GALOIS-TYPE EXTENSIONS AND EQUIVARIANT PROJECTIVITY Tomasz Brzezin´ski Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. E-mail: [email protected] 9 0 0 Piotr M. Hajac 2 Instytut Matematyczny, Polska Akademia Nauk n a ul. S´niadeckich 8, Warszawa, 00-956 Poland J 1 http://www.impan.gov.pl/ pmh and ] A Katedra Metod Matematycznych Fizyki, Uneiwersytet Warszawski Q ul. Hoz˙a 74, Warszawa, 00-682 Poland . h t a m [ Abstract 1 v The theory of general Galois-type extensions is presented, including the interre- 1 lations between coalgebra extensions and algebra (co)extensions, properties of cor- 4 1 responding (co)translation maps, and rudiments of entwinings and factorisations. 0 To achieve broad perspective, this theory is placed in the context of far reaching . 1 generalisations of the Galois condition to the setting of corings. At the same time, 0 9 tobringtogetherK-theoryandgeneralGaloistheory, theequivariant projectivity of 0 extensions is assumed resulting in the centrepiece concept of a principal extension. : v Motivated by noncommutative geometry, we employ such extensions as replace- i X ments of principal bundles. This brings about the notion of a strong connection r and yields finitely generated projective associated modules, which play the role of a noncommutative vector bundles. Subsequently, the theory of strong connections is developed. It is purported as a basic ingredient in the construction of the Chern character for Galois-type extensions (called the Chern-Galois character). Acknowledgements This work was partially supported by the European Commission grant MKTD-CT-2004- 509794 and the Polish Government grants 115/E-343/SPB/6.PR UE/DIE 50/2005 - 2008 and N201 1770 33. 1 Contents 1 Introduction 3 1.1 General conventions and standing assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Equivariant projectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Galois-type extensions and coextensions 5 2.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Coalgebra-Galoisextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Quotient-coalgebraand homogeneous Galois extensions . . . . . . . . . . . . . . . . . . . 8 2.1.3 Algebra-Galois coextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.4 Algebra-Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 The (co)translation map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 Coalgebra extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Algebra coextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.3 Algebra extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Entwining and factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.1 Entwining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2 Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Principal extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.1 Extensions by coseparable coalgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.2 Hopf fibrations over the Podle´s quantum 2-spheres . . . . . . . . . . . . . . . . . . . . . . 30 2.5 The Galois condition in the setting of corings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.1 The structure theorems for entwined modules . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.2 Corings and Galois comodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.3 Quantum groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Connections and associated modules 43 3.1 General coalgebra-Galoisextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.1 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.2 Associated modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.3 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Strong connections on principal extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.1 Covariant derivatives on associated modules . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.2 Strong connections on pullback constructions . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.3 Strong connections on extensions by coseparable coalgebras . . . . . . . . . . . . . . . . . 58 3.2.4 Strong connections on homogeneous Galois extensions . . . . . . . . . . . . . . . . . . . . 59 3.2.5 Dirac monopoles over the Podle´s 2-spheres . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2 1 Introduction Taking advantageof Peter-Weyl theory, principal comodule algebras (faithfullyflat Hopf-Galois extensionswithbijectiveantipodes)havebeenshown[2]togeneralisecompactprincipalbundles in the sense of Henri Cartan (no local triviality assumed). On the other hand, there are examples of quantum spaces which, classically, correspond to principal bundles, yet do not fit the Hopf-Galoisframework. More specifically, a natural source of examples of principal bundles is provided by homogeneous spaces. These can always be defined as quotients of a group by its subgroup. In the case of Hopf algebras understood as quantum groups, however, there is a rather limited number of quantum subgroups (given by surjections of Hopf algebras). As a result, not every quantum homogeneous space is a quotient of a quantum group by its quantum subgroup. For example, only one member of the family of quantum 2-spheres defined in [71] can be obtained as a quotient of SU (2) by U(1). The theory of Hopf-Galois extensions can q only describe quantum homogeneous spaces that are quotients of quantum groups by quantum subgroups. Thus it appears necessary to consider a wider class of extensions that, on one hand, would be close enough to principal comodule algebras, yet general enough to include examples coming from quantum homogeneous spaces. The basic idea is to replace a Hopf algebra in a Hopf- Galois extension by a coalgebra. This point of view for the first time was taken seriously in [21], where the studies of coalgebra principal bundles were initiated. Over the recent years and in a significant number of papers, the theory of coalgebra principal bundles or coalgebra- Galois extensions [17] has been developed and refined both in purely algebraic and differential geometric directions. On the algebraic side it has led to revival of the coring theory and provided new points of view on areas such as noncommutative descent theory [25]. On the differential geometric side, it has culminated in the introduction of principal extensions as noncommutative objects most closely describing principal bundles, and in the development of Chern-Weil theory for such extensions [18]. Most importantly, the abstract theory of principal extensions generalising principal comodule algebras was supported by new interesting examples such as noncommutative or quantum instanton bundles going beyond Hopf-Galois theory. It seems that the theory of coalgebra-Galois and principal extensions has achieved a level of maturity at which it could be profitable to review recent progress and present it in a unified manner. This is the aim of the current article. The article consists of two parts. In the first part we analyse the algebraic side of coalgebra-Galois extensions. We give basic definitions and properties, we look at dual ways of defining Galois-type extensions (by algebras or by coalgebras), we also put Galois-type extensions in a wider framework of corings and quantum groupoids. The second partisdevoted to geometrymotivatedaspects ofthecoalgebra-Galoistheory. In particular, we define modules associated to Galois-typeextensions via corepresentations oftheir structure comodule coalgebras. They can be understood as modules of sections of associated noncommutative vector bundles. We describe basic elements of the theory of connections and strong connections, and derive consequences of the definition of a principal extension. The key idea here is that the concept of equivariant projectivity replaces that of faithful flatness used in Hopf-Galois theory. These two concepts are equivalent in the Hopf-Galois setting (bijective antipode assumed) but only the implication “equivariant projectivity” ⇒ “faithfull flatness” is known in general. Therefore, we build our theory on equivariant projectivity which guarantees that the aforementioned associated modules are finitely generated projective for any finite- dimensional corepresentation of the structure coalgebra. This way we arrive at the K-theory 3 of the coaction-invariant subalgebra. Now, we can apply the noncommutative Chern character mapping the K -group to the even cyclic homology. 