As a natural starting point, bundles are defined and topological properties are described. The principal map is defined and shown that injectivity is equivalent to the freeness condition. The image of the canonical map is deduced and necessary conditions are imposed to ensure the bijectivity of this map. The detailed account of the material presented in this section can be found in [6].

Definition 2.1 A bundle is a triple where and are topological spaces and is a continuous surjective map. Here is called the base space, the total space and the projection of the bundle.

For each , the fibre over is the topological space , i.e., the points on the total space which are projected, under , onto the point in the base space. A bundle whose fibres are homeomorphic which satisfies a condition known as local triviality are known as fibre bundles. This is formally expressed in the next definition.

Definition 2.2 A fibre bundle is a triple where is bundle and is a topological space such that are homeomorphic to for each . Furthermore, satisfies the local triviality condition.

The local triviality condition is satisfied if for each , there is an open neighourhood such that is homeomorphic to the product space , in such a way that carries over to the projection onto the first factor. That is the following diagram commutes:

The map is the natural projection and is a homeomorphism.

Example 2.3 An example of a fibre bundle which is non-trivial, i.e., not a global product space, is the Möbius strip. It has a circle that runs lengthwise through the centre of the strip as a base B and a line segment running vertically for the fibre F. The line segments are in fact copies of the real line, hence each is homeomorphic to and the Mobius strip is a fibre bundle.

Let be a topological space which is compact and satisfies the Hausdorff property and G a compact topological group. Suppose there is a right action of on and write .

Definition 2.4 An action of on is said to be free if for any implies that , the group identity.

With an eye on algebraic formulation of freeness, the principal map is defined as .

Proposition 2.5 acts freely on if and only if is injective.

Proof. “" Suppose the action is free, hence implies that . If , then and . Applying the action of to both sides of we get , which implies by the freeness property, concluding and is injective as required.

“" Suppose is injective, so or implies and . Since from the properties of the action, if then from the injectivity property.

Since acts on we can define the quotient space ,

The sets are called the orbits of the points . They are defined as the set of elements in to which can be moved by the action of elements of . The set of orbits of under the action of forms a partition of , hence we can define the equivalence relation on as,

The equivalence relation is the same as saying and are in the same orbit, i.e., . Given any quotient space, then there is a canonical surjective map

which maps elements in to their orbits. We define the pull-back along this map to be the set

As described above, the image of the principal map contains elements of in the first leg and the action of on in the second leg. To put it another way, the image records elements of in the first leg and all the elements in the same orbit as this in the second leg. Hence we can identify the image of the canonical map as the pull back along , namely . This is formally proved as a part of the following proposition.

Proposition 2.6 acts freely on if and only if the map

is bijective.

Proof. First note that the map is well-defined since the elements and are in the same orbit and hence map to the same equivalence class under . Using Proposition 2.5 we can deduce that the injectivity of is equivalent to the freeness of the action. Hence if we can show that is surjective the proof is complete.

Take . This means , which implies and are in the same equivalence class, which in turn means they are in the same orbit. We can therefore deduce that for some . So, implying . Hence completing the proof.

Definition 2.7 An action of on is said to be principal if the map is both injective and continuous (and such that the inverse image of a compact subset is compact in a case of locally compact spaces).

Since the injectivity and freeness condition are equivalent, we can interpret principal actions as both free and continuous actions. We can also deduce that these types of actions give rise to homeomorphisms from onto the space . Principal actions lead to the concept of topological principle bundles.

Definition 2.8 A principal bundle is a quadruple such that

The first two properties tell us that principal bundles are bundles admitting a principal action of a group on the total space , i.e., principal bundles correspond to principal actions. By Definition , principal actions occur when the principal map is both injective and continuous, or equivalently, when the action is free and continuous. The third property ensures that the fibres of the bundle correspond to the orbits coming from the action and the final property implies that the quotient space can topologically be viewed as the base space of the bundle.

