Verlinde modules and quantization

VERLINDE MODULES AND QUANTIZA TION V AR GHESE MA THAI Abstract. Giv en a compact simple Lie group G and a pr imitiv e degr ee 3 twist η , we define a monoidal ca tegory C ( G, η ) with a May structure given b y disjoin t union and fusion pro duct. An ob ject in the ca tegory C ( G, η ) is a pair ( X , f ), where X is a co mpa ct G -manifold and f : X → G a smo o th G -map with res p ect to the c o njugation action of G on itself. Such a n ob ject determines a mo dule, the t wisted equiv ariant K-ho mology group K G ( X, f ∗ ( η )), for the V erlinde algebra, ter med a V erlinde module, where the mo dule action is induced b y the G -action on X . In order to under stand which ob jects in C ( G, η ) can b e quantized, we define the closely related monoidal c a tegory D ( G, η ) consisting o f equiv ariant twisted g eometric K-cycles, which also has a May structure given by disjoint union and fusion pro duct. There is a fo rgetful functor D ( G, η ) → C ( G, η ), showing that an o b ject in D ( G, η ) deter mines a V er linde mo dule. Every ob ject in the categor y D ( G, η ) also ha s a quantization, v a lued in the V erlinde algebra . Finally , the quantization functor induces an isomor phism b etw een the geometric equiv ariant twisted K- homology r ing K G geo ( G, η ), and the V erlinde a lgebra. Intr oduction This w or k is partly inspired b y a theorem of F r eed, Hopkins and T eleman [9, 1 0, 11], whic h iden tifies the t wisted G - equiv ariant K-homolog y group K G ( G, η ) of a compact Lie gro up G , with the V erlinde algebra R ℓ ( G ) of G at a lev el ℓ determined by the tw ist η . Firstly , they sho w that K G ( G, η ) is an algebra, with pro duct induced by the multiplic ation m : G × G − → G on the group G , and that their isomorphism “explains” the com binatoria lly complicated fusion pro duct in the V erlinde algebra. Another consequenc e of their theorem is that the V erlinde algebra R ℓ ( G ) has the same functorial prop erties as K G ( G, η ). Recall that the imp o rtance of the V erlinde algebra is that it encodes the selection rules for the op erator pro duct expansion in certain rat io nal conformal field theories suc h as the WZW-mo del. That is, they enco de the dimensions of spaces of conf o rmal blo c ks of these rationa l confo rmal field theories, i.e. dimensions o f certain spaces o f generalized theta functions (cf. [22]). These dimensions and their p olynomial b ehav iour are of fundamen tal impo r tance in confo r ma l field theory . Key wor ds and phr ases. V erlinde a lgebra, V er linde mo dules, quantization, twisted equiv a riant K - homology , May structure, monoidal categ ory , qua si-hamiltonian manifolds, group v alued moment maps. A cknow le dgements. This resea rch was supp or ted under Austr a lian Resea rch Council’s Discovery Pro jects funding sc heme (pro ject num b er DP0 8781 84). V.M. is the recipient of an Australian Research Council Australian Pr ofessor ial F ellowship (pro ject num b er DP077092 7). V.M. thanks P . Hekmati and S. Mahanta for int eresting discussio ns. 1 Another source of inspiration is the recen t w o r k b y Meinrenk en [17 , 18] on the relation of quasi-Hamiltonian manifo lds to the w ork of F reed, Hopkins and T eleman. T o ev ery compact quasi-Hamiltonian manifold ( M , ω , Φ) with group-v alued momen t map Φ : M → G (which satisfies Φ ∗ ( η ) = dω ), Meinrenk en defines the quan tizatio n Q ( M ) to be the eleme n t of the V erlinde algebra Φ ∗ ([ M ]) ∈ K G ( G, η ) ∼ = R ℓ ( G ), where [ M ] denotes the equiv arian t fundamen tal class of the compact G -manifold M , whic h is an elemen t in K G ( M , Cliff ( T M )) , since he shows that M has an equiv arian t tw isted Spinc structure, (explained later in the section). He then establishes sev eral v ery interes ting prop erties of his quan t izat io n pro cedure, as we ll as calculations of it. An approach related to Meinrenk en’s, but whic h uses equiv ariant bundle gerb e mo dules instead, is due to Carey-W a ng [7]. W e b egin by pro ving some v ery general results , phrased using equiv ariant (op erat o r) K- homology , whic h is equiv a rian t K-homolo gy on the category of separable G - C ∗ -algebras. It reduces to equiv ariant to p ological K-homology on compact top olog ical spaces. Equiv ariant op erator K-homology is a particular case of Kasparov’s equiv ariant biv a r ia n t K-theory (or KK G -theory) whic h has a cup-cap pro duct, making it into a p o w erful to ol. W e will utilise this cup-cap pro duct a nd its prop erties, to prov e our first general result, whic h sa ys that if A is a separable G - C ∗ -algebra which is also a G -coa lg ebra, t hen the Kasparov equiv ariant K - ho mology group KK G ( A, C ) (as defined in [13, 6]), a dmits a ring structure induced by the com ultiplication on A , see Prop osition 3.1. As a corollary , w e see that for a compact C ∗ -quan tum gr o up A , the K-homology group K K ( A, C ) admits a ring structure induced b y the com ultiplication on A , see Corollary 3.3. As another corollary , w e obtain a new pro of o f a theorem in [9, 1 0, 11 ] show ing that the equiv ariant tw isted K-homology K G ( G, η ), where G acts on itself b y conjugation and η is a primitiv e degree 3 t wist on G , is has a ring structure induced by the m ultiplication map m : G × G → G on G . The next general result says that if A 1 is a separable G - C ∗ -algebra whic h is also a G - coalgebra, and A 2 is another separable G - C ∗ -algebra which is also a G -como dule for A 1 , then the ab elian group K G ( A 2 ) = K K G ( A 2 , C ) is a mo dule for the algebra K G ( A 1 ) = K K G ( A 1 , C ), see Prop osition 3.4. As a corollary (see Corollary 3.7) w e see that if A 1 is a compact C ∗ -quan tum group and A 2 is a separable C ∗ -algebra which is also a comodule for A 1 , then the ab elian g roup KK( A 2 , C ) is a mo dule fo r the algebra KK( A 1 , C ). W e next define a category C ( G, η ) whose ob jects are pairs ( X, f ), where X is a compact G -manifold and f : X → G is a smo oth G -equiv arian t map, with resp ect to the conjugation action of G acting on G . Corollary 3.5 of Prop osition 3.4 describ ed ab ov e, establishes that the group action a : G × X − → X determines the mo dule K G ( X , f ∗ ( η )) for the V erlinde algebra R ℓ ( G ). W e term suc h a mo dule a V erlind e mo dule , explaining pa r t of the title of the pap er. Moreov er, any morphism in the catego r y C ( G, η ) determines a morphism o f V erlinde mo dules o v er the V erlinde algebra. In f a ct, we v erify that C ( G, η ) is a strict mono idal category t ha t has a May structur e , which essen tially says t ha t t he op erations of disjoin t 2 union and f usion pro duct are compatible, and can b e view ed as an analogue of an alg ebra structure for categories. Here the fusion pro duct ⊛ is defined a s ( X 1 , f 1 ) ⊛ ( X 2 , f 2 ) = ( X 1 × X 2 , m ◦ ( f 1 × f 2 )) , for ob jects ( X j , f j ) , j = 1 , 2 in C ( G, η ). Denote the category of V erlinde mo dules b y VMo d( R ℓ ( G )), whic h is a subcategory of the category of all mo dules Mo d( R ℓ ( G )) o v er the V erlinde a lgebra. W e establish sev eral in teresting prop erties of VMo d( R ℓ ( G )). More formally , we sho w that the lax monoidal functor F : C ( G, η ) − → Mo d( R ℓ ( G )) , defined by F ( X , f ) = K G ( X , f ∗ ( η )), is compat ible with the May structures on b oth cate- gories. These can b e view ed as the cen tra l results in t he pap er. Next w e w a n t to understand whic h ob jects in C ( G , η ) can b e quan tized. T o achiev e this goal, w e define a closely related category D ( G, η ), whic h ob jects are triples ( X , E , f ) where ( X , f ) is an ob ject in the catego r y C ( G , η ) and E is a G - equiv arian t (complex) v ector bundle o ver X . In addition, w e assume that X has an e quivarian t twiste d Spinc structur e , that is, there is a given equiv ariant isomorphism, f ∗ ( K η ) ∼ = Cliff ( T X ) ⊗ K , (1) where K η is the algebra bundle of compact op erators on G determine d b y η , Cliff ( T X ) denotes the Clifford alg ebra bundle asso ciated to t he tang ent bundle of X , and K denotes the algebra of compact op erators on a G -Hilb ert space. The ob jects of D ( G, η ) are an equiv ariant analogue of tw isted geometric K-cycle s in [4, 23]. The k ey observ ation made here is that in this sp ecial case, this category has a ric her structure than usual, giv en by the Ma y structure defined b elow. ( G, 1 , id : G → G ) is a final ob ject in the category D ( G, η ), where 1 is the trivial line bundle o ver G . The morphisms of D ( G, η ) are explicitly describ ed in the text. In particular, a compact quasi-Hamiltonian G -manifold ( M , ω , Φ) determines the ob ject ( M , 1 , Φ) in D ( G , η ), b y a result in [2]. Clearly D ( G, η ) is m uc h larger, and it is closed under disjoin t union ` , a dual op eration, the fusion pro duct ⊛ and a lso G -ve ctor bundle mo dification, all of which will b e explained in the text. He re w e mention that the fusion pro duct is ( X 1 , E 1 , f 1 ) ⊛ ( X 2 , E 2 , f 2 ) = ( X 1 × X 2 , E 1 ⊠ E 2 , m ◦ ( f 1 × f 2 )) , for ob jects ( X j , E j , f j ) , j = 1 , 2 in D ( G , η ). W e ve rify t hat D ( G, η ) is a strict monoidal category that has a May structure giv en b y ` and ⊛ . Ev ery ob ject ( X , E , f ) in D ( G, η ) has a fundamental class [ X ] ∈ K G ( X , Cliff ( T X )), b y a construction of Kasparo v [13]. The quantization Q ( X, E , f ) ∈ R ℓ ( G ) is defined as Q ( X, E , f ) = f ∗ ([ E ] ∩ [ X ]), whic h in the sp ecial case when X itself is an equiv arian t Spinc manifold (i.e. W G 3 ( X ) = W 3 ( X G ) = 0), reduces to Q ( X, E , f ) = [ f ∗ ( ð X ⊗ E )], where ð X is 3 the equiv a rian t Spinc Dirac op erat o r on X , and ð X ⊗ E denotes the coupled op erator. W e will show that the quan tizatio n then determines a monoidal functor, Q : D ( G, η ) − → K G ( G, η ) ∼ = R ℓ ( G ) , resp ecting t he May structures on b oth categories. In particular, an ob ject ( X, f ) in C ( G, η ) has a quantization if X has an equiv ariant twisted Spinc structure, in whic h case ( X , 1 , f ) defines a n ob ject in D ( G, η ). Consider the equiv alence relation ∼ on ob jects in D ( G, η ) generated isomorphism a nd by the f ollo wing three elemen ta ry mov es explained in the text, (1) direct sum-disjoin t union; (2) G -b o r dism o ver ( G, η ); (3) G -v ector bundle mo dification. The geometric equiv ariant tw isted K-homology group K G g eo ( G, η ) is defined to b e the ab elian group whic h is the quotient D ( G, η ) / ∼ , with addition induced b y disjoint union-direct sum. Our first observ atio n is that K G g eo ( G, η ) has a ring structure induced b y the fusion pro duct ⊛ o n D ( G, η ). The map induced by quan tization is an isomorphism of r ing s, Q : K G g eo ( G, η ) ∼ = − → K G ( G, η ) ∼ = R ℓ ( G ) . (2) This is a sp ecial case of a more g eneral theorem whic h will b e pro v ed elsewhere, whic h uses a hybrid of tec hniques in [4, 23 ] and [5]. An impact of this result is that the Ma y structure on the category D ( G, η ) induces the algebra structure o n K G ( G, η ), whic h b y [9, 10, 11] is just the V erlinde algebra R ℓ ( G ). 1. The ca tegor y C ( G, η ) Let G b e a connected, simply connected and simple compact Lie group with the m ultipli- cation map m : G × G → G . Let G act on G b y the conjugation. Then it is kno wn that ev ery cohomology class [ η ] ∈ H 3 G ( G, Z ) has a primitiv e represen tative, i.e., there is η ∈ Ω 3 Z ( G ) a closed 3-form on G with inte gral p erio ds, suc h that m ∗ ( η ) = p ∗ 1 ( η ) + p ∗ 2 ( η ). Let C ( G, η ) denote the category , whose ob jects are pairs ( X , f ), where X is a compact G -manifold and f : X → G is a G - equiv ariant map with resp ect to the conjugation action of G a cting on G . A morphism θ b et w een the ob jects ( X 1 , f 1 ) and ( X 2 , f 2 ) in the category C ( G, η ) is a (p ointe d) smo oth G -map θ : X 1 → X 2 compatible with the structure maps, i.e., the f ollo wing 4 diagram commutes X 1 f 1   2 2 2 2 2 2 2 2 2 2 2 2 2 2 θ / / X 2 f 2                 G (3) In particular, the ob jects ( X 1 , f 1 ) and ( X 2 , f 2 ) in the category C ( G, η ) are said to b e iso- morphic if t here is an isomorphism θ : ( X 1 , f 1 ) → ( X 2 , f 2 ) in C ( G, η ). Observ e that ( G, id : G → G ) is a final ob ject in this category . The chos en η determines a canonical class in H 3 ( X , Z ) for any ob ject ( X , f ) giv en b y the pullbac k along the structure map f ∗ ( η ). W e list b elow sev eral in teresting ob jects of the category C ( G, η ). More examples will b e give n in the app endix. Example 1. (T rivial action ) L et X b e a G -sp a c e w i th the trivial G -action, and f : X → G b e a c on tinuous map such that the imag e of f lies in the c entr e of G . Then f is e quiva ria nt with r esp e ct to the adjoint action of G on G , so that ( X , f ) ∈ C ( G, η ) for an y G -ve ctor bund le E over X . Example 2. (F r e e action) L et X b e a fr e e G -s p ac e, and f : X/G → G/G b e a c ontinuous map to the sp ac e o f c on jugacy c l a sses G/G in G . The n f lifts to an e quivariant map ˜ f : X → G with r esp e ct to the ad joint action of G on G , so that ( X , f ) ∈ C ( G, η ) . Example 3. (Hamiltonian G - s p ac es) L et ( X , ω ) b e a Hamiltonian G -sp ac e with mom ent map µ : X − → g ∗ . If G has an A d-inv a riant metric, then g ∗ ∼ = g , w h ich wil l b e assume d her e. Then ( X, f ) ∈ C ( G, η ) , wher e f = exp( µ ) , wher e exp : g − → G den o tes the exp onential map of the Lie gr oup G . Example 4. (Inverse) L et ( X , f ) ∈ C ( G, η ) . Then ( X , 1 /f ) ∈ C ( G, η ) . Example 5. (F usion pr o duct) L et ( X 1 , f 1 ) , ( X 2 , f 2 ) b e obje cts in C ( G, η ) . Then ( X 1 × X 2 , f 1 × f 2 ) ∈ C ( G × G, p ∗ 1 ( η ) + p ∗ 2 ( η )) , and ( X 1 × X 2 , m ◦ ( f 1 × f 2 )) ∈ C ( G, η ) is c al le d the fusion of ( X 1 , f 1 ) and ( X 2 , f 2 ) and is denote d by ( X 1 , f 1 ) ⊛ ( X 2 , f 2 ) . Her e we use the fact that m ∗ ( η ) = p ∗ 1 ( η ) + p ∗ 2 ( η ) . Example 6. (Disjoint union) L et ( X 1 , f 1 ) , ( X 2 , f 2 ) b e obje cts in C ( G, η ) . The d isjoint union ( X 1 ` X 2 , f 1 ` f 2 ) ∈ C ( G, η ) . 