0 Furthermore, strong connections give explicit formulae for idempotents. Although these formulae depend on the choice of strong connections, corresponding elements of the K -group 0 are connection independent. Thus we obtain an explicit map from the Grothendieck group of isomorphism classes of finite-dimensional corepresentations of the structure coalgebra to the even cyclic homology of the coaction-invariant subalgebra. We call it the Chern-Galois character, and view as noncommutative Chern-Weil theory. 1.1 General conventions and standing assumptions All (co)algebras are (co)unital and over a field k. We use the standard Heynemann-Sweedler notation (with the summation symbol suppressed) for coproducts and coactions, and ∗ for the convolution product of maps from a coalgebra to an algebra. The coproduct, counit, multiplication, and antipode are denoted by ∆, ε, m, and S, respectively. The kernel of the multiplication map A⊗A → A is written as Ω1A, and called the space of universal differential 1-forms. The formula da := 1⊗a−a⊗1 defines the universal differential A → Ω1A. Our typical notation for a left and a right coaction on a vector space V is ∆ and ∆ , or V̺ V V and ̺V, respectively. For actions on V, we use symbols like µ or m . For an algebra B and V V a coalgebra C, the symbol MC stands for the category of left B-modules that are also right B C-comodules with B-linear coactions. Morphisms in MC are left B-linear right C-colinear B maps. The space of all colinear homomorphisms is denoted by HomC. Analogous symbols denote other categories of left (co)modules right (co)modules with the left and right structures being compatible and other homomorphism spaces. 1.2 Equivariant projectivity The notion of equivariant projectivity of a (left) B-module P occurs whenever P has additional algebraic structure, compatible with the B-module structure. In this case we might like to require the properties of projectivity (such as the splitting of the product map) to respect this additional structure. A typical situation of key importance to the theory of principal extensions can be described as follows. As in [18], an object P ∈ MC is called a C-equivariantly projective left B-module if for B any two objects M,N and morphisms π : M → N, f : P → N in MC, together with a right B C-colinear splitting i : N → M of π there exists a morphism g : P → M in MC such that the B following diagram commutes: π // M N (1.1) ``Boo OO Bi B f ∃g B P Similarly to projective modules, the C-equivariant projectivity can be fully characterised by the splitting property of the multiplication map. Lemma 1.1. An object P ∈ MC is a C-equivariantly projective left B-module if and only if B there exists a left B-module right C-comodule section s of the product map B ⊗P → P. Here B ⊗P is a right C-comodule with the tensor product coaction id ⊗∆ . B P 4 Proof. Given a section s of the multiplication map m : B⊗P → P, and M, N, f, i and π as P in the diagram above, one defines the map g : P → M by g = m ◦(id ⊗(i◦f))◦s, where M B m : B ⊗M → M is the B-multiplication map for M. Conversely, in the defining diagram of M a C-equivariantly projective B-module P take M = B ⊗P, N = P, π = m , i : P → B ⊗P, P p 7→ 1 ⊗p and f the identity map. Then g constructed through such diagram is the required B splitting of the multiplication map. In an analogous way, one calls a (B,A)-bimodule P an A-equivariantly projective left B- module if for any two (B,A)-bimodules M,N and (B,A)-bilinear maps π : M → N, f : P → N in M , together with a right A-linear splitting i : N → M of π there exists a (B,A)-bilinear B A map g : P → M such that π ◦g = f. This is equivalent to the existence of a (B,A)-bilinear splitting of the multiplication map B ⊗P → P. Since any right C-comodule is a left module of the convolution algebra C∗, any object P ∈ MC is a (B,A)-bimodule, where A = C∗op. In this case, P is a C-equivariantly projective B left B-module if and only if it is an A-equivariantly projective left B-module (since there is a bijective correspondence between C-colinear and A-linear maps). The notion of equivariant projectivity should be contrasted with that of relative projectivity. Given an algebra map ι : A → B, any left B-module is also a left A-module via ι and the multiplication in B. In this situation, one often says that B is an A-ring or an algebra over A and that P is a module over an A-ring. The product map B ⊗P → P descends to the map m : B⊗ P. P is called an A-relatively projective left B-module provided the map m has P|A A P|A a left B-linear section. An equivariantly projective left B-module (be it A-equivariantly or C-equivariantly) is al- ways a projective left B-module (a (B,A)-linear splitting of the multiplication map is, in particular, left B-linear). Not every (B,A)-bimodule P that is projective as a left B-module is an equivariantly projective module. For an A-ring B, a projective left B-module is always an A-relatively projective left B-module, but the relative projectivity of P does not imply the pro- jectivity of P (however, when A is a separable algebra the A-relative projectivity is equivalent to the projectivity of P). 2 Galois-type extensions and coextensions This section is devoted to the definition and description of basic algebraic properties of general Galois-type extensions. We start in Section 2.1 by introducing the notion of equivariant pro- jectivity, then give the definition of coalgebra-Galois extensions and two other types of algebra- Galois (co)extensions. Every such extension is determined by the existence of a (co)translation map, the properties of which are studied in Section 2.2. Furthemore, any coalgebra-Galois extension or an algebra-Galois coextension gives rise to an algebraic structure, which encodes the symmetries of extension and is known as an entwining structure. This is closely related (by semi-dualisation) to factorisation of algebras. Both are described in Section 2.3. Section 2.4 is devoted to the definition of a principal extension [18] which generalises the concept of a faith- fully flat Hopf-Galois extension with bijective antipode and forms a cornerstone of the theory of noncommutative principal bundles. Representations of entwining structures are given in terms of entwined modules. These unify many categories of modules studied previously in Hopf algebra theory. Rudimentary properties of entwined modules are described in Section 2.5. In this section it is also shown, how the properties of such modules and Galois-type extensions can 5 be derived from the properties of corings and their comodules. The latter provide a conceptual and algebraic framework for Galois-type extensions. 2.1 Definitions and basic properties 2.1.1 Coalgebra-Galois extensions Let C be a coalgebra over a field k and P a k-algebra and right C-comodule with a comodule structure map ∆ : P → P ⊗C. In attempting to define a coalgebra-Galois extension one first P has to address the problem of defining the coaction invariants. Recall that for Hopf-Galois extensions coinvariant elements are defined as p ∈ P such that ∆ (p) = p⊗1, using the fact that the unit of a Hopf algebra is group-like. Since there might P not necessarily exist such a group-like element in the coalgebra C, we can no longer obtain coaction invariants of a C-comodule P in this way. Instead, we define the coaction invariants of P by1 PcoC := {b ∈ P | ∀ p ∈ P : ∆ (bp) = b∆ (p)}. (2.2) P P First observe that PcoC is a subalgebra of P. Indeed, for all b,b′ ∈ PcoC and p ∈ P, ∆ (bb′p) = b∆ (b′p) = bb′∆ (p). (2.3) P P P Thus bb′ ∈ PcoC, and since 1 ∈ PcoC, we conclude that PcoC is a