The Pullbacks of Principal Coactions

THE PULLBA CKS OF PRINCIP AL CO A CTIONS Piotr M. Haja c Inst ytut Matematyczn y , P olsk a Ak ademia Nauk ul. ´ Sniadeckic h 8, W arsza wa, 00-95 6 Poland h ttp: //www .impa n.pl/ e pmh and Katedra Meto d Matem atyczn y c h Fizyki, Uniwersytet W arszawski ul. Ho ˙ za 74, W arszaw a, 00-6 8 2 P ola nd Elmar W a gner Institut o de F ´ ısica y Matem ´ atica s Universidad Mic hoacana de San Nico l´ as de Hi dalg o Ediﬁcio C-3, Cd. Universitari a, 58040 Mo r elia, Mi c hoac´ an, M ´ exico e-mail : elmar @ifm.u mich.mx Abstract: W e pro v e that the class of principal co actions is closed under one-surjectiv e pullbac ks in an appropriate category of algebras equipp ed with left and righ t coactio ns. This allo ws us to handle cases of C ∗ -algebras lac king tw o diﬀeren t non-trivial ideals. It also allo ws us to go b eyo nd the catego ry of comod ule algebras. As an example of the former, we carry out an in dex compu tation for noncomm utativ e line bund les o ver the standard Podle ´ s sp here using the Ma y er-Vietoris typ e arguments aﬀorded by a one-surjectiv e p ullbac k p r esen tation of the C ∗ -algebra of th is quantum sphere. T o instan tiate the latter, w e deﬁn e a family of coalge braic noncomm utativ e deformations of the U(1)- principal bund le S 7 → C P 3 . Con ten ts 1 In tro duction and prelimi naries 2 1.1 Pullback diagr ams and ﬁbre pro ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Odd-to-even connecting homomorphism in K -theor y . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Principal coactions and ass o ciated pro jective modules . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Standard Hopf ﬁbration of quantum SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Principalit y of one-surjectiv e pul lbac ks 12 2.1 Principality of images and pr eimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 2.2 The one-surjective pullbacks of principal coactions are principal . . . . . . . . . . . . . . . . . . . 16 3 The pull bac k picture of the standard qua ntum Hopf ﬁbration 18 3.1 Pullback como dule algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1 3.2 Equiv alence o f the pullback and standa r d constructions . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Index pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 4 Examples of piecewise principal coalg ebra coactions 25 4.1 Piecewise principal coactions fro m a noncommutativ e join construction . . . . . . . . . . . . . . . 2 6 4.2 Quantum complex pro jective spaces C P 3 q,s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 In tro duction and p reliminaries The idea of decomp osing a complicated ob ject into simpler pieces and connecting data is a fundamen tal computational principle throughout mathematics . In the case of (co)homology theory , it yields the May er-Vietoris long exact sequence whose signiﬁcance and usefulness can hardly b e o v erestimated. The categorical underpinning of all this are pullbac k diagrams: in a giv en catego ry they giv e a rigorous meaning to putting together tw o ob jects o v er a third one. The goal of this pap er is to pro v e a general pullbac k theorem fo r principal coactions that signiﬁcan tly generalizes t he ma in result of [13] restricted t o como dule algebras and pullbacks of surjections. More precisely , our main result is that the pullback of principal coactions o v er morphisms of whic h at least one is surjectiv e is again a principal coaction. It ma y b e view ed as a non-linear ve rsion of the Milnor construction yielding a n o dd-to- ev en connecting homomorphism in alg ebraic K -theory [20]. Indeed, linearizing our pullbac k theorem with the help of a corepresen tation giv es precisely the o dd-to-eve n construction of a pro jectiv e mo dule deﬁning the connecting homomorphism in K - theory . W e apply this new result in t w o cases. In the ﬁrst case, we k eep the como dule-algebra setting but tak e a one-surjectiv e pullback dia gram (only one of the deﬁning morphisms is surjectiv e). In the second case, w e pro ceed the other w ay r o und, that is, w e tak e a pullback diagram given b y tw o surjections but take coactions that are not alg ebra homomorphisms . The pullbac k picture of the standard quan tum Hopf ﬁbratio n giv es us the ﬁrst-case example. It pro vides a new w ay of computing the index pairing for the asso ciated quan tum Hopf line bundles (cf. [28]). This index pairing was computed in [1 2] using a noncomm utativ e index form ula, and re- deriv ed in [22]. Here w e giv e yet another metho d to compute it. This simple example sho ws the nee d to generalize from t w o-surjectiv e to one-surjectiv e pullbac k diagrams, and the pullbac k metho d of index computation seems attractiv e due to its inheren t simplicit y . T o obtain the second-case example, w e ﬁrst sho w how the piecewise structure [13 ] of a non- comm utativ e join construction [9] a llows one to deﬁne a certain class of piecewise principal coac- tions. Although this class o f examples can also b e handled by earlier metho ds, it deﬁnitely sho ws that t here are inte resting piecewise principal coa ctions that ar e not algebra homomorphisms. T o obtain a concrete example, w e tak e Pﬂaum’s instanton bundle S 7 q → S 4 q [23] a s the noncom- m utativ e join o f SU q (2) a nd turn it into the coalg ebraic quan tum principal bundle S 7 q → C P 3 q ,s . W e do it with the help o f the canonical surjections π : O (SU q (2)) → O (SU q (2)) /J q ,s determined b y the coideals righ t ideals J q ,s := ( O (S 2 q ,s ) ∩ k er ε ) O (SU q (2)), where S 2 q ,s is a generic Podle ´ s quan tum sphere [24] and k er ε is the k ernel of the counit ma p. 2 The pap er is organized as fo llo ws. First, to make our exp osition self-con tained and to establish no t a tion, we recall fundamen tal concepts tha t w e use la ter o n. The k ey Section 2 is dev oted to the general pullbac k theorem for principal coactions o f coalgebras on algebras, Section 3 is on deriving the index pairing for quan tum Hopf line bundles as a corollary to the pullbac k presen tation o f the standard Hopf ﬁbration of SU q (2), and the ﬁnal Section 4 presen ts new examples of piec ewise principal coa ctions that go beyond Hopf- G alois theory . Throughout the pap er, we work with a lg ebras and coalgebras o v er a ﬁeld. The unadorned tensor pro duct stands fo r the algebraic tensor pro duct ov er t his ﬁeld. W e employ the Heyneman- Sw eedler ty p e notation (with the summation sym b ol suppres sed) for the com ultiplication ∆( c ) = c (1) ⊗ c (2) ∈ C ⊗ C and fo r coactions ∆ V ( v ) = v (0) ⊗ v (1) ∈ V ⊗ C , V ∆( v ) = v ( − 1) ⊗ v (0) ∈ C ⊗ V . The con v olution pro duct of t w o linear maps from a coalgebra to an algebra is denoted b y ∗ : ( f ∗ g )( c ) := f ( c (1) ) g ( c (2) ). The set of natural num b ers includes 0, that is, N = { 0 , 1 , 2 , . . . } . 1.1 Pullbac k diagrams and ﬁbre pro duc ts The purp o se of this section is t o collect some elemen tary facts a b out ﬁbre pro ducts. W e consider the category of v ector spaces as it will be the ambie nt category f or all our pullbac k diagrams. Let π 1 : A 1 → A 12 and π 2 : A 2 → A 12 b e linear maps. The ﬁbr e p r o duct of these maps is deﬁned b y A 1 × ( π 1 ,π 2 ) A 2 := { ( a 1 , a 2 ) ∈ A 1 × A 2 | π 1 ( a 1 ) = π 2 ( a 2 ) } . (1.1) T ogether with the canonical pr o jections pr 1 : A 1 × ( π 1 ,π 2 ) A 2 − → A 1 , pr 2 : A 1 × ( π 1 ,π 2 ) A 2 − → A 2 (1.2) it forms a unive rsal construction completing the initially g iven t w o linear maps in to t he follo wing comm utativ e diagram: A 1 × ( π 1 ,π 2 ) A 2 pr 2 − − − → A 2 pr 1   y π 2   y A 1 π 1 − − − → A 12 . (1.3) Suc h diagrams a re called pul l b ack diagr ams , and ﬁbre pro ducts are often referred to a s pullbac ks. Next, if π 1 : A 1 → A 12 and π 2 : A 2 → A 12 are morphisms of * - algebras, then the ﬁbre pro duct A 1 × ( π 1 ,π 2 ) A 2 is a *-subalgebra of A 1 × A 2 . F urthermore, if we consider the pullbac k diagram (1.3) in the category of (unital) C ∗ -algebras, then A 1 × ( π 1 ,π 2 ) A 2 with its component wise m ultiplication and *-structure is a (unital) C ∗ -algebra. Muc h the same, if B is an algebra and π 1 : A 1 → A 12 and π 2 : A 2 → A 12 are morphisms o f left B -mo dules, then the ﬁbre pro duct A 1 × ( π 1 ,π 2 ) A 2 is a left B -mo dule via the comp o nen t wise left actio n b. ( a 1 , a 2 ) = ( b.a 1 , b.a 2 ). 3 1.2 Odd-to-ev en conn ecting homomorphism in K -theory Consider a pullbac k diagram A w w ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ' ' ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ A 1 π 1 & & & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ A 2 π 2 x x ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ A 12 (1.4) in the category of unital algebras, and assume that one of the deﬁning mo r phisms (here w e c ho ose π 1 ) is surjectiv e. Then there exists a long exact seque nce in algebraic K -theory [2 0] · · · − → K 1 ( A 12 ) odd-to-even / / K 0 ( A ) − → K 0 ( A 1 ⊕ A 2 ) − → K 0 ( A 12 ) . (1.5) The mapping K 1 ( A 12 ) odd-to-even / / K 0 ( A ) is obtained as follo ws. First, giv en left A i -mo dules E i , i = 1 , 2, w e obtain left A 12 -mo dules π i ∗ E i deﬁned b y A 12 ⊗ A i E i . Since A 12 is unital, there are canonical morphisms π i ∗ : E i → π i ∗ E i , π i ∗ ( e ) = 1 ⊗ A i e . The mo dules E i and π i ∗ E i can also b e considered as left mo dules o v er t he ﬁbre pro duct algebra A via the left actions giv en b y a.e i = pr i ( a ) .e i , for e i ∈ E i , and a.f i = π i (pr i ( a )) .f i , for f i ∈ π i ∗ E i . Assume no w that h : π 1 ∗ E 1 → π 2 ∗ E 2 is a morphism of left A 12 -mo dules. Then h ◦ π 1 ∗ : E 1 → π 2 ∗ E 2 and π 2 ∗ : E 2 → π 2 ∗ E 2 can b e lifted to morphisms of left A - mo dules, and w e can consider their pullbac k diagram in the category of left A -mo dules: E 1 × ( h ◦ π 1 ∗ ,π 2 ∗ ) E 2 pr 1 x x q q q q q q q q q q pr 2 & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ E 1 π 1 ∗   E 2 π 2 ∗   π 1 ∗ E 1 h / / π 2 ∗ E 2 . (1.6) In [2 0, Section 2], it is pro v en