5 1.1. Morphisms in the category C ( G, η ) and more examples. A morphism φ : ( X 1 , f 1 ) − → ( X 2 , f 2 ), where ( X i , f i ) ∈ C ( G i , η i ) , i = 1 , 2 , is give n b y a gro up homomorphism φ G : G 1 − → G 2 suc h that φ ∗ G ( η 2 ) = η 1 and a smo o th map φ X : X 1 − → X 2 satisfying f 2 ◦ φ X = φ G ◦ f 1 , φ X ( g .x ) = φ G ( g ) .φ X ( x ) for all g ∈ G 1 and x ∈ X 1 . In particular, when G 1 = G 2 = G and η 1 = η 2 = η , w e obtain the mo r phisms in the category C ( G, η ). Example 7. (Fixe d p oint set) Supp o s e that a c omp act Lie gr oup K acts by automorphisms on ( X, f ) ∈ C ( G, η ) . Then the fixe d- p oin t set X K and the r estriction of f , f K : X K → G K determines an obje ct ( X K , f K ) ∈ C ( G K , η K ) and is such that the inclusion ι : X K ֒ → X is a mo rp hism. Example 8. (Fixe d p oint set - sp e cial c ase) As a sp e cia l c ase of the example ab ove, supp ose that ( X , f ) is an obje ct in C ( G, η ) and g ∈ G . Then the c omp onen ts of the fixe d p oint set of g , ( X g , f g ) ∈ C ( G g , η g ) is such that the inclusion ι : X g ֒ → X is a mo rp hism. Example 9. (Finite c overs ) Supp o s e that ( X , f ) is an obje ct in C ( G, η ) and p : e G → G a fin ite c over of G . Consider the fibr e pr o duct e X = { ( x, ˜ g ) ∈ X × e G   f ( x ) = p ( ˜ g ) } and e f = p ∗ 1 f , wher e p 1 : e X → X is the pr oj e ction onto the first factor. Then ( e X , e f ) is an obje ct in C ( G, η ) such that the quotient map e X 7→ X is a morphism in C ( G, η ) . Example 10. (Quasi-Hamiltonian G -sp ac es a n d Hamiltonian LG -sp ac es) We r e c a l l the definition fr om [1 ] of the definition of a gr oup-value d mo m ent map for a quasi-Hamiltonian G -sp a c e. A quasi-Hamiltonian G -manifold is a G -manifold M with an invaria nt 2-f o rm ω ∈ Ω 2 ( M ) G and an e quivariant m a p Φ ∈ C ∞ ( M , G ) G such that (1) dω = Φ ∗ ( η ) ; (2) The map Φ satisfies ι ( v ξ ) = k 2 Φ ∗ h θ + ¯ θ , ξ i , wher e v ξ is the ve ctor fi eld on M gen er ate d by ξ ∈ g , θ is the left invariant Cartan-Maur er form, ¯ θ is the right invari a nt Cartan- Maur er form; (3) At e ach x ∈ M , the kernel of ω x is given by k er( ω x ) = { v ξ | ξ ∈ k er ( Ad Φ( x )+1 ) } . The ma p Φ is c al le d the gr oup value d moment map of the quasi-Hamiltonian G - manifold M . It is pr ove d in [2] that ( M , Φ) is an element in C ( G, η ) . Basic ex a mples of quasi-Hamiltonian G -sp ac es ar e p r ovid e d by pr o ducts of c on j uga c y classes C ⊂ G as in [1] . Mor e gener al ly, a bije ctive c orr esp ondenc e b etwe en Hamiltonian lo op gr oup manifolds with pr op er moment map an d quasi-Hamiltonian G -man i f o ld is establishe d in that p ap er. They c onstitute a lar ge numb er of o bje cts in C ( G, η ) . 6 Example 11. (Disjoint union) The disjoint union of a p air of obje cts ( X i , f i ) , i = 1 , 2 in C ( G, η ) is defi ne d as the disj o int union, ( X 1 , f 1 ) ⊕ ( X 2 , f 2 ) = ( X 1 a X 2 , f 1 a f 2 ) Example 12. (Extension by maps) L et ( X , f ) b e an obje ct in C ( G, η ) and h : Y − → X b e a G -map. Then ( Y , f ◦ h ) ∈ C ( G, η ) . F or examp le, let W b e an e quivaria n t ve ctor bund le over X . Then ( b X , f ◦ π ) is again an obje ct in C ( G, η ) . Her e b X denotes the unit spher e bund le S (( X × R ) ⊕ W ) , π : b X − → X is the pr oje ction. 1.2. The fusion monoidal st r ucture on C ( G, η ) . Recall that a monoidal category is a category in whic h asso ciated to each pair of ob jects A and B there exists a pro duct ob ject A ⊗ B , and there is an identit y ob ject 1 , suc h t ha t 1 ⊗ A ∼ = A ∼ = A ⊗ 1 , together with asso ciator isomorphisms Φ = Φ A,B ,C : A ⊗ ( B ⊗ C ) → ( A ⊗ B ) ⊗ C for a n y three ob jects A , B a nd C , satisfying the p entagonal identity : A ⊗ ( B ⊗ ( C ⊗ D )) 1 A ⊗ Φ B,C, D u u j j j j j j j j j j j j j j j j j j j j Φ A,B, C ⊗ D ) ) T T T T T T T T T T T T T T T T T T T T A ⊗ (( B ⊗ C ) ⊗ D ) Φ A,B ⊗ C, D   @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ ( A ⊗ B ) ⊗ ( C ⊗ D ) Φ A ⊗ B,C, D   ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( A ⊗ ( B ⊗ C )) ⊗ D Φ A,B, C ⊗ 1 D / / (( A ⊗ B ) ⊗ C ) ⊗ D with each arrow the appropriate map Φ, and the triangle relation: A ⊗ ( 1 ⊗ B ) ∼ = A A A A A A A A A A A A A A A A A Φ A, 1 ,B / / ( A ⊗ 1 ) ⊗ B ∼ = ~ ~ } } } } } } } } } } } } } } } } } A ⊗ B MacL an e ’ s c oher enc e the or em ensures that these conditions a r e sufficien t to g uaran t ee con- sistency of all other rebrac k etings. Recall the fusion product in C ( G , η ) from Example 5. Giv en ( X 1 , f 1 ) and ( X 2 , f 2 ) the follo wing diagram describ es the fusion pro duct ( X 1 , f 1 ) ⊛ ( X 2 , f 2 ) = ( X 1 × X 2 , m ◦ ( f 1 × f 2 )) . 7 X 1 × X 2 p X 1 ! ! p X 2 ' ' f 1 × f 2 L L L L % % L L L L G × G m # # G G G G G G G G G X 2 f 2   X 1 f 1 / / G Prop osition 1.1. C ( G, η ) is a strict monoidal c ate gory with pr o duct given by the fusion pr o duct ⊛ (as in Exam ple 5) and i d entity ele m ent ( e, I) , wher e e ∈ G is the identity e le ment and I : e ֒ → G is the inclusion map, which is e quivariant unde r the adjoint action of G o n itself and the trivial action of G on e . Pr o of. W e need to v erify that C ( G, η ) satisfies the axioms of a monoidal category . Let ( X , f ) ∈ C ( G, η ) and consider the fusion pro duct ( X , f ) ⊛ ( e, I) = ( X × e, m ◦ ( f × I)) whic h is clearly isomorphic to ( X , f ). Similarly , ( e, I) ⊛ ( X , f ) is also isomorphic to ( X , f ). Therefore ( e, I) ∈ C ( G, η ) serv es as an identit y in C ( G, η ). Next, let ( X i , f i ) ∈ C ( G, η ) , i = 1 , 2 , 3, and Φ 1 , 2 , 3 : ( X 1 , f 1 ) ⊛ (( X 2 , f 2 ) ⊛ ( X 3 , f 3 )) → (( X 1 , f 1 ) ⊛ ( X 2 , f 2 )) ⊛ ( X 3 , f 3 ) . The LHS is canonically isomorphic to ( X 1 × X 2 × X 3 , m ◦ ( f 1 × m ◦ ( f 2 × f 3 ))) , whereas the RHS is canonically isomorphic to ( X 1 × X 2 × X 3 , m ◦ ( m ◦ ( f 1 × f 2 ) × f 3 )) . The equalit y m ◦ ( f 1 × m ◦ ( f 2 × f 3 )) = m ◦ ( m ◦ ( f 1 × f 2 ) × f 3 ) follows fro m the asso ciativit y of the m ultiplication m on G . In particular, Φ 1 , 2 , 3 = id and C ( G, η ) is a strict category . In particular, the pentagonal identit y is auto ma t ically satisfied, and b y MacLane’s coherence theorem, C ( G , η ) is a strict monoidal category .  1.3. Ma y structure. Motiv ated b y the study o f infinite lo op spaces and alg ebraic K-theory J. P . May in tro duced the mac hinery of (bi)p ermutativ e categories [15, 16]. These are, roughly , categor ies equipp ed with t wo monoidal structures satisfyin g certain compatibility conditions. W e hav e seen b efore the fusion monoidal structure on C ( G, η ). 