given by − ( qζ , q ∇ ) := ( qζ, q ∇ ) and on morphisms is given by − ( A ) := id Γ( qM,qV M ) ◦ A ◦ id − qM,qV M ) if A : Γ( qM, qV M ) −→ Γ( qM, qV M ) . Given two qvbs over qM , qζ i = (Γ( qM, qV M i ) , + , · ) ( i = 1, 2), we define the direct sum(or the Whitney sum) of qvbs as qζ ⊕ qζ = (Γ( qM, qV M ) ⊕ Γ( qM, qV M ) , + ⊕ , · ⊕ ) . On the other hand, if (Ω • ( M ) , d, ∗ ) is an admissible differential ∗ –calculus on qζ i ( i = 1,2), the map (considering the corresponding isomorphism) σ ⊕ := σ ⊕ σ guarantees us that (Ω • ( M ) , d, ∗ ) is an admissible differential ∗ –calculus on qζ ⊕ qζ as well,where σ i is the map given in Equation (5) for each qvb. Furthermore, for a qlc q ∇ i on qζ i ,we define the direct sum of qlcs by means of q ∇ ⊕ := q ∇ ⊕ q ∇ which is a qlc on qζ ⊕ qζ . UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 13
Definition 2.17 (Direct sum functor) . The direct sum functor on qvbs with qlcs is thegraded–preserving covariant functor M : qVB q ∇ Z qM −→ qVB q ∇ qM such that on objects is defined by M (( qζ , q ∇ ) , ( qζ , q ∇ )) := ( qζ ⊕ qζ , q ∇ ⊕ ) and on morphisms is defined by M ( A, A ′ ) := A ⊕ A ′ if ( A, A ′ ) has degree , and M ( A, A ′ ) := A ⊕ T A ′ if ( A, A ′ ) has degree , where A ⊕ T A ′ is the twisted direct sum of A with A ′ . Finally if one takes two qvbs over qM , qζ i = (Γ( qM, qV M i ) , + , · ) ( i = 1, 2), the qvb qζ ⊗ qζ = (Γ( qM, qV M ) ⊗ M Γ( qM, qV M ) , + ⊗ , · ⊗ )receives the name of the tensor product of qvbs. Also taking an admissible differential ∗ –calculus on qζ i ( i = 1, 2), (Ω • ( M ) , d, ∗ ), we get that it is an admissible differential ∗ –calculuson qζ ⊗ qζ to by means of σ ⊗ := (id Γ( qM,qV M ) ⊗ M σ ) ◦ ( σ ⊗ M id Γ( qM,qV M ) ) . Moreover given a qlc q ∇ i on qζ i , the qlc on qζ ⊗ qζ q ∇ ⊗ : Γ( qM, qV M ) ⊗ M Γ( qM, qV M ) −→ Ω ( M ) ⊗ M (Γ( qM, qV M ) ⊗ M Γ( qM, qV M ))such that q ∇ ⊗ ( x ⊗ M x ) = q ∇ ( x ) ⊗ M x + ( σ − ⊗ M id Γ( qM,qV M ) )( x ⊗ M q ∇ ( x ))receives the name of the tensor product of qlcs. Definition 2.18 (Tensor product functor) . The tensor product functor on qvbs with qlcs isdefined as the graded–preserving covariant functor O : qVB q ∇ Z qM −→ qVB q ∇ qM such that on objects is given by O (( qζ , q ∇ ) , ( qζ , q ∇ )) := ( qζ ⊗ qζ , q ∇ ⊗ ) and on morphisms is defined by O ( A, A ′ ) := A ⊗ M A ′ if ( A, A ′ ) has degree , and O ( A, A ′ ) := A ⊗ T M A ′ if ( A, A ′ ) has degree . It is well–defined by the right Leibniz rule.
It is easy to see that tensor products are associative and distributive over direct sums.Also we have to notice that we could define a qlc as a linear map from Γ( qM, qV M ) toΓ( qM, qV M ) ⊗ M Ω • ( M ) and with all necessary changes we could recreate all theory presentedhere. Furthermore all our functors have an extension to qVB q ∇ and qVB q ∇ Z .2.3. Quantum Principal Bundles and Quantum Principal Connections.
This sub-section will be based on the theory developed by M. Durdevich in the text [SZ] written byS. Sonz (especially because we going to use the notation of this book with little changes).Also one can check this theory in the original work [D1], [D2].
Definition 2.19 (Quantum principal qG –bundle) . Let qM = ( M , · , , ∗ ) be a quantum spaceand let qG be a cqg. A quantum principal qG –bundle (qpqgb) over qM is a quantum structureformally represented by the triplet qζ = ( qGM, qM, GM Φ) , where qGM = ( GM , · , ∗ ) is a quantum space called the total quantum space, with qM asquantum subspace which receives the name of base quantum space, and GM Φ :
GM −→ GM ⊗ G is a ∗ –algebra morphism that satisfies (1) GM Φ is a qG –representation. (2) GM Φ( x ) = x ⊗ if and only if x ∈ M . (3) The linear map β : GM ⊗ GM −→ GM ⊗ G given by β ( x ⊗ y ) := x · GM Φ( y ) = ( x ⊗ ) · GM Φ( y ) is surjective. One has to notice that in this situation qM appears as a secondary object: right invari-ant elements . There are a lot of extra structure that we have to add in order to get anoncommutative version of the concept of principal connections. First of all Definition 2.20 (Differential calculus) . Given a qpqgb over qM , qζ , a graded differentialcalculus on it is (1) A graded differential ∗ –algebra over GM (Ω • ( GM ) , d, ∗ ) , such that it is generatedas graded differential ∗ –algebra by Ω ( GM ) = GM (quantum differential forms on qGM ). (2) A bicovariant ∗ –FODC over G (Γ , d ) . (3) The map GM Φ is extendible to a graded differential ∗ –algebra morphism Ω Ψ : Ω • ( GM ) −→ Ω • ( GM ) ⊗ Γ ∧ , where (Γ ∧ , d ) is the universal differential envelope ∗ –calculus of the ∗ –FODC (Γ , d ) (which is just called universal differential calculus in [SZ] ). Second
Definition 2.21 (Horizontal forms) . Let qζ be a qpqgb over qM with a graded differentialcalculus. We define the space of horizontal forms as Hor • GM := { ϕ ∈ Ω • ( GM ) | Ω Ψ( ϕ ) ∈ Ω • ( GM ) ⊗ G} . UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 15
It can be proven that Hor • GM is a graded ∗ –subalgebra of Ω • ( GM ) and Ω Ψ(Hor • GM ) ⊆ Hor • GM ⊗ G , so H Φ := Ω Ψ | Hor • GM turns into a qG –representation [SZ]. Also we have Definition 2.22 (Base forms) . Let qζ be a qpqgb over qM with a graded differential calculus.We define the space of base forms as Ω • ( M ) := { µ ∈ Ω • ( GM ) | Ω Ψ( µ ) = µ ⊗ } . Ω • ( M ) is a graded differential ∗ –subalgebra of Ω • ( GM ) [SZ]. It is important to mentionthat in general, Ω • ( M ) is not generated as graded differential ∗ –algebras by Ω ( M ) = M .Furthermore, it turns out that the graded ∗ –algebra Hor • GM is generally not generated by GM and Hor GM [SZ]. Definition 2.23 (Vertical forms) . Let qζ be a qpqgb over qM with a graded differentialcalculus. We define the space of vertical forms as Ver • GM := GM ⊗ inv Γ ∧ , where inv Γ ∧ := { θ ∈ Γ ∧ | Φ Γ ∧ ( θ ) = ⊗ θ } , with Φ Γ ∧ the extension of the canonical corepresentation of G on Γ . Even more, since inv Γ ∧ is a graded differential ∗ –subalgebra of Γ ∧ , Ver • GM has a natural structure of graded vectorspace and defining the operations ( x ⊗ θ )( y ⊗ ˆ θ ) := xy (0) ⊗ ( θ ◦ y (1) )ˆ θ, ( x ⊗ θ ) ∗ := x (0) ∗ ⊗ ( θ ∗ ◦ x (1) ∗ ) and d v ( x ⊗ θ ) = x ⊗ dθ + x (0) ⊗ π ( x (1) ) θ, (Ver • GM , d v , ∗ ) is a graded differential ∗ –algebra generated by Ver GM = GM ⊗ C = GM ,where π : G −→ inv
Γ := inv Γ ∧ is the quantum germs map, π ( g ) ◦ g ′ = π ( gg ′ − ǫ ( g ) g ′ ) and GM Φ( x ) = x (0) ⊗ x (1) (we are using Sweedler’s notation) [SZ] . It is really important to emphasize that unlike the classical case, here in the noncommu-tative case there are not canonical calculus over the spaces. This gives us a richer theory.