in detail that, if E i is a ﬁnitely generated pro jectiv e mo dule o v er A i , i = 1 , 2, and h is an isomorphism, then the ﬁbre pro duct M := E 1 × ( h ◦ π 1 ∗ , π 2 ∗ ) E 2 is a ﬁnitely generated pro jectiv e A -mo dule. F urthermore, up to isomorphism, ev ery ﬁnitely gener- ated pro jectiv e mo dule ov er A has this form, and the A i -mo dules E i and pr i ∗ M := A i ⊗ A M , i = 1 , 2, are naturally isomorphic. In particular, if E 1 ∼ = A n 1 and E 2 ∼ = A n 2 , the isomor phism h : π 1 ∗ E 1 → π 2 ∗ E 2 is given b y an inv ertible matrix U ∈ GL n ( A 12 ). Using the canonical em b edding GL n ( A 12 ) ⊆ G L ∞ ( A 12 ), we get a map GL ∞ ( A 12 ) ∋ U 7− → M ∈ Pro j( A ) (1.7) giv en by the pullbac k diagram M u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ) ) ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ A n 1 π 1 ( ( ( ( P P P P P P P P A n 2 π 2 v v ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ A n 12 U ∼ = A n 12 . (1.8) 4 This map induce s an o dd-t o -ev en connecting homomorphism on t he lev el of b o th algebraic [20] and C ∗ -algebraic [15] K -theory . An explicit description of the mo dule M is as follo ws. Assume that π 1 : A 1 → A 12 is surjectiv e. Then there exist liftings c, d ∈ Mat n ( A 1 ) suc h that ev aluating π 1 on c and d comp onent wise yields U − 1 and U resp ectiv ely . Applying [11, Theorem 2.1] to o ur situation yields E 1 × ( h ◦ π 1 ∗ , π 2 ∗ ) E 2 ∼ = A 2 n p , whe re p =  ( c (2 − dc ) d, 1) ( c (2 − dc )(1 − dc ) , 0 ) ((1 − dc ) d, 0) ((1 − dc ) 2 , 0)  ∈ Mat 2 n ( A ) . (1.9) 1.3 Principal coactions and asso ciated pro jectiv e mo dules Recall ﬁrst the general deﬁnition of an en twining structure. Let C be a coalgebra with comu l- tiplication ∆ and counit ε , and let A b e an a lg ebra with m ultiplication m and unit η . A linear map ψ : C ⊗ A − → A ⊗ C (1.10) is called an en twining structur e if and only if it is unital, counita l, and distributiv e with resp ect to b oth the m ultiplication and com ultiplication: ψ ◦ (id ⊗ m ) = ( m ⊗ id) ◦ (id ⊗ ψ ) ◦ ( ψ ⊗ id ) , ψ ◦ (id ⊗ η ) = ( η ⊗ id) ◦ ﬂip , (1.11) (id ⊗ ∆) ◦ ψ = ( ψ ⊗ id) ◦ (id ⊗ ψ ) ◦ (∆ ⊗ id ) , (id ⊗ ε ) ◦ ψ = ﬂip ◦ ( ε ⊗ id ) . (1.12) If ψ is an en tw ining of a coalgebra C and an algebra A , a nd M is a right C -como dule and a righ t A -mo dule, we call M an entwine d mo dule [4] when it satisﬁes the compatibilit y condition ( ma ) (0) ⊗ ( ma ) (1) = m (0) ψ ( m (1) ⊗ a ) . (1.13) Next, let P b e an algebra equipp ed with a coaction ∆ P : P → P ⊗ C of a coalgebra C . Deﬁne the coaction-in v a rian t subalgebra of P b y B := P co C := { b ∈ P | ∆ P ( bp ) = b ∆ P ( p ) for all p ∈ P } . (1.14) W e call the inclusion B ⊆ P a C -extension. W e call it a c o algebr a-Galois C -extension when the canonical left P -mo dule right C -como dule map can : P ⊗ B P − → P ⊗ C , p ⊗ B p ′ 7− → p ∆ P ( p ′ ) , (1.15) is bijectiv e [5]. Note that the bijectivit y of can a llows us to deﬁne the so-called translation map τ : C − → P ⊗ B P , τ ( c ) := can − 1 (1 ⊗ c ) . (1.16) Moreo v er, ev ery coalgebra-Galo is C - extension comes naturally equipp ed with a unique en t win- ing structure tha t mak es P a ( P , C )-en t wined mo dule in the sense of (1.13). It is called the canonical ent wining structure [5], and is v ery useful in calculations o r further constructions. Explicitly , it can b e written as: ψ ( c ⊗ p ) = can ( can − 1 (1 ⊗ c ) p ) . (1.17) 5 An alg ebra P with a righ t C -coaction ∆ P is said to b e e -c o augmen te d if and only if t here ex- ists a gr o up-lik e elemen t e ∈ C suc h that ∆ P (1) = 1 ⊗ e . W e call the C -extension B := P co C ⊆ P e -coaugmen ted. (Muc h t he same w ay , o ne deﬁnes the coaugmen tation of left coactions.) F or the e -coaugmente d coalgebra-Galois C -extensions, o ne can show that the coa ctio n- in v ariant subalgebra deﬁn ed in (1.14) can be express ed as P co C = { p ∈ P | ∆ P ( p ) = p ⊗ e } . (1.18) Indeed, F ormu la (1.17) allows us to express the right coaction in terms of the en t wining ∆ P ( p ) = ψ ( e ⊗ p ) , (1.19) and Equation (1.11) yields the righ t-in-left inclus ion. The o pp osite inclus ion is ob vious. Next, if ψ is in v ertible, one can use (1.12) to sho w that the f orm ula P ∆( p ) := ψ − 1 ( p ⊗ e ) (1.20) deﬁnes a left coaction P ∆ : P → C ⊗ P . W e deﬁne the left coaction-in v ar ian t subalgebra co C P as in (1.14), and deriv e the left-sided v ersion of (1.17). Hence, for an y e -coaugmented coalgebra-Galois C -extension with invertible c anonic al entwining , the right coaction-in v ar ia n t subalgebra coincid es with the left coaction-in v ariant subalgebra: P co C = { p ∈ P | ∆ P ( p ) = p ⊗ e } = { p ∈ P | P ∆( p ) = e ⊗ p } = co C P . (1.21) Finally , w e need to assume one more condition on C -extensions to obtain a suitable deﬁnition of a principal coaction: e quivariant pr oje c tivity . It is a piv otal prop ert y that guara ntees the pro jectivit y of a sso ciated mo dules, a nd thu s leads to index pairings b etw een K -theory and K -homology . Putting t o gether the aforemen tioned four conditions, w e sa y that a coalgebra C -extension B ⊆ P is princip al [6] if: (i) The canonical ma p ca n : P ⊗ B P → P ⊗ C , p ⊗ B p ′ 7→ p ∆ P ( p ′ ), is bijectiv e (G alois condition). (ii) The righ t coaction is e - coaugmen ted f or some group-like e ∈ C , i.e. ∆ P (1) = 1 ⊗ e . (iii) The canonical en t wining ψ : C ⊗ P → P ⊗ C , c ⊗ p 7→ can ( can − 1 (1 ⊗ c ) p ), is bijectiv e. (iv) The algebra P is C -equiv a rian tly pro jectiv e as a left B -mo dule, i.e. there exists a left B -linear and rig h t C -colinear splitting of the multiplication map B ⊗ P → P . In the framew ork of coalgebra extensions, the ro le of connections on principal bundles is pla y ed b y strong connections [6 ]. Let P b e an a lgebra and b oth a left and right e -coaugmen ted C -como dule. (Note that the left and righ t coactions nee d not comm ute.) A str ong c onne ction is a linear map ℓ : C → P ⊗ P satisfying f can ◦ ℓ = 1 ⊗ id , (id ⊗ ∆ P ) ◦ ℓ = ( ℓ ⊗ id ) ◦ ∆ , ( P ∆ ⊗ id ) ◦ ℓ = (id ⊗ ℓ ) ◦ ∆ , ℓ ( e ) = 1 ⊗ 1 . (1.22 ) 6 Here f can : P ⊗ P → P ⊗ C is the lifting of can to P ⊗ P . Assuming that there exists an in v ertible en t wining ψ : C ⊗ P → P ⊗ C making P an en twined mo dule, the ﬁrst three equations of (1.22 ) read in the Heyneman-Sw eedler ty p e notation c 7→ ℓ ( c ) h 1 i ⊗ ℓ ( c ) h 2 i as f ollo ws: ℓ ( c ) h 1 i ψ ( e ⊗ ℓ ( c ) h 2 i ) = ℓ ( c ) h 1 i ℓ ( c ) h 2 i (0) ⊗ ℓ ( c ) h 2 i (1) = 1 ⊗ c, (1.23) ℓ ( c ) h 1 i ⊗ ψ ( e ⊗ ℓ ( c ) h 2 i ) = ℓ ( c ) h 1 i ⊗ ℓ ( c ) h 2 i (0) ⊗ ℓ ( c ) h 2 i (1) = ℓ ( c (1) ) h 1 i ⊗ ℓ ( c (1) ) h 2 i ⊗ c (2) , (1.24) ψ − 1 ( ℓ ( c ) h 1 i ⊗ e ) ⊗ ℓ ( c ) h 2 i = ℓ ( c ) h 1 i ( − 1) ⊗ ℓ ( c ) h 1 i (0) ⊗ ℓ ( c ) h 2 i = c (1) ⊗ ℓ ( c (2) ) h 1 i ⊗ ℓ ( c (2) ) h 2 i . (1.25) Applying id ⊗ ε to (1 .23) yields a useful formu la ℓ ( c ) h 1 i ℓ ( c ) h 2 i = ε ( c ) . (1.26) It is w orth while to observ e the left-right symmetry of principal extensions. W e already noted (se e (1.21)) the equalit y of the left and righ t coaction-in v ariant subalgebras. No w let us deﬁne the left canonical map as can L : P ⊗ B P ∋ p ⊗ q 7− → p ( − 1) ⊗ p (0) q ∈ C ⊗ P . (1.27) One can c hec k t hat it is r elat ed to the right canonical map can by the f orm ula [7] ψ ◦ can L = can . (1.28) Also, if ℓ is a strong connection a nd f can L := (id ⊗ m ) ◦ ( P ∆ ⊗ id) is the lifted left canonical map, then f can L ◦ ℓ = id ⊗ 1. Hence c ⊗ p 7− → ℓ ( c ) h 1 i ⊗ ℓ ( c ) h 2 i p (1.29) is a splitting of f can L just a s p ⊗ c 7− → p ℓ ( c ) h 1 i ⊗ ℓ ( c ) h 2 i (1.30) is a splitting of f can . Lemma 1.1. L et P b e an o b je ct in the c ate gory C e Alg C e of al l unital alg ebr as with e -c o augmente d left and right C -c o actions. Assume that ther e exists an invertible entwining ψ : C ⊗ P → P ⊗ C making P an entwine d mo dule. Then, if P admits a str ong c o n ne ction ℓ , it is princip al. Pr o of. F ollowing [6], ﬁrst w e argue that σ : P ∋ p 7− → p (0) ℓ ( p (1) ) h 1 i ⊗ ℓ ( p (1) ) h 2 i ∈ B ⊗ P (1.31) is a left B -linear splitting of the m ultiplication map. Indeed, m ◦ σ = id b ecause of (1 .26), a nd the calculatio n ψ ( e ⊗ p (0) ℓ ( p (1) ) h 1 i ) ⊗ ℓ ( p (1) ) h 2 i = p (0) ℓ ( p (1) ) h 1 i ⊗ e ⊗ ℓ ( p (1) ) h 2 i (1.32) obtained using (1.11) prov es that σ ( P ) ⊆ B ⊗ P . This splitting is eviden tly right C -colinear, so that its existence pro v es the equiv aria n t pro jectivit y . 7 Next, let us c heck that the f o rm ula can − 1 : P ⊗ C − → P ⊗ B P , p ⊗ c 7− → pℓ ( c ) h 1 i ⊗ B ℓ ( c ) h 2 i , (1.33) deﬁnes the inv erse of the canonical map can , so that the coaction of C is Galois. It follows from (1.23) that can ( can − 1 ( p ⊗ c )) = p ℓ ( c ) h 1 i ℓ ( c ) h 2 i (0) ⊗ ℓ ( c ) h 2 i (1) = p ⊗ c . (1.34) On the other ha nd, taking adv an tage of (1.2 6) and (1.31), w e see that can − 1 ( can ( p ⊗ B q )) = pq (0) ℓ ( q (1) ) h 1 i ⊗ B ℓ ( q (1) ) h 2 i = p ⊗ B q (0) ℓ ( q (1) ) h 1 i ℓ ( q (1) ) h 2 i = p ⊗ B q . (1.35) Th us the conditions (i) and (iv) o f the principalit y of a C -extens ion are satisﬁed. F inally , Condition (ii) is simply assumed, and Condition (iii) fo llo ws fro m t he uniqueness of an en tw ining that makes P an en t wined module. Note that, if there exists a strong connection ℓ , then (1.33) yields τ ( c ) = ℓ ( c ) h 1 i ⊗ B ℓ ( c ) h 2 i . (1.36) In the Heyneman-Sw eedler type nota tion, w e write τ ( c ) = τ ( c ) [1] ⊗ B τ ( c ) [2] . Then the canonical en t wining reads ψ ( c ⊗ p ) = τ ( c ) [1] ( τ ( c ) [2] p ) (0) ⊗ ( τ ( c ) [2] p ) (1) = ℓ ( c ) h 1 i ( ℓ ( c ) h 2 i p ) (0) ⊗ ( ℓ ( c ) h 2 i p ) (1) . (1.37) Remark 1.2. In [6], there is the conv erse statemen t: if P is principal, it admits a strong connection. Th us principal extensions can b e characterize d as these that a dmit a strong con- nection. Recall now that classical principal bundles can b e view ed a s functors transforming ﬁnite- dimensional vec tor space s into asso ciated v ector bundles. Analogously , o ne can prov e that a principal C -extension B ⊆ P deﬁnes a functor from the catego ry