8 Definition 1.2. A May structur e o n a c ate gory C is a p air of monoidal structur es ⊕ , ⊗ on C , such that for al l X , Y , Z ∈ C ther e is a natur al (distributivity) isom o rphism X ⊗ ( Y ⊕ Z ) ∼ = ( X ⊗ Y ) ⊕ ( X ⊗ Z ) . We c al l a c ate gory e quipp e d with a May structur e as a May c ate gory. A May functor F : C − → D b etwe en May c ate gories is one which is lax monoidal with r esp e ct to b oth ⊕ a n d ⊗ , such that, in addition, the fol lowing diag r am c om mutes: F ( X ) ⊗ F ( Y ) ⊕ F ( X ) ⊗ F ( Z ) / /   F ( X ⊗ Y ) ⊕ F ( X ⊗ Z ) / / F (( X ⊗ Y ) ⊕ ( X ⊗ Z ))   F ( X ) ⊗ ( F ( Y ) ⊕ F ( Z )) / / F ( X ) ⊗ F ( Y ⊕ Z ) / / F ( X ⊗ ( Y ⊕ Z )) Her e the vertic al maps ar e the distributivity isomorphism s a nd the horizontal m aps ar e fur- nishe d by the lax monoidal structur es. Example 13. The c ate gory C ( G, η ) endowe d with ⊕ = ` and ⊗ = ⊛ is a May c ate go ry. We have alr e ady se en that disjoin t unio n and fusion p r o duct ar e monoidal structur e s on C ( G, η ) . The distributivity c ondition is r e ad i l y verifie d. 2. Continuous tra ce C ∗ -algebras on G and G -coalgebra structure Let G b e a compact Lie group and let X b e a compact G -space. Let K denote the C ∗ -algebra of compact op erators o n a separable G -Hilb ert space. A G -equiv ariant Dix mier– Douady (D D) bund le on X is a G -equiv ar ia n t algebra bundle on X , whose fibres are K with the pro jectiv e unitary group PU as the structure group. Giv en a n y G - equiv ariant principal PU bundle P ov er X , one can construct an a D D-bundle o v er X as an associated bundle K P = P × PU K . The equiv ariant Dixmier-Douady class of a G - equiv ariant Dixmier–D o uady (DD) bundle on X is a cohomology class in H 3 G ( X , Z ). A r ecen t theorem of A tiyah-Segal [3] sa ys that ev ery equiv a rian t cohomology class in H 3 G ( X , Z ) determines a G -equiv ariant PU- bundle P o ver X up to G -equiv arian t isomorphism, whic h determines the G -equiv arian t D D- bundle K P , th us establishin g the conv erse. In fact, the G -equiv aria n t isomorphism class es of DD- bundles K P o ver X form an ab elian gro up under direct sum, called the equiv ariant Brauer group, Br G ( X ). Since the equiv ariant Dixmer-Douady class of a direct sum of G - equiv ariant DD- bundles is equal to the sum of the equiv aria n t Dixmer-Do ua dy classes of the G -equiv ariant DD -bundles, there is a natural isomorphism of groups, Br G ( X ) ∼ = H 3 G ( X , Z ). Consider a catego ry whose ob jects are pairs ( X , P ), where X is a compact G -space and P is a fixed c hoice of a G -equiv aria n t principal PU-bundle on X . A morphism ( X, P ) → ( X ′ , P ′ ) in this category is a con tin uous G -ma p f : X → X ′ , suc h that f ∗ ( P ′ ) = P . Let G, X b e as ab ov e and let P b e a G -equiv aria nt principal PU-bundle on X . Let CT( X, P ) denote the stable contin uous trace C ∗ -algebra of a ll con tin uous sections of the a sso ciated equiv ariant DD bundle K P o ver X , that v anish at infinit y . The induced G -action on CT( X , P ) mak es it into a 9 G - C ∗ -algebra, whic h enables us to construct t he crossed pro duct C ∗ -algebras CT( X , P ) ⋊ G . The asso ciation ( X, P ) 7→ CT( X , P ) ⋊ G is f unctorial with resp ect to the ab o ve-men tio ned morphisms of pairs. F urthermore, if ( X , P ) and ( X , P ′ ) are tw o pairs, suc h that the Dixmier– Douady in v ariants of P and P ′ determine the same class in H 3 G ( X , Z ), then the equiv arian t Dixmier–Douady Theorem say s t hat the C ∗ -algebras CT( X , P ) ⋊ G and CT( X , P ′ ) ⋊ G are stably isomorphic [3]. Giv en a compact simple Lie group G with multiplication map m : G × G → G , a G - equiv ariant DD -bundle K P on G is said to b e primitive if m ∗ ( K P ) ∼ = p ∗ 1 ( K P ) ⊗ p ∗ 2 ( K P ) a s G -equiv ariant bundles on G × G . The equiv ariant Dixmier-Douady class η of a primitiv e G - equiv ariant DD- bundle K P satisfies the cor r esp o nding primitivity prop erty , m ∗ ( η ) = p ∗ 1 ( η ) + p ∗ 2 ( η ). There are man y natural w ays to construct primitiv e G -equiv ariant DD-bundle K P on G . One of these uses p ositiv e energy represen tat io ns [3, 9, 10, 11] a nd another more constructiv e metho d is in [18]. In this case, w e denote the equiv a r ia n t contin uous trace algebra by CT( G, η ). If ( X , f ) is an ob ject in the catego ry C ( G, η ), t hen w e denote the corresp onding equiv ariant contin uous trace algebra by CT( X , f ∗ ( η )). W e next prov e some fundamen t a l facts ab o ut the equiv ariant con tin uous trace alg ebras CT( G, η ) and CT( X , f ∗ ( η )). Lemma 2.1. L et G b e a c om p a c t gr o up, and η b e a primitive DD-bund le on G . Then the C ∗ -algebr as CT( G, η ) and CT( G, η ) ⋊ G b oth c arry a natur al c o algebr a structur e i n duc e d by the m ultiplic ation on G . Pr o of. The m ult iplicatio n map m : G × G → G induces a ∗ - homomorphism m ∗ : CT( G, η ) → CT( G × G, m ∗ ( η )) ∼ = CT( G × G, p ∗ 1 ( η ) + p ∗ 2 ( η )) ∼ = CT( G, η ) ⊗ CT( G, η ) . Since the m ultiplication map m and the C ∗ -tensor pro duct ⊗ are asso ciativ e, it follows that the induced ∗ - homomorphism m ∗ is coa sso ciativ e, that is, the diagram b elo w commute s, CT( G, η ) ⊗ CT( G, η ) 1 ⊗ m ∗   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? CT( G, η ) m ∗   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? m ∗                       CT( G, η ) ⊗ CT( G, η ) ⊗ CT( G, η ) CT( G, η ) ⊗ CT( G, η ) m ∗ ⊗ 1                       10 Moreo ve r, the m ultiplication map m : G × G → G is G -equiv arian t under the adjoint action of G . Therefore, it induces a ∗ -homomorphism o f G - C ∗ -algebras and in turn that of the cro ssed pro duct algebras, m ∗ : CT( G, η ) ⋊ G → (CT( G, η ) ⋊ G ) ⊗ (CT( G, η ) ⋊ G ) . Coasso ciativit y of m ∗ follo ws by similar reasoning as ab ov e.  Using a nalogous argumen ts one easily prov es t he following result: Lemma 2.2. L et ( X, f ) b e an o b j e ct in the c ate gory C ( G, η ) . Then CT( X , f ∗ ( η )) (r e s p. CT( X , f ∗ ( η )) ⋊ G ) is a c omo dule over the c o alg e b r a CT( G, η ) (r esp. CT( G, η ) ⋊ G ) , which is induc e d b y the gr oup action map. Pr o of. The group action map a : G × X → X is a G -map under the adjoint action of G , therefore it induces a ∗ -homomor phism o f G - C ∗ -algebras, a ∗ : CT( X , f ∗ ( η )) → CT( G × X, a ∗ f ∗ ( η )) ∼ = CT( G × X, p ∗ 1 η + p ∗ 2 f ∗ ( η )) ∼ = CT( G, η ) ⊗ CT( X, f ∗ ( η )) since a ∗ ( f ∗ η ) = p ∗ 1 ( η ) + p ∗ 2 f ∗ ( η ) b y the primitivity assumption o n η . The defining prop ert y of the action map a sho ws that a ∗ is a como dule map, that is, the diagram b elow comm utes, CT( G, η ) ⊗ CT( X , f ∗ ( η )) 1 ⊗ a ∗   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? CT( X , f ∗ ( η )) a ∗   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? a ∗                       CT( G, η ) ⊗ CT( G, η ) ⊗ CT( X , f ∗ ( η )) CT( G, η ) ⊗ CT( X , f ∗ ( η )) a ∗ ⊗ 1                       Moreo ve r, the group action map a : G × X → X is G -equiv a rian t under the adjoin t action of G . Therefore, it induces a ∗ - homomorphism of G - C ∗ -algebras and in t ur n tha t of t he crossed pr o duct a lg ebras, a ∗ : CT( X, f ∗ ( η )) ⋊ G → (CT( G, η ) ⋊ G ) ⊗ (CT( X , f ∗ ( η )) ⋊ G ) . Coasso ciativit y of a ∗ follo ws by similar reasoning as ab ov e.  11 3. The ring structure on K G ( G, η ) and Verlinde modules W e will deriv e the ring structure on K G ( G, η ) as a sp ecial instance of a more general result. W e will a lso sho w t ha t for ( X , f ) ∈ C ( G, η ), K G ( X , f ∗ ( η )) is a mo dule for the ring K G ( G, η ) as a sp ecial case of a more general result. The f unctor F : C ( G , η ) − → Mo