Definition 2.24 (Quantum principal connection) . Let qζ be a qpqgb over qM with a gradeddifferential calculus. A linear map qω : inv Γ −→ Ω ( GM ) is a quantum principal connection (qpc) if it satisfies (1) qω ( θ ∗ ) = qω ( θ ) ∗ (2) Ω Ψ( qω ( θ )) = ( qω ⊗ id G ) ad ( θ ) + ⊗ θ, where ad : inv Γ −→ inv Γ ⊗ G is the right adjoin qG –representation. A qpqgb with a qpc will be denote by ( qζ , qω ) .A qpc is called regular if for all ϕ ∈ Hor k GM and θ ∈ inv Γ , we have qω ( θ ) ϕ = ( − k ϕ (0) qω ( θ ◦ ϕ (1) ) , where H Φ( ϕ ) = ϕ (0) ⊗ ϕ (1) . A qpc is called multiplicative if qω ( π ( g (1) )) qω ( π ( g (2) )) = 0 for all g ∈ R with φ ( g ) = g (1) ⊗ g (2) , where we are considering that R ⊆
Ker( ǫ ) is the right ideal of G associated to thebicovariant ∗ –FODC (Γ , d ) [SZ] . A really useful characteristic of regular qpc is that any homogeneous element of Ω • ( M )graded–commutes with all elements of Im( qω ). In the whole paper we will assume that allour qpcs are regular, so we will omit the word regular . Definition 2.25 (Curvature) . Taking a qpqgb with a qpc ( qζ , qω ) , we define the curvatureof qω as R qω := d ◦ qω − h ω, ω i with h ω, ω i : inv Γ −→ Ω ( GM ) given by h ω, ω i ( θ ) = m Ω • ( GM ) ◦ ( qω ⊗ qω ) ◦ δ, where δ : inv Γ −→ inv Γ ⊗ inv Γ is an embedded differential ( [D2] ) and m Ω • ( GM ) is the multipli-cation map of Ω • ( GM ) . A really important property of multiplicative qpcs is the fact that for these connectionsthe curvature does not depend on the map δ [D2]. Definition 2.26 (Covariant derivative) . For a given qpqgb with a qpc ( qζ , qω ) , the first–orderlinear map D qω : Hor • GM −→
Hor • GM such that for every ϕ ∈ Hor k GM D qω ( ϕ ) = dϕ − ( − k ϕ (0) qω ( π ( ϕ (1) )) with H Φ( ϕ ) = ϕ (0) ⊗ ϕ (1) is called the covariant derivative of qω . We have to remark that the last definition is not the most general way to define thecovariant derivative ([D2]), but it will be enough for our purposes. Furthermore one canprove that D qω satisfies(14) D qω ∈ Mor Rep ∞ qG ( H Φ , H Φ) . and(15) D qω ( ϕ ψ ) = D qω ( ϕ ) ψ + ( − k ϕ D qω ( ψ ) , D qω ◦ ∗ = ∗ ◦ D qω , D qω | Ω • ( M ) = d | Ω • ( M ) where ϕ ∈ Hor k GM and ψ ∈ Hor • GM [SZ]. UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 17
Remark 2.27.
Let T a complete set of mutually nonequivalent irreducible finite dimensional qG –representations. From this point until the end of this paper we going to consider for agiven qζ = ( qGM, qM, GM Φ) and each qα ∈ T that there exists { T L k } dk =1 ⊆ Mor Rep ∞ qG ( qα, GM Φ) such that d X k =1 x qα ∗ ki x qαkj = δ ij , with x qαki := T L k ( e i ) , where { e i } ni =1 is some fixed basis of the vector space V qα on which qα coacts; and the following relation holds ( Z qα X qα ) T X qα ∗ = Id n , where X qα = ( x qαij ) ∈ M d × n ( GM ) , Id n is the identity element of M n ( GM ) and Z qα = ( z qαij ) ∈ M d ( C ) is a strictly positive element [D2] . Given a qpqgb, qζ = ( qGM, qM, GM Φ), let us consider a graded ∗ –algebra (Ω • H , , ∗ ) suchthat Ω = GM with a graded ∗ –subalgebra Ω •M with structure of graded differential ∗ –algebra generated by its degree 0 elements Ω M = M and a qG –representation Φ coactingon Ω • H such that Ω •M is exactly the set of all Φ–invariant elements, where Φ is also a graded ∗ –algebra morphism which extends GM Φ. Now we going to denote by(16)
Der the space of all first–order linear maps D that satisfies Equations (14), (15) with respect to { Φ , Ω • H , (Ω •M , d, ∗ ) } . It can be proven that there exists a bicovariant ∗ –FODC over G , (Γ , d ) such that together { Φ , Ω • H , (Ω •M , d, ∗ ) } one can get a differential calculus on qζ [D4]. Even more, Ω • H is the spaceof horizontal forms of this calculus, Ω •M corresponds to the space of base forms and qpcs(relative to this differential calculus) on qζ are in bijection with elements of Der in a naturalway: for every qpc qω , D qω ∈ Der and for every D ∈ Der there exists a unique qω such that D qω = D [D4]. In other words, we just need Der to get all the previous structures that wehave just presented in this subsection. Another important result of this particular way toget the differential calculus is that every qpc is multiplicative as well.
