Axioms 2012, 1(2), 201-225; doi:10.3390/axioms1020201

Article
Bundles over Quantum Real Weighted Projective Spaces
Tomasz Brzezinński * and Simon A. Fairfax
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK; Email: 201102@swansea.ac.uk
*
Author to whom correspondence should be addressed; Email: T.Brzezinski@swansea.ac.uk; Tel.: +44-1792-295460; Fax: +44-1792-295843.
Received: 10 July 2012; in revised form: 21 August 2012 / Accepted: 23 August 2012 /
Published: 17 September 2012

Abstract

: The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U (1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.
Keywords:
quantum real weighted projective space; principal comodule algebra; noncommutative line bundle

1 Introduction

In an algebraic setup an action of a circle on a quantum space corresponds to a coaction of a Hopf algebra of Laurent polynomials in one variable on the noncommutative coordinate algebra of the quantum space. Such a coaction can equivalently be understood as a Axioms 01 00201 i002-grading of this coordinate algebra. A typical Axioms 01 00201 i002-grading assigns degree ±1 to every generator of this algebra (different from the identity). The degree zero part forms a subalgebra which in particular cases corresponds to quantum complex or real projective spaces (grading of coordinate algebras of quantum spheres [1] or prolonged quantum spheres [2]). Often this grading is strong, meaning that the product of Axioms 01 00201 i003-graded parts is equal to the Axioms 01 00201 i004-part of the total algebra. In geometric terms this reflects the freeness of the circle action.

In two recent papers [4,3] circle actions on three-dimensional (and, briefly, higher dimensional) quantum spaces were revisited. Rather than assigning a uniform grade to each generator, separate generators were given degree by pairwise coprime integers. The zero part of such a grading of the coordinate algebra of the quantum odd-dimensional sphere corresponds to the quantum weighted projective space, while the zero part of such a grading of the algebra of the prolonged even dimensional quantum sphere leads to quantum real weighted projective spaces.

In this paper we focus on two classes of algebras Axioms 01 00201 i005( Axioms 01 00201 i006a positive integer) and Axioms 01 00201 i007( Axioms 01 00201 i006an odd positive integer) identified in [3] as fixed points of weighted circle actions on the coordinate algebra Axioms 01 00201 i008of a non-orientable quantum Seifert manifold described in [2]. Our aim is to construct quantum Axioms 01 00201 i009-principal bundles over the corresponding quantum spaces Axioms 01 00201 i011and describe associated line bundles. Recently, the importance of such bundles in non-commutative geometry was once again brought to the fore in [5], where the non-commutative Thom construction was outlined. As a further consequence of the principality of Axioms 01 00201 i009-coactions we also deduce that Axioms 01 00201 i012can be understood as quotients of Axioms 01 00201 i013by almost free Axioms 01 00201 i014-actions.

We begin in Section 2 by reviewing elements of algebraic approach to classical and quantum bundles. We then proceed to describe algebras Axioms 01 00201 i016in Section 3. Section 4 contains main results including construction of principal comodule algebras over Axioms 01 00201 i018. We observe that constructions albeit very similar in each case yield significantly different results. The principal comodule algebra over Axioms 01 00201 i005is non-trivial while that over Axioms 01 00201 i007turns out to be trivial (this means that all associated bundles are trivial, hence we do not mention them in the text). Whether it is a consequence of our particular construction or there is a deeper (topological or geometric) obstruction to constructing non-trivial principal circle bundles over Axioms 01 00201 i019remains an interesting open question.

Throughout we work with involutive algebras over the field of complex numbers (but the algebraic results remain true for all fields of characteristic 0). All algebras are associative and have identity, we use the standard Hopf algebra notation and terminology and we always assume that the antipode of a Hopf algebra is bijective. All topological spaces are assumed to be Hausdorff.

2 Review of Bundles in Non-Commutative Geometry

The aim of this section is to set out the topological concepts in relation to topological bundles, in particular principal bundles. The classical connection is made for interpreting topological concepts in an algebraic setting, providing a manageable methodology for performing calculations. In particular, the connection between principal bundles in topology and the algebraic Hopf–Galois condition is described. The reader familiar with classical theory of bundles can proceed directly to Definition 2.14.

2.1. Topological Aspects of Bundles

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 Axioms 01 00201 i020where Axioms 01 00201 i021and Axioms 01 00201 i022are topological spaces and Axioms 01 00201 i023is a continuous surjective map. Here Axioms 01 00201 i022is called the base space, Axioms 01 00201 i021the total space and Axioms 01 00201 i024the projection of the bundle.

For each Axioms 01 00201 i025, the fibre over Axioms 01 00201 i026is the topological space Axioms 01 00201 i027, i.e., the points on the total space which are projected, under Axioms 01 00201 i024, onto the point Axioms 01 00201 i026in 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 Axioms 01 00201 i028where Axioms 01 00201 i020is bundle and Axioms 01 00201 i029is a topological space such that Axioms 01 00201 i027are homeomorphic to Axioms 01 00201 i029for each Axioms 01 00201 i025. Furthermore, Axioms 01 00201 i024satisfies the local triviality condition.

The local triviality condition is satisfied if for each Axioms 01 00201 i030, there is an open neighourhood Axioms 01 00201 i031such that Axioms 01 00201 i032is homeomorphic to the product space Axioms 01 00201 i033, in such a way that Axioms 01 00201 i024carries over to the projection onto the first factor. That is the following diagram commutes:

Axioms 01 00201 i035

The map Axioms 01 00201 i036is the natural projection Axioms 01 00201 i037and Axioms 01 00201 i038is 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 Axioms 01 00201 i027is homeomorphic to Axioms 01 00201 i039and the Mobius strip is a fibre bundle.

Let Axioms 01 00201 i040be a topological space which is compact and satisfies the Hausdorff property and G a compact topological group. Suppose there is a right action Axioms 01 00201 i041of Axioms 01 00201 i042on Axioms 01 00201 i040and write Axioms 01 00201 i043.

Definition 2.4 An action of Axioms 01 00201 i042on Axioms 01 00201 i040is said to be free if Axioms 01 00201 i044for any Axioms 01 00201 i045implies that Axioms 01 00201 i046, the group identity.

With an eye on algebraic formulation of freeness, the principal map Axioms 01 00201 i047is defined as Axioms 01 00201 i048.

Proposition 2.5 Axioms 01 00201 i042acts freely on Axioms 01 00201 i040if and only if Axioms 01 00201 i049is injective.

Proof. Axioms 01 00201 i050" Suppose the action is free, hence Axioms 01 00201 i051implies that Axioms 01 00201 i052. If Axioms 01 00201 i053, then Axioms 01 00201 i054and Axioms 01 00201 i055. Applying the action of Axioms 01 00201 i056to both sides of Axioms 01 00201 i055we get Axioms 01 00201 i057, which implies Axioms 01 00201 i058by the freeness property, concluding Axioms 01 00201 i059and Axioms 01 00201 i060is injective as required.

Axioms 01 00201 i061" Suppose Axioms 01 00201 i060is injective, so Axioms 01 00201 i062or Axioms 01 00201 i053implies Axioms 01 00201 i054and Axioms 01 00201 i059. Since Axioms 01 00201 i063from the properties of the action, if Axioms 01 00201 i064then Axioms 01 00201 i052from the injectivity property.