of ﬁnite-dimensional left C - como dules in to the category of ﬁnitely generated pro jective left B - mo dules [6]. Explicitly , if V is a left C -como dule with coaction V ∆, t his functor assigns to it the cotensor pro duct P ✷ C V := { P i p i ⊗ v i ∈ P ⊗ V | P i ∆ P ( p i ) ⊗ v i = P i p i ⊗ V ∆( v i ) } . (1.38) In particular, if g ∈ C is a group-lik e elemen t, the for mula C ∆(1) := g ⊗ 1 deﬁnes a 1- dimen- sional corepres en tation, and P ✷ C C = { p ∈ P | ∆ P ( p ) = p ⊗ g } =: P g (1.39) can b e view ed as a noncomm utativ e ass o ciated complex line bundle. Then the general formula for computing an idemp otent E g of the asso ciated mo dule P g out of a corepresen tation and a strong connection b ecomes a v ery simple sp ecial case of [6, Theorem 3.1 ]: P g ∼ = B n E g , ( E g ) n i,j =1 :=  g R i g L j  n i,j =1 , ℓ ( g ) =: n X k =1 g L k ⊗ g R k ∈ P g − 1 ⊗ P g . (1.40) 8 A fundamen tal sp ecial case of principal extensions is provide d by princip al c omo dule alge- br as . One assumes then that C = H is a Hopf a lg ebra with bijectiv e an tip o de S , the canonical map is bijectiv e, and P is a n H -equiv arian tly pro jectiv e left B -mo dule. This brings us in touc h with compact quan tum groups. Assume that ¯ H is the C ∗ -algebra of a compact quan tum group in the sense of W oronowic z [30, 32], and H is its dense Hopf *- subalgebra spanned b y the ma- trix co eﬃcien ts of the irreducible unitary corepresen tations. Let ¯ P b e a unital C ∗ -algebra and δ : ¯ P → ¯ P ¯ ⊗ ¯ H an injectiv e C ∗ -algebraic righ t coa ction of ¯ H on ¯ P . (See [1, Deﬁnition 0.2] for a general deﬁnition and [3, Deﬁnition 1] for the special case of compact quan tum groups.) Here ¯ ⊗ denotes the minimal C ∗ -completion of the algebraic tens or pro duct ¯ P ⊗ ¯ H . T o extend W orono wicz’s P eter-W eyl theory [3 2 ] from compact quan tum g roups to compact quan tum principal bundles, one deﬁnes [2] the subalgebra P δ ( ¯ P ) ⊆ ¯ P of elemen ts for whic h the coaction lands in ¯ P ⊗ H , i.e. P δ ( ¯ P ) := { p ∈ ¯ P | δ ( p ) ∈ P ⊗ H } . (1.41) One easily c hec ks that it is an H - como dule algebra. W e call P δ ( ¯ P ) the Peter-Weyl c omo dule algebr a asso ciated to the C ∗ -coaction δ . It follo ws from results of [3 ] and [25] that P δ ( ¯ P ) is dense in ¯ P . Also, it is straightforw ar d to v erify [2] tha t the op eration ¯ P 7→ P δ ( ¯ P ) giv es a functor comm uting with taking ﬁbre pro ducts (pullbac ks), and that P δ ( ¯ P ) co H coincides with the C ∗ -algebra ¯ P co ¯ H . Finally , let us remark that , for a compact Hausdorﬀ top olog ical group G and a unital C ∗ - algebra A , w e can use the isomorphism A ¯ ⊗ C ( G ) ∼ = C ( G, A ) (e.g. see [29, Corollar y T.6.17]) to tra nslate a righ t C ( G )- coaction δ in to a G - action χ : G ∋ g 7→ χ g ∈ Aut( A ) as follo ws: δ : A − → A ¯ ⊗ C ( G ) ∼ = C ( G, A ) , δ ( a )( g ) =: χ g ( a ) . (1.42) Th us we can use the terminolog y of right C ( G )-como dule C ∗ -algebras a nd G - C ∗ -algebras syn- on ymously . It is imp o rtan t to b ear in mind that the P eter-W eyl functor maps G -equiv ar ia n t *-homomorphisms to colinear homomorphisms of righ t O ( G )-como dule algebras [2]. 1.4 Standard Hopf ﬁ b ration of quan tum SU(2) The standard quan tum Hopf ﬁbration is giv en b y an action of U(1 ) on the quan tum gr o up SU q (2), q ∈ (0 , 1 ). The co ordinate ring of O (SU q (2)) is generated b y α , β , γ , δ with relations αβ = q β α, αγ = q γ α , β δ = q δ β , γ δ = q δ γ , β γ = γ β , (1.43) αδ − q β γ = 1 , δ α − q − 1 β γ = 1 , (1.44) and in v olution α ∗ = δ , β ∗ = − qγ . It is a Hopf * - algebra with com ultiplication ∆, counit ε , and an tip o de S giv en b y ∆( α ) = α ⊗ α + β ⊗ γ , ∆( β ) = α ⊗ β + β ⊗ δ , (1.45) ∆( γ ) = γ ⊗ α + δ ⊗ γ , ∆( δ ) = γ ⊗ β + δ ⊗ δ , (1.46) ε ( α ) = ε ( δ ) = 1 , ε ( β ) = ε ( γ ) = 0 , (1.47) S ( α ) = δ , S ( β ) = − q − 1 β , S ( γ ) = − q γ , S ( δ ) = α. (1.48) 9 Let O (U(1)) denote the comm utativ e and co comm utativ e P eter-W eyl Hopf *-algebra of U(1), and let u stand fo r its unitary gro up-lik e generator. Note that the counit ε and the an tip o de S satisfy ε ( u ) = 1 and S ( u ) = u ∗ . There is a Ho pf *-alg ebra surjection π : O (SU q (2)) − → O (U(1 )) , π ( α ) = u, π ( δ ) = u − 1 , π ( β ) = π ( γ ) = 0 . (1.49) Setting ∆ R := (id ⊗ π ) ◦ ∆, w e obtain a righ t O (U(1))- coaction on O (SU q (2)). On generators, the coa ctio n reads ∆ R ( α ) = α ⊗ u , ∆ R ( β ) = β ⊗ u − 1 , ∆ R ( γ ) = γ ⊗ u, ∆ R ( δ ) = δ ⊗ u − 1 . (1.50) The *-subalgebra of coaction inv arian ts deﬁnes the co ordinate ring of the standar d P o dle ´ s quan tum sphere: O (S 2 q ) := O ( SU q (2)) co O (U(1)) = { a ∈ O (SU q (2)) | ∆ R ( a ) = a ⊗ 1 } . (1.51) One can prov e (see [24]) that O (S 2 q ) is isomorphic to the *-algebra generated b y B and hermitean A satisfying the relations AB = q 2 B A, B ∗ B = A − A 2 , B B ∗ = q 2 A − q 4 A 2 . (1.52) An isomorphism is explicitly given b y the form ulas A = − q − 1 β γ and B = − β α . The irreducible Hilb ert space represen tations of O (S 2 q ) are giv en b y ρ 0 ( A ) = ρ 0 ( B ) = 0 , ρ 0 (1) = 1 on H = C , (1.53) ρ + ( A ) e n = q 2 n e n , ρ + ( B ) e n = q n (1 − q 2 n ) 1 / 2 e n − 1 on H = ℓ 2 ( N ) , (1.54) where { e n | n = 0 , 1 , . . . } is an orthono rmal basis of ℓ 2 ( N ). Recall that the univ ersal C ∗ -algebra of a complex *- algebra is the C ∗ -completion with resp ect to the univ ersal C ∗ -norm g iv en by the suprem um (if it exists) of the op erator norms o v er all b ounded *-represen tations. Let C (S 2 q ) denote the univ ersal C ∗ -algebra generated b y A and B satisfying (1.52). F ro m the ab ov e represen ta tions, it follows that C (S 2 q ) is the minimal unitalization of K ( ℓ 2 ( N )), that is, C (S 2 q ) ∼ = K ( ℓ 2 ( N )) ⊕ C ⊆ B ( ℓ 2 ( N )) , (1.55) ( k + α )( k ′ + α ′ ) = ( k k ′ + α ′ k + αk ′ ) + αα ′ , k , k ′ ∈ K ( ℓ 2 ( N )) , α, α ′ ∈ C . (1.56 ) Here K ( ℓ 2 ( N )) and B ( ℓ 2 ( N )) denote the C ∗ -algebras of compact and b ounded op erators resp ec- tiv ely on the Hilb ert space ℓ 2 ( N ). The isomorphism (1 .5 5) implies tha t K 0 ( C (S 2 q )) ∼ = Z ⊕ Z , where one generator of K -theory is g iv en b y the class of the unit 1 ∈ C (S 2 q ), and the other by the class of the 1-dimensional pro jection on to C e 0 ⊆ ℓ 2 ( N ). F urthermore, K 0 ( C (S 2 q )) ∼ = Z ⊕ Z . W e iden tify o ne generator of K - homology with the class of the pair of represen tations [(id , ε )], where id( k + α ) = k + α a nd ε ( k + α ) = α f or all k ∈ K ( ℓ 2 ( N )) and α ∈ C . The other generator can b e given b y the class of the pair of represen tations [( ε, ε 0 )] with the (non-unital) repres en tation ε 0 of K ( ℓ 2 ( N )) ⊕ C deﬁned by ε 0 ( k + α ) = α SS ∗ , whe re S : ℓ 2 ( N ) − → ℓ 2 ( N ) , S e n = e n +1 , (1.57) 10 denotes the unilateral shift on ℓ 2 ( N ). (See [19] for a detailed treatmen t o f the K -homology and K -theory of P o dle ´ s quan tum spheres .) W e shall also consider the co ordinate ring of the quan tum disc O (D q ) generated by z and z ∗ with the relation z ∗ z − q 2 z z ∗ = 1 − q 2 . (1.58) Its b ounded ir r educible Hilb ert space represen tations are giv en by µ θ ( z ) = e i θ on H = C , θ ∈ [0 , 2 π ) , (1.59) µ ( z ) e n = (1 − q 2( n +1) ) 1 / 2 e n +1 on H = ℓ 2 ( N ) . (1.60) It has been sho wn in [1 6] that the univ ersal C ∗ -algebra of O ( D q ) is isomorphic to the T o eplitz algebra given as the C ∗ -algebra g enerated by the unilateral shift S of Equation (1.57). The represen t a tion µ deﬁnes then an em b edding of O ( D q ) into T . Let C (U(1)) denote the C ∗ -algebra of con t inuous functions o n the unit circle S 1 , and let u b e its unitar y generator. The T o eplitz algebra giv es rise to the follow ing short exact seque nce of C ∗ -algebras: 0 − → K ( ℓ 2 ( N )) − → T σ − → C (U(1 )) − → 0 . (1.61) Here the so-called sym b ol map σ : T → C (U(1)) is giv en b y σ (S) = u . Since S − µ ( z ) b elongs to K ( ℓ 2 ( N )), it follow s in particular that σ ( µ ( z )) = u . No w let us consider the asso ciated quan tum line bundles as ﬁnitely generated pro jectiv e mo dules. They are deﬁned b y the 1- dimensional corepres en tations C ∋ 1 7→ u N ⊗ 1, N ∈ Z , as cotensor pro ducts (1.39): M N := { p ∈ O (SU q (2)) | ∆ R ( p ) = p ⊗ u N } . (1.62) Since ∆ R is a morphism of algebras, M N is an O (S 2 q )-bimo dule. O ur next step is to determine explicitly pro jections describing these pro jectiv e mo dules. F o r l ∈ 1 2 N and i, j = − l, − l + 1 , . . . , l , let t l i,j denote the matrix elemen ts of the irreducib le unitary corepres en tations of O (SU q (2)), tha t is, ∆( t l i,j ) = l X k = − l t l i,k ⊗ t l k ,j , l X k = − l t l ∗ k ,i t l k ,j = l X k = − l t l i,k t l ∗ j,k = δ ij . (1.63) By the P eter-W eyl theorem f o r compact quan tum groups [31], O (SU q (2)) = ⊕ l ∈ 1 2 N ⊕ l i,j = − l C t l i,j . F ro m the explicit desc ription of t l i,j [17, Section 4 .2.4] and the deﬁnition of ∆ R , it follows that ∆ R ( t l i,j ) = t l i,j ⊗ u − 2 j , so that t l i, − j ∈ M 2 j . It can b e sho wn [14, 26 ] that t | j | i, − j , i = −| j | , . . . , | j | generate M 2 j as a left O (S 2 q )-mo dule and M 2 j ∼ = O (S 2 q ) 2 | j | +1 E 2 j for a ll j ∈ 1 2 Z , where E 2 j =     t | j | −| j | , − j . . . t | j | | j | , − j      t | j |∗ −| j | , − j , · · · , t | j |∗ | j | , − j  ∈ Mat 2 | j | +1 ( O (S 2 q )) . (1.64) It is clear that E 2 2 j = E 2 j due to (1.63) and E ∗ 2 j = E 2 j , so that E 2 j is a pro jection. 