d( R ℓ ( G )) defined b y F ( X , f ) = K G ( X , f ∗ ( η )) is show n to b e a lax monoidal whic h resp ects the May structures on b oth categor ies. The v ery general result that w e will prov e in this section this describ ed as fo llows. Recall that if A is a separable G - C ∗ -algebra, then the equiv ariant K-homology K G ( A ) is in general only an ab elian group. Ho w ev er, if A is also a G -coa lg ebra, then we will sho w that K G ( A ) is a ring , with pro duct induced b y t he comultiplication on A . In particular, w e deduce that the K-homology of a C ∗ -quan tum group is an r ing , whic h app ears t o b e new. W e also prov e in an analo gous w ay that the equiv aria n t K- theory K G ( A ) is a coring, with copro duct induced b y the com ultiplicatio n on A . In particular, w e deduce that the K- theory of a C ∗ -quan tum group is a coring, which also app ears to b e new. Finally , if A is also K-or ien ted in equiv arian t K-theory , then b ot h K G ( A ) and K G ( A ) are bialgebras in a natural w a y . A key step in defining the product in K G ( A ) us es Kasparo v’s cup-cap pro duct. More precisely , let A 1 , A 2 , D , B 1 , B 2 b e se parable G - C ∗ -algebras, where G is a lo cally compact group. Then the cup-c ap pr o duct , (Definition 2.12 of [13]) ⊗ D : KK G ( A 1 , B 1 ⊗ D ) ⊗ KK G ( D ⊗ A 2 , B 2 ) − → KK G ( A 1 ⊗ A 2 , B 1 ⊗ B 2 ) (4) Prop osition 3.1. L et A b e a sep ar able G - C ∗ -algebr a which is also a G -c o algebr a. The n the ab e lian gr oup KK G ( A, C ) admits a ring s tructur e induc e d by the c omultiplic ation on A . Pr o of. When B 1 = C = B 2 = D and A 1 = A 2 = A , the cup-cap pro duct (4 ) reduces to ⊗ C : KK G ( A, C ) ⊗ KK G ( A, C ) − → KK G ( A ⊗ A, C ) . (5) The com ultiplication ∆ on A is a G - ∗ -homomorphism ∆ : A − → A ⊗ A. (6) Since equiv ariant-K-homology is con t ra v ariant with resp ect to G - ∗ -homomo r phisms, w e get ∆ ∗ : KK G ( A ⊗ A, C ) − → KK G ( A, C ) . (7) The comp osition of the mor phisms in equations ( 5) and (7) give s ◦ : KK G ( A, C ) ⊗ KK G ( A, C ) − → KK G ( A, C ) , whic h is a pr o duct ◦ o n equiv a rian t K- homology KK G ( A, C ) induced by the com ultiplication. The associativity of t he pro duct ◦ follo ws from the asso ciativit y of the cup-cap product (Theorem 2.14 of [13]), the coasso ciat ivity of the com ultiplicatio n ∆ on A and the naturality of the cup-cap pro duct under G - C ∗ -homomorphisms. 12 More precisely , the asso ciativit y of Kasparo v’s cup-cap pro duct sa ys t ha t the following diagram commutes , K G ( A ⊗ A ) ⊗ K G ( A ) ⊗ C   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? K G ( A ) ⊗ K G ( A ) ⊗ K G ( A ) 1 ⊗⊗ C   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ⊗ C ⊗ 1                       K G ( A ⊗ A ⊗ A ) K G ( A ) ⊗ K G ( A ⊗ A ) ⊗ C                       where K G ( A ) denotes KK G ( A, C ). On the other hand, the coassociativity of the com ultiplication ∆ sa ys t ha t the following diagram commutes , A ⊗ A 1 ⊗ ∆   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? A ∆   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ∆                       A ⊗ A ⊗ A A ⊗ A ∆ ⊗ 1                       Therefore the induced diagram in equiv a rian t K- homology comm utes 13 K G ( A ⊗ A ) ∆ ∗   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? K G ( A ⊗ A ⊗ A ) 1 ⊗ ∆ ∗   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ∆ ∗ ⊗ 1                       K G ( A ) K G ( A ⊗ A ) ∆ ∗                       Therefore one has the comm utat ive diagram, K G ( A ) ⊗ K G ( A ) ⊗ K G ( A ) 1 ⊗ ( ⊗ C ) / / ( ⊗ C ) ⊗ 1   K G ( A ) ⊗ K G ( A ⊗ A ) 1 ⊗ ∆ ∗ / / ⊗ C   K G ( A ) ⊗ K G ( A ) ⊗ C   K G ( A ⊗ A ) ⊗ K G ( A ) ⊗ C / / ∆ ∗ ⊗ 1   K G ( A ⊗ A ⊗ A ) 1 ⊗ ∆ ∗ / / ∆ ∗ ⊗ 1   K G ( A ⊗ A ) ∆ ∗   K G ( A ) ⊗ K G ( A ) ⊗ C / / K G ( A ⊗ A ) ∆ ∗ / / K G ( A ) The top left hand square comm utes since the cup-cap pro duct is asso ciativ e, the b ottom righ t hand square comm utes since the com ultiplication is coasso ciativ e, while t he r emaining squares commute b ecause the cup-cap pro duct is functorial under G - C ∗ -homomorphisms. Therefore one has the comm utat ive diagram, 14 K G ( A ) ⊗ K G ( A ) ◦   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? K G ( A ) ⊗ K G ( A ) ⊗ K G ( A ) 1 ⊗◦   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ◦⊗ 1                       K G ( A ) K G ( A ) ⊗ K G ( A ) ◦                       whic h precisely says that pro duct ◦ is asso ciativ e.  The following corollar y was w as first established in [9, 10, 11]. See a lso [21]. Corollary 3.2. L et G b e a c omp act Lie gr oup, and c onside r the c ongugation a c tion of G on itself. L et [ η ] ∈ H 3 G ( G, Z ) a p rimitive c ohom o lo g y class. Then the ab elian gr oup K G ( G, η ) = KK G (CT( G, η ) , C ) admits a ring structu r e induc e d by the multiplic ation m : G × G − → G on G . Pr o of. Setting A = CT( G, η ) in Prop osition 3.1, w e need t o sho w that A is a G -coalgebra. The m ultiplication map m : G × G → G is equiv arian t under the adjo in t action of G , therefore it induces a ∗ -homomor phism o f G - C ∗ -algebras whic h is a com ultiplicatio n, m ∗ : CT( G, η ) → CT( G, η ) ⊗ CT( G, η ) . Since the m ultiplication map m and the C ∗ -tensor pro duct ⊗ are asso ciativ e, it follows that the induced ∗ - homomorphism m ∗ is coa sso ciativ e.  The following is an immediate corollary of Prop osition 3.1. Corollary 3.3. L et A b e a c om p ac t C ∗ -quantum g r oup. Then the ab elian gr oup KK( A, C ) admits a ring s tructur e induc e d b y the c omultiplic ation on A . W e next pro ve another general result. Prop osition 3.4. L et A 1 b e a se p ar able G - C ∗ -algebr a which is als o a G -c o al g ebr a . L et A 2 b e another sep ar able G - C ∗ -algebr a which is also a G -c omo dule for A 1 . Then the ab elian gr oup K G ( A 2 ) = KK G ( A 2 , C ) i s a mo dule for the algebr a K G ( A 1 ) = KK G ( A 1 , C ) . 15 Pr o of. A sp ecial case of the cup-cap pro duct in equation (4), reduces when B 1 = C = B 2 = D to ⊗ C : KK G ( A 1 , C ) ⊗ KK G ( A 2 , C ) − → KK G ( A 1 ⊗ A 2 , C ) . (8) The G -como dule action map a : A 2 − → A 1 ⊗ A 2 . induces a ∗ -homomorphism of G - C ∗ -algebras. Using the fact that K-homology is con trav ari- an t with resp ect to ∗ -homomo r phisms, we get a canonical ab elian gr o up homomorphism a ∗ : KK G ( A 1 ⊗ A 2 , C ) − → KK G ( A 2 , C ) . (9) Comp osing ⊗ C with ∆ ∗ , w e obta in ◦ : KK G ( A 1 , C ) ⊗ KK G ( A 2 , C ) − → KK G ( A 2 , C ) . The fact that ◦ is a n action fo llo ws from the asso ciativit y o f the cup-cap pro duct (Theorem 2.14 of [13 ]) and the defining prop erty of the como dule map ∆. More precisely , the asso ciativit y of Kasparo v’s cup-cap pro duct sa ys t ha t the following diagram commutes , K G ( A 1 ⊗ A 1 ) ⊗ K G ( A 2 ) ⊗ C   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? K G ( A 1 ) ⊗ K G ( A 1 ) ⊗ K G ( A 2 ) 1 ⊗⊗ C   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ⊗ C ⊗ 1                       K G ( A 1 ⊗ A 1 ⊗ A 2 ) K G ( A 1 ) ⊗ K G ( A 1 ⊗ A 2 ) ⊗ C                       where K G ( A ) denotes KK G ( A, C ), A 1 = CT( G, η ) and A 2 = CT( X , f ∗ ( η )). 