Since Axioms 01 00201 i065acts on Axioms 01 00201 i066we can define the quotient space Axioms 01 00201 i067,

Axioms 01 00201 i068

The sets Axioms 01 00201 i069are called the orbits of the points Axioms 01 00201 i070. They are defined as the set of elements in Axioms 01 00201 i066to which Axioms 01 00201 i070can be moved by the action of elements of Axioms 01 00201 i065. The set of orbits of Axioms 01 00201 i066under the action of Axioms 01 00201 i065forms a partition of Axioms 01 00201 i066, hence we can define the equivalence relation on Axioms 01 00201 i066as,

Axioms 01 00201 i071

The equivalence relation is the same as saying Axioms 01 00201 i070and Axioms 01 00201 i072are in the same orbit, i.e., Axioms 01 00201 i073. Given any quotient space, then there is a canonical surjective map

Axioms 01 00201 i074

which maps elements in Axioms 01 00201 i075to their orbits. We define the pull-back along this map Axioms 01 00201 i076to be the set

Axioms 01 00201 i077

As described above, the image of the principal map Axioms 01 00201 i078contains elements of Axioms 01 00201 i075in the first leg and the action of Axioms 01 00201 i079on Axioms 01 00201 i080in the second leg. To put it another way, the image records elements of Axioms 01 00201 i081in the first leg and all the elements in the same orbit as this Axioms 01 00201 i080in the second leg. Hence we can identify the image of the canonical map as the pull back along Axioms 01 00201 i076, namely Axioms 01 00201 i082. This is formally proved as a part of the following proposition.

Proposition 2.6 Axioms 01 00201 i083acts freely on Axioms 01 00201 i084if and only if the map

Axioms 01 00201 i085

is bijective.

Proof. First note that the map Axioms 01 00201 i086is well-defined since the elements Axioms 01 00201 i087and Axioms 01 00201 i088are in the same orbit and hence map to the same equivalence class under Axioms 01 00201 i089. Using Proposition 2.5 we can deduce that the injectivity of Axioms 01 00201 i086is equivalent to the freeness of the action. Hence if we can show that Axioms 01 00201 i086is surjective the proof is complete.

Take Axioms 01 00201 i090. This means Axioms 01 00201 i091, which implies Axioms 01 00201 i087and Axioms 01 00201 i092are in the same equivalence class, which in turn means they are in the same orbit. We can therefore deduce that Axioms 01 00201 i093for some Axioms 01 00201 i094. So, Axioms 01 00201 i095implying Axioms 01 00201 i096. Hence Axioms 01 00201 i097completing the proof.

Definition 2.7 An action of Axioms 01 00201 i083on Axioms 01 00201 i084is said to be principal if the map Axioms 01 00201 i098is 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 Axioms 01 00201 i086from Axioms 01 00201 i099onto the space Axioms 01 00201 i100. Principal actions lead to the concept of topological principle bundles.

Definition 2.8 A principal bundle is a quadruple Axioms 01 00201 i101such that

  • (a) Axioms 01 00201 i102is a bundle and Axioms 01 00201 i083is a topological group acting continuously on Axioms 01 00201 i084with action Axioms 01 00201 i103, Axioms 01 00201 i104;

  • (b) the action Axioms 01 00201 i105is principal;

  • (c) Axioms 01 00201 i106such that Axioms 01 00201 i107;

  • (d) the induced map Axioms 01 00201 i108is a homeomorphism.

The first two properties tell us that principal bundles are bundles admitting a principal action of a group Axioms 01 00201 i042on the total space Axioms 01 00201 i040, i.e., principal bundles correspond to principal actions. By Definition Axioms 01 00201 i109, 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 Axioms 01 00201 i040is a topological space and Axioms 01 00201 i042a topological group which acts on Axioms 01 00201 i040from the right. The triple Axioms 01 00201 i110where Axioms 01 00201 i111is the orbit space and Axioms 01 00201 i024the natural projection is a bundle. A principal action of Axioms 01 00201 i042on Axioms 01 00201 i040makes the quadruple Axioms 01 00201 i112a principal bundle.

We describe a principal bundle Axioms 01 00201 i113as a Axioms 01 00201 i042-principal bundle over Axioms 01 00201 i114, or Axioms 01 00201 i040as a Axioms 01 00201 i042-principal bundle over Axioms 01 00201 i115.

Definition 2.10 A vector bundle is a bundle Axioms 01 00201 i020where each fibre Axioms 01 00201 i027is 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 Axioms 01 00201 i042-principal bundle Axioms 01 00201 i116and let Axioms 01 00201 i117be a representation space of Axioms 01 00201 i042, i.e., a (topological) vector space with a (continuous) left Axioms 01 00201 i042-action Axioms 01 00201 i118, Axioms 01 00201 i119. Then Axioms 01 00201 i042acts from the right on Axioms 01 00201 i120by

Axioms 01 00201 i121

We can define Axioms 01 00201 i122and a surjective (continuous map) Axioms 01 00201 i123, Axioms 01 00201 i124and thus have a fibre bundle Axioms 01 00201 i125. In the case where Axioms 01 00201 i117is a vector space, we assume that Axioms 01 00201 i042acts linearly on Axioms 01 00201 i117.

Definition 2.11 A section of a bundle Axioms 01 00201 i126is a continuous map Axioms 01 00201 i127such that, for all Axioms 01 00201 i128,

Axioms 01 00201 i129

i.e., a section is simply a section of the morphism Axioms 01 00201 i130. The set of sections of Axioms 01 00201 i021is denoted by Axioms 01 00201 i131.

Proposition 2.12 Sections in a fibre bundle Axioms 01 00201 i125associated to a principal Axioms 01 00201 i042-bundle Axioms 01 00201 i040are in bijective correspondence with (continuous) maps Axioms 01 00201 i132such that

Axioms 01 00201 i133

All such Axioms 01 00201 i042-equivariant maps are denoted by Axioms 01 00201 i134.

Proof. Remember that Axioms 01 00201 i135. Given a map Axioms 01 00201 i136, define the section Axioms 01 00201 i137, Axioms 01 00201 i138

Conversely, given Axioms 01 00201 i139, define Axioms 01 00201 i140by assigning to Axioms 01 00201 i045a unique Axioms 01 00201 i141such that Axioms 01 00201 i142. Note that Axioms 01 00201 i143is unique, since if Axioms 01 00201 i144, then Axioms 01 00201 i044and Axioms 01 00201 i145. Freeness implies that Axioms 01 00201 i046, hence Axioms 01 00201 i146. The map Axioms 01 00201 i147has the required equivariance property, since the element of Axioms 01 00201 i148corresponding to Axioms 01 00201 i149is Axioms 01 00201 i150.

2.2. Non-Commutative Principal and Associated Bundles

To make the transition from algebraic formulation of principal and associated bundles to non-commutative setup more transparent, we assume that Axioms 01 00201 i040is a complex affine variety with an action of an affine algebraic group Axioms 01 00201 i042and set Axioms 01 00201 i151(all with the usual Euclidean topology). Let Axioms 01 00201 i152, Axioms 01 00201 i153and Axioms 01 00201 i154be the corresponding coordinate rings. Put Axioms 01 00201 i155and Axioms 01 00201 i156and note the identification Axioms 01 00201 i157. Through this identification, Axioms 01 00201 i158is a Hopf algebra with comultiplication: Axioms 01 00201 i159 Axioms 01 00201 i160, counit Axioms 01 00201 i161, Axioms 01 00201 i162, and the antipode Axioms 01 00201 i163, Axioms 01 00201 i164.