subalgebra of P. Another, and perhaps more intuitive, definition of coaction invariants is possible, if there exists a group-like element e in the coalgebra C such that ∆ (1) = 1⊗e. (We call coactions P enjoying this property e-coaugmented.) Then one can define the set of e-coaction invariants as PcoC := {p ∈ P | ∆ (p) = p⊗e}. (2.4) e P Note, however, that it is not always true that PcoC is a subalgebra of P, although it is a subset e of P which contains 1. These two types of coaction invariants are related in the following way. Lemma 2.1. Let C be a coalgebra with a group-like element e, and let P be an algebra and a right C-comodule such that ∆ (1) = 1⊗e. Then PcoC ⊆ PcoC. P e Proof. If b ∈ PcoC, then ∆ (b) = ∆ (b·1) = b∆ (1) = b·(1⊗e) = b⊗e, i.e. b ∈ PcoC. P P P e Althoughthisisnotimmediatelyapparent, bothdefinitionsofcoactioninvariantsarerelated toagroup-likeelement. This is, however, notagroup-likeelement inC but agroup-likeelement in P⊗C, understood as a coalgebra over P or a coring. More information aboutcorings is given below, and the role of group-like elements is explained in Remark 2.49 (cf. Proposition 2.23). We call an extension of algebras B⊆ P a C-extension if B = PcoC. The definition of PcoC immediately implies that the coaction of a right C-comodule P is a left PcoC-linear map. This observation allows us to define when a coaction of a coalgebra on an algebra is Galois, and thus to generalise the notion of a Hopf-Galois extension. 1We owe this definition to M. Takeuchi. 6 Definition 2.2 ([17]). Let C be a coalgebra and B⊆ P a C-extension of algebras. We call the left P-module and right C-comodule homomorphism can : P ⊗ P −→ P ⊗C, p⊗p′ 7−→ p∆ (p′), (2.5) B P the canonical map of the C-extension B ⊆ P. We say that this extension is a coalgebra-Galois extension if the canonical map is bijective. Furthermore, if there exists a group-like element e such that ∆ (1) = 1⊗e, we call B⊆ P an e-coaugmented coalgebra-Galois C-extension. P A straightforward generalisation of [49] provides us with an alternative definition of a coalgebra-Galois extension. Proposition 2.3. Let C be a coalgebra and B⊆ P a C-extension of algebras. The extension is a coalgebra-Galois extension if and only if the following sequence is exact: 0 −→ P(Ω1B)P −→ Ω1P cga−n|→Ω1P P ⊗C+ −→ 0. (2.6) can Here can : P ⊗ P → P ⊗ P → P ⊗ C is the natural lifting of the canonical map, and B C+ := Ker ε is the augmentation ideal of C. g Proof. Consider first the following commutative diagram (of left P-modules) with exact rows and columns: 0 // Ker can|Ω1P // Ker can // 0 (2.7) g(cid:15)(cid:15) (cid:15)(cid:15)g m (cid:15)(cid:15) 0 //Ω1P //P ⊗P // P //0 cgan|Ω1P cgan (cid:15)(cid:15) (cid:15)(cid:15) id⊗ε (cid:15)(cid:15) 0 //P ⊗C+ //P ⊗C // P //0 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) Coker can|Ω1P //Coker can // 0 //0. Applying the Snake Lemma to the above diagram we obtain the exact sequence g g 0 −→ Kercan| −→ Kercan −→ 0 −→ Cokercan| −→ Cokercan −→ 0. (2.8) Ω1P Ω1P It follows from the exactness of this sequence that g g g g Kercan| = Kercan, Cokercan| = Cokercan. (2.9) Ω1P Ω1P On the other hand, the Snake Lemma applied to g g g g 0 //P(Ω1B)P // Kercan //Kercan (2.10) (cid:15)(cid:15) (cid:15)(cid:15)g (cid:15)(cid:15) 0 //P(Ω1B)P //P ⊗P //P ⊗ P //0 B cgan can (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) 0 //0 //P ⊗C //P ⊗C // 0 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) 0 // Cokercan //Cokercan // 0 7 g yields the following exact sequence: 0 → P(Ω1B)P → Kercan → Kercan → 0 → Cokercan → Cokercan → 0. (2.11) Assume nowthatB⊆ P isgacoalgebra-GaloisC-extensiogn. Then Kercan = 0 = Cokercan, and, from the exactness of (2.11), we can infer that Cokercan = 0 and Kercan = P(Ω1B)P. Combining this with (2.9), we conclude that (2.6) is exact. g g Conversely, assume that the sequence (2.6) is exact. Then Kercan = Kercan| = Ω1P P(Ω1B)P, and Cokercan = Cokercan| = 0. Consequently, again due to the exactness of Ω1P (2.11), we have that Kercan = 0 = Cokercan, i.e. B⊆ P is a coalgebra-Galois extension. g g g g Let X and X′ be total spaces of principal bundles with the same base and structure group. Recall that any map X → X′ inducing identity on the base and commuting with the group action has to be bijective. We end this section with a coalgebra-Galois incarnation of this fact. It is a straightforward generalisation of [81]. Lemma 2.4. Let P and P′ be coalgebra-Galois C-extensions of B, and let P′ be right faithfully flat over B. Then any left B-linear right C-colinear map F : P → P′ is an isomorphism. Proof. Consider P′ as a right module over P via F. The composition P′ ⊗ P ⊗ P −→ P′ ⊗ P i−d⊗→F P′ ⊗ P′ −ca→n′ P′ ⊗C −→ P′ ⊗ P ⊗C (2.12) P B B B P coincides with id⊗ can : P′ ⊗ P ⊗ P −→ P′ ⊗ P ⊗C, (2.13) P P B P where can and can′ are the respective canonical maps. Hence id⊗ F : P′ ⊗ P → P′ ⊗ P′ B B B is an isomorphism. Therefore, so is F by the right faithful flatness of P′ over B. Remark 2.5. As a special case of coalgebra-Galois extensions, obtained by replacing Hopf alge- bras in Hopf-Galois extensions by braided groups one can consider braided Hopf-Galois exten- sions. These provide an intermediate step in between the H- and C-Galois, and allows one to develop a braided group gauge theory [65]. 2.1.2 Quotient-coalgebra and homogeneous Galois extensions Though it is demonstrated in the previous and following sections that one can get away with the lack of a group-like element in defining and developing some general aspects of coalgebra- Galois theory, throughout this section all extensions will be coaugmented by some group-like element e. The reason is that we are not aware of interesting examples of non-coaugmented extensions, and co-augmentation seems indispensable to prove some of the desired technical results. On the other hand, a very interesting class of examples comes from the theory of Hopf- algebra quotients that is elaborated in [79], and that already in 1990 gave birth to coalgebra- Galois theory [81]. The setting is as follows. Let H be a Hopf algebra, P be a right H-comodule algebra and I a right ideal coideal of H. Then the composite map P −∆→P P ⊗H −→ P ⊗(H/I) (2.14) 8 defines a right coaction of the quotient coalgebra H/I on P. Demanding this coaction to be Galois defines a ¯1-coaugmented coalgebra-Galois H/I-extension [81]. (Here ¯1 is the class of 1 in H/I.) Thus the coaugmentation of such extensions comes automatically from the Hopf-algebra symmetry that is fundamental in this definition. We call such extensions quotient-coalgebra Galois extensions. The above construction is parallel to what happens in differential geometry. Let us explain it on the example of the principal instanton bundle S7 → S4. The sphere S7 is a homogeneous space of SU(4). Viewing SU(2) as a block-diagonal subgroup ofSU(4) gives anaction of SU(2) on S7 that defines the principal instanton fibration: S7/SU(2) ∼= S4. The most sophisticated example of a quotient-coalgebra Galois extension that we know of is a noncommutative defor- mation of the instanton bundle [11]. Here one starts with the Soibelman-Vaksman quantum sphere S7 [92], which is a homogeneous space ofSU (4), andthen, following the insight given by q q Poisson geometry [12], oneconstructs a coideal right ideal I of theHopf algebra O(SU (4)) such q that the canonical surjection O(SU (4)) → O(SU (4))/I corresponds to the block-diagonal in- q q clusion of SU(2) in SU(4) and the induced coaction is Galois yielding O(S4) as the coaction q invariant subalgebra [11]. One of the reasons why this example is interesting is that it uses the full generality of quotient-coalgebra Galois extensions, i.e., we have P 6= H and I 6= 0. Observe that for I = 0 we recover as a special case Hopf-Galois theory, whereas for P = H we obtain what is called homogeneous coalgebra-Galois extensions. We devote the remainder of this section to the latter case. This is the case which deals with quantum homogeneous spaces or left coideal subalgebras (thus justifying the name “homogeneous coalgebra-Galois extension”). The aim is to try and reproduce in the general noncommutative setting a classical construction in which a homogeneous space M of a group G is viewed as a base for a principal bundle with the total space G. Let P be a Hopf algebra and I a coideal right ideal of P, so that P/I is a coalgebra and a right P-module. View P as a right P/I-comodule via the induced coaction ∆ := (id⊗π )◦∆, P −π→I P/I. (2.15) P I The corresponding P/I-extension B⊆ P is called a homogeneous P/I-extension. The impor- tance of extensions of this type stems from the fact that B is a quantum homogeneous space or a left coideal subalgebra of B. Let us disucss this in more detail. Since 1 is a group-like element in a Hopf algebraP, its coalgebra projection π (1) is a group- I like element in P/I. Furthermore, ∆ (1) = 1 ⊗π (1). Observe then that for a homogeneous P I P/I-extension, the coaction-invariant subalgebra B is equal to the subalgebra of π (1)-coaction I P/I P/I invariants P = {b ∈ P | ∆ (b) = b⊗π (1)}. Indeed, B⊆ P by Lemma 2.1. Conversely, πI(1) P I πI(1) if ∆ (b) = b⊗π (1), then, for all p ∈ P, P I ∆ (bp) = b p ⊗π (b p ) = b p ⊗π (b )p P (1) (1) I (2) (2) (1) (1) I (2) (2) = bp ⊗π (1)p = bp ⊗π (p ) = b∆ (p), (1) I (2) (1) I (2) P where we have used the fact that π is a right P-module map. Thus, b ∈ B as required. I Next, using the coassociativity of the coaction ∆ and the description of B as π (1)-coaction P I invariants, apply ∆ ⊗ id to equation b ⊗ π (b ) = b ⊗ π (1) to deduce that for all b ∈ B, (1) I (2) I ((id⊗∆ )◦∆)(b) = ∆(b)⊗π (1). This implies that P I ∀b ∈ B, b ⊗b ∈ P ⊗B, (2.16) (1) (2) 9 i.e., ∆(B)⊆ P ⊗ B, so that B is a left coideal subalgebra of P or a quantum homogeneous space of P. Thus homogeneous P/I-extensions provide one with a suitable set-up for principal bundles over quantumhomogeneousspaces. Toexploit thisfully, however, weneedtoaddress aquestion when a homogeneous P/I-extension is a coalgebra-Galois extension. The answer turns out to determine the structure of I completely (cf. Lemma 5.2 in [20]). Theorem 2.6. Let B⊆ P be a homogeneous P/I-extension. Then this extension is Galois if and only if I = B+P, where B+ := B ∩Kerε. Proof. Assume first that I = B+P. Taking advantage of (2.16), for any b ∈ B+, p ∈ P, we compute: S(b p )⊗ b p = S(b p )b ⊗ p = S(p )ε(b)⊗ p = 0. (2.17) (1) (1) B (2) (2) (1) (1) (2) B (2) (1) B (2) Hence there is a well-defined map T : P ⊗(P/I) −→ P ⊗ P, T(p⊗[p′] ) := pS(p′ )⊗ p′ . (2.18) B I (1) B (2) It is straightforward to verify that T is the inverse of the canonical map can. Consequently, P is a coalgebra-Galois P/I-extension. To show the converse, let us first prove the following: Lemma 2.7. Let P, I and B be as above. Then B⊆ P is a coalgebra-Galois P/I-extension if and only if (π ◦(S⊗id)◦∆)(I) = 0, where π : P ⊗P → P ⊗ P is the canonical surjection. B B B Proof. ByProposition2.3,P isacoalgebra-GaloisP/I-extensionofB ifandonlyifthefollowing sequence 0 −→ P(Ω1B)P −→ P ⊗P −cga→n P ⊗P/I −→ 0 (2.19) is exact. One can check that (can◦(S ⊗id)◦∆)(I) = 0. Hence, it follows from the exactness of (2.19) that ((S ⊗id)◦∆)(I)⊆ P(Ω1B)P. Consequently, (π ◦(S ⊗id)◦∆)(I) = 0 due to B the exactness of the sequence g 0 −→ P(Ω1B)P −→ P ⊗P −π→B P ⊗ P −→ 0. (2.20) B To prove the converse, one can proceed as in the considerations preceding this lemma. Corollary 2.8. Let B⊆ P be a coalgebra-Galois P/I-extension as above. Then the translation map τ := can−1(1⊗·) is given by the formula: τ([p] ) := S(p )⊗ p . I (1) B (2) Assume now that P is a coalgebra-Galois P/I-extension of B. It follows from the above corollary and (2.17) that τ([B+P] ) = 0. Hence, by the injectivity of τ, we have B+P ⊆ I. I Furthermore, there is a well-defined map can′ : P ⊗ P −→ P ⊗(P/B+P), p⊗ p′ 7−→ pp′ ⊗[p′ ] . (2.21) B B (1) (2) B+P Indeed, taking again advantage of (2.16), we obtain p⊗bp′ 7→ pb p′ ⊗[(b −ε(b ))p′ +ε(b )p′ ] (1) (1) (2) (2) (2) (2) (2) B+P = pb p′ ⊗ε(b )[p′ ] (1) (1) (2) (2) B+P = pbp′ ⊗[p′ ] . (2.22) (1) (2) B+P 10