11 2 Principalit y of one-surjec t iv e pu l l bac ks W e begin b y deﬁning an ambien t category for pullbac k diagrams app earing in the second part of this section. Let P b e a unital algebra equipp ed with b oth a right coaction ∆ P : P → P ⊗ C and a left coaction P ∆ : P → C ⊗ P of the same coalgebra C . W e do not assume that these coactions comm ute, but w e do assume tha t they are coaugmented b y the same group- lik e elemen t e ∈ C , i.e., ∆ P (1) = 1 ⊗ e and P ∆(1) = e ⊗ 1. F or a ﬁxed coalgebra C and a group-lik e e ∈ C , w e consider t he catego r y C e Alg C e of a ll suc h unital algebras with e -coaugmen ted left and righ t C -coactions. Here morphisms are bicolinear algebra homomorphisms. Since we w ork o v er a ﬁeld, this category is eviden tly closed under an y pullback s. If π 1 : P 1 → P 12 and π 2 : P 2 → P 12 are mo r phisms in C e Alg C e , then the ﬁbre pro duct a lge- bra P := P 1 × ( π 1 ,π 2 ) P 2 b ecomes a right C -como dule via ∆ P ( p, q ) = ( p (0) , 0) ⊗ p (1) + (0 , q (0) ) ⊗ q (1) , (2.1) and a left C -como dule via P ∆( p, q ) = p ( − 1) ⊗ ( p (0) , 0) + q ( − 1) ⊗ (0 , q (0) ) . (2.2) Also, it is clear that ∆ P (1 , 1) = (1 , 1) ⊗ e and P ∆(1 , 1) = e ⊗ (1 , 1). 2.1 Principalit y of images and preimages In the following lemma, w e prov e that any surjectiv e morphism in C e Alg C e whose domain is a principal extension can b e split by a left colinear map and b y a right colinear map (not necessarily by a bicolinear map). Note that the ﬁrst part of the lemma is pro v ed m uc h the same wa y as in the Hopf-Galois case [13, Lemma 3.1]: Lemma 2.1. L et π : P → Q b e a surje ctive morph ism in the c ate gory C e Alg C e of unital alg e b r as with e -c o augmente d left a n d right C -c o actions. If P is p rincip al, then: (i) The induc e d map π co C : P co C → Q co C is surje ctive. (ii) Ther e exists a unital righ t C -c oline ar splitting of π . (iii) Ther e exists a unital le f t C -c oline ar s plitting of π . (iv) Q is princip al. F urthermor e, if Q ′ ∈ C e Alg C e , Q ′ ⊆ Q , is princip al, then so is π − 1 ( Q ′ ) . Pr o of. It follo ws from the rig h t colinearit y and surjectivit y of π that π ( P co C ) ⊆ Q co C . T o pro v e the con v erse inclusion, w e take adv antage of the left P co C -linear retractio n of the inclusion P co C ⊆ P that was used to prov e [6, Theorem 2.5( 3)]: σ ϕ : P − → P co C , σ ϕ ( p ) := p (0) ℓ ( p (1) ) h 1 i ϕ ( ℓ ( p (1) ) h 2 i ) . (2.3) 12 Here ℓ is a strong connection o n P a nd ϕ is a n y unital linear functional on P . It follo ws from (1.31) that σ ϕ ( p ) ∈ P co C . If π ( p ) ∈ Q co C , then σ ϕ ( p ) is a desired elemen t of P co C that is mapp ed b y π to π ( p ). Indeed, since π ( p (0) ) ⊗ p (1) = π ( p ) (0) ⊗ π ( p ) (1) = π ( p ) ⊗ e , using the unitalit y of π , ϕ , a nd ℓ ( e ) = 1 ⊗ 1 , w e compute π ( σ ϕ ( p )) = π ( p (0) ) π ( ℓ ( p (1) ) h 1 i ) ϕ ( ℓ ( p (1) ) h 2 i ) = π ( p ) . (2.4) T o sho w the second assertion, let us choose an y unital k -linear splitting of π ↾ P co C and denote it by α co C . W e w an t to prov e that the formula α R ( q ) := α co C ( q (0) π ( ℓ ( q (1) ) h 1 i )) ℓ ( q (1) ) h 2 i (2.5) deﬁnes a unital right colinear splitting of π . Since π is surjectiv e, w e can write q = π ( p ). Then, using pro p erties o f π , w e o bt a in: q (0) π ( ℓ ( q (1) ) h 1 i ) ⊗ ℓ ( q (1) ) h 2 i = π ( p ) (0) π ( ℓ ( π ( p ) (1) ) h 1 i ) ⊗ ℓ ( π ( p ) (1) ) h 2 i = π ( p (0) ) π ( ℓ ( p (1) ) h 1 i ) ⊗ ℓ ( p (1) ) h 2 i = π ( p (0) ℓ ( p (1) ) h 1 i ) ⊗ ℓ ( p (1) ) h 2 i . (2.6) No w it follow s from (1.31) that the ab ov e tensor is in Q co C ⊗ P . Hence α R is w ell deﬁned. It is straightforw ar d to ve rify that α R is unital, right colinear, and splits π . (Note t ha t, since q ∈ Q co C implies q (0) ⊗ q (1) = q ⊗ e , w e ha v e α co C = α R ↾ Q co C .) The third assertion can b e pro v en in an analogous manner. T o prov e ( iv ), we ﬁrst sho w that the inv erse of the canonical map can Q : Q ⊗ Q co C Q → Q ⊗ C (see (1.15 )) is giv en b y can − 1 Q : Q ⊗ C − → Q ⊗ Q co C Q, q ⊗ c 7− → q π ( ℓ ( c ) h 1 i ) ⊗ Q co C π ( ℓ ( c ) h 2 i ) . (2.7) Using the prop erties of π and ℓ , we get ( can Q ◦ can − 1 Q ) ( π ( p ) ⊗ c ) = can Q  π ( pℓ ( c ) h 1 i ) ⊗ Q co C π ( ℓ ( c ) h 2 i )  = π  p ℓ ( c ) h 1 i ℓ ( c ) h 2 i (0)  ⊗ ℓ ( c ) h 2 i (1) = π ( p ) ⊗ c . (2.8) Similarly , ( can − 1 Q ◦ can Q )  π ( p ) ⊗ Q co C π ( p ′ )  = can − 1 Q  π ( pp ′ (0) ) ⊗ p ′ (1)  = π ( pp ′ (0) ℓ ( p ′ (1) ) h 1 i ) ⊗ Q co C π ( ℓ ( p ′ (1) ) h 2 i ) (2.9) = π ( p ) ⊗ Q co C π ( p ′ (0) ℓ ( p ′ (1) ) h 1 i ℓ ( p ′ (1) ) h 2 i ) = π ( p ) ⊗ Q co C π ( p ′ ) . (2.10) Here we used the fact that π ( p ′ (0) ℓ ( p ′ (1) ) h 1 i ) ⊗ ℓ ( p ′ (1) ) h 2 i ∈ Q co C ⊗ P . Hence the extension Q co C ⊆ Q is G alois, and w e hav e the canonical ent wining ψ Q : C ⊗ Q → Q ⊗ C . 13 Our next aim is to pro v e t ha t ψ Q is bijectiv e. W e kno w by assumption that the canonical en t wining ψ P : C ⊗ P → P ⊗ C is in v ertible. T o determine it s in v erse, r ecall that the left and righ t coactions are giv en b y ψ − 1 P ( p ⊗ e ) and ψ P ( e ⊗ p ), resp ectiv ely . Then apply (1.11) to compute ψ P  ( p ℓ ( c ) h 1 i ) ( − 1) ⊗ ( p ℓ ( c ) h 1 i ) (0) ℓ ( c ) h 2 i  = p ℓ ( c ) h 1 i ψ P  e ⊗ ℓ ( c ) h 2 i  = p ℓ ( c ) h 1 i ℓ ( c ) h 2 i (0) ⊗ ℓ ( c ) h 2 i (1) = p ⊗ c. (2.11) Hence ψ − 1 P ( p ⊗ c ) = ( p ℓ ( c ) h 1 i ) ( − 1) ⊗ ( p ℓ ( c ) h 1 i ) (0) ℓ ( c ) h 2 i . On the other hand, ψ Q ( c ⊗ π ( p )) = π ( ℓ ( c ) h 1 i )  π ( ℓ ( c ) h 2 i ) π ( p )  (0) ⊗  π ( ℓ ( c ) h 2 i ) π ( p )  (1) = π  ℓ ( c ) h 1 i  π  ( ℓ ( c ) h 2 i p ) (0)  ⊗ ( ℓ ( c ) h 2 i p ) (1) = ( π ⊗ id)  ψ P ( c ⊗ p )  , (2.12) (id ⊗ π )  ψ − 1 P ( p ⊗ c )  = ( p ℓ ( c ) h 1 i ) ( − 1) ⊗ π  ( p ℓ ( c ) h 1 i ) (0)  π  ℓ ( c ) h 2 i  =  π ( p ℓ ( c ) h 1 i )  ( − 1) ⊗  π ( p ℓ ( c ) h 1 i )  (0) π ( ℓ ( c ) h 2 i ) = Q ∆ ( π ( p ) π ( ℓ ( c ) h 1 i ) ) π ( ℓ ( c ) h 2 i ) . (2.13) The second part of the ab o v e computation implies that the assignmen t ψ − 1 Q : Q ⊗ C − → C ⊗ Q, π ( p ) ⊗ c 7− → (id ⊗ π )( ψ − 1 P ( p ⊗ c )) (2.14) is well deﬁned. Now it follo ws from the ﬁrst part that ψ − 1 Q is the in v erse of ψ Q : ψ Q  ψ − 1 Q  π ( p ) ⊗ c   = ψ Q  (id ⊗ π )  ψ − 1 P ( p ⊗ c )   = ( π ⊗ id)  ψ P  ψ − 1 P ( p ⊗ c )   = π ( p ) ⊗ c, (2.15) ψ − 1 Q  ψ Q  c ⊗ π ( p )   = ψ − 1 Q  ( π ⊗ id)  ψ P ( c ⊗ p )   = (id ⊗ π )  ψ − 1 P  ψ P ( c ⊗ p )   = c ⊗ π ( p ) . (2.16) On the other ha nd, w e observ e that ( π ⊗ π ) ◦ ℓ is a strong connection on Q . Combine d with the j ust prov en existence of a bijectiv e en tw ining that mak es Q an en twine d mo dule, it allows us to apply Lemma 1.1 and conclude the pro o f of ( iv ). T o pro v e the ﬁnal statement of the lemma, note ﬁrst that π − 1 ( Q ′ ) ∈ C e Alg C e . Next, observ e that, if ℓ ′ : C → Q ′ ⊗ Q ′ is a strong connection on Q ′ , then it is also a strong connection on Q . No w, it f ollo ws from (1.37 ) that for an y q ∈ Q ′ ψ Q ( c ⊗ q ) = ℓ ′ ( c ) h 1 i  ℓ ′ ( c ) h 2 i q  (0) ⊗  ℓ ′ ( c ) h 2 i q  (1) ∈ Q ′ ⊗ C . (2.17) Muc h the same w a y , it follo ws from the Q -analog of the form ula following (2.1 1) that ψ − 1 Q ( Q ′ ⊗ C ) ⊆ C ⊗ Q ′ . Hence to see that ψ P and ψ − 1 P restrict to π − 1 ( Q ′ ), w e can apply (2.12) and (2.14), respectiv ely . A k ey step no w is to construct a strong connection on π − 1 ( Q ′ ). Let α R and α L b e, respec- tiv ely , righ t and left colinear unital splittings of π . Their ex istence is guaran teed b y the already pro v en ( ii ) and ( iii ). The map ( α L ⊗ α R ) ◦ ℓ ′ : C → π − 1 ( Q ) ⊗ π − 1 ( Q ) is bicolinear and satisﬁes α L ( ℓ ′ ( e ) h 1 i ) ⊗ α R ( ℓ ′ ( e ) h 2 i ) = 1 ⊗ 1 . (2.18) 14 Ho w ev er, 1 ⊗ c −  f can ◦ ( α L ⊗ α R ) ◦ ℓ ′  ( c ) = 1 ⊗ c − α L ( ℓ ′ ( c (1) ) h 1 i ) α R ( ℓ ′ ( c (1) ) h 2 i ) ⊗ c (2) 6 = 0 . (2.19) T o solve this problem, w e a pply to it t he splitting of the lifted canonical map given b y a strong connection ℓ (see (1.3 0)), a nd add to ( α L ⊗ α R ) ◦ ℓ ′ : ℓ R ( c ) := ( α L ⊗ α R )( ℓ ′ ( c )) + ℓ ( c ) − α L ( ℓ ′ ( c (1) ) h 1 i ) α R ( ℓ ′ ( c (1) ) h 2 i ) ℓ ( c h 1 i (2) ) ⊗ ℓ ( c h 2 i (2) ) . (2.20) No w f can ◦ ℓ R = 1 ⊗ id, as needed. Also, ℓ R ( e ) = 1 ⊗ 1 and (( π ⊗ id) ◦ ℓ R )( C ) ⊆ Q ′ ⊗ P . The righ t colinearity of ℓ R is clear. T o c hec k the left colinearity of ℓ R , using the fact that P is a ψ P en t wined and e -coaugmen ted mo dule, we sho w that ( m P ◦ ( α L ⊗ α R ) ◦ ℓ ′ ) ∗ ℓ is left colinear. (Here m P is the m ultiplication of P .) First we note t hat ( P ∆ ⊗ id) ◦ ( ( m P ◦ ( α L ⊗ α R ) ◦ ℓ ′ ) ∗ ℓ ) = ( id ⊗ ( m P ◦ ( α L ⊗ α R ) ◦ ℓ ′ ) ∗ ℓ ) ◦ ∆ (2.21) is equiv a lent t o α L ( ℓ ′ ( c (1) ) h 1 i ) α R ( ℓ ′ ( c (1) ) h 2 i ) ℓ ( c (2) ) h 1 i ⊗ e ⊗ ℓ ( c (2) ) h 2 i = ψ P  c (1) ⊗ α L ( ℓ ′ ( c (2) ) h 1 i ) α R ( ℓ ′ ( c (2) ) h 2 i ) ℓ ( c (3) ) h 1 i  ⊗ ℓ ( c (3) ) h 2 i . (2.22) Since c (1) ⊗ α L ( ℓ ′ ( c (2) ) h 1 i ) ⊗ ℓ ′ ( c (2) ) h 2 i = ψ − 1 P  α L ( ℓ ′ ( c ) h 1 i ) ⊗ e  ⊗ ℓ ′ ( c ) h 2 i , w e obtain ψ P  c (1) ⊗ α L ( ℓ ′ ( c (2) ) h 1 i ) α R ( ℓ ′ ( c (2) ) h 2 i ) ℓ ( c (3) ) h 1 i  ⊗ ℓ ( c (3) ) h 2 i = α L ( ℓ ′ ( c (1) ) h 1 i ) ψ P  e ⊗ α R ( ℓ ′ ( c (1) ) h 2 i ) ℓ ( c (2) ) h 1 i  ⊗ ℓ ( c (2) ) h 2 i = α L ( ℓ ′ ( c (1) ) h 1 i ) α R ( ℓ ′ ( c (1) ) h 2 i ) ψ P  c (2) ⊗ ℓ ( c (3) ) h 1 i  ⊗ ℓ ( c (3) ) h 2 i = α L ( ℓ ′ ( c (1) ) h 1 i ) α R ( ℓ ′ ( c (1) ) h 2 i ) ψ P  ψ − 1 P  ℓ ( c (2) ) h 1 i ⊗ e   ⊗ ℓ ( c (2) ) h 2 i = α L ( ℓ ′ ( c (1) ) h 1 i ) α R ( ℓ ′ ( c (1) ) h 2 i ) ℓ ( c (2) ) h 1 i ⊗ e ⊗ ℓ ( c (2) ) h 2 i . (2.23) Hence ℓ R is a strong connection with the prop ert y ℓ R ( C ) ⊆ π − 1 ( Q ′ ) ⊗ P . In a similar manner, we construct a strong connection ℓ L with t he pro p ert y ℓ L ( C ) ⊂ P ⊗ π − 1 ( Q ′ ). No w w e need to apply the splitting of the left lifted canonical map giv en b y ℓ (se e ( 1 .29)) to derive the form ula ℓ L := ( α L ⊗ α R ) ◦ ℓ ′ + ℓ − ℓ ∗ ( m P ◦ ( α L ⊗ α R ) ◦ ℓ ′ ) . (2.24) It is clear that ℓ L ( e ) = 1 ⊗ 1 and ℓ L ( C ) ⊆ P ⊗ π − 1 ( Q ′ ). A computation similar to (2.23) sho ws the righ t colinearity of ℓ L . Since furthermore ψ P (1 ⊗ c ) = c ⊗ 1 for an y c ∈ C and f can = ψ P ◦ f can L , w e obtain f can ( ℓ L ( c )) = ψ P  f can L ( ℓ ( c ))  = ψ P ( c ⊗ 1 ) = 1 ⊗ c. (2.25) Hence ℓ L is a desired strong connection. Plugging it in to (2.20 ) instead of ℓ , w e get a strong connection ℓ LR = ( α L ⊗ α R ) ◦ ℓ ′ + ℓ L − ( m p ◦ ( α L ⊗ α R ) ◦ ℓ ′ ) ∗ ℓ L (2.26) with the prop ert y ℓ LR ⊆ π − 1 ( Q ′ ) ⊗ π − 1 ( Q ′ ). Applying no w Lemma 1.1 ends the pro o f of this lemma. 