Using the fact that Axioms 01 00201 i042acts on Axioms 01 00201 i040we can construct a right coaction of Axioms 01 00201 i165on Axioms 01 00201 i166by Axioms 01 00201 i167, Axioms 01 00201 i168. 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 Axioms 01 00201 i066and Axioms 01 00201 i065, next we view the space of functions on Y, Axioms 01 00201 i169, where Axioms 01 00201 i170. Axioms 01 00201 i171is a subalgebra of Axioms 01 00201 i172by

Axioms 01 00201 i173

where Axioms 01 00201 i076is the canonical surjection defined above. The map Axioms 01 00201 i174is injective, since Axioms 01 00201 i175in Axioms 01 00201 i176means there exists at least one orbit Axioms 01 00201 i177such that Axioms 01 00201 i178, but Axioms 01 00201 i179, so Axioms 01 00201 i180which implies Axioms 01 00201 i181. Therefore, we can identify Axioms 01 00201 i171with Axioms 01 00201 i182. Furthermore, Axioms 01 00201 i183if and only if

Axioms 01 00201 i184

for all Axioms 01 00201 i185, Axioms 01 00201 i186. This is the same as

Axioms 01 00201 i187

for all Axioms 01 00201 i081, Axioms 01 00201 i079, where Axioms 01 00201 i188is the unit function Axioms 01 00201 i189(the identity element of Axioms 01 00201 i191). Thus we can identify Axioms 01 00201 i192with the coinvariants of the coaction Axioms 01 00201 i193:

Axioms 01 00201 i194

Since Axioms 01 00201 i192is a subalgebra of Axioms 01 00201 i195, it acts on Axioms 01 00201 i195via the inclusion map Axioms 01 00201 i196, Axioms 01 00201 i197. We can identify Axioms 01 00201 i198with Axioms 01 00201 i199by the map

Axioms 01 00201 i200

Note that Axioms 01 00201 i201is well defined because Axioms 01 00201 i202. Proposition 2.6 immediately yields

Proposition 2.13 The action of Axioms 01 00201 i083on Axioms 01 00201 i084is free if and only if Axioms 01 00201 i203, Axioms 01 00201 i204is bijective.

In view of the definition of the coaction of Axioms 01 00201 i205on Axioms 01 00201 i206, we can identify Axioms 01 00201 i207with the canonical map

Axioms 01 00201 i208

Thus the action of Axioms 01 00201 i083on Axioms 01 00201 i084is free if and only if this purely algebraic map is bijective. In the classical geometry case we take Axioms 01 00201 i210, Axioms 01 00201 i211and Axioms 01 00201 i212, 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 Axioms 01 00201 i165be a Hopf algebra and Axioms 01 00201 i206a right Axioms 01 00201 i165-comodule algebra with coaction Axioms 01 00201 i213. Let Axioms 01 00201 i215, the coinvariant subalgebra of Axioms 01 00201 i206. We say that Axioms 01 00201 i216is a Hopf–Galois extension if the left Axioms 01 00201 i206-module, right Axioms 01 00201 i165-comodule map

Axioms 01 00201 i217

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 Axioms 01 00201 i165be a Hopf algebra with bijective antipode and let Axioms 01 00201 i206be a right Axioms 01 00201 i165-comodule algebra with coaction Axioms 01 00201 i213. Let Axioms 01 00201 i219denote the coinvariant subalgebra of Axioms 01 00201 i206. We say that Axioms 01 00201 i206is a principal Axioms 01 00201 i165-comodule algebra if:

  • (a) Axioms 01 00201 i216is a Hopf–Galois extension;

  • (b) the multiplication map Axioms 01 00201 i220, Axioms 01 00201 i221, splits as a left Axioms 01 00201 i219-module and right Axioms 01 00201 i165-comodule map (the equivariant projectivity condition).

As indicated already in [8,7,9], principal comodule algebras should be understood as principal bundles in noncommutative geometry. In particular, if Axioms 01 00201 i165is the Hopf algebra associated to a Axioms 01 00201 i222-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 [12,11] gives an effective method for proving the principality of coaction.

Proposition 2.16 A right Axioms 01 00201 i165-comodule algebra Axioms 01 00201 i166with coaction Axioms 01 00201 i167is principal if and only if it admits a strong connection form, that is if there exists a map Axioms 01 00201 i223such that

Axioms 01 00201 i224
Axioms 01 00201 i226
Axioms 01 00201 i227
Axioms 01 00201 i228

Here Axioms 01 00201 i229denotes the multiplication map, Axioms 01 00201 i230is the unit map, Axioms 01 00201 i231is the comultiplication, Axioms 01 00201 i232counit and Axioms 01 00201 i163the (bijective) antipode of the Hopf algebra Axioms 01 00201 i165, and Axioms 01 00201 i233is the flip.

Proof. If a strong connection form Axioms 01 00201 i234exists, then the inverse of the canonical map Axioms 01 00201 i235(see Definition 2.14 ) is the composite

Axioms 01 00201 i236

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

Axioms 01 00201 i238

Conversely, if Axioms 01 00201 i240is a principal comodule algebra, then Axioms 01 00201 i234is the composite

Axioms 01 00201 i241

where Axioms 01 00201 i243is the left Axioms 01 00201 i244-linear right Axioms 01 00201 i165-colinear splitting of the multiplication Axioms 01 00201 i245.

Example 2.17 Let Axioms 01 00201 i166be a right Axioms 01 00201 i165-comodule algebra. The space of Axioms 01 00201 i246-linear maps Axioms 01 00201 i247is an algebra with the convolution product

Axioms 01 00201 i248

and unit Axioms 01 00201 i249. Axioms 01 00201 i172is said to be cleft if there exists a right Axioms 01 00201 i250-colinear map Axioms 01 00201 i251that has an inverse in the convolution algebra Axioms 01 00201 i252and is normalised so that Axioms 01 00201 i253. Writing Axioms 01 00201 i254for the convolution inverse of Axioms 01 00201 i255, one easily observes that

Axioms 01 00201 i256

is a strong connection form. Hence a cleft comodule algebra is an example of a principal comodule algebra. The map Axioms 01 00201 i255is called a cleaving map or a normalised total integral.

In particular, if Axioms 01 00201 i251is an Axioms 01 00201 i250-colinear algebra map, then it is automatically convolution invertible (as Axioms 01 00201 i257) and normalised. A comodule algebra Axioms 01 00201 i172admitting such a map is termed a trivial principal comodule algebra.

Example 2.18 Let Axioms 01 00201 i250be a Hopf algebra of the compact quantum group. By the Woronowicz theorem [10], Axioms 01 00201 i250admits an invariant Haar measure, i.e., a linear map Axioms 01 00201 i258such that, for all Axioms 01 00201 i259,

Axioms 01 00201 i260

where Axioms 01 00201 i261is the Sweedler notation for the comultiplication. Next, assume that the lifted canonical map:

Axioms 01 00201 i263

is surjective, and write

Axioms 01 00201 i264

for the Axioms 01 00201 i265-linear map such that Axioms 01 00201 i266, for all Axioms 01 00201 i267. Then, by the Schneider theorem [8], Axioms 01 00201 i195is a principal Axioms 01 00201 i191-comodule algebra. Explicitly, a strong connection form is

Axioms 01 00201 i268

where the coaction is denoted by the Sweedler notation Axioms 01 00201 i269; 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 Axioms 01 00201 i270associated to a principal Axioms 01 00201 i083-bundle Axioms 01 00201 i084. Since Axioms 01 00201 i271is a vector representation space of Axioms 01 00201 i083, also the set Axioms 01 00201 i272is a vector space. Consequently Axioms 01 00201 i273is a vector space. Furthermore, Axioms 01 00201 i274is a left module of Axioms 01 00201 i275with the action Axioms 01 00201 i276To understand better the way in which Axioms 01 00201 i219-module Axioms 01 00201 i273is associated to the principal comodule algebra Axioms 01 00201 i277we recall the notion of the cotensor product.

Definition 2.19 Given a Hopf algebra Axioms 01 00201 i165, right Axioms 01 00201 i165-comodule Axioms 01 00201 i206with coaction Axioms 01 00201 i278and left Axioms 01 00201 i165-comodule Axioms 01 00201 i271with coaction Axioms 01 00201 i279, the cotensor product is defined as an equaliser:

Axioms 01 00201 i280

If Axioms 01 00201 i206is an Axioms 01 00201 i165-comodule algebra, and Axioms 01 00201 i282, the Axioms 01 00201 i283is a left Axioms 01 00201 i219-module with the action Axioms 01 00201 i286In particular, in the case of a principal Axioms 01 00201 i083-bundle Axioms 01 00201 i084over Axioms 01 00201 i287, for any left Axioms 01 00201 i288-comodule Axioms 01 00201 i271the cotensor product Axioms 01 00201 i290is a left Axioms 01 00201 i291-module.