Example 2.9 Suppose is a topological space and a topological group which acts on from the right. The triple where is the orbit space and the natural projection is a bundle. A principal action of on makes the quadruple a principal bundle.

We describe a principal bundle as a -principal bundle over , or as a -principal bundle over .

Definition 2.10 A vector bundle is a bundle where each fibre is endowed with a vector space structure such that addition and scalar multiplication are continuous maps.

Any vector bundle can be understood as a bundle associated to a principal bundle in the following way. Consider a -principal bundle and let be a representation space of , i.e., a (topological) vector space with a (continuous) left -action , . Then acts from the right on by

We can define and a surjective (continuous map) , and thus have a fibre bundle . In the case where is a vector space, we assume that acts linearly on .

Definition 2.11 A section of a bundle is a continuous map such that, for all ,

i.e., a section is simply a section of the morphism . The set of sections of is denoted by .

Proposition 2.12 Sections in a fibre bundle associated to a principal -bundle are in bijective correspondence with (continuous) maps such that

All such -equivariant maps are denoted by .

Proof. Remember that . Given a map , define the section ,

Conversely, given , define by assigning to a unique such that . Note that is unique, since if , then and . Freeness implies that , hence . The map has the required equivariance property, since the element of corresponding to is .

To make the transition from algebraic formulation of principal and associated bundles to non-commutative setup more transparent, we assume that is a complex affine variety with an action of an affine algebraic group and set (all with the usual Euclidean topology). Let , and be the corresponding coordinate rings. Put and and note the identification . Through this identification, is a Hopf algebra with comultiplication: , counit , , and the antipode , .

Using the fact that acts on we can construct a right coaction of on by , . This coaction is an algebra map due to the commutativity of the algebras of functions involved.

We have viewed the spaces of polynomial functions on and , next we view the space of functions on Y, , where . is a subalgebra of by

where is the canonical surjection defined above. The map is injective, since in means there exists at least one orbit such that , but , so which implies . Therefore, we can identify with . Furthermore, if and only if

for all , . This is the same as

for all , , where is the unit function (the identity element of ). Thus we can identify with the coinvariants of the coaction :

Since is a subalgebra of , it acts on via the inclusion map , . We can identify with by the map

Note that is well defined because . Proposition 2.6 immediately yields

Proposition 2.13 The action of on is free if and only if , is bijective.

In view of the definition of the coaction of on , we can identify with the canonical map

Thus the action of on is free if and only if this purely algebraic map is bijective. In the classical geometry case we take , and , but in general there is no need to restrict oneself to commutative algebras (of functions on topological spaces). In full generality this leads to the following definition.

Definition 2.14 (Hopf–Galois Extensions) Let be a Hopf algebra and a right -comodule algebra with coaction . Let , the coinvariant subalgebra of . We say that is a Hopf–Galois extension if the left -module, right -comodule map

is an isomorphism.

Proposition 2.13 tells us that when viewing bundles from an algebraic perspective, the freeness condition is equivalent to the Hopf–Galois extension property. Hence, the Hopf–Galois extension condition is a necessary condition to ensure a bundle is principal. Not all information about a topological space is encoded in a coordinate algebra, so to make a fuller reflection of the richness of the classical notion of a principal bundle we need to require conditions additional to the Hopf–Galois property.

Definition 2.15 Let be a Hopf algebra with bijective antipode and let be a right -comodule algebra with coaction . Let denote the coinvariant subalgebra of . We say that is a principal -comodule algebra if:

As indicated already in [7,8,9], principal comodule algebras should be understood as principal bundles in noncommutative geometry. In particular, if is the Hopf algebra associated to a -algebra of functions on a quantum group [10], then the existence of the Haar measure together with the results of [8] mean that condition (a) in Definition 2.15 implies condition (b) (i.e., the freeness of the coaction implies its principality).

The following characterisation of principal comodule algebras [11,12] gives an effective method for proving the principality of coaction.