15 2.2 The one-surjectiv e pullbac ks of principal coactions are p rincipal Our goal no w is to sho w that the sub categor y o f principal extensions is closed under one- surjectiv e pullbacks . Here the right coa ctio n is the coaction deﬁning a principal extension and the left coaction is the one deﬁned by the inv erse of the canonical ent wining (see (1.20)). With this structure, principal extensions f orm a full sub categor y o f C e Alg C e . The following theorem is the main result of this pap er g eneralizing the t heorem of [13] on the pullback of surjections of principal como dule algebras: Theorem 2.2. L et C b e a c o algebr a, e ∈ C a gr oup-like element, and P the pul lb ack of π 1 : P 1 → P 12 and π 2 : P 2 → P 12 in the c ate gory C e Alg C e of unital algebr as with e -c o augmente d left and right C -c o actions. If π 1 or π 2 is surje c tive and b o th P 1 and P 2 ar e princip al e -c o augmente d C -extensions, then also P is a princip al e -c o augmente d C -extension. Pr o of. Without loss o f generality , w e assume that π 1 is surjectiv e. W e ﬁrst sho w that P inherits an ent wined structure from P 1 and P 2 . Lemma 2.3. L et ψ 1 and ψ 2 denote the entwini n g structur e s of P 1 and P 2 , r es p e c tive l y. Then P is an entwine d mo dule with an inve rtible entwining structur e ψ = ψ 1 ◦ (id ⊗ pr 1 ) + ψ 2 ◦ (id ⊗ pr 2 ) . (2.27) Her e pr 1 and pr 2 ar e morphi sms of the pul lb ack diagr am as in (1.3) . Pr o of. Our strategy is to construct a bijectiv e map ˜ ψ : C ⊗ ( P 1 × P 2 ) → ( P 1 × P 2 ) ⊗ C , and to sho w that it res tricts to a bijectiv e en t wining on C ⊗ P . W e put ˜ ψ := ψ 1 ◦ (id ⊗ ˜ pr 1 ) + ψ 2 ◦ (id ⊗ ˜ pr 2 ) . (2.28) The sym b ols ˜ pr 1 and ˜ pr 2 stand for resp ectiv e compo nen t wise pro jections. Their restrictions t o P yield pr 1 and pr 2 . It is easy to c hec k that t he inv erse of ˜ ψ is giv en b y ˜ ψ − 1 = ψ − 1 1 ◦ ( ˜ pr 1 ⊗ id ) + ψ − 1 2 ◦ ( ˜ pr 2 ⊗ id ) (2.29) T o sho w that ˜ ψ ( C ⊗ P ) ⊆ P ⊗ C and ˜ ψ − 1 ( P ⊗ C ) ⊆ C ⊗ P , w e note ﬁrst that P 12 and π 2 ( P 2 ) are principal b y Lemma 2.1(iv). Conseque ntly , their canonical ent winings ψ 12 and ψ π 2 ( P 2 ) are bijectiv e. F urthermore, arguing as in the pro of of Lemma 2.1, we see that ψ π 2 ( P 2 ) = ψ 12 ↾ C ⊗ π 2 ( P 2 ) and ψ − 1 π 2 ( P 2 ) = ψ − 1 12 ↾ π 2 ( P 2 ) ⊗ C . An adv antage of ha ving b oth summands in terms of ψ 12 is tha t we can apply (2.12 ) to compute  ( π 1 ◦ ˜ pr 1 − π 2 ◦ ˜ pr 2 ) ⊗ id  ◦ ˜ ψ = ( π 1 ◦ ˜ pr 1 ⊗ id) ◦ ψ 1 ◦ (id ⊗ ˜ pr 1 ) − ( π 2 ◦ ˜ pr 2 ⊗ id) ◦ ψ 1 ◦ (id ⊗ ˜ pr 1 ) + ( π 1 ◦ ˜ pr 1 ⊗ id) ◦ ψ 2 ◦ (id ⊗ ˜ pr 2 ) − ( π 2 ◦ ˜ pr 2 ⊗ id) ◦ ψ 2 ◦ (id ⊗ ˜ pr 2 ) = ( π 1 ⊗ id) ◦ ψ 1 ◦ (id ⊗ ˜ pr 1 ) − ( π 2 ⊗ id) ◦ ψ 2 ◦ (id ⊗ ˜ pr 2 ) = ψ 12 ◦ (id ⊗ π 1 ) ◦ (id ⊗ ˜ pr 1 ) − ψ π 2 ( P 2 ) ◦ (id ⊗ π 2 ) ◦ (id ⊗ ˜ pr 2 ) = ψ 12 ◦  id ⊗ ( π 1 ◦ ˜ pr 1 − π 2 ◦ ˜ pr 2 )  . (2.30) Hence ˜ ψ ( C ⊗ P ) ⊆ P ⊗ C . Muc h t he same w a y , using (2.14) instead of (2.12), w e show that the bijection ˜ ψ − 1 ( P ⊗ C ) ⊆ C ⊗ P . 16 It remains to v erify that the bijection ψ = ˜ ψ ↾ C ⊗ P is an en t wining that mak es P an en tw ined mo dule. The former is pro v en b y a direct c heck ing of (1 .1 1) and (1 .12). The la tter f o llo ws from the f a ct that P 1 and P 2 are, resp ectiv ely , ψ 1 and ψ 2 en t wined mo dules: ∆ P ( pq ) = ∆ P 1 (pr 1 ( p )pr 1 ( q )) + ∆ P 2 (pr 2 ( p )pr 2 ( q )) = pr 1 ( p (0) ) ψ 1 ( p (1) ⊗ pr 1 ( q )) + pr 2 ( p (0) ) ψ 2 ( p (1) ⊗ pr 2 ( q )) =  pr 1 ( p (0) ) + pr 2 ( p (0) )  ψ 1 ( p (1) ⊗ pr 1 ( q )) + ψ 2 ( p (1) ⊗ pr 2 ( q ))  = p (0) ψ ( p (1) ⊗ q ) . (2 .31) This prov es the lemma. Let α 1 L and α 1 R b e a unital left colinear splitting and a unital right colinear splitting of π 1 , resp ectiv ely . Also, let α 2 R b e a right colinear splitting of π 2 view ed as a map on to π 2 ( P 2 ). Suc h maps exist b y Lemma 2.1. On the other hand, by [6, Lemma 2.2], since P 1 and P 2 are principal, they admit strong connections ℓ 1 and ℓ 2 , respectiv ely . F or brevit y , let us in tro duce the notation α 12 L := α 1 L ◦ π 2 , α 12 R := α 1 R ◦ π 2 , α 21 R := α 2 R ◦ π 1 ↾ π − 1 1 ( π 2 ( P 2 )) , L := m P 1 ◦ ( α 12 L ⊗ α 12 R ) ◦ ℓ 2 , (2.3 2) where m P 1 is the m ultiplication of P 1 . The situation is illustrated in the follo wing diagram: (2.33) C L   P pr 1 v v ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ pr 2 ( ( ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ π − 1 1 ( π 2 ( P 2 )) E e s s ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ α 21 R + + ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ P 1 ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ P 2 . α 12 L o o α 12 R o o ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ π 2 s s s s ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ π 1 ' ' ' ' P P P P P P P P P P P P P P P P P π 2 ( P 2 ) α 2 R 3 3 ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ _    π 2 w w ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ P 12 α 1 L g g ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ α 1 R g g ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Our prov es hinges on constructing a strong connection on P out of strong connections on P 1 and P 2 . Roughly sp eaking, the basic idea is to tak e a strong connection o n P 2 , induce a strong connection on the the common par t P 12 , and prolo ng ate it to P 1 . T o this end, w e chec k that ( α 12 L + id) ⊗ ( α 12 R + id) is a unital bicolinear map from P 2 ⊗ P 2 to P ⊗ P . Therefore, as a ﬁrst a ppro ximation for constructing a strong connection o n P , w e choose the fo rm ula ℓ I :=  ( α 12 L + id) ⊗ ( α 12 R + id)  ◦ ℓ 2 . (2.34) It is a bicolinear map from C to P ⊗ P satisfying ℓ I ( e ) = 1 ⊗ 1 as needed. Ho w ev er, it do es not split the lifted canonical map: ( f can ◦ ℓ I )( c ) − 1 ⊗ c = α 12 L ( ℓ 2 ( c ) h 1 i ) α 12 R ( ℓ 2 ( c ) h 2 i ) (0) ⊗ α 12 R ( ℓ 2 ( c ) h 2 i ) (1) + ℓ 2 ( c ) h 1 i ℓ 2 ( c ) h 2 i (0) ⊗ ℓ 2 ( c ) h 2 i (1) − 1 ⊗ c = α 12 L ( ℓ 2 ( c (1) ) h 1 i ) α 12 R ( ℓ 2 ( c (1) ) h 2 i ) ⊗ c (2) + (0 , 1) ⊗ c − 1 ⊗ c = L ( c (1) ) ⊗ c (2) − (1 , 0) ⊗ c ∈ P 1 ⊗ C . (2.35) 17 W e correct it b y adding to ℓ I ( c ) the splitting of the lifted cano nical map on P 1 ⊗ P 1 aﬀorded b y ℓ 1 and applied to (1 , 0) ⊗ c − L ( c (1) ) ⊗ c (2) : ℓ I I ( c ) = ℓ I ( c ) + ℓ 1 ( c ) h 1 i ⊗ ℓ 1 ( c ) h 2 i − L ( c (1) ) ℓ 1 ( c (2) ) h 1 i ⊗ ℓ 1 ( c (2) ) h 2 i = ( ℓ I + ℓ 1 − L ∗ ℓ 1 )( c ) . (2.36) The abov e approxim ation to a strong connection on P is clearly right colinear. Using the fa ct that P 1 is a ψ 1 -en t wined and e - coaugmen ted module, we follow the lines of (2 .21)–(2.23) t o sho w that L ∗ ℓ 1 is left colinear. Hence ℓ I I is bicolinear. It also satisﬁes ℓ I I ( e ) = 1 ⊗ 1. How ever, the price we pay for having ℓ I I ( c ) h 1 i ℓ I I ( c ) h 2 i (0) ⊗ ℓ I I ( c ) h 2 i (1) = 1 ⊗ c is t hat the imag e of ℓ I I is no longer in P ⊗ P . The troublesome term ℓ 1 − L ∗ ℓ 1 tak es v a lues in P ⊗ P 1 . No w one w ould lik e to comp o se id ⊗ (id + α 21 R ) with ℓ 1 − L ∗ ℓ 1 to force it taking v alues in P ⊗ P . Ho w ev er, since α 21 R is deﬁned only on π − 1 1 ( π 2 ( P 2 )), w e need to replace an arbitra ry strong connection ℓ 1 b y a strong connection taking v alues in P 1 ⊗ π − 1 1 ( π 2 ( P 2 )). Suc h a strong connection is prov ided for us b y (2.24): ˜ ℓ 1 := ( α 12 L ⊗ α 12 R ) ◦ ℓ 2 + ℓ 1 − ℓ 1 ∗ L. (2.37) Inserting ˜ ℓ 1 in place of ℓ 1 allo ws us to apply the correction map id ⊗ (id + α 21 R ) to obtain ℓ I I I = ℓ I +  id ⊗ (id + α 21 R )  ◦ ( ˜ ℓ 1 − L ∗ ˜ ℓ 1 ) . ( 2 .38) T o end the pro of, let us che c k that ℓ I I I is indeed a strong connection on P . First, since ℓ I ( C ) ⊆ P ⊗ P and (id + α 21 R )( π − 1 1 ( π 2 ( P 2 ))) ⊆ P , w e conclude that ℓ I I I tak es v alues in P ⊗ P . Next, it is bicolinear because α 21 R is righ t colinear. Also, it is clearly unital. T o v erify that ℓ I I I splits the canonical map, ﬁrst we note that f can ◦ ( id ⊗ α 21 R ) ◦ ( ˜ ℓ 1 − L ∗ ˜ ℓ 1 ) = 0 b ecause m P 1 × P 2 ( p 1 ⊗ p 2 ) = 0 for all p 1 ∈ P 1 and p 2 ∈ P 2 . Com bining this with the fact that f can ◦ ( ℓ ′ 1 − L ∗ ℓ ′ 1 ) do es not dep end on the c hoice of a strong connection ℓ ′ 1 , we infer that f can ◦ ℓ I I I = f can ◦ ℓ I I = 1 ⊗ id. Th us ℓ I I I is a strong connection on P , as desired. Com bining this fact with Lemma 2.3 and Lemma 1 .1 prov es the theorem. Putting the form ulas in the pro of of Theorem 2.2 together, w e obtain the fo llo wing strong connection on P : ℓ = ( ( α 12 L + id) ⊗ ( α 12 R + id) ) ◦ ℓ 2 (2.39) + ( η 1 ◦ ε − L ) ∗ ( ( id ⊗ ( id + α 21 R ) ) ◦ ( ℓ 1 − ℓ 1 ∗ L + ( α 12 L ⊗ α 12 R ) ◦ ℓ 2 ) ) . 3 The pullbac k picture of the standard quan tum Hopf ﬁbration Recall that the classical Hopf ﬁbration is a U(1)-principal bundle giv en by the maps π : S 3 = { ( z 1 , z 2 ) ∈ C 2 | | z 1 | 2 + | z 2 | 2 = 1 } − → S 2 ∼ = C P 1 , π (( z 1 , z 2 )) = [( z 1 : z 2 )] , S 3 × U(1) − → S 3 , ( z 1 , z 2 ) ⊳ u = ( z 1 u, z 2 u ) . (3.1) 18 T o unra v el the structure of this non-trivial ﬁbration, w e split S 3 in to t w o disjoint parts: S 3 = { ( z 1 , z 2 ) ∈ C 2 | | z 1 | 2 < 1 , | z 2 | 2 = 1 − | z 1 | 2 } ∪ { ( z 1 , 0) | | z 1 | = 1 } . (3.2) Note that b oth sets are inv arian t under the U(1)-action. The second set is U(1), a nd ﬁrst set is U(1)-equiv arian tly homeomorphic to the in terior o f the solid torus D × U(1) equipp ed with the diagonal action. (Here D = { z ∈ C | | z | ≤ 1 } .) By an appropriate U(1)-equiv a r ia n t gluing of the b oundary torus of D × U(1 ) with U(1), w e reco v er S 3 with its U(1)-action: (3.3) S 3 φ 1 ( z , v ) = ( z , v p 1 − | z | 2 ) φ 2 ( u ) = ( u, 0 ) D × U(1) φ 1 = = ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ U(1) φ 2 ^ ^ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ( ι, id)( u, v ) = ( u, v ) pr 1 ( u, v ) = u . U(1) × U(1) pr 1 ? ?          ( ι, id) b b ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ Ho w ev er, to view D × U(1) as a trivial U(1)-principal bundle, w e need to gauge the di- agonal action to the action on the r ig h t slot. This is ac hiev ed with the help o f the follo wing homeomorphism in tert wining thes e t w o actions: Ψ : D × U(1) − → D × U(1) , Ψ( x, v ) := ( xv , v ) , Ψ( x, v u ) = Ψ( x, v ) ⊳ u. (3.4) Let us denote the restriction of Ψ to U(1) × U(1) b y the same sym b ol. No w w e can extend the ab ov e pushout diagra m to t he commutativ e dia g ram (3.5) S 3 D × U(1) Ψ / / D × U(1) φ 1 ; ; ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ U(1) φ 2 ` ` ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ U(1) id o o U(1) × U(1) pr 1 > > ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ( ι, id) c c ● ● ● ● ● ● ● ● ● ● U(1) × U(1) , ( ι, id) f f ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ Ψ O O m ; ; ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ where m is the m ultiplication map. The outer diagram is again a pushout diagram of U(1)- spaces, but no w its deﬁning U(1)-spaces a re trivial U(1)-principal bundles. It is the outer pushout diag ram that we shall use to analyse a noncomm utativ e deformation of the Hopf ﬁbration. 3.1 Pullbac k como dule algebra W e consider the tensor pro ducts P 1 := T ⊗ O (U(1)), P 2 = C ⊗ O (U( 1 )) ∼ = O (U(1)) and P 12 := C (U(1)) ⊗ O (U(1)). These algebras are rig h t O (U(1 ))-como dule algebras f or the coaction 19 x ⊗ u N 7→ x ⊗ u N ⊗ u N , N ∈ Z . Moreov er, P 1 and P 2 are trivially principal with strong connections ℓ i : O ( U(1)) → P i ⊗ P i giv en b y ℓ i ( u N ) = (1 ⊗ u N ∗ ) ⊗ (1 ⊗ u N ), i = 1 , 2. T o construct a pullbac k of P 1 and P 2 , we deﬁne the followin g morphisms of righ t O (U(1))- como dule algebras: π 1 : T ⊗ O (U(1) ) − → C (U(1)) ⊗ O (U(1)) , π 1 ( t ⊗ w ) = σ ( t ) ⊗ w , (3.6) π 2 : O ( U(1)) − → C (U(1)) ⊗ O ( U(1)) , π 2 ( w ) = ∆( w ) . (3.7) Then the ﬁbre pro duct P := T ⊗ O (U(1)) × ( π 1 ,π 2 ) O (U(1)) deﬁne d b y the pullbac k diagram T ⊗ O (U(1)) × ( π 1 ,π 2 ) O (U(1)) pr 1 v v ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ pr 2 ( ( P P P P P P P P P P P P P T ⊗ O (U(1)) π 1 ) ) ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ O (U(1)) π 2 v v ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ C (U(1)) ⊗ O ( U(1 )) (3.8) is a righ t O (U(1) )-como dule a lgebra. By Prop osition 2.2, it is principal. F urthermore, deﬁne unital resp ectiv ely left colinear and righ t colinear splittings of π 1 b y α 1 L ( f ⊗ u N ) = α 1 R ( f ⊗ u N ) = T f ⊗ u N , N ∈ Z . (3.9) Here f ∈ C (U(1)) and T f denotes the T o eplitz o p erator with sym b ol f . In particular, T u N = S N and T u ∗ N = S ∗ N . A righ t colinear splitting of the map π 2 : O ( U(1)) → π 2 ( O (U(1))) is giv en b y α 2 R ( u N ⊗ u N ) = u N , N ∈ Z . (3.10) Inserting the deﬁnitions of α 1 L , α i R and ℓ i , i = 1 , 2, in to (2 .3 2) and (2.3 9), w e obtain the f o llo wing strong connection on P : ℓ ( u N ) = (S ∗ N ⊗ u ∗ N , u ∗ N ) ⊗ ( S N ⊗ u N , u N ) , (3.11) ℓ ( u ∗ N ) = (S N ⊗ u N , u N ) ⊗ (S ∗ N ⊗ u ∗ N , u ∗ N ) + ((1 − S N S ∗ N ) ⊗ u N , 0) ⊗ ((1 − S N S ∗ N ) ⊗ u ∗ N , 0) , N ∈ N . (3.12) Note nex t that, b y construction, w e hav e P = n P k ( t k ⊗ u k , α k u k ) ∈  T ⊗ O (U(1))  × O (U(1))    σ ( t k ) = α k u k o , (3.13) where α k ∈ C . F or C ∆(1) = u N ⊗ 1, let L N := P  O (U(1)) C ∼ = { p ∈ P | ∆ P ( p ) = p ⊗ u N } . (3.14) Then L 0 = P co O (U(1)) , eac h L N is a left P co O (U(1)) -mo dule a nd P = L N ∈ Z L N . F r om ∆ P  X k ( t k ⊗ u k , α k u k )  = X k ( t k ⊗ u k , α k u k ) ⊗ u k , (3.15) 20 it follows that L N = n ( t ⊗ u N , αu N ) ∈  T ⊗ O (U(1))  × O (U(1))    σ ( t ) = α u N o . (3.16) The next prop osition show s that L 0 ∼ = T × ( σ , 1) C is isomorphic to the C ∗ -algebra o f the standard P o dle ´ s sphere and that L N ∼ = T × ( u − N σ , 1) C , (3.17) where T × ( u N σ , 1) C is giv en b y the pullbac k diagram T × ( u − N σ , 1) C pr 1 x x q q q q q q q q q q pr 2 & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ T σ   C α 7→ α 1   C (U(1)) f 7→ u − N f / / C (U(1)) . (3.18) Prop osition 3.1. The ﬁbr e pr o duct T × ( σ , 1) C is isomo rphic to the C ∗ -algebr a C (S 2 q ) , and L N is isomorphic to T × ( u − N σ , 1) C as a left C (S 2 q ) -mo dule with r esp e ct to the left C (S 2 q ) -action on T × ( u − N σ , 1) C given by ( t, α ) · ( h, β ) = ( th, αβ ) . Pr o of. F or N = 0, t he mappings T ∋ t 7→ σ ( t ) ∈ C (U(1)) and C ∋ α 7→ α 1 ∈ C (U(1)) are mo r phisms of C ∗ -algebras, so that T × ( σ , 1) C is a C ∗ -algebra. Next, recall that C (S 2 q ) ∼ = K ( ℓ 2 ( N )) ⊕ C (see (1.56 )), and deﬁne φ : T × ( σ , 1) C − → K ( ℓ 2 ( N )) ⊕ C , φ ( t, α ) = t, (3.19) φ − 1 : K ( ℓ 2 ( N )) ⊕ C − → T × ( σ , 1) C , φ − 1 ( k + α ) = ( k + α , α ) . (3.20) Clearly , φ : T × ( σ , 1) C → B ( ℓ 2 ( N )) is a morphism of C ∗ -algebras. Since φ ( t, α ) = ( t − α ) + α , and σ ( t − α ) = 0 b y the pullbac k diagr a m (3 .18), it follows from the short exact sequence (1.61) that t − α ∈ K ( ℓ 2 ( N )), so that φ ( t, α ) ∈ K ( ℓ 2 ( N )) ⊕ C . One easily sees that φ − 1 is its inv erse so that T × ( σ , 1) C ∼ = K ( ℓ 2 ( N )) ⊕ C . The fact that T × ( u − N σ , 1) C with the given C (S 2 q )-action is a left C (S 2 q )-mo dule follo ws f r o m the discussion preceding the pullbac k diagra m ( 1.6) with the free rank 1 mo dules E 1 = T , E 2 = C and π 1 ∗ E 1 = π 2 ∗ E 2 = C (U(1)). Obv iously , L N ∋ ( t ⊗ u N , αu N ) 7→ ( t, α ) ∈ T × ( u − N σ , 1) C deﬁnes an isomorphism o f left C (S 2 q )-mo dules. 3.2 Equiv alence of the pullbac k and standard constructions Let us view U(1) as a compact quantum group. W e consider its C ∗ -algebra C (U(1)) of all contin uous function to gether with the ob vious copro duct, counit a nd a n tip o de giv en b y ∆( f )( x, y ) = f ( xy ), ε ( f ) = f (1 ) and S ( f )( x ) = f ( x − 1 ), resp ectiv ely . F ur t hermore, let ¯ ⊗ 21 stand f or the completed tensor pro duct of C ∗ -algebras. In o ur case it is unique b ecause o f the n uclearit y of the in v olv ed C ∗ -algebras. No w let π 2 : C (U(1)) → C (U(1)) ¯ ⊗ C (U(1)) b e giv en by the copro duct, i.e., π 2 ( f )( x, y ) = (∆ f )( x, y ) = f ( xy ), and let σ ⊗ id denote the tensor pr o duct of t he sy m b ol map σ : T → C (U(1)) and id : C (U(1)) → C (U(1)). Then ¯ P := T ¯ ⊗ C (U(1)) × ( π 1 ,π 2 ) C (U(1)) is deﬁned by the pullback diagra m T ¯ ⊗ C (U(1)) × ( π 1 ,π 2 ) C (U(1)) pr 1 v v ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ pr 2 ( ( P P P P P P P P P P P P P T ¯ ⊗ C (U(1)) π 1 = σ ⊗ i d ) ) ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ C (U(1)) π 2 =∆ v v ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ C (U(1)) ¯ ⊗ C (U(1)) . (3.21) With the C (U(1))-coaction giv en by the copro duct ∆ on the rig h t tensor factor C (U(1)), π 1 and π 2 are morphisms in the category of right C ( U(1 ))-como dule C ∗ -algebras. Equiv alen tly , w e can view this diag r am as a diagram in the category of U(1 )- C ∗ -algebras (see Section 1.3). Therefore, ¯ P inherits the structure of a right U(1)- C ∗ -algebra. T o determine t he Pe ter-W eyl como dule algebra P ∆ ( ¯ P ), w e ﬁrst note that O (U(1) ) is the dense Hopf *-subalgebra of C (U(1)) spanned b y the matrix co eﬃcien ts of the irreducible unitary corepresen tations. Using the counit ε : C (U(1)) → C and the fact that the P eter-W eyl functor comm utes with t a king pullbac ks, w e easily conclude that P ∆ ( ¯ P ) = T ⊗ O (U(1) ) × ( π 1 ,π 2 ) O (U(1)), so P ∆ ( ¯ P ) is the como dule alg ebra P of Section 3.1. Consider nex t the *- representation of O (SU q (2)) o n ℓ 2 ( N ) given by [27] ρ ( α ) e n = (1 − q 2 n ) 1 / 2 e n − 1 , ρ ( β ) e n = − q n +1 e n , ρ ( γ ) e n = q n e n , ρ ( δ ) e n = (1 − q 2( n +1) ) 1 / 2 e n +1 . (3.22) Note tha t ρ ( β ) , ρ ( γ ) ∈ K ( ℓ 2 ( N )). Comparing ρ with the represen tation µ of O (D q ) f rom (1.60), one readily sees that ρ ( O ( SU q (2))) ⊆ T . F urthermore, the sym b ol map σ yields σ ( ρ ( β )) = σ ( ρ ( γ )) = 0. Using an appropria t e diagonal compact operat o r k , we also obtain σ ( ρ ( α )) = σ ( ρ ( α ) − S ∗ ) + σ (S ∗ ) = σ ( k S ∗ ) + σ (S ∗ ) = u − 1 , σ ( ρ ( δ )) = σ ( ρ ( α )) ∗ = u. (3.23) Th us w e obtain a U(1)-equiv aria n t *-algebra morphism ι : O (SU q (2)) → P by setting ι ( α ) = ( ρ ( α ) ⊗ u, u ) , ι ( γ ) = ( ρ ( γ ) ⊗ u , 0) . (3.24) One easily c heck s that the image of a P oincar´ e-Birkhoﬀ-Witt basis of O (SU q (2)) remains linearly indep enden t, so that ι is injectiv e and w e can consider O (SU q (2)) as a subalgebra of P . In particular, we ha v e ι ( M N ) ⊆ L N as left O ( S 2 q )-mo dules. (See Section 1.4 and Section 3.1 for the deﬁnitions of M N and L N , respectiv ely .) The main ob jective of this section is to establish an U(1)- C ∗ -algebra isomorphism b et w een C (SU q (2)) and ¯ P . The univ ersal C ∗ -algebra C (SU q (2)) of O ( SU q (2)) has b een studied in [18] 22 and [31]. Here w e shall use the fact fro m [18, Corollary 2.3] that a faithful *- represen t a tion ˆ ρ of C (SU q (2)) o n the Hilb ert space ℓ 2 ( N ) ¯ ⊗ ℓ 2 ( Z ) is giv en b y ˆ ρ ( α )( e n ⊗ b k ) = (1 − q 2 n ) 1 / 2 e n − 1 ⊗ b k − 1 , ˆ ρ ( γ )( e n ⊗ b k ) = q n e n ⊗ b k − 1 , (3.25) where { e n } n ∈ N and { b k } k ∈ Z denote the standard bases of ℓ 2 ( N ) and ℓ 2 ( Z ), resp ectiv ely . T o compare ( 3 .25) with [18, Coro llary 2.3], one has to apply the unitary transformation T : ℓ 2 ( N ) ¯ ⊗ ℓ 2 ( Z ) → ℓ 2 ( N ) ¯ ⊗ ℓ 2 ( Z ) , T ( e n ⊗ b k ) = e n ⊗ b k − n . (3.26) A r igh t C (U(1))-coactio n o n C (SU q (2)) is given by (id ⊗ ¯ π ) ◦ ∆, where ∆ denotes the copro duct of the compact quan tum group C (SU q (2)) and ¯ π is the extension of the Hopf *-alg ebra surjection π : O ( SU q (2)) → O (U(1) ) deﬁned in (1.4 9) to C (SU q (2)). Using the fait hf ulness of ˆ ρ , we can transfer ¯ π to ˆ ρ ( C (SU q (2))). In [18], it is shown that ¯ π giv es rise to the C ∗ -algebra exte nsion 0 / / K ( ℓ 2 ( N )) ¯ ⊗ C (U(1))   / / ˆ ρ ( C (SU q (2))) ¯ π / / C (U(1)) / / 0 . (3.27) Theorem 3.2. The U(1) - C ∗ -algebr as C (SU q (2)) and ¯ P ar e isomorph i c . Pr o of. First note that k er(pr 1 ) = { (0 , y ) ∈ ¯ P | π 2 ( y ) = ∆( y ) = 0 } = { 0 } . Hence w e can iden tify ¯ P with t he image of pr 1 in T ¯ ⊗ C (U(1)). W e will pro v e the theorem b y applying the ﬁv e lemma to the follo wing comm utativ e diagram