Remark 2.28.
From this point until the end of this paper we shall assume that every qpc isgiven by the above conditions.
Definition 2.29 (Quantum principal bundle morphism) . Let qζ i = ( qGM i , qM i , GM Φ i ) be aquantum principal qG i –bundle with a qpc qω i ( i = 1 , ). A parallel quantum principal bundlemorphism of type II (pqpb morphism II ) is a pair ( h, F ) , where h : G −→ G is a ∗ –hopf algebra morphism and F : Hor • GM −→ Hor • GM is a graded ∗ –algebra morphism such that (17) Hor • GM H Φ −−−−−−−−−−→ Hor • GM ⊗ G F y (cid:9) y F ⊗ h Hor • GM −−−−−−−−−−→ H Φ Hor • GM ⊗ G . with F ◦ D qω = D qω ◦ F. If (Ω • ( M ) , d, ∗ ) = (Ω • ( M ) , d, ∗ ) , a pqpb morphism of type I or a pqpb morphism is a pqpbmorphism II with F | Ω • ( M ) = id Ω • ( M ) . It is clear that qpbs with qpcs and pqpb morphisms become into a category where compo-sition of morphisms is composition of maps and the identity morphism is id
Hor • GM . In thisway Definition 2.30 (The category of quantum principal qG –bundles with quantum principalconnections) . We define qPB qω as the category whose objects are triplets ( qG, qζ , qω ) , where qG is cqg and ( qζ , qω ) is a qpqgb with a qpc; and whose morphisms are pqpb morphisms II.We will denote by qPB qωqM the category whose objects are the same as before but qζ is a qpqgbover a fixed quantum space qM and whose morphisms are pqpb morphisms. The functor qAss qωqζ
In this section we going to define the association functor in the framework of noncomuu-tative differential geometry using the theory presented in last section and we going to give acharacterization of it.3.1.
Construction.
Let qα ∈ T coacting on the vector space V qα and let qζ = ( qGM, qM, GM Φ) be a qpqgb. Let us define(18) Γ( qM, qV qα M ) := Mor qG ( qα, GM Φ) . Notice that Γ( qM, qV qα M ) is a M –bimodule by means of · : M ⊗ Γ( qM, qV qα M ) −→ Γ( qM, qV qα M ) p ⊗ T pT · : Γ( qM, qV qα M ) ⊗ M −→ Γ( qM, qV qα M ) T ⊗ p T p.
Furthermore, for every T ∈ Γ( qM, qV qα M ), T = d X k =1 p Tk T L k = d X k =1 T R k ˆ p Tk , where T R k = d X i =1 z ki T L i and p Tk = n X i =1 T ( e i ) x qα ∗ ki , ˆ p Tk = d,n X i,j =1 y qαik w qα ∗ ij T ( e j ) ∈ M UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 19 with ( w qαij ) = Z qα X qα and Y qα = ( y qαij ) ∈ M d ( C ) the inverse of Z qα (see Remark 2.27)[D2]. These sets of M –generators help us to show that Γ( qM, qV qα M ) is actually a finitelygenerated projective left–right M –module [D2]; so qζ qα = (Γ( qM, qV qα M ) , + , · )is a qvb over qM .Now let us fix a qpc qω on qζ . The mapsΥ − qα : Ω • ( M ) ⊗ M Γ( qM, qV qα M ) −→ Mor Rep ∞ qG ( qα, H Φ)ˆΥ − qα : Γ( qM, qV qα M ) ⊗ M Ω • ( M ) −→ Mor Rep ∞ qG ( qα, H Φ)such that Υ − qα ( µ ⊗ M T ) = µ T and ˆΥ − qα ( T ⊗ M µ ) = T µ are M –bimodule isomorphisms, where Mor Rep ∞ qG ( qα, H Φ) has the M –bimodule structuresimilar to the one of Γ( qM, qV qα M ). Specifically their inverses are given by(19) Υ qα ( τ ) = d X k =1 µ τK ⊗ M T L k and ˆΥ qα ( τ ) = d X k =1 T R k ⊗ M ˆ µ τk , with µ τk = n X i =1 τ ( e i ) x qα ∗ ki , ˆ µ τk = d,n X i,j =1 y qαik w qα ∗ ij τ ( e j ) ∈ Ω( M ) . In this way, taking(20) σ qα := ˆΥ qα ◦ Υ − qα we obtain that the space of base forms (Ω • ( M ) , d, ∗ ) is an admissible differential ∗ –calculusfor qζ qα and the linear map ∇ qωqα : Γ( qM, qV qα M ) −→ Ω ( M ) ⊗ M Γ( qM, qV qα M ) T Υ qα ◦ D qω ◦ T, (21)is a qlc on qζ qα . In summary ( qζ qα , ∇ qωqα ) ∈ Obj ( qVB q ∇ qM ) provided that qα ∈ T . Proposition 3.1. If qα i ∈ T coacts on V i (for i = 1 , ) and f ∈ Mor Rep qG ( qα , qα ) , thenthe map A f : Γ( qM, qV M ) −→ Γ( qM, qV M ) T T ◦ f is an element of Mor qVB q ∇ qM (( qζ qα , ∇ qωqα ) , ( qζ qα , ∇ qωqα )) . Also if f ∈ Mor Rep qG ( qα , qα ) ,the map A ∗ f : Γ( qM, qV M ) −→ Γ( qM, qV M ) T T ∗ ◦ f is an element of Mor qVB q ∇ qM (( qζ qα , ∇ qωqα ) , ( qζ qα , ∇ qωqα )) , where T ∗ : V −→ GM is given by T ∗ ( v ) = T ( v ) ∗ . Proof.