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

Proposition 2.20 Assume that the fibre Axioms 01 00201 i271of a vector bundle Axioms 01 00201 i270associated to a principal Axioms 01 00201 i083-bundle Axioms 01 00201 i084is finite dimensional. View Axioms 01 00201 i271as a left comodule of Axioms 01 00201 i292with the coaction Axioms 01 00201 i293 (summation implicit) determined by Axioms 01 00201 i294Then the left Axioms 01 00201 i291-module of sections Axioms 01 00201 i273is isomorphic to the left Axioms 01 00201 i291-module Axioms 01 00201 i296.

Proof. First identify Axioms 01 00201 i297with Axioms 01 00201 i298. Let Axioms 01 00201 i299be a (finite) dual basis. Take Axioms 01 00201 i300, and define Axioms 01 00201 i302.

In the converse direction, define a left Axioms 01 00201 i153-module map

Axioms 01 00201 i305

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 Axioms 01 00201 i166be a principal Axioms 01 00201 i165-comodule algebra. Set Axioms 01 00201 i306and let Axioms 01 00201 i117be a left Axioms 01 00201 i165-comodule. The left Axioms 01 00201 i244-module Axioms 01 00201 i308is called a module associated to the principal comodule algebra Axioms 01 00201 i166.

Axioms 01 00201 i309is a projective left Axioms 01 00201 i244-module, and if Axioms 01 00201 i117is a finite dimensional vector space, then Axioms 01 00201 i309is a finitely generated projective left Axioms 01 00201 i244-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 Axioms 01 00201 i310-group of Axioms 01 00201 i244. If Axioms 01 00201 i166is a cleft principal comodule algebra, then every associated module is free, since Axioms 01 00201 i311as a left Axioms 01 00201 i244-module and right Axioms 01 00201 i165-comodule, so that

Axioms 01 00201 i313

3 Weighted Circle Actions on Prolonged Spheres.

In this section we recall the definitions of algebras we study in the sequel.

3.1. Circle Actions and Axioms 01 00201 i314-Gradings.

The coordinate algebra of the circle or the group Axioms 01 00201 i315, Axioms 01 00201 i316can be identified with the Axioms 01 00201 i317-algebra Axioms 01 00201 i318of Laurent polynomials in a unitary variable Axioms 01 00201 i319(unitary means Axioms 01 00201 i320). As a Hopf Axioms 01 00201 i317-algebra Axioms 01 00201 i318, is generated by the grouplike element Axioms 01 00201 i319, i.e.,

Axioms 01 00201 i321

and thus it can be understood as the group algebra Axioms 01 00201 i322. As a consequence of this interpretation of Axioms 01 00201 i318, an algebra Axioms 01 00201 i166is a Axioms 01 00201 i318-comodule algebra if and only if Axioms 01 00201 i166is a Axioms 01 00201 i323-graded algebra,

Axioms 01 00201 i324

Axioms 01 00201 i325is the coinvariant subalgebra of Axioms 01 00201 i166. Since Axioms 01 00201 i318is spanned by grouplike elements, any convolution invertible map Axioms 01 00201 i326must assign a unit (invertible element) of Axioms 01 00201 i166to Axioms 01 00201 i327. Furthermore, colinear maps are simply the Axioms 01 00201 i323-degree preserving maps, where Axioms 01 00201 i328. Put together, convolution invertible colinear maps Axioms 01 00201 i329are in one-to-one correspondence with sequences

Axioms 01 00201 i330

3.2. The Axioms 01 00201 i331and Axioms 01 00201 i332Coordinate Algebras

Let Axioms 01 00201 i333be a real number, Axioms 01 00201 i334. The coordinate algebra Axioms 01 00201 i335of the even-dimensional quantum sphere is the unital complex Axioms 01 00201 i336-algebra with generators Axioms 01 00201 i337, subject to the following relations:

Axioms 01 00201 i338
Axioms 01 00201 i340

Axioms 01 00201 i341is a Axioms 01 00201 i342-graded algebra with Axioms 01 00201 i343and so is Axioms 01 00201 i344(with Axioms 01 00201 i345). In other words, Axioms 01 00201 i341is a right Axioms 01 00201 i346-comodule algebra and Axioms 01 00201 i344is a left Axioms 01 00201 i346-comodule algebra, hence one can consider the cotensor product algebra Axioms 01 00201 i348. It was shown in [2] that, as a unital Axioms 01 00201 i336-algebra, Axioms 01 00201 i349has generators Axioms 01 00201 i350and a central unitary Axioms 01 00201 i351which are related in the following way:

Axioms 01 00201 i352
Axioms 01 00201 i354

For any choice of Axioms 01 00201 i355pairwise coprime numbers Axioms 01 00201 i356one can define the coaction of the Hopf algebra Axioms 01 00201 i357on Axioms 01 00201 i358as

Axioms 01 00201 i359

for Axioms 01 00201 i360. This coaction is then extended to the whole of Axioms 01 00201 i361so that Axioms 01 00201 i361is a right Axioms 01 00201 i362-comodule algebra.

The algebra of coordinate functions on the quantum real weighted projective space is now defined as the subalgebra of Axioms 01 00201 i363containing all coinvariant elements, i.e.,

Axioms 01 00201 i364

3.3. The 2D Quantum Real Projective Space Axioms 01 00201 i365

In this paper we consider two-dimensional quantum real weighted projective spaces, i.e., the algebras obtained from the coordinate algebra Axioms 01 00201 i008which is generated by Axioms 01 00201 i366and central unitary Axioms 01 00201 i367such that

Axioms 01 00201 i368
Axioms 01 00201 i369

The linear basis of Axioms 01 00201 i008is

Axioms 01 00201 i370

For a pair Axioms 01 00201 i371of coprime positive integers, the coaction Axioms 01 00201 i372is given on generators by

Axioms 01 00201 i373

and extended to the whole of Axioms 01 00201 i374so that the coaction is a Axioms 01 00201 i317-algebra map. We denote the comodule algebra Axioms 01 00201 i374with coaction Axioms 01 00201 i375by Axioms 01 00201 i377.

It turns out that the two dimensional quantum real projective spaces split into two cases depending on not wholly the parameter Axioms 01 00201 i378but instead whether Axioms 01 00201 i378is either even or odd, and hence only cases Axioms 01 00201 i379and Axioms 01 00201 i380need to be considered [3]. We describe these cases presently.

3.3.1. The Odd or Negative Case

For Axioms 01 00201 i379, Axioms 01 00201 i381is a polynomial Axioms 01 00201 i317-algebra generated by Axioms 01 00201 i382, Axioms 01 00201 i383, Axioms 01 00201 i384which satisfy the relations:

Axioms 01 00201 i385
Axioms 01 00201 i387
Axioms 01 00201 i389
Axioms 01 00201 i391

The embedding of generators of Axioms 01 00201 i381into Axioms 01 00201 i392or the isomorphism of Axioms 01 00201 i381with the coinvariants of Axioms 01 00201 i393is provided by

Axioms 01 00201 i394

Up to equivalence Axioms 01 00201 i381has the following irreducible Axioms 01 00201 i317-representations. There is a family of one-dimensional representations labelled by Axioms 01 00201 i395and given by

Axioms 01 00201 i396

All other representations are infinite dimensional, labelled by Axioms 01 00201 i397, and given by

Axioms 01 00201 i398
Axioms 01 00201 i400

where Axioms 01 00201 i402, Axioms 01 00201 i403, is an orthonormal basis for the representation space Axioms 01 00201 i404.