16 On the other hand, the defining prop ert y of the como dule map ∆ sa ys that t he following diagram commutes , A 1 ⊗ A 2 1 ⊗ a   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? A 2 a   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? a                       A 1 ⊗ A 1 ⊗ A 2 A 1 ⊗ A 2 a ⊗ 1                       Therefore the induced diagram in equiv a rian t K- homology comm utes K G ( A 1 ⊗ A 2 ) a ∗   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? K G ( A 1 ⊗ A 1 ⊗ A 2 ) 1 ⊗ a ∗   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? a ∗ ⊗ 1                       K G ( A 2 ) K G ( A 1 ⊗ A 2 ) a ∗                       Therefore one has the comm utat ive diagram, K G ( A 1 ) ⊗ K G ( A 1 ) ⊗ K G ( A 2 ) 1 ⊗ ( ⊗ C ) / / ( ⊗ C ) ⊗ 1   K G ( A 1 ) ⊗ K G ( A 1 ⊗ A 2 ) 1 ⊗ a ∗ / / ⊗ C   K G ( A 1 ) ⊗ K G ( A 2 ) ⊗ C   K G ( A 1 ⊗ A 1 ) ⊗ K G ( A 2 ) ⊗ C / / a ∗ ⊗ 1   K G ( A 1 ⊗ A 1 ⊗ A 2 ) 1 ⊗ a ∗ / / a ∗ ⊗ 1   K G ( A 1 ⊗ A 2 ) a ∗   K G ( A 1 ) ⊗ K G ( A 2 ) ⊗ C / / K G ( A 1 ⊗ A 2 ) a ∗ / / K G ( A 2 ) 17 The top left hand square comm utes since the cup-cap pro duct is asso ciativ e, the b ottom righ t hand square comm utes since the coa ctio n is coasso ciative , while the remaining squares comm ute b ecause the cup-cap pro duct is functorial under G - C ∗ -homomorphisms. Therefore the pro duct ◦ satisfies the comm utative diagra m K G ( A 1 ) ⊗ K G ( A 2 ) ◦   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? K G ( A 1 ) ⊗ K G ( A 1 ) ⊗ K G ( A 2 ) 1 ⊗◦   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ◦⊗ 1                       K G ( A 2 ) K G ( A 1 ) ⊗ K G ( A 2 ) ◦                       whic h precisely says that ◦ is an action.  Corollary 3.5. L et G b e a c omp act Lie gr oup, and [ η ] ∈ H 3 G ( G, Z ) a p rimitive c ohom olo gy class. L et ( X , f ) ∈ C ( G, η ) . Then the ab elian gr oup K G ( X , f ∗ ( η )) = KK G (CT( X , f ∗ ( η )) , C ) admits a K G ( G, η ) = KK G (CT( G, η ) , C ) -mo dule structur e , induc e d by the gr o up ac tion map a : G × X → X . Pr o of. Let ( X , f ) ∈ C ( G, η ). Setting A 1 = CT( G, η ) and A 2 = CT( X , f ∗ ( η )), w e see tha t ⊗ C : KK G (CT( G, η ) , C ) ⊗ KK G (CT( X, f ∗ ( η )) , C ) − → KK G (CT( G, η ) ⊗ CT( X , f ∗ ( η )) , C ) . The group action map a : G × X → X is a G -map under the adjoint action of G , therefore it induces a ∗ -homo mo r phism of G - C ∗ -algebras, a ∗ : CT( X , f ∗ ( η )) → CT( G × X, a ∗ f ∗ ( η )) ∼ = CT( G × X, p ∗ 1 η + p ∗ 2 f ∗ ( η )) ∼ = CT( G, η ) ⊗ CT( X, f ∗ ( η )) since a ∗ ( f ∗ η ) = p ∗ 1 ( η ) + p ∗ 2 f ∗ ( η ) b y the primitivity assumption o n η . The defining prop ert y of the action map a sho ws that a ∗ is a como dule ma p. The corollary is pro v ed b y applying Prop osition 3.4.  Com bining the ab o v e with the result of F reed-Hopkins-T eleman [9, 10, 1 1], we hav e, Corollary 3.6. L et ( X, f ) ∈ C ( G, η ) , wher e G is a c omp act simple Lie gr oup. The n K G ( X , f ∗ ( η )) is a mo dule over the V erlinde algebr a R ℓ ( G ) , wher e ℓ is the level determine d by twist η . 18 W e call such a mo dule ov er R ℓ ( G ), a V erlinde m o d ule . The following is an immediate corollary of Prop osition 3.4. Corollary 3.7. L et A 1 b e a c omp act C ∗ -quantum gr oup an d A 2 b e a sep ar able C ∗ -algebr a which is also a c omo dule for A 1 . Then the ab elia n gr oup KK( A 2 , C ) is a mo dule for the algebr a KK( A 1 , C ) . 3.1. Morphisms of V erlinde mo dules. Here w e study mo r phisms in t he category C ( G , η ), and o ur main result here is that an y morphism in the category C ( G, η ), determines a mor- phism of V erlinde mo dules o v er t he V erlinde algebra. Prop osition 3.8. L et φ : ( X 1 , f 1 ) − → ( X 2 , f 2 ) b e a morphism in C ( G, η ) , wher e G is a c om p a c t simple Lie gr oup. Then φ ∗ : K G ( X 1 , f ∗ 1 ( η )) − → K G ( X 2 , f ∗ 2 ( η )) i s a morphism of V erli nde mo dules. Pr o of. Let φ : ( X 1 , f 1 ) − → ( X 2 , f 2 ) b e a morphism in C ( G, η ). That is, φ : X 1 → X 2 is an equiv ariant map suc h that the following diagra m comm utes, X 1 f 1   2 2 2 2 2 2 2 2 2 2 2 2 2 2 φ / / X 2 f 2                 G That is, f 1 = f 2 ◦ φ . Then it induces a morphism of groups φ ∗ : K G ( X 1 , f ∗ 1 ( η )) − → K G ( X 2 , f ∗ 2 ( η )) . W e will sho w that φ ∗ is actually a morphism o f V erlinde mo dules o ve r the V erlinde algebra. That is, the following diagram comm utes, K G ( G, η ) ⊗ K G ( X 1 , f ∗ 1 ( η )) ◦ 1 / / id ⊗ φ ∗   K G ( X 1 , f ∗ 1 ( η )) φ ∗   K G ( G, η ) ⊗ K G ( X 2 , f ∗ 2 ( η )) ◦ 2 / / K G ( X 2 , f ∗ 2 ( η )) (10) That is, for ξ ∈ K G ( G, η ) and x 1 ∈ K G ( X 1 , f ∗ 1 ( η )), the comm utativit y of the diagra m ab ov e say s that φ ∗ ( ξ ◦ 1 x 1 ) = ξ ◦ 2 φ ∗ ( x 1 ) , 19 where w e define ◦ j b elo w. Now we ha ve the comm utativ e diagram, G × X 1 a 1 / / id × φ   X 1 φ   G × X 2 a 2 / / X 2 (11) Therefore φ ◦ a 1 = a 2 ◦ φ . So the induced maps on equiv aria n t K - homology satisfy φ ∗ ◦ a 1 ∗ = a 2 ∗ ◦ φ ∗ . As observ ed earlier, since η is primitiv e, a ∗ j ( f ∗ j ( η )) = p ∗ 1 ( η ) + p ∗ 2 ( f ∗ ( η )), j = 1 , 2 . Consider the dia g ram, K G ( G, η ) ⊗ K G ( X 1 , f ∗ 1 ( η )) ⊗ C / / id ⊗ φ ∗   K G ( G × X 1 , a ∗ 1 f ∗ 1 ( η )) id × φ ∗   a 1 ∗ / / K G ( X 1 , f ∗ 1 ( η )) φ ∗   K G ( G, η ) ⊗ K G ( X 2 , f ∗ 2 ( η )) ⊗ C / / K G ( G × X 2 , a ∗ 2 f ∗ 2 ( η )) a 2 ∗ / / K G ( X 2 , f ∗ 2 ( η )) The commutativit y of the rig h t square follo ws from the comm utativit y of the diag ram in equation (11). The commu tativit y of the left square fo llo ws b y naturality of the cup-cap pro duct of Kasparo v [13 , 6]. Therefore w e ha v e justified the comm utativity of t he diagram in equation (1 0), proving the prop o sition.  3.2. The Ma y structure on C ( G, η ) and a lax Ma y functor. Let Mo d(R ℓ (G)) denote the ab elian category of all mo dules ov er the V erlinde ring R ℓ ( G ). W e endo w Mo d(R ℓ (G)) with the sy mmetric monoidal structure giv en by the t ensor pro duct ⊗ = ⊗ Z of ab elian groups, a s w ell as the symmetric monoidal structure giv en b y the direct sum ⊕ , g iving it a Ma y structure. Recall that a functor F : ( C , ⊗ C , I C ) − → ( D , ⊗ D , I D ) b et w een strict monoidal categories is called lax mo n oidal if there are natural maps F ( C ) ⊗ D F ( D ) → F ( C ⊗ C D ) and I D → F ( I C ) for eve ry pair of ob jects C , D ∈ C satisfying certain predictable compatibility conditions. Theorem 3.9. The functor F : ( C ( G , η ) , ` , ⊛ ) − → (Mo d( R ℓ ( G )) , ⊕ , ⊗ ) , define d by F ( X , f ) = K G ( X , f ∗ ( η )) is lax monoidal, r esp e cting the May structur es on b oth c ate gories. 