It is clear that A f is a M –bimodule morphism. For every T ∈ Γ( qM, qV M )Υ qα ( D qω ◦ T ) = ∇ qωqα ( T ) = d X k =1 µ k ⊗ M T L2 k , for some µ k ∈ Ω( M ) such that D qω ◦ T = d X k =1 µ k T L2 k with { T L2 k } d k =1 ⊆ Γ( qM, qV M ) the setof left generators; so (id Ω ( M ) ⊗ M A f ) ∇ qωqα ( T ) = d X k =1 µ k ⊗ M T L2 k ◦ f. On the other handΥ qα ( D qω ◦ A f ( T )) = ∇ qωqα ( A f ( T )) = d X k =1 µ k ⊗ M T L1 k , with µ k ∈ Ω( M ) such that D qω ◦ A f ( T ) = D qω ◦ T ◦ f = d X k =1 µ k T L1 k where { T L1 k } d k =1 ⊆ Γ( qM, qV M ) is the set of left generators. Applying Υ − qα in both last equalities it is easyto show that they are the same element, so A f satisfies Diagram (9) for F = id Ω • ( M ) . In asimilar way it can be proven that A f satisfies Diagram (10) for F = id Ω • ( M ) and hence A f ∈ Mor qVB q ∇ qM (( qζ qα , ∇ qωqα ) , ( qζ qα , ∇ qωqα )).Analogously one can prove that A ∗ f ∈ Mor qVB q ∇ qM (( qζ qα , ∇ qωqα ) , ( qζ qα , ∇ qωqα )). For exam-ple, for all T ∈ Γ( qM, qV M )ˆΥ − qα (( ∗ ⊗ T M A ∗ f ) ∇ qωqα ( T )) = d X k =1 ( T L2 ∗ k ◦ f ) µ ∗ k ;while ( ˆΥ − qα ◦ σ qα ◦ ∇ qωqα )( A ∗ f ( T )) = D qω ( T ∗ ◦ f );so evaluating on any basis it is easy to see that they are the same element and therefore A ∗ f satisfies Diagram (11). (cid:4) Due to the fact that every qG –representation is (isomorphic with degree 0 morphismsto) a direct sum of irreducible ones [MVD], we can extend in a really natural way all ourprevious results using the functor L (see Definitions 2.7, 2.17). Thus taking ( qζ , qω ) ∈ Obj ( qPB qωqG, qM ), we define the association quantum vector bundle of qζ with respect to any qG –representation qα as the qvb over qM , qζ qα , formally represented by its space of smoothsections (Γ( qM, qV qα M ) , + , · ) , where Γ( qM, qV qα M ) := Mor qG ( qα, GM Φ); and we define the induced quantum linear con-nection of qω on qζ qα as the linear map ∇ qωqα : Γ( qM, qV qα M ) −→ Ω ( M ) ⊗ M Γ( qM, qV qα M )given by Equation (21) (using L ). UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 21
Remark 3.2.
Definition and interpretation of qζ qα = (Γ( qM, qV qα M ) , + , · ) is in accordancewith Gauge Principle [KMS] , [SW] . The Gauge Principle also gives us a characterizationof vector–bundle differential forms which we can observe with the isomorphisms Υ qα , ˆΥ qα .Finally it also tells us that induced connections on associated bundles are given by covariantderivatives of principal connections, which is in clear resonance with our definition. Proposition 3.1 also can be extended using L , so finally we have Definition 3.3 (Quantum association functor) . Let ( qζ , qω ) be a qpqgb over qM with aqpc. Then we define the quantum association functor as the graded–preserving contravariantfunctor qAss qωqζ : Rep qG −→ qVB q ∇ qM such that on objects is given by qAss qωqζ qα = ( qζ qα , ∇ qωqα ) and on morphisms is given by qAss qωqζ ( f ) = A f if f has degree , and qAss qωqζ ( f ) = A ∗ f if f has degree . Properties.
Now that we have our desired functor, we going to prove some propertiesof it before the proof of the categorical equivalence. First of all notice that qAss qωqζ is linearin morphisms no matters the grade. Second
Proposition 3.4. qAss qωqζ qα triv V = ( qζ qα triv V , ∇ qωqα triv V ) is a trivial qvb with a trivial qlc (seeExamples 2.2, 2.11)Proof. Let { e i } ni =1 be a basis of V and { f i } ni =1 its dual basis. Then the set { T f i } ni =1 , where T f i : V −→ M v −→ f i ( v ) , is a left and right M –basis of Γ( qM, qV qα M ), so qζ qα triv V is a trivial qvb. With this basis itis easy to show that ∇ qωqα triv V is trivial. (cid:4) Corollary 3.5. qζ qα triv C ∼ = ( M , + , · ) and under this isomorphism the induced qlc is just D qω | M = d | M . Moreover, σ qα triv C = id Ω • ( M ) . By construction, it should be clear that qAss qωqζ commutes with L , but also Proposition 3.6.
For every qα ∈ Obj ( Rep qG ) , there is a natural isomorphism between qAss qωqζ − and − qAss qωqζ (see Definitions 2.6, 2.16). Moreover, the isomorphism given hasdegree .Proof. Notice that it is enough to prove the proposition for elements of T ; so let qα ∈ T and qζ qα = (Γ( qM, qV M ) , + , · ) , qζ qα = (Γ( qM, qV qα M ) , + , · ) . Actually we have Γ( qM, qV M ) =
Mor Rep ∞ qG ( qα, GM Φ) . Let us consider the M –bimodule isomorphism A qα : Γ( qM, qV M ) −→ Γ( qM, qV qα M ) T T ∗ . If ∇ qωqα is the induced connection on qζ qα and ∇ qωqα is the induced connection on qζ qα , thenfor all T ∈ Γ( qM, qV qα M ) ∇ qωqα ( A qα ( T )) = ∇ qωqα ( T ∗ ) = d X k =1 (id Γ( qM,qV qα M ) ⊗ T M ∗ )( σ qα ( µ D qω ◦ T ∗ k ⊗ M T L k ))= d X k =1 (id Γ( qM,qV qα M ) ⊗ T M ∗ )( T R k ⊗ M ˆ µ D qω ◦ T ∗ k )= d X k =1 (ˆ µ D qω ◦ T ∗ k ) ∗ ⊗ M T R k = d X k =1 (ˆ µ ( D qω ◦ T ) ∗ k ) ∗ ⊗ M T R k where in the ultimate equality we have used Equation (15) and the Equation (19) for qα taking into account that Mor Rep ∞ qG ( qα, H Φ) =
Mor Rep ∞ qG ( qα, H Φ). Also we get((id Ω ( M ) ⊗ M A qα ) ◦ ∇ qωqα )( T ) = d X k =1 (id Ω ( M ) ⊗ M A qα )[(ˆ µ ( D qω ◦ T ) ∗ k ) ∗ ⊗ M T R ∗ k ]= d X k =1 (ˆ µ ( D qω ◦ T ) ∗ k ) ∗ ⊗ M T R k and hence A qα satisfies Diagram (9) for F = id Ω ( M ) . A similar calculation shows that(( A qα ⊗ M id Ω • ( M ) ) ◦ σ qα )( µ ⊗ M T ) = d X k =1 T L k ⊗ M ( µ T ∗ µ ∗ ) ∗ = ( σ qα ◦ (id Ω • ( M ) ⊗ M A qα ))( µ ⊗ M T )and by linearity we conclude that A qα fulfills Diagram (10); soqAss qωqζ − qα ∼ = − qAss qωqζ qα. Finally, a quick calculation shows that for any f ∈ Mor
Rep qG ( qα , qα )(22) Γ( qM, qV M ) qAss qωqζ − ( f ) −−−−−−−−−−→ Γ( qM, qV M ) A qα y (cid:9) y A qα Γ( qM, qV M ) −−−−−−−−−−→ − qAss qωqζ ( f ) Γ( qM, qV M )and proposition follows. (cid:4) and UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 23
Proposition 3.7.