The Axioms 01 00201 i405-algebra of continuous functions on Axioms 01 00201 i406, obtained as the completion of these bounded representations, can be identified with the pullback of Axioms 01 00201 i407-copies of the quantum real projective plane Axioms 01 00201 i408introduced in [14].

3.3.2. The Even or Positive Case

For Axioms 01 00201 i409and hence Axioms 01 00201 i407odd, Axioms 01 00201 i411is a polynomial Axioms 01 00201 i336-algebra generated by Axioms 01 00201 i412, Axioms 01 00201 i413which satisfy the relations:

Axioms 01 00201 i414
Axioms 01 00201 i416

The embedding of generators of Axioms 01 00201 i417into Axioms 01 00201 i418or the isomorphism of Axioms 01 00201 i417with the coinvariants of Axioms 01 00201 i419is provided by

Axioms 01 00201 i420

Similarly to the odd Axioms 01 00201 i422case, there is a family of one-dimensional representations of Axioms 01 00201 i417labelled by Axioms 01 00201 i423and given by

Axioms 01 00201 i424

All other representations are infinite dimensional, labelled by Axioms 01 00201 i425, and given by

Axioms 01 00201 i427

where Axioms 01 00201 i428, Axioms 01 00201 i429is an orthonormal basis for the representation space Axioms 01 00201 i430.

The Axioms 01 00201 i431-algebra Axioms 01 00201 i432of continuous functions on Axioms 01 00201 i019, obtained as the completion of these bounded representations, can be identified with the pullback of Axioms 01 00201 i006-copies of the quantum disk Axioms 01 00201 i433introduced in [15]. Furthermore, Axioms 01 00201 i434can also be understood as the quantum double suspension of Axioms 01 00201 i006points in the sense of [16, Definition 6.1].

4 Quantum Real Weighted Projective Spaces and Quantum Principal Bundles

The general aim of this paper is to construct quantum principal bundles with base spaces given by Axioms 01 00201 i435and fibre structures given by the circle Hopf algebra Axioms 01 00201 i436. The question arises as to which quantum space (i.e., a Axioms 01 00201 i437-comodule algebra with coinvariants isomorphic to Axioms 01 00201 i018) we should consider as the total space within this construction. We look first at the coactions of Axioms 01 00201 i437on Axioms 01 00201 i008that define Axioms 01 00201 i438, i.e., at the comodule algebras Axioms 01 00201 i376.

4.1. The (Non-)Principality of Axioms 01 00201 i439

Theorem 4.1 Axioms 01 00201 i440is a principal comodule algebra if and only if Axioms 01 00201 i441.

Proof. As explained in [2] Axioms 01 00201 i442is a prolongation of the Axioms 01 00201 i443-comodule algebra Axioms 01 00201 i444. The latter is a principal comodule algebra (over the quantum real projective plane Axioms 01 00201 i445 [14]) and since a prolongation of a principal comodule algebra is a principal comodule algebra [8, Remark 3.11], the coaction Axioms 01 00201 i446is 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 Axioms 01 00201 i447, i.e., it cannot be surjective since we know Axioms 01 00201 i447is in the codomain. We begin by identifying a basis for the algebra Axioms 01 00201 i448; observing the relations in Equations (6a) and (6b) it is clear that a basis for Axioms 01 00201 i449is given by elements of the form

Axioms 01 00201 i450
Axioms 01 00201 i451

noting that all powers are non-negative. Hence a basis for Axioms 01 00201 i452is given by elements of the form Axioms 01 00201 i453, where Axioms 01 00201 i454. Applying the canonial map gives

Axioms 01 00201 i456

where Axioms 01 00201 i457means Axioms 01 00201 i458for simplicity of notation. The next stage is to construct all possible elements in Axioms 01 00201 i459which map to Axioms 01 00201 i447. To obtain the identity in the first leg we must use one of the following relations:

Axioms 01 00201 i460
Axioms 01 00201 i461

or

Axioms 01 00201 i462

We see that to obtain identity in the first leg we require the powers of Axioms 01 00201 i463and Axioms 01 00201 i464to be equal. We now construct all possible elements of the domain which map to Axioms 01 00201 i447after applying the canonical map.

Case 1: use the first relation to obtain Axioms 01 00201 i465( Axioms 01 00201 i466); this can be done in fours ways. First, using Axioms 01 00201 i467, Axioms 01 00201 i468, Axioms 01 00201 i469and Axioms 01 00201 i470. Now,

Axioms 01 00201 i472

and

Axioms 01 00201 i473

hence no possible terms. A similar calculation for the three other cases shows that Axioms 01 00201 i474cannot be obtained as an element of the image of the canonical map in this case.

Case 2: use the second relation to obtain Axioms 01 00201 i475( Axioms 01 00201 i476); this can be done in four ways Axioms 01 00201 i477, Axioms 01 00201 i478, Axioms 01 00201 i478and Axioms 01 00201 i479. Now,

Axioms 01 00201 i481

and

Axioms 01 00201 i482

Note that Axioms 01 00201 i483is not a problem provided Axioms 01 00201 i484is not equal to Axioms 01 00201 i485. This is reviewed at the next stage of the proof. The same conclusion is reached in all four cases.

In all possibilities Axioms 01 00201 i486appears only when Axioms 01 00201 i487, in which case the relation simplifies to Axioms 01 00201 i488, so the next stage involves constructing elements in the domain which map to Axioms 01 00201 i489. There are eight possibilities altogether to be checked: Axioms 01 00201 i490, Axioms 01 00201 i491, Axioms 01 00201 i492, Axioms 01 00201 i493, Axioms 01 00201 i494, Axioms 01 00201 i495, Axioms 01 00201 i496and Axioms 01 00201 i497. The first case gives:

Axioms 01 00201 i498

and

Axioms 01 00201 i499

Hence Axioms 01 00201 i500cannot be obtained as an element in the image in this case. Similar calculations for the remaining possibilities show that either Axioms 01 00201 i500is not in the image of the canonical map, or that if Axioms 01 00201 i500is in the image then Axioms 01 00201 i501.

Case 3: finally, it seems possible that Axioms 01 00201 i502, 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: Axioms 01 00201 i503, Axioms 01 00201 i504, Axioms 01 00201 i505, Axioms 01 00201 i506, Axioms 01 00201 i507, Axioms 01 00201 i508, Axioms 01 00201 i509and Axioms 01 00201 i510. The first possibility comes out as

Axioms 01 00201 i512

Also

Axioms 01 00201 i513

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

This concludes that Axioms 01 00201 i502, which is contained in Axioms 01 00201 i514, is not in the image of the canonical map, proving that this map is not surjective and ultimately not an isomorphism when Axioms 01 00201 i515and Axioms 01 00201 i006are both not simultaneously equal to Axioms 01 00201 i516, completing the proof that Axioms 01 00201 i517is not a principal comodule algebra in this case.

Theorem 4.1 tells us that if we use Axioms 01 00201 i376as our total space, then we are forced to put Axioms 01 00201 i441to ensure that the required Hopf–Galois condition does not fail. A consequence of this would be the generators Axioms 01 00201 i518and Axioms 01 00201 i519would have Axioms 01 00201 i002-degree Axioms 01 00201 i520. This suggests that the comodule algebra Axioms 01 00201 i517is too restrictive as there is no freedom with the weights Axioms 01 00201 i378or Axioms 01 00201 i006, and that we should in fact consider a subalgebra of Axioms 01 00201 i008which admits a Axioms 01 00201 i521-coaction that would offer some choice. Theorem 4.1 indicates that the desired subalgebra should have generators with grades Axioms 01 00201 i520to ensure the Hopf–Galois condition is satisfied. This process is similar to that followed in [4], where the bundles over the quantum teardrops Axioms 01 00201 i522have the total spaces provided by the quantum lens spaces and structure groups provided by the circle group Axioms 01 00201 i009. We follow a similar approach in the sense that we view Axioms 01 00201 i376as a right Axioms 01 00201 i165-comodule algebra, where Axioms 01 00201 i165is the Hopf algebra of a suitable cyclic group.