20 Pr o of. Let ( X 1 , f 1 ) , ( X 2 , f 2 ) ∈ C ( G, η ) and recall that F ( X i , f i ) = K G (CT( X i , f ∗ i ( η ))) for i = 1 , 2. Recall that K G ( X , f ∗ ( η )) = K G (CT( X , f ∗ ( η ))) . Then F (( X 1 , f 1 ) ⊛ ( X 2 , f 2 )) = K G ( X 1 × X 2 , ( m ◦ ( f 1 × f 2 )) ∗ ( η )) The cup-cap pro duct in equation (8) with A 1 = C T( X 1 , f ∗ 1 ( η )) and A 2 = C T( X 2 , f ∗ 2 ( η )) giv es ⊗ C : K G ( X 1 , f ∗ 1 ( η )) ⊗ K G ( X 2 , f ∗ 2 ( η )) − → K G ( X 1 × X 2 , ( f 1 ◦ p 1 ) ∗ ( η ) + ( f 2 ◦ p 2 ) ∗ ( η )) ∼ = K G ( X 1 × X 2 , ( f 1 × f 2 ) ∗ ◦ m ∗ ( η )) ∼ = K G ( X 1 × X 2 , ( m ◦ ( f 1 × f 2 )) ∗ ( η )) where w e hav e used the primitivity of η for the last equality . Therefore w e g et a natural map F ( X 1 , f 1 ) ⊗ F ( X 2 , f 2 ) − → F (( X 1 , f 1 ) ⊛ ( X 2 , f 2 )) There is also a canonical map relating the unit ob jects as follow s: Z → K ( e, I) = K G ( e ) , whic h is the unique unital ring homomorphism. Note that Z is the unit ob ject in the monoidal category Mo d(R ℓ (G)) with resp ect to ⊗ Z . Also F (( X 1 , f 1 ) a ( X 2 , f 2 )) = K G ( X 1 a X 2 , ( f 1 a f 2 ) ∗ ( η )) = K G ( X 1 , f ∗ 1 ( η )) ⊕ K G ( X 2 , f ∗ 2 ( η )) = F ( X 1 , f 1 ) ⊕ F ( X 2 , f 2 ) . The distributiv e prop ert y is clear, completing the pro of of the theorem.  Remark 3.10. I f A is a G - C ∗ -algebr a w h ich is a G -c o alg ebr a , define C ( A ) to b e the c ate gory of al l G - C ∗ -algebr as that ar e G -c omo dules ov er A . T hen ther e is a fusion pr o duct ⊛ on C ( A ) , an d a lax monoidal functor F : ( C ( A ) , ⊕ , ⊛ ) − → (Mo d(K G ( A )) , ⊕ , ⊗ ) define d by F ( B ) = K G ( B ) . T he pr o of i s similar to ab ove and wil l b e analyse d in futur e work. 4. The ca tegor y D ( G, η ) and quantiza tion In this section, w e explore when ob jects in the category C ( G, η ) can b e quantize d. T o ac hiev e this goal, w e define a closely related category D ( G, η ), whic h ob jects are triples ( X , E , f ) where ( X , f ) is a n ob ject in the category C ( G, η ) and E is a G -equiv arian t (complex) v ector bundle o v er X . In a dditio n, w e assume that X has an e quivariant twiste d Spinc 21 structur e , that is, the follo wing diagram commutes , X G ν / / f G   B S O π   G G η G / / K ( Z , 3) . (12) Here K ( Z , 3) is the 3rd Eilen b erg-Maclane space, ν is a con tinuous map classifying the stable normal bundle o f the Borel construction X G = E G × G X or equiv a lently , classifying the equiv ariant stable normal bundle of X . Similarly G G = E G × G G where G acts on itself b y conjugatio n and f G is the map induced b y f . Moreov er π is a con t inuous map determined b y the Stieffel-Whitney class, up to homotopy . Suc h a ch oice is fixed. This implies that f ∗ G ( η G ) + W 3 ( X G ) = 0 , (13) where η G is the induced t wisting on G G . This is t he analogue of the F reed-Witten anom- aly cancellation condition fo r D-branes in type I I sup erstring theory , [12]. Geometrically , equation (13) means t ha t there is an equiv ariant isomorphism, f ∗ ( K η ) ∼ = Cliff ( T X ) ⊗ K , (14) where K η is the algebra bundle of compact op erators on G determine d b y η , Cliff ( T X ) denotes the Clifford alg ebra bundle asso ciated to t he tang ent bundle of X , and K denotes the algebra of compact op erators on a G -Hilb ert space. The ob jects of D ( G, η ) are an equiv ariant ana lo gue of tw isted geometric K -cycles in [4, 23]. The k ey observ atio n made here is that in this sp ecial case, this category has a ric her structure than usual, giv en by the Ma y structure. ( G, 1 , id : G → G ) is a final ob ject in the category D ( G, η ), where 1 is the trivial line bundle o ver G . The morphisms of D ( G, η ) are explicitly describ ed in the text. In particular, a compact quasi-Hamiltonian G -manifold ( M , ω , Φ) determines the ob ject ( M , 1 , Φ) in D ( G , η ), b y a result in [2]. Clearly D ( G, η ) is m uc h larger, and it is closed under disjoin t union ` , a dual op eration, the fusion pro duct ⊛ and a lso G -ve ctor bundle mo dification, all of which will b e explained in the text. Here w e men tio n that the f usion pro duct is ( X 1 , E 1 , f 1 ) ⊛ ( X 2 , E 2 , f 2 ) = ( X 1 × X 2 , E 1 ⊠ E 2 , m ◦ ( f 1 × f 2 )), for ob jects ( X j , E j , f j ) j = 1 , 2 in D ( G, η ). W e v erify that D ( G, η ) is a strict monoidal category that ha s a Ma y structure giv en b y ` and ⊛ . Let ( X 1 , E 1 , f 1 ) and ( X 2 , E 2 , f 2 ) denote equiv arian t t wisted geometric K- cycles in D ( G, η ). They a re said to b e isomorp h ic if there is an equiv ariant diffeomeophism φ : X 1 → X 2 suc h 22 that the following diagra m comm utes, X 1 f 1   2 2 2 2 2 2 2 2 2 2 2 2 2 2 φ / / X 2 f 2                 G That is, f 1 = f 2 ◦ φ . Moreo ve r, it is assumed that there is an equiv arian t isomorphism φ ∗ ( E 2 ) ∼ = E 1 . W e now imp ose an equiv alence relation ∼ o n D ( G, η ), generated by isomorphism and the follo wing three elemen t a ry relatio ns: (1) Direct sum - disjoin t union. Let ( X , E 1 , f ) and ( X , E 2 , f ) denote eq uiv aria n t t wisted g eometric K- cycles in D ( G, η ) with the same equiv a rian t t wisted Spinc struc- ture, then their disjoin t union is the equiv arian t t wisted geometric K-cycle give n b y the dir ect sum, ( X , E 1 , f ) a ( X , E 2 , f ) ∼ ( X , E 1 ⊕ E 2 , f ) . (2) Equiv arian t b ordism. Giv en tw o equiv ariant t wisted g eometric K-cycles ( X 1 , E 1 , f 1 ) and ( X 2 , E 2 , f 2 ) suc h that t here exists an equiv ariant tw isted Spinc manifold with b oundary W , an equiv ariant v ector bundle E o v er W and a G -map f : W → G suc h that ∂ W = − X 1 a X 2 , ∂ E = E 1 a E 2 , f   ∂ W = f 1 a f 2 . Here − X denotes the G - manif old X with the opp o site equiv arian t twis ted Spinc structure. Then ( W , E , f ) is said to b e an equiv ariant b ordism b et w een the equiv ari- an t tw isted geometric K-cycles ( X 1 , E 1 , f 1 ) and ( X 2 , E 2 , f 2 ). (3) Equiv arian t Spinc vector bundle mo dification. Let ( X , E , f ) b e an equiv ariant t wisted geometric K-cycle and V an equiv arian t a Spinc vec tor bundle o ve r X with ev en dimensional fib ers. Denote b y R the trivial rank one real ve ctor bundle. Cho ose an inv arian t R iemannian metric on V ⊕ R , let b X = S ( V ⊕ R ) b e the total space of the sphere bundle of V ⊕ R , whic h is a G -manifold. Then the v ertical tangent bundle T ver t ( b X ) of b X admits a natural equiv arian t Spinc structure with an asso ciated Z 2 -graded equiv arian t spinor bundle S + V ⊕ S − V . Denote b y π : b X → X the pro jection, whic h is equiv ariantly K- orien ted. Then the equiv ariant Spinc 23 v ector bundle mo dification of ( X, E , f ) along the equiv ariant Spinc v ector bundle V , is the equiv a rian t twisted geometric K- cycle ( b X , π ∗ E ⊗ S + V , f ◦ π ). Definition 4.1. Denote by K G g eo, • ( G, η ) = D ( G , η ) / ∼ the ge ometric e quivariant twiste d K- homolo gy. A d d ition in K G g eo, • ( G, η ) is given by disjoint union - dir e ct sum r elation. Note that the e quivalenc e r ela tion ∼ pr eserves the p arity of the dimension of the underlying e quivariant twiste d Spinc manifold. L et K G g eo ( G, η ) denote the sub gr oup of K G g eo, • ( G, η ) determine d by al l ge o metric cycles with eve n dimensional e quivari a nt twiste d Spinc m anifolds. Define the fusion pro duct ⊛ of eq uiv ariant tw isted geometric K - cycles ( X 1 , E 1 , f 1 ) and ( X 2 , E 2 , f 2 ) as ( X 1 , E 1 , f 1 ) ⊛ ( X 2 , E 2 , f 2 ) = ( X 1 × X 2 , E 1 ⊠ E 2 , m ◦ ( f 1 × f 2 )) . Here m : G × G → G denotes the multiplic ation on the group, whic h is