of U(1)- C ∗ -algebras: 0 / / K ( ℓ 2 ( N )) ¯ ⊗ C (U(1)) id     / / ˆ ρ ( C (SU q (2))) ¯ π / / τ   C (U(1)) id   / / 0 0 / / K ( ℓ 2 ( N )) ¯ ⊗ C (U(1))   / / pr 1 ( ¯ P ) ω / / C (U(1)) / / 0 . (3.28) T o deﬁne τ , w e realize C (U(1)) as a concrete C ∗ -algebra of b ounded op erat ors on ℓ 2 ( Z ) by setting u ( b k ) = b k − 1 . Then ˆ ρ ( α ) = ρ ( α ) ⊗ u = pr 1 ( ρ ( α ) ⊗ u, u ) ∈ pr 1 ( ¯ P ) , ˆ ρ ( γ ) = ρ ( γ ) ⊗ u = pr 1 ( ρ ( γ ) ⊗ u, 0) ∈ pr 1 ( ¯ P ) . (3.29) Since C (SU q (2)) is generated b y α and γ , w e tak e τ to b e the inclus ion ˆ ρ ( C (SU q (2))) ⊂ pr 1 ( ¯ P ). W e deﬁne the U(1)- C ∗ -algebra morphism ω b y ω : pr 1 ( ¯ P ) − → C (U(1)) , ω := ( ε ◦ σ ) ⊗ id , (3.30) where ε denotes the counit of C (U(1)). The surjectiv ity of ω follow s from u k = ω ( ρ ( α k ) ⊗ u k ) and u − k = ω ( ρ ( α ∗ k ) ⊗ u ∗ k ) for all k ∈ N and taking the closure of O (U(1)) in C (U(1)). T o pro v e the exactness o f the lo w er ro w, note that K ( ℓ 2 ( N )) ¯ ⊗ C (U(1)) = ke r( σ ) ¯ ⊗ C (U(1)) ⊂ k er( ω ). No w let f ∈ pr 1 ( ¯ P ) \ k er ( σ ) ¯ ⊗ C (U(1)). Then ( σ ⊗ id)( f ) 6 = 0. By the comm utativ e diagram (3.2 1), there exists a non-zero ele men t g ∈ C (U(1)) suc h that ( σ ⊗ id)( f ) = ∆( g ). Hence ω ( f ) = ( ε ⊗ id) ◦ ∆( g ) = g 6 = 0 whic h pro v es k er( ω ) = K ( ℓ 2 ( N )) ¯ ⊗ C (U(1)). It remains to sho w that the diagra m (3.28 ) is comm utativ e. This is clear for the left part since τ is just the inclusion. The commutativit y o f the righ t part follows f rom ω  τ  ˆ ρ ( α )   = ε  σ  ρ ( α )   ⊗ u = ε ( u ) u = u = ¯ π ( ˆ ρ ( α )) , (3.31) ω  τ  ˆ ρ ( γ )   = ε  σ  ρ ( γ )   ⊗ u = 0 = ¯ π ( ˆ ρ ( γ )) (3.32) 23 b y ta king limits since ˆ α a nd ˆ γ generate C (SU q (2)). Therefore, by the ﬁv e lemma, τ is an isomorphism of U(1)- C ∗ -algebra. By the ﬁnal r emark of Section 1.3, we conclude from Theorem 3.2 that the Peter-W eyl como dule algebras P ∆ ( C (SU q (2))) a nd P are isomorphic. W e use this isomorphism to iden tify asso ciated pro jectiv e mo dules. F o r N ∈ Z and the left O (U(1) )-coaction on C g iv en b y C ∆(1) = u N ⊗ 1, w e deﬁne a “completed” v ersion of M N (see (1.62)): ¯ M N := P ∆ ( C (SU q (2)))  O (U(1)) C = { p ∈ P ∆ ( C (SU q (2))) | ((id ⊗ ¯ π ) ◦ ∆)( p ) = p ⊗ u N } . (3.33) No w it follows from Equation (3.14) that ¯ M N ∼ = L N . Applying the same arguments as at the end of Section 1.4, we infer that ¯ M N ∼ = C (S 2 q ) N +1 E N , with E N b eing the pro j ection matrix o f Equation ( 1 .64). T a king a dv antage o f these isomorphisms of mo dules, w e pro v e: Lemma 3.3. Identifying C (S 2 q ) w ith K ( ℓ 2 ( N )) ⊕ C , we obtain the fol lowing iso m orphisms of left C (S 2 q ) -mo dules: C (S 2 q ) N +1 E N ∼ = C (S 2 q ) p N , p N := S N S N ∗ , N ≥ 0 , (3.34) C (S 2 q ) | N | +1 E N ∼ = C (S 2 q ) 2 p N , p N :=  1 0 0 1 − S | N | S | N |∗  , N < 0 . (3.35) Pr o of. W e apply (1.40) to construct pro jections P N , N ∈ Z , from the strong connection giv en in (3.11 ) and ( 3.12). F or N < 0, w e obtain ( P N ) 11 = (S ∗| N | ⊗ u N , u N )(S | N | ⊗ u | N | , u | N | ) = (1 ⊗ 1 , 1 ) , (3.36) ( P N ) 12 = ( P N ) ∗ 21 = (S ∗| N | ⊗ u N , u N )((1 − S | N | S ∗| N | ) ⊗ u | N | , 0) = 0 , (3.37) ( P N ) 22 = ((1 − S | N | S ∗| N | ) ⊗ u N , 0)((1 − S | N | S ∗| N | ) ⊗ u | N | , 0) = ((1 − S | N | S ∗| N | ) ⊗ 1 , 0) . (3.38) Analogously , for N ≥ 0, w e get ( P N ) 11 = (S N ⊗ u N , u N )(S ∗ N ⊗ u ∗ N , u ∗ N ) = (S N S ∗ N ⊗ 1 , 1) . (3.39) Finally , applying the isomorphism (3.19) comp onen t wise to P N , N ∈ Z , yields the result. The pro jections p N of Lemma 3.3 can also b e obtained fro m the o dd-to-eve n construction in Section 1.2. First let N < 0. Since L N ∼ = T × ( u − N σ , 1) C (see (3.17)), w e can apply the formula (1.9) b y taking E 1 = T , E 2 = C , π 1 ∗ E 1 = π 2 ∗ E 2 = C (U(1)), and c ho o sing h in (1.6) to b e the isomorphism giv en b y the multiplication with u | N | . As the sym b ol map σ applied to S is u (see (1.61)), w e can lift u | N | and its in v erse u −| N | to S | N | and S | N |∗ resp ectiv ely . Inserting c = S | N |∗ and d = S | N | in to (1.9) give s T × ( u − N σ , 1) C ∼ = ( T × ( σ , 1) C ) 2 Q N , whe re Q N =  (1 , 1) (0 , 0) (0 , 0) (1 − S | N | S | N |∗ , 0)  . (3.40) Finally , applying the isomorphism (3 .19) yields the pro jection in (3.35). Similarly , for N ≥ 0, w e insert c = S N and d = S N ∗ in to (1.9). Since S N ∗ S N = 1, we obtain T × ( u − N σ , 1) C ∼ = ( T × ( σ , 1) C ) 2 Q N with Q N =  (S N S N ∗ , 1) (0 , 0) (0 , 0) (0 , 0)  , (3.41) whic h is equiv alen t to T × ( u − N σ , 1) C ∼ = C (S 2 q ) S N S N ∗ . 24 3.3 Index pairing Recall that for a C ∗ -algebra A , a pro jection p ∈ Mat n ( A ), and * -represen tations ρ + and ρ − of A on a Hilb ert space H suc h that [( ρ + , ρ − )] ∈ K 0 ( A ) (e.g. see [10, Chapter 4]), one has t he follo wing: If the op erator T r Mat n ( ρ + − ρ − )( p ) is trace class, then the fo rm ula h [( ρ + , ρ − )] , [ p ] i = T r H (T r Mat n ( ρ + − ρ − )( p )) (3.42) yields a pairing betw een K 0 ( A ) and K 0 ( A ). In this section, w e compute the pairing b et w een the K 0 -classes o f the pro jectiv e C (S 2 q )- mo dules describing quantum line bundles and the tw o generators of K 0 ( A ). By Lemma 3.3, w e can take the pro jections p N as represen tativ es of resp ective K 0 -classes. Their simple form mak es it v ery easy to compute the index pairing. Theorem 3.4. L et ¯ M N b e the asso ciate d mo dules of ( 3 .33) , and l e t [( id , ε )] and [( ε, ε 0 )] denote the gener ators of K 0 ( C (S 2 q )) given in Se ction 1.4. Then, for al l N ∈ Z , h [( ε, ε 0 )] , [ ¯ M N ] i = 1 , h [(id , ε )] , [ ¯ M N ] i = − N . (3.43) Pr o of. Let N ≥ 0. Then p N = S N S N ∗ = (S N S N ∗ − 1) + 1, so that ε ( p N ) = 1 and ε 0 ( p N ) = SS ∗ . F urthermore, since for a n y N ∈ N \ { 0 } , the image of the pro jection 1 − S N S N ∗ is span { e 0 , . . . , e N − 1 } ⊂ ℓ 2 ( N ), t he pro jection 1 − S N S N ∗ is trace class. Moreov er, with the help of Lemma 3.3 and F orm ula (3.42), it implie s that h [( ε, ε 0 )] , [ ¯ M N ] i = T r ℓ 2 ( N ) ( ε − ε 0 )( p N ) = T r ℓ 2 ( N ) (1 − SS ∗ ) = 1 , (3.44) h [(id , ε )] , [ ¯ M N ] i = T r ℓ 2 ( N ) (id − ε )( p N ) = T r ℓ 2 ( N ) (S N S N ∗ − 1) = − N . (3.45) F o r N < 0, w e ha v e T r Mat 2 ( p N ) = 2 − S | N | S | N |∗ = 2 − p | N | . Com bining this with the abov e index pairing for p | N | , the form ulas ( ε − ε 0 )(2) = 2(1 − SS ∗ ) and (id − ε )(2) = 0, and (3.42), w e obtain h [( ε, ε 0 )] , [ ¯ M N ] i = T r ℓ 2 ( N ) ( ε − ε 0 )(2 − p | N | ) = T r ℓ 2 ( N ) (1 − SS ∗ ) = 1 , (3.46) h [(id , ε )] , [ ¯ M N ] i = T r ℓ 2 ( N ) (id − ε )(2 − p | N | ) = T r ℓ 2 ( N ) (1 − S | N | S | N |∗ ) = − N . (3.47) This completes the pro o f. The ab ov e theorem agrees with the classical situation. Indeed, the pairing h [( ε, ε 0 )] , [ ¯ M N ] i yields the rank of the line bundles, and h [(id , ε )] , [ ¯ M N ] i computes t he winding num b er of the map u − N : S 1 → S 1 . 4 Examples of piecew ise pri ncipal coalgebr a coactions W e b egin b y recalling the piecewise structure [13] of a noncomm utativ e join construction pro- p osed b y [9]. Then w e instantiate it to SU q (2) to obtain a quan tum instan ton bundle S 7 q → S 4 q [23] as a piece wise trivial principal comodule algebra. A k ey step is then to replace SU q (2) 25 b y quotienting the Hopf algebra O ( SU q (2)) b y a coideal righ t ideal ( O (S 2 q ,s ) ∩ k er ε ) O (SU q (2)) pro vided by the generic P o dle ´ s qu antum spheres S 2 q ,s , s 6 = 0 [24 ]. The quotient coalgebra is isomorphic with O (U(1)) [21]. Applying our main theorem, w e will pro v e that the induce d righ t coa ctio n of O (U(1)) is principal. 4.1 Piecewise p rincipal c oactions from a noncomm utativ e join con- struction Let ¯ H b e the C ∗ -algebra of a compact quan tum group and H its P eter-W eyl Hopf a lgebra [30, 3 2]. W e ta k e the algebra o f nor m con tin uous functions C ([ a, b ] , ¯ H ) from a closed interv al [ a, b ] to the C ∗ -algebra ¯ H , and deﬁne P 1 := { f ∈ C ([0 , 1 2 ] , ¯ H ) ⊗ H | f (0) ∈ ∆( H ) } , (4.1) P 2 := { f ∈ C ([ 1 2 , 1] , ¯ H ) ⊗ H | f (1) ∈ C ⊗ H } . (4.2) Here w e iden tify elemen ts of C ([ a, b ] , ¯ H ) ⊗ H with functions [ a, b ] → ¯ H ⊗ H . The P i ’s are righ t H -como dule algebras f or the coaction ∆ P i = id C ([ a i ,b i ] , ¯ H ) ⊗ ∆, where ∆ stands for the copro duct of H . The subalgebras of coaction in v arian ts can b e iden tiﬁed with B 1 := { f ∈ C ([0 , 1 2 ] , ¯ H ) | f ( 0 ) ∈ C } , B 2 := { f ∈ C ([ 1 2 , 1] , ¯ H ) | f (1) ∈ C } . The como dule algebra P 2 is eviden t ly the same as B 2 ⊗ H . Unlik e P 2 , the como dule algebra P 1 do es not coincide with B 1 ⊗ H . How eve r, there is a clea ving map j : H → P 1 b y j ( h )( t ) :=  t 7→ h (1)  ⊗ h (2) , that is, j ( h )( t ) := ∆( h ) for all t ∈ [0 , 1 2 ]. Since j is an algebra homomorphism, it iden tiﬁes the como dule alg ebra P 1 with a smash pro duct B 1 # H . No w one can deﬁne the noncomm utativ e join of ¯ H as the pullbac k righ t H -como dule algebra P := { ( p, q ) ∈ P 1 ⊕ P 2 | π 1 ( p ) = π 2 ( q ) } (4.3) giv en by the ev aluation maps π 1 := ev 1 2 ⊗ id : P 1 → P 12 := ¯ H ⊗ H , π 2 := ev 1 2 ⊗ id : P 2 → P 12 := ¯ H ⊗ H , (4.4) where ev t is given by the ev aluation of functions of C ([ a, b ] , ¯ H ) in t ∈ [ a, b ]. Our goal no w is to replace H by a quotient coalgebra without lo o sing principalit y . Using [7, Example 2.29], it is straigh tforw ard to v erify the follo wing lemma. Lemma 4.1. L et H b e a Hopf algeb r a with bi j e ctive antip o de, let ∆ P : P → P ⊗ H b e a c o action making P a right H -c omo d ule algebr a a nd J a c oide al ri g ht ide al of H . Then C := H /J is a c o algebr a c o ac ting on P via ρ R := (id ⊗ π ) ◦ ∆ P , π : H → C the c a n onic a l surje ction, and the formula Ψ : C ⊗ P ∋ ¯ π ( h ) ⊗ p 7− → p (0) ⊗ π ( hp (1) ) ∈ P ⊗ C (4.5) deﬁnes a bije ctive entwining making P an entwine d mo dule. The inverse of Ψ is given by Ψ − 1 ( p ⊗ π ( h )) = π ( hS − 1 ( p (1) )) ⊗ p (0) , (4.6) 26 and deﬁnes a left o action