4.2. The Negative Case Axioms 01 00201 i523

4.2.1. The Principal Axioms 01 00201 i524-Comodule Algebra over Axioms 01 00201 i381

Take the group Hopf Axioms 01 00201 i317-algebra Axioms 01 00201 i526which is generated by unitary grouplike element Axioms 01 00201 i527and satisfies the relation Axioms 01 00201 i528. The algebra Axioms 01 00201 i392is a right Axioms 01 00201 i529-comodule Axioms 01 00201 i317-algebra with coaction

Axioms 01 00201 i530

Note that the Axioms 01 00201 i531-degree of the generator Axioms 01 00201 i532is determined by the degree of Axioms 01 00201 i533: the relation Axioms 01 00201 i534and that the coaction must be compatible with all relations imply that Axioms 01 00201 i535. Since Axioms 01 00201 i533has degree zero, Axioms 01 00201 i532must also have degree zero.

The next stage of the process is to find the coinvariant elements of Axioms 01 00201 i392given the coaction defined above.

Proposition 4.2 The fixed point subalgebra of the above coaction is isomorphic to the algebra Axioms 01 00201 i536, generated by Axioms 01 00201 i537, Axioms 01 00201 i538and Axioms 01 00201 i539subject to the following relations

Axioms 01 00201 i541

and Axioms 01 00201 i539is central unitary. The embedding of Axioms 01 00201 i536into Axioms 01 00201 i392is given by Axioms 01 00201 i542, Axioms 01 00201 i543and Axioms 01 00201 i544

Proof. Clearly Axioms 01 00201 i533, Axioms 01 00201 i532, Axioms 01 00201 i545and Axioms 01 00201 i546are coinvariant elements of Axioms 01 00201 i392. Apply the coaction to the basis (7) to obtain

Axioms 01 00201 i547

These elements are coinvariant, provided Axioms 01 00201 i548. Hence every coinvariant element is a polynomial in Axioms 01 00201 i533, Axioms 01 00201 i532, Axioms 01 00201 i545and Axioms 01 00201 i546. Equations (20) are now easily derived from Equations (6) and (18).

The algebra Axioms 01 00201 i536is a right Axioms 01 00201 i524-comodule coalgebra with coaction defined as

Axioms 01 00201 i550

Note in passing that the second and third relations in Equations (20) tell us that the grade of Axioms 01 00201 i539must be double the grade of Axioms 01 00201 i551since Axioms 01 00201 i552and Axioms 01 00201 i553have degree zero, and so

Axioms 01 00201 i555

Proposition 4.3 The algebra Axioms 01 00201 i556of invariant elements under the coaction Axioms 01 00201 i557is isomorphic to the Axioms 01 00201 i558.

Proof. We aim to show that the Axioms 01 00201 i317-subalgebra of Axioms 01 00201 i559of elements which are invariant under the coaction is generated by Axioms 01 00201 i560, Axioms 01 00201 i561and Axioms 01 00201 i562. The isomorphism of Axioms 01 00201 i563with Axioms 01 00201 i564is then obtained by using the embedding of Axioms 01 00201 i536in Axioms 01 00201 i392described in Proposition 4.2, i.e., Axioms 01 00201 i565, Axioms 01 00201 i566and Axioms 01 00201 i567.

The algebra Axioms 01 00201 i568is spanned by elements of the type Axioms 01 00201 i569, Axioms 01 00201 i570, where Axioms 01 00201 i571and Axioms 01 00201 i572. Applying the coaction Axioms 01 00201 i573to these basis elements gives Axioms 01 00201 i574Hence Axioms 01 00201 i569is Axioms 01 00201 i573-invariant if and only if Axioms 01 00201 i575. If Axioms 01 00201 i576is even, then Axioms 01 00201 i577is even and

Axioms 01 00201 i578

If Axioms 01 00201 i576is odd, then so is Axioms 01 00201 i577and

Axioms 01 00201 i579

The case of Axioms 01 00201 i580is dealt with similarly, thus proving that all coinvariants of Axioms 01 00201 i573are polynomials in Axioms 01 00201 i581, Axioms 01 00201 i582, Axioms 01 00201 i583and their Axioms 01 00201 i336-conjugates.

The main result of this section is contained in the following theorem.

Theorem 4.4 Axioms 01 00201 i584is a non-cleft principal Axioms 01 00201 i585-comodule algebra over Axioms 01 00201 i586via the coaction Axioms 01 00201 i573.

Proof. To prove that Axioms 01 00201 i584is a principal Axioms 01 00201 i585-comodule algebra over Axioms 01 00201 i586we employ Proposition 2.16 and construct a strong connection form as follows.

Define Axioms 01 00201 i587recursively as follows.

Axioms 01 00201 i588
Axioms 01 00201 i590
Axioms 01 00201 i592

where Axioms 01 00201 i429and, for all Axioms 01 00201 i593, the deformed or q-binomial coefficients Axioms 01 00201 i594are defined by the following polynomial equality in indeterminate Axioms 01 00201 i595

Axioms 01 00201 i597

The map Axioms 01 00201 i234has been designed such that normalisation property, Equation (1a), is automatically satisfied. To check Equation (1b) for Axioms 01 00201 i234given by Equation (22b) and (22c) takes a bit more work. We use proof by induction, but first have to derive an identity to assist with the calculation. Set Axioms 01 00201 i598, Axioms 01 00201 i599in Equation (23) to arrive at

Axioms 01 00201 i600

which, using Equations (20), simplifies to

Axioms 01 00201 i603

Now to start the induction process we consider the case Axioms 01 00201 i604. By Equation (24) Axioms 01 00201 i605providing the basis. Next, we assume that the relation holds for Axioms 01 00201 i606, that is Axioms 01 00201 i607, and consider the case Axioms 01 00201 i608,

Axioms 01 00201 i610

applying the multiplication map to both sides and using the induction hypothesis,

Axioms 01 00201 i612

showing Equation (1b) holds for all Axioms 01 00201 i613, where Axioms 01 00201 i614. To show this property holds for each Axioms 01 00201 i615we adopt the same strategy; this is omitted from the proof as it does not provide further insight, instead repetition of similar arguments.

Equation (1c): this is again proven by induction. Applying Axioms 01 00201 i616to Axioms 01 00201 i617gives

Axioms 01 00201 i618

This shows that Equation (1c) holds for Axioms 01 00201 i234given by Equation (22b) when Axioms 01 00201 i604. We now assume the property holds for Axioms 01 00201 i622, hence Axioms 01 00201 i623, and consider the case Axioms 01 00201 i606.

Axioms 01 00201 i624

hence Equation (1c) is satisfied for all Axioms 01 00201 i626where Axioms 01 00201 i403. The case for Axioms 01 00201 i627is proved in a similar manner, as is Equation (1d). Again, the details are omitted as the process is identical. This completes the proof that Axioms 01 00201 i628is a strong connection form, hence Axioms 01 00201 i629is a principal comodule algebra.

Following the discussion of Section 3.1, to determine whether the constructed comodule algebra is cleft we need to identify invertible elements in Axioms 01 00201 i630. Since

Axioms 01 00201 i632

and the only invertible elements in the algebraic tensor Axioms 01 00201 i633are scalar multiples of Axioms 01 00201 i634for Axioms 01 00201 i403, we can conclude that the only invertible elements in Axioms 01 00201 i636are the elements of the form Axioms 01 00201 i634. These elements correspond to the elements Axioms 01 00201 i637in Axioms 01 00201 i638, which in turn correspond to Axioms 01 00201 i639in Axioms 01 00201 i640.