Proposition 2.16 A right -comodule algebra with coaction is principal if and only if it admits a strong connection form, that is if there exists a map such that

Here denotes the multiplication map, is the unit map, is the comultiplication, counit and the (bijective) antipode of the Hopf algebra , and is the flip.

Proof. If a strong connection form exists, then the inverse of the canonical map (see Definition 2.14 ) is the composite

while the splitting of the multiplication map (see Definition 2.15 (b)) is given by

Conversely, if is a principal comodule algebra, then is the composite

where is the left -linear right -colinear splitting of the multiplication .

Example 2.17 Let be a right -comodule algebra. The space of -linear maps is an algebra with the convolution product

and unit . is said to be cleft if there exists a right -colinear map that has an inverse in the convolution algebra and is normalised so that . Writing for the convolution inverse of , one easily observes that

is a strong connection form. Hence a cleft comodule algebra is an example of a principal comodule algebra. The map is called a cleaving map or a normalised total integral.

In particular, if is an -colinear algebra map, then it is automatically convolution invertible (as ) and normalised. A comodule algebra admitting such a map is termed a trivial principal comodule algebra.

Example 2.18 Let be a Hopf algebra of the compact quantum group. By the Woronowicz theorem [10], admits an invariant Haar measure, i.e., a linear map such that, for all ,

where is the Sweedler notation for the comultiplication. Next, assume that the lifted canonical map:

is surjective, and write

for the -linear map such that , for all . Then, by the Schneider theorem [8], is a principal -comodule algebra. Explicitly, a strong connection form is

where the coaction is denoted by the Sweedler notation ; see [13].

Having described non-commutative principal bundles, we can look at the associated vector bundles. First we look at the classical case and try to understand it purely algebraically. Start with a vector bundle associated to a principal -bundle . Since is a vector representation space of , also the set is a vector space. Consequently is a vector space. Furthermore, is a left module of with the action To understand better the way in which -module is associated to the principal comodule algebra we recall the notion of the cotensor product.

Definition 2.19 Given a Hopf algebra , right -comodule with coaction and left -comodule with coaction , the cotensor product is defined as an equaliser:

If is an -comodule algebra, and , theis a left -module with the action In particular, in the case of a principal -bundle over , for any left -comodule the cotensor product is a left -module.

The following proposition indicates the way in which cotensor products enter description of associated bundles.

Proposition 2.20 Assume that the fibre of a vector bundle associated to a principal -bundle is finite dimensional. View as a left comodule of with the coaction (summation implicit) determined by Then the left -module of sections is isomorphic to the left -module .

Proof. First identify with . Let be a (finite) dual basis. Take , and define .

In the converse direction, define a left -module map

One easily checks that the constructed map are mutual inverses.

Moving away from commutative algebras of functions on topological spaces one uses Proposition 2.20 as the motivation for the following definition.

Definition 2.21 Let be a principal -comodule algebra. Set and let be a left -comodule. The left -module is called a module associated to the principal comodule algebra .

is a projective left -module, and if is a finite dimensional vector space, then is a finitely generated projective left -module. In this case it has the meaning of a module of sections over a non-commutative vector bundle. Furthermore, its class gives an element in the -group of . If is a cleft principal comodule algebra, then every associated module is free, since as a left -module and right -comodule, so that

The coordinate algebra of the circle or the group , can be identified with the -algebra of Laurent polynomials in a unitary variable (unitary means ). As a Hopf -algebra , is generated by the grouplike element , i.e.,

and thus it can be understood as the group algebra . As a consequence of this interpretation of , an algebra is a -comodule algebra if and only if is a -graded algebra,

is the coinvariant subalgebra of . Since is spanned by grouplike elements, any convolution invertible map must assign a unit (invertible element) of to . Furthermore, colinear maps are simply the -degree preserving maps, where . Put together, convolution invertible colinear maps are in one-to-one correspondence with sequences

Let be a real number, . The coordinate algebra of the even-dimensional quantum sphere is the unital complex -algebra with generators , subject to the following relations:

is a -graded algebra with and so is (with ). In other words, is a right -comodule algebra and is a left -comodule algebra, hence one can consider the cotensor product algebra . It was shown in [2] that, as a unital -algebra, has generators and a central unitary which are related in the following way:

For any choice of pairwise coprime numbers one can define the coaction of the Hopf algebra on as

for . This coaction is then extended to the whole of so that is a right -comodule algebra.