For every ( qα , qα ) ∈ Obj ( Rep Z qG ) , there is a natural isomorphism be-tween qAss qωqζ N and N (qAss qωqζ , qAss qωqζ ) (see Definitions 2.8, 2.18), defining (qAss qωqζ , qAss qωqζ ) : Rep Z qG −→ qVB q ∇ Z qM in the obviously way. In addition, the isomorphism given has degree .Proof. As before, it is enough to prove the proposition for elements of T ; so let us take qα i ∈ T coacting on V i for i = 1, 2. Using the sets of M –generators shown in Remark 2.27 onecan prove that the map A − qα ,qα : Γ( qM, qV M ) ⊗ M Γ( qM, qV M ) −→ Γ( qM, q ( V ⊗ V ) M )such that A − qα ,qα ( T ⊗ M T ) : V ⊗ V −→ GM is given by A − qα ,qα ( T ⊗ M T )( v ⊗ v ) = T ( v ) T ( v )is a M –bimodule isomorphism, where Γ( qM, qV i M ) := Mor Rep ∞ qG ( qα i , GM Φ) with i = 1,2 and Γ( qM, q ( V ⊗ V ) M ) := Mor Rep ∞ qG ( qα ⊗ qα , GM Φ). Using A qα ,qα , Υ qα ⊗ qα andˆΥ qα ⊗ qα we can induced M –bimodule isomorphisms Mor
Rep ∞ qG ( qα ⊗ qα , H Φ) ∼ = Ω • ( M ) ⊗ M Γ( qM, qV M ) ⊗ M Γ( qM, qV M ) Mor
Rep ∞ qG ( qα ⊗ qα , H Φ) ∼ = Γ( qM, qV M ) ⊗ M Γ( qM, qV M ) ⊗ M Ω • ( M )which we going to denote by Λ and ˆΛ, respectively. With this a direct calculation like in thelast proposition proves thatΛ − ◦ (id Ω ( M ) ⊗ M A qα ,qα ) ◦ ∇ qωqα ⊗ qα = Λ − ◦ ∇ ⊗ ◦ A qα ,qα , ˆΛ − ◦ ( A qα ,qα ⊗ M id Ω • ( M ) ) ◦ σ qα ⊗ qα = ˆΛ − ◦ σ ⊗ ◦ (id Ω • ( M ) ⊗ M A qα ,qα ) . Thus A qα ,qα is a pqvb isomorphism of degree 0 betweenqAss qωqζ O ( qα , qα ) and O (qAss qωqζ qα , qAss qωqζ qα );and also one can prove(23) Γ( qM, q ( V ⊗ V ′ ) M ) qAss qωqζ ⊗ ( f,f ′ ) −−−−−−−−−−→ Γ( qM, q ( V ⊗ V ′ ) M ) A qα ,qα ′ y (cid:9) y A qα ,qα ′ Γ( qM, qV M ) ⊗ M Γ( qM, qV ′ M ) −−−−−−−−−−→ ⊗ (qAss qωqζ ( f ) , qAss qωqζ ( f ′ )) Γ( qM, qV M ) ⊗ M Γ( qM, qV ′ M )for any ( f, f ′ ) ∈ Mor
Rep Z qG (( qα , qα ′ ) , ( qα , qα ′ )). (cid:4) It is easy to show that the set of isomorphisms { A qα ,qα } satisfies Proposition 3.8.
In the context of last proposition ( A qα ,qα ⊗ M id V M ) ◦ A qα ,qα ⊗ qα = (id Γ( qM,qV M ) ⊗ M A qα ,qα ) ◦ A qα ,qα ⊗ qα . In other words, qAss qωqζ preserves associativity of N . A well–known result in the theory of qG –representations is that for any two irreduciblefinite dimensional corepresentations qα , qβ coacting on V and W respectively, Mor Rep qG ( qα, qβ ) = { } or Mor Rep qG ( qα, qβ ) = { λ f | λ ∈ C } , with f : V −→ W a linear isomorphism [MVD]. Also we have the same result for degree 1morphisms: Mor Rep qG ( qα, qβ ) = { } or Mor Rep qG ( qα, qβ ) = { λ f | λ ∈ C } , with f : V −→ W an antilinear isomorphism. Proposition 3.9.
The functor qAss qωqζ sends degree k monomorphisms into degree k epimor-phisms and vice versa, for k = 0 , .Proof. Let f ∈ Mor
Rep qG ( qα, qβ ). Of course if f is surjective, qAss qωqζ ( f ) is injective.Let us take a degree k injective morphism f . We know that there exists irreducible qG –representations qα i , qβ i coacting on V i and W i respectively such that [MVD] qα = n M i =1 qα i and qβ = m M j =1 qβ j . If π j : W −→ W j is the canonical projection and ι i : V i −→ V is the canonical embedding,then π j ◦ f ◦ ι i ∈ Mor k Rep qG ( qα i , qβ j ), so ( f ◦ ι i )( V i ) = W j or Im(( f ◦ ι i )( V i )) ∩ W j = { } .Without loss of generality, we shall assume ( f ◦ ι i )( V i ) = W i for i = 1 , ..., n . Fixing T ∈ Γ( qM, qV M ) we can choose T j ∈ Γ( qM, qW j M ) for j > n and in this way, for k = 0 thelinear map T ext : W = W × · · · × W n × W n +1 × · · · × W m −→ GM ( w , ... , w n , w n +1 , ... , w n ) n X j =1 ( T ◦ f − )( w j ) + m X j = n +1 T j ( w j )is an element of Γ( qM, qW M ) which satisfies A f ( T ext ) = T ;and for k = 1 T ext : W = W × · · · × W n × W n +1 × · · · × W m −→ GM ( w , ... , w n , w n +1 , ... , w n ) n X j =1 ( T ∗ ◦ f − )( w j ) + m X j = n +1 T j ( w j )is also an element of Γ( qM, qW M ) that fulfills A ∗ f ( T ext ) = T. Hence qAss qωqζ ( f ) is surjective. (cid:4) Corollary 3.10.
The functor qAss qωqζ is exact.
UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 25
For any qpggb qζ = ( qGM, qM, GM Φ) and qα ∈ T that coacts on V qα , let GM qα ⊆ GM be the multiple irreducible subspace corresponding to this qG –representation. Each GM qα is a M –bimodule and(24) GM ∼ = M qα ∈ T GM qα . as M –bimodules. Furthermore, GM qα ∼ = Γ( qM, qV qα M ) ⊗ V qα as M –bimodules via the map T ( v ) ←→ T ⊗ v, with qζ qα = (Γ( qM, qV qα M ) , + , · ). This identification is a degree 0 corepresentation isomor-phism between [D2] GM Φ | GM qα and id Γ( qM,qV qα M ) ⊗ qα. According to Corollary 3.5, there is a canonical inclusion of M in the right side of Equation(24) since qα triv C ∈ T . Even more, using Proposition 3.7 we can get a ∗ –algebra structure onthe right side of Equation (24) by means of T · T := A − qα ,qα ( T ⊗ M T );and the ∗ operation is defined such that[qAss qωqζ ( f )( T )] ∗ = T ◦ f for any degree 1 corepresentation morphism f . Since qG coacts on the right side of Equation(24) with the direct sum of id Γ( qM,qV qα M ) ⊗ qα , Equation (24) holds as qpbs [D3]. In otherwords, given qAss qωqζ ,we can recreate qζ .On the other hand, we can always define a contravariant functor between Rep qG and thecategory of graded Ω • ( M )–bimodules (with their corresponding graded morphisms of degree1) qAss H qζ : Rep qG −→ Ω • ( M ) − GradBimod such that in objects is qAss H qζ qα = Mor Rep ∞ qG ( qα, H Φ)and in morphisms is qAss H qζ ( f ) = A H f if f has degree 0 and qAss H qζ ( f ) = A ∗ H f if f has degree 1, where A H f and A ∗ H f are defined in a similar way that A f and A ∗ f . Due tothe fact that(25) Hor • GM ∼ = M qα ∈ T Hor • GM qα , as graded Ω • ( M )–bimodules withHor • GM qα ∼ = Mor Rep ∞ qG ( qα, H Φ) ⊗ V qα one can use qAss H qζ to rebuilt the graded ∗ –algebra Hor • GM and the coaction H Φ like wehave just done for GM and GM Φ. According to [D4], using { GM Φ , Hor • GM , (Ω • ( M ) , d, ∗ ) } one can recreate the whole differential calculus on qζ (see Definition 2.20). Furthermore, inthis context qpcs are in bijection with elements of Der (see Equation (16)).Considering that the graded Ω • ( M )–bimodule structure on Ω • ( M ) ⊗ M Γ( qM, qV qα M )is given by Equations (6), (7); Υ qα becomes into a graded–preserving Ω • ( M )–bimoduleisomorphism (clearly we have an analogous result for Γ( qM, qV qα M ) ⊗ M Ω • ( M ) and ˆΥ qα ).With this in mind, for every qα ∈ T we define a first–order linear map D qα : Mor Rep ∞ qG ( qα, H Φ) −→ Mor Rep ∞ qG ( qα, H Φ)given by(26) D qα := Υ − qα ◦ d ∇ qωqα L ◦ Υ qα : Mor Rep ∞ qG ( qα, H Φ) −→ Mor Rep ∞ qG ( qα, H Φ)(see Equation (8)). One has to notice that in this case, D qα ( τ ) = D qω ◦ τ (see Definition2.26). D qα satisfies(27) D qα triv C = d | Ω • ( M ) , D qα ⊗ qα ( τ · τ ) = D qα ( τ ) τ + ( − k τ D qα ( τ ) and D qα ◦ qAss H qζ ( f ) = qAss H qζ ( f ) ◦ D qα , for any f ∈ Mor
Rep qG ( qα , qα ). Using these properties one can induce a first–order an-tiderivation on the right side of Equation (25) which coincides with D qω and according to[D4], one can recovery qω from D qω . In summary, just using qAss ( qζ,qω ) and qAss H qζ it ispossible to rebuilt ( qζ , qω ).To conclude this subsection let us notice that (see Equation (8)) d ∇ qωqα L = Υ qα ◦ D qω ◦ Υ − qα , and d ∇ qωqα R = ˆΥ qα ◦ D qω ◦ ˆΥ − qα ;which implies d ∇ qωqα R = σ qα ◦ d ∇ qωqα L ◦ σ − qα . According to [D3], we conclude that the curvature of the induced connection (see Definitions2.12, 2.25) is R ∇ qωqα ( T ) = Υ qα ◦ D qω ◦ T = − Υ qα ◦ T R qω ( π ( T )) , where T : V −→ GM and T : V −→ G are given by (using Sweedler notation) GM Φ( T ( v )) = T ( v ) ⊗ T ( v ) . The Equivalence.
Now we will proceed to prove the categorical equivalence. We shallbegin with the following result.
Theorem 3.11.
Let qF : Rep qG −→ qVB q ∇ qM be a graded–preserving contravariant functor. For every qα ∈ Rep qG that coacts on V qα , let qF qα =: ( b qζ qα , q b ∇ qα ) with b qζ qα = ( \ Γ( qM, qV qα M ) , + , · ) and qF ( f ) =: b A f UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 27 if f has degree and qF ( f ) =: b A ∗ f if f has degree . The existence of q b ∇ qα implies the existence of a M –bimodule isomorphismwhich we will denote by b σ qα : Ω • ( M ) ⊗ M \ Γ( qM, qV qα M ) −→ \ Γ( qM, qV qα M ) ⊗ M Ω • ( M ) . Let us assume that qF satisfies (1) qF qα triv C ∼ = (( M , + , · ) , d | M ) . (2) For every ( qα , qα ) ∈ Obj ( Rep Z qG ) with qα , qα ∈ T , there is a natural isomor-phism between qF N and N ( qF , qF ) , defining ( qF , qF ) : Rep Z qG −→ qVB q ∇ Z qM in the obviously way. All isomorphisms given have degree . (3) If { υ qα,qβ } is the set of all previous isomorphisms, then ( υ qα ,qα ⊗ M id V qα M ) ◦ υ qα ,qα ⊗ qα = (id Γ( qM,qV qα M ) ⊗ M υ qα ,qα ) ◦ υ qα ,qα ⊗ qα . (4) For every qα ∈ T there exists a basis { e k } nk =1 of V qα and a set { x L k } dk =1 ⊆ Γ( qM, qV qα M ) such that d X k =1 υ − qα,qα ( b A ∗ id − V qα ( x L k ) ⊗ M x L k ) ⊗ ( e i ⊗ e j ) = δ ij ⊗ ∈ M ⊗ C , and there exists a strictly positive matrix Z qα = ( z qαij ) ∈ M d ( C ) such that X k,l,t υ − qα,qα ( z qαkl x L l c ti ⊗ M b A ∗ id − V qα ( x L k )) ⊗ ( e t ⊗ e j ) = δ ij ⊗ ∈ M ⊗ C , where C = ( c ij ) ∈ M n ( C ) is the canonical degree corepresentation morphism between qα and its second contragradiant representation. (5) For every qα ∈ T b σ qα ( ⊗ M x ) = x ⊗ M and d q b ∇ qα R = b σ qα ◦ d q b ∇ qα L ◦ b σ − qα . Then there exists a quantum principal qG –bundle over qM qζ such that qF is naturallyisomorphic to qAss qωqζ .Proof. Define a graded–preserving contravariant functor qF H : Rep qG −→ Ω • ( M ) − GradBimod such that in objects is qF H qα = Ω • ( M ) ⊗ M \ Γ( qM, qV qα M )(considering the Ω • ( M )–bimodule structure on Ω • ( M ) ⊗ M \ Γ( qM, qV qα M ) given by Equa-tions (6), (7)); and in morphisms is qF H ( f ) = b A H f := id Ω • ( M ) ⊗ M b A f , qF H ( f ) = b A ∗ H f := b σ − qα ◦ ( ∗ ⊗ T M b A ∗ f ) , depending on the degree of f ∈ Mor