on P via ρ L : P ∋ p 7− → Ψ − 1 ( p ⊗ π (1)) = π ( S − 1 ( p (1) ) ⊗ p (0) ∈ C ⊗ P . ( 4 .7) Lemma 4.2. L et P b e a p rincip a l H -c omo dule algebr a for ∆ P : P → P ⊗ H . A lso, let J b e a c oide al right ide al of H deﬁning a c o algebr a C := H / J , let ρ R := (id ⊗ π ) ◦ ∆ P , π : H → C the c anonic al surje ction, b e its ri g ht c o a c tion on P , and let i : C → H b e a unital (i.e., i ( π (1)) = 1 ) C -bic oline ar (for the c o actions ∆ H := (id ⊗ π ) ◦ ∆ a n d H ∆ := ( π ⊗ id) ◦ ∆ ) splitting (i.e., π ◦ i = id ). Then P is princip al for the c o action ρ R . Pr o of. Let ℓ : H → P ⊗ P be a strong connection on P . One can easily chec k tha t ℓ ◦ i : C → P ⊗ P is a strong connection on P for the righ t coaction ρ R := (id ⊗ π ) ◦ ∆ P and the left coaction ρ L := ( π ⊗ id) ◦ P ∆, where P ∆( p ) = S − 1 ( p (1) ) ⊗ p (0) is the left H -coaction o n P view ed a s a principal H -como dule algebra. On the ot her hand, it follow s from Lemma 4.1 that Ψ : C ⊗ P ∋ π ( h ) ⊗ p 7→ p (0) ⊗ π ( hp (1) ) ∈ P ⊗ C is a bij ective en tw ining making P an en t wined mo dule. Therefore, since ρ R (1) = 1 ⊗ π (1 ) , ρ L (1) = π (1) ⊗ 1, and ρ L ( p ) = Ψ − 1 ( p ⊗ π (1)) for all p ∈ P b y (4.7), the principality of P for the C -coaction ρ R follo ws from Lemma 1.1. Com bining Lemma 4.2 with Theorem 2.2 yields the follo wing result. Theorem 4.3. L et ¯ H b e the C ∗ -algebr a of a c omp ac t quantum gr oup, H its Peter-Weyl Hopf algebr a, J a c oi d e al rig ht ide al of H an d π : H → C := H/ J the c annonic al surje ction . A lso let P 1 := { f ∈ C ([0 , 1 2 ] , ¯ H ) ⊗ H | (ev 0 ⊗ id)( f ) ∈ ∆( H ) } , (4.8) P 2 := { f ∈ C ([ 1 2 , 1] , ¯ H ) ⊗ H | (ev 1 ⊗ id)( f ) ∈ C ⊗ H } (4.9) b e right a n d left C -c omo dules for the right and left c o actions ρ i R := (id ⊗ π ) ◦ ∆ P i , ρ i L := ( π ⊗ id) ◦ P i ∆ , i = 1 , 2 , (4 .1 0) r esp e ctively. Her e ∆ P i := id ⊗ ∆ and P i ∆ := ( S − 1 ⊗ id) ◦ ﬂip ◦ ∆ P i . Then, if ther e exists a unital bic oline ar splitting i : C → H of π : H → C , the pul lb ac k C -c omo dule P := { ( p 1 , p 2 ) ∈ P 1 × P 2 | (ev 1 2 ⊗ id )( p 1 ) = (ev 1 2 ⊗ id)( p 2 ) } (4.11) is princip al. Let us no w tak e a closer lo ok a t t he compatibility of strong connections on principal comod- ules as app earing in the ab ov e theorem. First w e observ e t hat, if b oth of π 1 and π 2 deﬁning the pullback diagra m (2.33) are surjective , then (2.39) simpliﬁes to ℓ = ( ( α 12 L + id) ⊗ ( α 12 R + id) ) ◦ ℓ 2 + ( η 1 ◦ ε − m P 1 ◦ ( α 12 L ⊗ α 12 R ) ◦ ℓ 2 ) ∗  ( id ⊗ (id + α 21 R ) ) ◦ ℓ 1  . (4.1 2) Indeed, since no w α 21 R is deﬁned on the whole P 1 , a special constructed connection ˜ ℓ 1 in (2.38 ) can b e replaced b y a n y strong connection ℓ 1 on P 1 . Note that sp ecializing ( 4.12) to como dule algebras coincides with what w as obtained in [1 3]. Next, we observ e that the formu lae α 1 : P 12 → P 1 , α 1 ( ¯ h ⊗ h ) = 2 t ¯ h ⊗ h + (1 − 2 t ) ¯ ε ( ¯ h ) h (1) ⊗ h (2) , (4.13) α 2 : P 12 → P 2 , α 2 ( ¯ h ⊗ h ) = 2(1 − t ) ¯ h ⊗ h + ( 2 t − 1) ¯ ε ( ¯ h ) ⊗ h, (4.14) 27 where ¯ ε is an y unital linear functional on ¯ H , deﬁne unital C -bicolinear splittings of π 1 and π 2 , resp ectiv ely . Hence we can tak e α 12 L = α 12 R = α 1 ◦ π 2 and α 21 R = α 2 ◦ π 1 . Comb ining this with the fact that a cleav ing map j deﬁnes a strong connection via ℓ := ( j − 1 ⊗ j ) ◦ ∆, w e obtain v ery explic it formulae for strong connections on P 1 and P 2 : ℓ 1 := ( j − 1 1 ⊗ j 1 ) ◦ ∆ ◦ i, ℓ 2 := ( j − 1 2 ⊗ j 2 ) ◦ ∆ ◦ i. (4.15) Here j 1 : H → P 1 , j 1 ( h ) := ( t 7→ h (1) ) ⊗ h (2) , and j 2 : H → P 2 , j 2 ( h ) := 1 ⊗ h are clea ving maps for P 1 and P 2 , respectiv ely . 4.2 Quan tum comp lex pro jectiv e spaces C P 3 q ,s Finally , w e instantiate ¯ H to b e C (SU q (2)), H = O (SU q (2)), J = ( O (S 2 q ,s ) ∩ k er ε ) O (SU q (2)), and ¯ ε : C (SU q (2)) → C to the counit. Here O (S 2 q ,s ) stands for the co ordinate algebra of a P o dle ´ s quantum sphe re S 2 q ,s , s ∈ [0 , 1], [8]. (Note that the case s = 0 brings us to the como dule- algebra setting.) The most in teresting part of this structure is the unital bicolinear splitting of π : O (SU q (2)) → O (SU q (2)) /J giv en b y [8, Propo sition 6.3]. All this deﬁnes a family of noncomm utativ e deformatio ns of the U(1)-principal bundle S 7 → C P 3 . More precisely , w e obtain deformatio ns of a U(1)- principal action o n S 7 giv en b y ( z 1 , z 2 , z 3 , z 4 )e i ϕ = ( z 1 e i ϕ , z 2 e − i ϕ , z 3 e i ϕ , z 4 e − i ϕ ) , | z 1 | 2 + | z 2 | 2 + | z 3 | 2 + | z 4 | 2 = 1 . (4.16) Ho w ev er, this is isomorphic with the diagonal action of U(1) o n S 7 , so that the quotien t space is again C P 3 . Th us out of Pﬂaum’s S 7 w e obtain a family of quantum pro jectiv e space s C P 3 q ,s . A v ery explicit Ma y er-Vietoris type fo rm ula for a strong connection on S 7 q → C P 3 q ,s should allo w us to study the K -t heory aspects of the tautological line bundle o v er C P 3 q ,s , but this is b ey ond the scop e of t his pap er. Ac kno wledgements The authors gratefully ac knowledge ﬁna ncial supp ort from t he follo wing res earc h grants: PIRSES-GA-2008- 230836, 1261/7 .PR UE/2009/7, N201 1770 33, and 189 /6.PR UE/2007/7. It is also a pleasure to thank T omasz Brzezi ´ nski for helping us with Section 2, and Ulrich Kr¨ ahmer for pro ofreading the man uscript. 28 References [1] Baa j, S., G . Sk andalis: Unitair es multiplic atifs et dualit´ e p our les pr o duits cr ois´ es de C ∗ - alg` ebr es . Ann. Sci. ´ Ecole Norm. Sup. (4) 26 (1993), 425 –488. [2] Baum, P . F., K. De Commer, P . M. Ha ja c: F r e e actions of c omp act quantum gr oups on unital C ∗ -algebr as , in preparation. [3] Bo ca, F. P .: Er go dic actions of c omp act m atrix pseudo gr oups on C ∗ -algebr as . In R e c ent advanc es in op er ator algebr as (Orl ´ eans, 1992), Ast´ erisque 232 (19 9 5), 93 –109. [4] Brzezi ´ nski, T.: On mo dules asso ciate d to c o algebr a Galois extensions. J. Algebra 215 (1999), 290–317. [5] Brzezi ´ nski, T., P . M. Ha jac: Co algebr a extensions an d algebr a c o ex tens i o ns of Galois typ e . Comm un. Algebra 27 (1999), 13 47–1367. [6] Brzezi ´ nski, T., P . M. Ha jac: The Chern - Galois char acter . C.R. Acad. Sci. P aris, Ser. I 338 (2004), 113–116 . [7] Brzezi ´ nski, T., P . M. Ha jac: Galoi s -typ e extensions and e quivariant pr oje ctivity . In Q uan- tum Symmetry in Noncomm utativ e Geometry , P . M. Ha jac (ed.), EMS Publishing House, to app ear. [8] Brzezi ´ nski, T., S. Ma j id: Quantum ge ometry of algebr a factorisations and c o algebr a bun- d les . Comm un. Math. Ph ys. 213 (2000), 491–521. [9] D ¸ abrowsk i, L ., T. Hadﬁeld, P . M. Ha jac: Princip al c o ac tions fr om a nonc ommutative join c onstruction , in preparatio n. [10] Connes, A.: Nonc ommutative ge ometry . Academic Press, Sa n D iego, 1 9 94. [11] D ¸ abro wski, L., T. Hadﬁeld, P . M. Ha ja c, R . Matthes, E. W agner: Index p a i rin gs for pul lb acks of C*-algebr as . Banac h Cen ter Publ. 98 (20 1 2), 67 –84. [12] Ha jac, P . M.: Bund les over quantum spher e an d nonc ommutative index the or em . K - Theory 21 (2000), 141–150 . [13] Ha jac, P . M., U. Kr¨ ahmer, R. Matthes, B. Z ieli ´ nski: Pie c ewise princip a l c o m o dule algebr as . J. Noncomm ut. Geom. 5 (201 1 ), 591–61 4 . [14] Ha jac, P . M., S. Ma jid: Pr oje ctive m o dule de scription of the q -monop ole . Comm un. Math. Ph ys. 206 (1999) , 247–264. [15] Ha jac, P . M., A. Rennie, B. Zieli ´ nski: The K-the ory of He e gaar d quantum lens sp ac es , to app ear in Journal of Noncommu tative Geometry . [16] Klimek, S., A. Lesniewsk i: A two-p ar ameter quantum deformation of the unit disc . J. F unct. Anal. 115 (1993), 1–23. [17] Klim yk, K. A., K. Sc hm ¨ udgen: Quantum Gr oups and Their R epr esentations . Springer, Berlin, 1 9 97. 29 [18] Masuda, T., Y. Nak a gami, J. W ata na b e: Nonc o mmutative Diﬀer ential Ge ometry o n the Quantum SU(2), I: An A lgebr aic Viewp oint . K -Theory 4 (19 90), 1 57–180. [19] Masuda, T., Y. Nak a g ami, J. W atanab e: Nonc ommutative diﬀer ential ge o metry on the quantum two spher e of Po d le ´ s. I: A n Algebr aic Viewp oint . K -Theory 5 (1991) , 151–175. [20] Milnor, J.: I ntr o duction to algebr aic K -the ory . Annals of Mathematics Studie s, No. 7 2, Princeton Univ ersity Press, Princeton, 1971. [21] M ¨ uller, E. F., H.-J. Schne ider: Quantum ho m o gen e ous sp ac es w ith faithful ly ﬂat mo dule structur es . Israel J. Math. 111 (1999), 157–190. [22] Nesh v ey ev, S., L. T uset: A lo c al i n dex formula for the quantum spher e . Comm un. Math. Ph ys. 254 (2005) , 323–341. [23] Pﬂaum, M.: Quantum gr oups on ﬁbr e bund les . Comm un. Math. Ph ys. 166 (1994) , 279–315 [24] P o dle ´ s, P .: Quantum Sph er es . L ett. Math. Ph ys. 14 (1987), 1 93–202. [25] P o dle ´ s, P .: Symmetries of quantum s p ac es. Sub gr oups and q uotient sp ac es of q uantum SU(2) and SO(3) gr oups . Comm un. Math. Ph ys. 170 (19 9 5), 1– 20. [26] Sc hm ¨ udgen, K., E. W agner: R epr esentations of cr oss pr o duct algebr as of Po d le ´ s’ quantum spher e s . J. Lie Theory 17 (20 07), 751– 7 90. [27] V aksman, L. L., Y a. S. So ˘ ıb el’man: A n algebr a of functions on the quantum gr oup SU( 2). (Russian) F unktsional. Anal. i Prilozhen. 22 , (1988), 1–14; t ranslation in F unct. Anal. Appl. 22 (1989), 170–181. [28] W agner, E.: Fibr e pr o duct appr o ach to in d e x p airings fo r the generic Hopf ﬁbr ation of SU q (2). J. K-Theory 7 (2011 ), 1–1 7 . [29] W egge-Olsen, N. E.: K -the o ry and C ∗ -algebr as. A friend ly appr o ach . Oxford Unive rsit y Press, New Y o r k, 1993. [30] W oro nowicz, S. L.: Comp act matrix pseudo gr oups . Comm un. Math. Ph ys. 111 (19 87), 613–665. [31] W oro nowicz, S. L.: T wiste d SU(2) gr oup. An example of a nonc omm utative diﬀer ential c alculus . Publ. R IMS, Ky oto Univ. 23 (1987), 117– 181. [32] W oro nowicz, S. L.: Comp act quantum gr oups . In Sym´ etries quantiques (Les Houc hes, 1995), A. Connes, K. Gaw edzki, J. Z inn-Justin (eds .), North-Holland 1998, 84 5–884. 30

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