Suppose Axioms 01 00201 i251is the cleaving map; to ensure the map is convolution invertible we are forced to put Axioms 01 00201 i641. Since Axioms 01 00201 i642has degree Axioms 01 00201 i485in Axioms 01 00201 i643and Axioms 01 00201 i644has degree Axioms 01 00201 i645in Axioms 01 00201 i640, the map Axioms 01 00201 i255fails to preserve the degrees, hence it is not colinear. Therefore, Axioms 01 00201 i646is a non-cleft principal comodule algebra.

4.2.2. Almost Freeness of the Coaction Axioms 01 00201 i647

At the classical limit, Axioms 01 00201 i648, the algebras Axioms 01 00201 i649represent singular manifolds or orbifolds. It is known that every orbifold can be obtained as a quotient of a manifold by an almost free action. The latter means that the action has finite (rather than trivial as in the free case) stabiliser groups. As explained in Section 2, on the algebraic level, freeness is encoded in the bijectivity of the canonical map Axioms 01 00201 i650, or, more precisely, in the surjectivity of the lifted canonical map Axioms 01 00201 i651(Equation (2)). The surjectivity of Axioms 01 00201 i651means the triviality of the cokernel of Axioms 01 00201 i651, thus the size of the cokernel of Axioms 01 00201 i651can be treated as a measure of the size of the stabiliser groups. This leads to the following notion proposed in [4].

Definition 4.5 Let Axioms 01 00201 i191be a Hopf algebra and let Axioms 01 00201 i195be a right Axioms 01 00201 i191-comodule algebra with coaction Axioms 01 00201 i652. We say that the coaction is almost free if the cokernel of the (lifted) canonical map

Axioms 01 00201 i653

is finitely generated as a left Axioms 01 00201 i195-module.

Although the coaction Axioms 01 00201 i654defined in the preceding section is free, at the classical limit Axioms 01 00201 i655 Axioms 01 00201 i656represents a singular manifold or an orbifold. On the other hand, at the same limit, Axioms 01 00201 i008corresponds to a genuine manifold, one of the Seifert three-dimensional non-orientable manifolds; see [17]. It is therefore natural to ask, whether the coaction Axioms 01 00201 i657of Axioms 01 00201 i658on Axioms 01 00201 i008which has Axioms 01 00201 i659as fixed points is almost free in the sense of Definition 4.5.

Proposition 4.6 The coaction Axioms 01 00201 i657is almost free.

Proof. Denote by Axioms 01 00201 i661, the Axioms 01 00201 i662-algebra embedding described in Proposition 4.2. One easily checks that the following diagram

Axioms 01 00201 i664

where Axioms 01 00201 i665, is commutative. The principality or freeness of Axioms 01 00201 i654proven in Theorem 4.4 implies that Axioms 01 00201 i666, Axioms 01 00201 i667, where Axioms 01 00201 i668is the (lifted) canonical map corresponding to coaction Axioms 01 00201 i657. This means that Axioms 01 00201 i669. Therefore, there is a short exact sequence of left Axioms 01 00201 i008-modules

Axioms 01 00201 i671

The left Axioms 01 00201 i008-module Axioms 01 00201 i672is finitely generated, hence so is Axioms 01 00201 i673.

4.2.3. Associated Modules or Sections of Line Bundles

One can construct modules associated to the principal comodule algebra Axioms 01 00201 i674following the procedure outlined at the end of Section 2.2; see Definition 2.21.

Every one-dimensional comodule of Axioms 01 00201 i675is determined by the grading of a basis element of Axioms 01 00201 i246, say Axioms 01 00201 i520. More precisely, for any integer Axioms 01 00201 i676, Axioms 01 00201 i246is a left Axioms 01 00201 i524-comodule with the coaction

Axioms 01 00201 i677

Identifying Axioms 01 00201 i678with Axioms 01 00201 i674we thus obtain, for each coaction Axioms 01 00201 i679

Axioms 01 00201 i681

In other words, Axioms 01 00201 i682consists of all elements of Axioms 01 00201 i674of Axioms 01 00201 i323-degree Axioms 01 00201 i676. In particular Axioms 01 00201 i683. Each of the Axioms 01 00201 i682is a finitely generated projective left Axioms 01 00201 i684-module, i.e., it represents the module of sections of the non-commutative line bundle over Axioms 01 00201 i685. The idempotent matrix Axioms 01 00201 i686defining Axioms 01 00201 i682can be computed explicitly from a strong connection form Axioms 01 00201 i234(see Equations (22) in the proof of Theorem 4.4) following the procedure described in [11]. Write Axioms 01 00201 i687. Then

Axioms 01 00201 i688

For example, for Axioms 01 00201 i689and Axioms 01 00201 i604, using Equations (22b) and (22a) as well as redistributing numerical coefficients we obtain

Axioms 01 00201 i691

Although the matrix Axioms 01 00201 i692is not hermitian, the left-upper Axioms 01 00201 i693block is hermitian. On the other hand, once Axioms 01 00201 i694is completed to the Axioms 01 00201 i222-algebra Axioms 01 00201 i695of continuous functions on Axioms 01 00201 i696(and then identified with the suitable pullback of two algebras of continuous functions over the quantum real projective space; see [3]), then a hermitian projector can be produced out of Axioms 01 00201 i692by using the Kaplansky formula; see [18, page 88].

The traces of tensor powers of each of the Axioms 01 00201 i686make up a cycle in the cyclic complex of Axioms 01 00201 i697, whose corresponding class in the cyclic homology Axioms 01 00201 i698is known as the Chern character of Axioms 01 00201 i682. Again, as an illustration of the usage of an explicit form of a strong connection form, we compute the traces of Axioms 01 00201 i686for general Axioms 01 00201 i006.

Lemma 4.7 The zero-component of the Chern character of Axioms 01 00201 i682is the class of the polynomial Axioms 01 00201 i699in generator Axioms 01 00201 i382of Axioms 01 00201 i697, given by the following recursive formula. First, Axioms 01 00201 i700, and then, for all positive Axioms 01 00201 i676,

Axioms 01 00201 i702
Axioms 01 00201 i704

Proof. We will prove the formula (27a) as (27b) is proven by similar arguments. Recall that Axioms 01 00201 i705. By normalisation (22a) of the strong connection Axioms 01 00201 i628, obviously Axioms 01 00201 i706. In view of Equation (22b) we obtain the following recursive formula

Axioms 01 00201 i708

In principle, Axioms 01 00201 i709could be a polynomial in Axioms 01 00201 i710and Axioms 01 00201 i711. However, the third of Equations (20) together with Equation (24) and identification of Axioms 01 00201 i412as Axioms 01 00201 i583yield

Axioms 01 00201 i714

that is a polynomial in Axioms 01 00201 i412only. As commuting Axioms 01 00201 i070and Axioms 01 00201 i072through a polynomial in Axioms 01 00201 i412in Equation (28) will produce a polynomial in Axioms 01 00201 i412again, we conclude that each of the Axioms 01 00201 i709is a polynomial in Axioms 01 00201 i412. The second of Equations (20), the centrality of Axioms 01 00201 i644and the identification of Axioms 01 00201 i412as Axioms 01 00201 i715imply that

Axioms 01 00201 i716

and in view of Equations (28) and (29) yield Equation (27a).