Rep qG ( qα , qα ). Notice that qF H qα triv C ∼ = Ω • ( M ) (as graded ∗ –algebras), b σ qα triv C = id Ω • ( M ) and d d | M L = d | Ω • ( M ) .On the other hand it is easy to prove that for every ( qα , qα ) ∈ Obj ( Rep Z qG ) with qα , qα ∈ T , there exists a natural isomorphism between qF H N and N ( qF H , qF H ) and theseisomorphisms are given by { id Ω • ( M ) ⊗ M υ qα,qβ } , which have degree 0. It follows that qF H fulfills the hypothesis (3) of this theorem.Let us define the M –bimodule GM := M qα ∈ T \ Γ( qM, qV qα M ) ⊗ V qα and the graded Ω • ( M )–bimoduleHor • GM := M qα ∈ T Ω • ( M ) ⊗ M \ Γ( qM, qV qα M ) ⊗ V qα , where now we are assuming that qα coacts on V qα . qG naturally coacts on GM and Hor • GM via GM Φ := M qα ∈ T id \ Γ( qM,qV qα M ) ⊗ qα and H Φ := M qα ∈ T id Ω • ( M ) ⊗ M id \ Γ( qM,qV qα M ) ⊗ qα. Notice that Hor GM = GM and H Φ extends GM Φ. Using the method presented before([D3]), one can equip to GM (Hor • GM ) with a (graded) ∗ –algebra structure such that GM Φ( H Φ) is a (graded) ∗ –algebra morphism and in such way that Ω • ( M ) is a graded ∗ –subalgebraof Hor • GM . This implies that qζ := ( qGM, qM, GM Φ) := (( GM , · , , ∗ ) , ( M , · , , ∗ ) , GM Φ)is a quantum principal qG –bundle and the set { H Φ , Hor • GM , (Ω • ( M ) , d, ∗ ) } provides us adifferential calculus on qζ (see Definition 2.20; [D4]). Hypothesis (4) guarantees us that qζ satisfies the written in Remark 2.27. Under these conditions we have \ Γ( qM, qV qα M ) ∼ = Mor Rep ∞ qG ( qα, GM Φ)as M –bimodules [D3]; andΩ • ( M ) ⊗ M \ Γ( qM, qV qα M ) ∼ = Mor Rep ∞ qG ( qα, H Φ) ∼ = \ Γ( qM, qV qα M ) ⊗ M Ω • ( M )as graded Ω • ( M )–bimodules . The first isomorphism agrees with the map µ ⊗ M T µ T (taking in consideration the corresponding structures). Denoting this map by Υ − qα , theisomorphism between \ Γ( qM, qV qα M ) ⊗ M Ω • ( M ) and Mor Rep ∞ qG ( qα, H Φ) is given by ˆΥ − qα :=Υ − qα ◦ b σ − qα , so Equation (20) is satisfied. Using hypothesis (5) one can show that ˆΥ − qα agreeswith the map T ⊗ M µ T µ . Moreover, under these identifications [D3] b A f = A f and b A H f = A H f , We are assuming that the graded Ω • ( M )–bimodule structure on \ Γ( qM, qV qα M ) ⊗ M Ω • ( M ) is similarto the one of Ω • ( M ) ⊗ M \ Γ( qM, qV qα M ). UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 29 for degree 0 corepresentation morphisms; and b A ∗ f = A ∗ f and b A ∗ H f = A ∗ H f , for degree 1 corepresentation morphisms.For every qα ∈ T we define a first–order linear map b D qα : Mor Rep ∞ qG ( qα, H Φ) −→ Mor Rep ∞ qG ( qα, H Φ)exactly like in Equation (26) (using d q b ∇ qα L instead of d ∇ qωqα L ) and one can prove that b D qα satisfiesEquation (27) as well. For example, since (see Equation (8)) d q b ∇ qα R = b σ qα ◦ d q b ∇ qα L ◦ b σ − qα and d q b ∇ qα L ( ψ µ ) = ( d q b ∇ qα L ψ ) µ + ( − k ψ ( dµ )for any ψ ∈ Ω k ( M ) ⊗ M \ Γ( qM, qV qα M ), it is posible to show that (see Definition 2.18) d q b ∇ ⊗ L ( ψ ⊗ Ω • ( M ) ψ ) = d q b ∇ qα L ( ψ ) ⊗ Ω • ( M ) ψ + ( − k ψ ⊗ Ω • ( M ) d q b ∇ qα L ( ψ )for any ψ ∈ Ω k ( M ) ⊗ M \ Γ( qM, qV qα M ), ψ ∈ Ω • ( M ) ⊗ M \ Γ( qM, qV qα M ), and then b D qα ⊗ qα ( τ · τ ) = b D qα ( τ ) τ + ( − k τ b D qα ( τ ) . In this way one can induce a first–order antiderivation on Hor • GM that satisfies Equations(14), (15). This implies the existence of a covariant derivative (see Definition 2.26) D qω : Hor • GM −→ Hor • GM, for a unique qpc qω ([D4]). Now it should be clear that by construction, qF and qAss qωqζ arenatural isomorphic. (cid:4) Objects in the category qGTS q ∇ qM of quantum gauge theory sectors on qM with quantumconnections are tuples ( qG, qF ) formed by a cqg qG and a contraviariant functor qF be-tween Rep qG and qVB q ∇ qM that satisfy hypothesis of Theorem 3.11. In qGTS q ∇ qM morphismsbetween two objects ( qG , qF ), ( qG , qF ) are pairs ( h, nt ) where h is a ∗ –Hopf algebramorphism and nt : qF −→ qF ˆ h is a natural transformation with ˆ h : Rep qG −→ Rep qG the graded–preserving covariant functor given byˆ h qα = (id V ⊗ h ) ◦ qα if qα coacts on V (notice that ˆ h qα coacts on V as well) andˆ h ( f ) = f for morphisms such that(1) nt qα is always a degree 0 morphism.(2) nt commutes with − and N . Now it is possible to interpret the quantum association functor qAss qωqζ as an another functor(see Definition 2.30) qAss : qPB qωqM −→ qGTS q ∇ qM such that on objects is defined by( qG, qζ , qω ) ( qG, qAss qωqζ )and morphisms by means of qAss( h, F ) := ( h, ˆ F ) , where ˆ F : qAss qω qζ −→ qAss qω qζ ˆ h is the natural transformation given byˆ F qα : qAss qω qζ qα −→ qAss qω qζ ˆ h qαT F ◦ T. The next theorem is the propose of this paper and finally we have the tools to prove it.On can check [SW] to compare this theorem with its classical version.
Theorem 3.12.