The algebra of coordinate functions on the quantum real weighted projective space is now defined as the subalgebra of containing all coinvariant elements, i.e.,

In this paper we consider two-dimensional quantum real weighted projective spaces, i.e., the algebras obtained from the coordinate algebra which is generated by and central unitary such that

The linear basis of is

For a pair of coprime positive integers, the coaction is given on generators by

and extended to the whole of so that the coaction is a -algebra map. We denote the comodule algebra with coaction by .

It turns out that the two dimensional quantum real projective spaces split into two cases depending on not wholly the parameter but instead whether is either even or odd, and hence only cases and need to be considered [3]. We describe these cases presently.

Theorem 4.1 is a principal comodule algebra if and only if .

Proof. As explained in [2] is a prolongation of the -comodule algebra . The latter is a principal comodule algebra (over the quantum real projective plane [14]) and since a prolongation of a principal comodule algebra is a principal comodule algebra [8, Remark 3.11], the coaction is principal as stated.

In the converse direction, we aim to show that the canonical map is not an isomorphism by showing that the image does not contain , i.e., it cannot be surjective since we know is in the codomain. We begin by identifying a basis for the algebra ; observing the relations in Equations (6a) and (6b) it is clear that a basis for is given by elements of the form

noting that all powers are non-negative. Hence a basis for is given by elements of the form , where . Applying the canonial map gives

where means for simplicity of notation. The next stage is to construct all possible elements in which map to . To obtain the identity in the first leg we must use one of the following relations:

or

We see that to obtain identity in the first leg we require the powers of and to be equal. We now construct all possible elements of the domain which map to after applying the canonical map.

Case 1: use the first relation to obtain (); this can be done in fours ways. First, using , , and . Now,

and

hence no possible terms. A similar calculation for the three other cases shows that cannot be obtained as an element of the image of the canonical map in this case.

Case 2: use the second relation to obtain (); this can be done in four ways , , and . Now,

and

Note that is not a problem provided is not equal to . This is reviewed at the next stage of the proof. The same conclusion is reached in all four cases.

In all possibilities appears only when , in which case the relation simplifies to , so the next stage involves constructing elements in the domain which map to . There are eight possibilities altogether to be checked: , , , , , , and . The first case gives:

and

Hence cannot be obtained as an element in the image in this case. Similar calculations for the remaining possibilities show that either is not in the image of the canonical map, or that if is in the image then .

Case 3: finally, it seems possible that , using the third relation, could be in the image of the canonical map. All possible elements in the domain which could potentially map to this element are constructed and investigated. There are eight possibilities: , , , , , , and . The first possibility comes out as

Also

which implies there are no terms. The same conclusion can be reached for the remaining relations.

This concludes that , which is contained in , is not in the image of the canonical map, proving that this map is not surjective and ultimately not an isomorphism when and are both not simultaneously equal to , completing the proof that is not a principal comodule algebra in this case.

Theorem 4.1 tells us that if we use as our total space, then we are forced to put to ensure that the required Hopf–Galois condition does not fail. A consequence of this would be the generators and would have -degree . This suggests that the comodule algebra is too restrictive as there is no freedom with the weights or , and that we should in fact consider a subalgebra of which admits a -coaction that would offer some choice. Theorem 4.1 indicates that the desired subalgebra should have generators with grades to ensure the Hopf–Galois condition is satisfied. This process is similar to that followed in [4], where the bundles over the quantum teardrops have the total spaces provided by the quantum lens spaces and structure groups provided by the circle group . We follow a similar approach in the sense that we view as a right -comodule algebra, where is the Hopf algebra of a suitable cyclic group.