4.3. The Positive Case Axioms 01 00201 i717

4.3.1. The Principal Axioms 01 00201 i585-Comodule Algebra over Axioms 01 00201 i417

In the same light as the negative case we aim to construct quantum principal bundles with base spaces Axioms 01 00201 i586, and proceed by viewing Axioms 01 00201 i418as a right Axioms 01 00201 i718-comodule algebra, where Axioms 01 00201 i718is a Hopf-algebra of a finite cyclic group. The aim is to construct the total space Axioms 01 00201 i719of the bundle over Axioms 01 00201 i720as the coinvariant subalgebra of Axioms 01 00201 i008. Axioms 01 00201 i721must contain generators Axioms 01 00201 i722and Axioms 01 00201 i723of Axioms 01 00201 i724. Suppose Axioms 01 00201 i725and Axioms 01 00201 i726is a coaction. We require Axioms 01 00201 i727to be compatible with the algebraic relations and to give zero Axioms 01 00201 i728-degree to Axioms 01 00201 i722and Axioms 01 00201 i723are zero. These requirements yield

Axioms 01 00201 i729

Bearing in mind that Axioms 01 00201 i006is odd, the simplest solution to these requirements is provided by Axioms 01 00201 i730, Axioms 01 00201 i731, Axioms 01 00201 i732, Axioms 01 00201 i733. This yields the coaction

Axioms 01 00201 i734

where Axioms 01 00201 i735( Axioms 01 00201 i736) is the unitary generator of Axioms 01 00201 i737. Axioms 01 00201 i738is extended to the whole of Axioms 01 00201 i008so that Axioms 01 00201 i738is an algebra map, making Axioms 01 00201 i008a right Axioms 01 00201 i737-comodule algebra.

Proposition 4.8 The fixed point subalgebra of the coaction Axioms 01 00201 i738is isomorphic to the Axioms 01 00201 i662-algebra Axioms 01 00201 i721generated by Axioms 01 00201 i739and central unitary Axioms 01 00201 i740subject to the following relations:

Axioms 01 00201 i741
Axioms 01 00201 i743

The isomorphism between Axioms 01 00201 i721and the coinvariant subalgebra of Axioms 01 00201 i008is given by Axioms 01 00201 i744, Axioms 01 00201 i745and Axioms 01 00201 i746.

Proof. Clearly Axioms 01 00201 i747, Axioms 01 00201 i748, Axioms 01 00201 i749and Axioms 01 00201 i750are coinvariant elements of Axioms 01 00201 i008. Apply the coaction Axioms 01 00201 i738to the basis (7) to obtain

Axioms 01 00201 i751

These elements are coinvariant, provided Axioms 01 00201 i752in the first case or Axioms 01 00201 i753in the second. Since Axioms 01 00201 i006is odd, Axioms 01 00201 i243must be even and then Axioms 01 00201 i548, hence the invariant elements must be of the form

Axioms 01 00201 i754

as required. Equations (30) are now easily derived from Equations (6) and (18).

The algebra Axioms 01 00201 i755is a right Axioms 01 00201 i524-comodule with coaction defined as,

Axioms 01 00201 i757

The first relation in Equations (30a) bears no information on the possible gradings of the generators of Axioms 01 00201 i755, however the second relation in Equations (30a) tells us that the grade of Axioms 01 00201 i758must be the same as that of Axioms 01 00201 i759since,

Axioms 01 00201 i760

hence,

Axioms 01 00201 i761

This is consistent with Equations (30b) since the left hand sides, Axioms 01 00201 i762and Axioms 01 00201 i763, have degree zero, as do the right had sides,

Axioms 01 00201 i764

The coaction Axioms 01 00201 i765is defined setting the grades of Axioms 01 00201 i766and Axioms 01 00201 i767as 1, and putting the grade of Axioms 01 00201 i759as Axioms 01 00201 i768to ensure the coaction is compatible with the relations of the algebra Axioms 01 00201 i755.

Proposition 4.9 The right Axioms 01 00201 i524-comodule algebra Axioms 01 00201 i755has Axioms 01 00201 i769as its subalgebra of coinvariant elements under the coaction Axioms 01 00201 i765.

Proof. The fixed points of the algebra Axioms 01 00201 i755under the coaction Axioms 01 00201 i765are found using the same method as in the odd Axioms 01 00201 i378case. A basis for the algebra Axioms 01 00201 i755is given by Axioms 01 00201 i770, Axioms 01 00201 i771, where Axioms 01 00201 i772and Axioms 01 00201 i773.

Applying the coaction Axioms 01 00201 i765to the first of these basis elements gives,

Axioms 01 00201 i775

Hence the invariance of Axioms 01 00201 i776is equivalent to Axioms 01 00201 i777. Simple substitution and re-arranging gives,

Axioms 01 00201 i778

i.e., Axioms 01 00201 i776is a polynomial in Axioms 01 00201 i779and Axioms 01 00201 i780. Repeating the process for the second type of basis element gives the Axioms 01 00201 i336-conjugates of Axioms 01 00201 i779and Axioms 01 00201 i780. Using Proposition 4.8 we can see that Axioms 01 00201 i781and Axioms 01 00201 i782.

In contrast to the odd Axioms 01 00201 i783case, although Axioms 01 00201 i784is a principal comodule algebra it yields trivial principal bundle over Axioms 01 00201 i785.

Proposition 4.10 The right Axioms 01 00201 i786-comodule algebra Axioms 01 00201 i784is trivial.

Proof. The cleaving map is given by,

Axioms 01 00201 i787

which is an algebra map since Axioms 01 00201 i788is central unitary in Axioms 01 00201 i789, hence must be convolution invertible. Also, Axioms 01 00201 i790is a right Axioms 01 00201 i585-comodule map since,

Axioms 01 00201 i791

completing the proof.

Since Axioms 01 00201 i792is a trivial principal comodule algebra, all associated Axioms 01 00201 i417-modules are free.

4.3.2. Almost Freeness of the Coaction Axioms 01 00201 i793

As was the case for Axioms 01 00201 i656, the principality of Axioms 01 00201 i721can be used to determine that the Axioms 01 00201 i658-coaction Axioms 01 00201 i793on Axioms 01 00201 i008that defines Axioms 01 00201 i794is almost free.

Proposition 4.11 The coaction Axioms 01 00201 i793is almost free.

Proof. Denote by Axioms 01 00201 i795the Axioms 01 00201 i662-algebra embedding described in Proposition 4.8. One easily checks that the following diagram

Axioms 01 00201 i797

where Axioms 01 00201 i798is commutative. By the arguments analogous to those in the proof of Proposition 4.6 one concludes that there is a short exact sequence of left Axioms 01 00201 i008-modules

Axioms 01 00201 i800

where Axioms 01 00201 i668is the lifted canonical map corresponding to coaction Axioms 01 00201 i793. The left Axioms 01 00201 i008-module Axioms 01 00201 i801is finitely generated, hence so is Axioms 01 00201 i673.

5 Conclusions

In this paper we discussed the principality of the Axioms 01 00201 i524-coactions on the coordinate algebra of the quantum Seifert manifold Axioms 01 00201 i392weighted by coprime integers Axioms 01 00201 i378and Axioms 01 00201 i006. We concluded that the coaction is principal if and only if Axioms 01 00201 i802, which corresponds to the case of a Axioms 01 00201 i315-bundle over the quantum real projective plane. In all other cases the coactions are almost free. We identified subalgebras of Axioms 01 00201 i803which admit principal Axioms 01 00201 i524-coactions, whose invariants are isomorphic to coordinate algebras Axioms 01 00201 i804of quantum real weighted projective spaces. The structure of these subalgebras depends on the parity of Axioms 01 00201 i378. For the odd Axioms 01 00201 i378case, the constructed principal comodule algebra Axioms 01 00201 i674is non-trivial, while for the even case, the corresponding principal comodule algebra Axioms 01 00201 i755turns out to be trivial. The triviality of Axioms 01 00201 i805is a disappointment. Whether a different nontrivial principal Axioms 01 00201 i524-comodule algebra over Axioms 01 00201 i806can be constructed or whether such a possibility is ruled out by deeper geometric, topological or algebraic reasons remains to be seen.

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