For every quantum space qM , the functor qAss provides us with an equiv-alence of categories from qPB qωqM to qGTS q ∇ qM .Proof. By Theorem 3.11, every ( qG, qF ) ∈ Obj ( qGTS q ∇ qM ) is isomorphic in qGTS q ∇ qM to( qG, qAss qωqζ ) for a suitable principal qG -bundle over qM qζ with a quantum principal con-nection qω ; so in order to prove the statement we just need to show q Ass induces a bijectionbetween
Mor qPB qωqM (( qG , qζ , qω ) , ( qG , qζ , qω ))and Mor qGTS q ∇ qM (( qG , qAss qω qζ ) , ( qG , qAss qω qζ ))for arbitrary ( qG , qζ , qω ), ( qG , qζ , qω ) ∈ Obj ( qPB qωqM ). According to our previousresults we know that GM i ∼ = M qα i ∈ T i Γ( qM, qV qα i M ) ⊗ V qα i , Hor • GM i ∼ = M qα i ∈ T i Ω • ( M ) ⊗ M Γ( qM, qV qα i M ) ⊗ V qα i , and GM Φ ∼ = M qα i ∈ T i id Γ( qM,qV qαi M ) ⊗ qα i , H Φ ∼ = M qα i ∈ T i id Ω • ( M ) ⊗ M id Γ( qM,qV qαi M ) ⊗ qα i , where we are assuming that qα i coacts on V qα i and T i is a complete set of mutually nonequiv-alent irreducible finite dimensional qG i –representations for i = 1, 2. Also the maps D qα i = Υ − qα i ◦ d ∇ qωiqαi L ◦ Υ qα i UNCTORIALITY OF QUANTUM PRINCIPAL BUNDLES AND QUANTUM CONNECTIONS 31 consistently recreate the covariant derivative D qω i and hence the qpc qω i ([D4]). Taking( h, nt ) ∈ Mor qGTS q ∇ qM (( qG , qAss qω qζ ) , ( qG , qAss qω qζ )), let us define F : M qα ∈ T Ω • ( M ) ⊗ M Γ( qM, qV qα M ) ⊗ V qα −→ M qα ∈ T Ω • ( M ) ⊗ M Γ( qM, qV qα M ) ⊗ V qα such that F ( µ ⊗ M T ⊗ v ) := M qα ∈T µ ⊗ M T qα ⊗ v qα , if T ∈ Γ( qM, qV qα M ) and nt qα ( T ) = M qα ∈T T qα ∈ Γ( qM, qV ˆ h qα M ) ⊆ M qα ∈T Γ( qM, qV qα M ) ,v = M qα ∈T v qα ∈ V qα = M qα ∈T V qα , where in the last three expressions we have used the same finite number of corepresentations { qα } ∈ T . Now a direct calculation using the defined ∗ –algebra structure and propertiesof nt proves that ( h, F ) ∈ Mor qPB qωqM (( qG , qζ , qω ) , ( qG , qζ , qω )) and by constructionˆ F = nt . This implies that qAss( h, F ) = ( h, nt ). On the other hand if qAss( h , F ) =( h , ˆ F ) = ( h , ˆ F ) = qAss( h , F ) it is clear that h = h and since ˆ F = ˆ F we get that F ◦ T = F ◦ T for all T ∈ Γ( qM, qV qα M ) and all qα ∈ T . Considering the decomposition ofHor • GM into the direct sum it follows that F = F , so ( h , F ) = ( h , F ). This completesthe proof. (cid:4) Concluding Comments
First of all we going to talk about the differential calculus that we have used on quantumgroups. In general, it is enough to consider a differential calculus (covering the ∗ –FODC(Γ , d )) that allows us to extend the comultiplication map φ : G −→ G ⊗ G . These kindsof differential calculus are bicovariant. For example, in [W2] we can see the definition ofanother differential calculus called the braided exterior calculus which leaves to extend φ aswell. It can be proven that the universal differential ∗ –calculus is maximal in this sense [D1],[SZ] and the braided exterior calculus is minimal [W2]. The maps getting by this universalproperties are such that they reducing to the identity on Γ and G . Also we have to highlightthat definition of the universal differential ∗ –calculus is independent of the quantum groupstructure on qG .Notice we have changed the traditional definitions of corepresentation morphism and qvbmorphism in order to define the ∗ operation in the right side of Equation (24). If we forgetthe ∗ structure in the whole work, i.e., if we just work with (associative unital) algebras, thischange in morphism would not be necessary. This is a recent point of view in noncommutativegeometry in which one can consider that the ∗ structure is not an essential initial condition[D7]. Of course, this gives us a richer and more general theory.All our conditions presented in Remark 2.27 are not just technical, specially the first one.For example, in [D5] one can check that there is a bijective relation between the existence of { T L k } dk =1 and d –classifying maps ( ρ, γ ). Reference [D6] shows a noncommutative–geometric generalization of classical Weil Theory of characteristic classes for qpqgb considereing reg-ular qpc and in the general case; in particular, one can fine the noncommutative counter-part of the Chern character for structures admitting regular connections. In addition, onealso uses this particular set of M –generators in the theory presented in [D4], in which webased as well; to say nothing of using the both set of M –generators it can be proven that Mor Rep ∞ qG ( qα, GM Φ) is a finitely generated projective left–right M –module [D2]. Even more,Remark 2.27 presents sufficiente condition for built the quantum association bundle and getthe categorical equivalence; in clearly difference with the classical case in which for eachprincipal bundle with a principal connection the association functor always exists [SW].Now we have to talk about the regularity condition that we have imposed in quantumprincipal connections. Regularity condition leaves us to prove that ∇ qωqα = Υ qα ◦ D qω ◦ T satisfies the right Leibniz rule as we defined it in Definition 2.10; also it is crucial in thetheory developed in [D4], which allows us to built quantum connections using just a fewstructures, not the complet differential calculus. Our definition of qpb morphisms is basedon this fact (see Definition 2.29) and we used it strongly in Theorem 3.11, so it intervenesimplicity in Theorem 3.12. If we remove this condition and if we still want to have aquantum association functor, we would have to consider qlc as linear maps that just satisfythe left Leibniz rule and the morphism σ would not be necessary, at least for this definition.In addition there would be certain properties that would not be fulfilled. For example,the quantum association functor would not satisfies Theorem 3.7 and we could not get aqpc in Theorem 3.11 since this reconstrunction is based on the regularity condition [D4].Nevertheless putting away all qpcs that are not regular would be a mistake that wouldseparate us from many interesting examples to study, even if we do not have a categoricalequivalence for these qpcs. It is important to emphasize that every (classical) principalconnection is regular and multiplicative.Finally and as we have mentioned before, the theory developed in this paper is an extentionof the one presented in [D3]: we have considered qpcs with the same importance that qpbs. In[D3] we can see some examples that show us this theory is not trivial. Clearly these examplesalso work if we consider regular qpcs on these quantum bundles. We must remark that all ourimposed conditions (or restrictions) are just properties that guarantees us noncommutativegeneralizations of classical conditions [SW]. The fact that we were able to recreate theclassical categorical equivalence could tell us that our definitions of qvb, qlc, qpb, qpc andassociated qvb are the correct ones. References [B]
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