an equiv arian t map with resp ect to the conjuga tion a ction of G on itself. Then ( D ( G, η ) , ` , ⊛ ) is a Ma y catego ry and w e ha v e, Prop osition 4.2. T h e ge ome tric e quivariant twiste d K-homolo gy gr oup K G g eo ( G, η ) is a ring, with p r o duct induc e d by the fusion p r o duct ⊛ . There is a qu an t izat io n functor Q : D ( G, η ) → K G ( G, η ) whic h w e recall here. Recall that ev ery equiv ariant t wisted geometric K-cycle ( X , E , f ) has a fundamen tal class [ X ] ∈ K G ( X , Cliff ( T X )) as defined in [13], whic h is a Dira c t yp e op erator. Then Q ( X , E , f ) = f ∗ ([ E ] ∩ [ X ]) ∈ K G ( G, η ). Then w e hav e Theorem 4.3. The quantization functor Q : D ( G, η ) − → K G ( G, η ) , define d by Q ( X, E , f ) = f ∗ ([ E ] ∩ [ X ]) is monoidal, r esp e c ting the May structur e s on b oth c ate gories. Pr o of. Let ( X 1 , E 1 , f 1 ) , ( X 2 , E 2 , f 2 ) ∈ D ( G, η ) and recall that Q ( X i , E i , f i ) = f i ∗ ([ E i ] ∩ [ X i ]) for i = 1 , 2. Then Q (( X 1 , E 1 , f 1 ) ⊛ ( X 2 , E 2 , f 2 )) = ( m ◦ ( f 1 × f 2 )) ∗ ([ E 1 ⊠ E 2 ] ∩ [ X 1 × X 2 ]) = m ∗ ( f 1 ∗ ([ E 1 ] ∩ [ X 1 ]) × f 2 ∗ ([ E 2 ] ∩ [ X 2 ])) = f 1 ∗ ([ E 1 ] ∩ [ X 1 ]) ◦ f 2 ∗ ([ E 2 ] ∩ [ X 2 ]) = Q ( X 1 , E 1 , f 1 ) ◦ Q ( X 2 , E 2 , f 2 ) There is also a cano nical map relating t he unit ob jects as follo ws: Q ( e, 1 , I) = I ∗ ( 1 ∩ [ e ]) = 1 whic h is the uniq ue unital ring homomorphism. Also by the disjoint union-direct sum 24 prop erty , one has Q (( X 1 , E 1 , f 1 ) a ( X 2 , E 2 , f 2 )) = Q (( X 1 a X 2 , E 1 a E 2 , f 1 a f 2 )) = ( f 1 a f 2 ) ∗ ([ E 1 a E 2 ] ∩ [ X 1 a X 2 ]) = ( f 1 ) ∗ ([ E 1 ] ∩ [ X 1 ]) a ( f 2 ) ∗ ([ E 2 ] ∩ [ X 2 ]) = Q ( X 1 , E 1 , f 1 ) + Q ( X 2 , E 2 , f 2 ) , since [ X 1 ` X 2 ] = [ X 1 ] ` [ X 2 ] a nd [ E 1 ` E 2 ] = [ E 1 ] ` [ E 2 ]. The distributiv e pro p ert y is clear, completing the pro of of the theorem.  Finally , the following is a sp ecial case of a more general result, Prop osition 4.4. The quantization functor Q in d uc es an isomo rp h ism o f e q uiva ri a nt K- homolo gy rings, K G g eo ( G, η ) ∼ = K G ( G, η ) . This is a special case of a more general theorem whic h will b e pro v ed elsewhe re, whic h uses a hy brid of tec hniques in [4, 23] and [5]. An impact of this result is that the Ma y structure on the category D ( G, η ) induces the algebra structure o n K G ( G, η ), whic h b y [9, 10, 11] is just the V erlinde algebra R ℓ ( G ). 5. Outlook and questions F or G as in the pap er, recall tha t the V erlinde algebra R ℓ ( G ) consists of equiv alence classes of p ositiv e energy repres entations (at a fixed lev el ℓ ) of the free lo op group LG , [19]. The theorem of F reed-Hopkins-T eleman [9, 10, 11] establishes an explicit isomorphism b et wee n K G ( G, η ) and R ℓ ( G ), where η is a degree 3- twist on G that determines the lev el ℓ . This isomorphism is given b y a pro jectiv e family of second quan tized sup ersymmetric Dirac op erators, coupled to a p o sitiv e energy represen tation. Giv en ( X , f ) an ob ject in C ( G, η ), is it p ossible to construct a represen tatio n theoretic group R ℓ ( X , f ), whic h is a mo dule ov er R ℓ ( G ), together with a n explicit isomorphism b e- t we en K G ( X , f ∗ ( η )) and R ℓ ( X , f ) that g eneralizes the F reed-Hopkins-T eleman isomorphism? The alg ebraic prop erties o f the V erlinde algebra R ℓ ( G ) a re we ll understo o d by now . Giv en ( X , f ) an ob ject in C ( G, η ), it w ould b e in teresting to understand the algebraic prop erties of the V erlinde mo dule K G ( X , f ∗ ( η )). 25 Appendix A. More examples of objects in the ca te gor y C ( G, η ) Example 14. (1st Iter ate d pr o duct) Consider G r = Hom( F r , G ) , wher e F r is the fr e e gr o up on r gener ators, with G -action given by the diagonal G -ac tion. Consider the smo oth m ap λ : G r − → G ( g 1 , . . . , g r ) − → n Y j =1 g j which is e quivarian t for the adjoint action of G on G . The r efor e ( G r , λ ) ∈ C ( G, η ) for any G -ve ctor bund le over G r . Be c a use η is a primitive class, by induction on r one se e s that λ ∗ ( η ) = p ∗ 1 ( η ) + · · · + p ∗ r ( η ) . wher e p j denotes the pr oje ction to the j -th factor of G r for j = 1 , . . . , r . Example 15. (2nd I ter ate d pr o duct) In the notation ab ove, c on s i d er G 2 r = Hom( F 2 r , G ) , with G -action given by the diagonal G -action. Consider the smo oth map τ : G 2 r − → G ( g 1 , h 1 . . . , g r , h r ) − → n Y j =1 [ g j , h j ] wher e [ g j , h j ] = g j h j g − 1 j h − 1 j denotes the gr oup c om m utator. Then τ is an e quivariant map for the adjoint action of G on G . Ther efor e ( G 2 r , τ ) ∈ C ( G, η ) . A ls o one c omp utes that τ ∗ ( η ) = 0 . Example 16. (Pr o ducts of sp ac es in C ( G, η ) p art 1) Supp o s e that ( X j , f j ) ∈ C ( G, η ) for j = 1 , . . . , r . Then the Cartesian p r o duct r Y j =1 f j : r Y j =1 X j − → G r . Comp osing with the map λ ab ov e, we se e that ( r Y j =1 X j , λ ◦ ( r Y j =1 f j )) ∈ C ( G , η ) and ( r Y j =1 X j , λ ◦ ( r Y j =1 f j )) ∈ C ( G, η ) . 26 Final ly, using the 1st iter a te d pr o duct exam ple, we se e that ( r Y j =1 f j ) ∗ λ ∗ ( η ) = f ∗ 1 p ∗ 1 ( η ) + . . . + f ∗ r p ∗ r ( η ) . Example 17. (Pr o ducts of sp ac es in C ( G, η ) p art 2) Supp o s e that ( X j , f j ) ar e obje cts in C ( G, η ) for j = 1 , . . . , 2 r . Then the Cartesian pr o d uct 2 r Y j =1 f j : 2 r Y j =1 X j − → G 2 r . Comp osing with the map τ ab ove, w e se e that ( r Y j =1 X j , τ ◦ ( r Y j =1 f j )) ∈ C ( G, η ) and ( r Y j =1 X j , τ ◦ ( r Y j =1 f j )) ∈ C ( G, η ) . Using the 2nd iter ate d pr o duct example ( 2 r Y j =1 f j ) ∗ τ ∗ ( η ) = 0 . Example 18. (Examples fr om Lie gr oups) L et G b e a c omp act Lie gr oup, and T a torus sub gr oup of G , for example the maxim a l torus. Then ther e is a c anon i c al map, p : G / T × T → G ( g T , t ) 7→ g tg − 1 The smo oth map p is e quivariant for the action of G on G/T × T given by, G × ( G/T × T ) − → G/T × T ( g 1 , ( g T , t )) − → ( g 1 g T , t ) and for the adjoint action of G on G . Ther efor e ( G/T × T , p ) ∈ C ( G, η ) . The princip al torus bund le T → G → G/T has first Chern class a ∈ H 2 ( G/T ; b T ) , wher e b T denotes the Pontryagin dual of T . Also, the universal c over of T determines a class b ∈ H 1 ( T ; b T ) . Then a c alc ulation shows that p ∗ ( η ) = p ∗ 1 ( a ) ∪ p ∗ 2 ( b ) ∈ H 3 ( G/T × T ; Z ) wher e we have use d the p airing b T × b T → Z given by the dot p r o duct, ( n , m ) → n.m . 27 Reference s [1] A. Alek s eev, A. Malkin and E. Meinrenken. Lie gr oup v a lued momen t maps. J. Different ial Geom. 48 (1998), no. 3 , 445- 495. [2] A. Alek s eev and E . Meinr enken. Dirac structures and Dixmier- Douady bundles, a rxiv:09 07.125 7. [3] M. Atiy ah and G. Sega l. 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W ang , Geometr ic cycles, index theor y a nd twisted K-homo lo gy . J. Nonc o mmu t. Geom. 2 (2008), no. 4 , 4 97-55 2. (V Mathai) Dep ar tment of P ure Ma thema tics, University of Adela ide, Adelaide, SA 5005, Australia E-mail addr ess : m athai .vargh ese@adelaide.edu.au 28