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Axioms 2012, 1(2), 201-225; doi:10.3390/axioms1020201
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.
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 -grading of this coordinate algebra. A typical -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  or prolonged quantum spheres ). Often this grading is strong, meaning that the product of -graded parts is equal to the -part of the total algebra. In geometric terms this reflects the freeness of the circle action.
In two recent papers [3,4] 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 ( a positive integer) and ( an odd positive integer) identified in  as fixed points of weighted circle actions on the coordinate algebra of a non-orientable quantum Seifert manifold described in . Our aim is to construct quantum -principal bundles over the corresponding quantum spaces and describe associated line bundles. Recently, the importance of such bundles in non-commutative geometry was once again brought to the fore in , where the non-commutative Thom construction was outlined. As a further consequence of the principality of -coactions we also deduce that can be understood as quotients of by almost free -actions.
We begin in Section 2 by reviewing elements of algebraic approach to classical and quantum bundles. We then proceed to describe algebras in Section 3. Section 4 contains main results including construction of principal comodule algebras over . We observe that constructions albeit very similar in each case yield significantly different results. The principal comodule algebra over is non-trivial while that over turns 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 remains 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 .
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
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
(a) is a bundle and is a topological group acting continuously on with action , ;
(b) the action is principal;
(c) such that ;
(d) the induced map is a homeomorphism.
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 .
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 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:
(a) is a Hopf–Galois extension;
(b) the multiplication map , , splits as a left -module and right -comodule map (the equivariant projectivity condition).
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 , then the existence of the Haar measure together with the results of  mean that condition (a) in Definition 2.15 implies condition (b) (i.e., the freeness of the coaction implies its principality).
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 , 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 , is a principal -comodule algebra. Explicitly, a strong connection form is
where the coaction is denoted by the Sweedler notation ; see .
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 , the is 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
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 -Gradings.
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
3.2. The and Coordinate Algebras
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  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.,
3.3. The 2D Quantum Real Projective Space
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 . We describe these cases presently.
3.3.1. The Odd or Negative Case
For , is a polynomial -algebra generated by , , which satisfy the relations:
The embedding of generators of into or the isomorphism of with the coinvariants of is provided by
Up to equivalence has the following irreducible -representations. There is a family of one-dimensional representations labelled by and given by
All other representations are infinite dimensional, labelled by , and given by
where , , is an orthonormal basis for the representation space .
The -algebra of continuous functions on , obtained as the completion of these bounded representations, can be identified with the pullback of -copies of the quantum real projective plane introduced in .
3.3.2. The Even or Positive Case
For and hence odd, is a polynomial -algebra generated by , which satisfy the relations:
The embedding of generators of into or the isomorphism of with the coinvariants of is provided by
Similarly to the odd case, there is a family of one-dimensional representations of labelled by and given by
All other representations are infinite dimensional, labelled by , and given by
where , is an orthonormal basis for the representation space .
The -algebra of continuous functions on , obtained as the completion of these bounded representations, can be identified with the pullback of -copies of the quantum disk introduced in . Furthermore, can also be understood as the quantum double suspension of points 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 and fibre structures given by the circle Hopf algebra . The question arises as to which quantum space (i.e., a -comodule algebra with coinvariants isomorphic to ) we should consider as the total space within this construction. We look first at the coactions of on that define , i.e., at the comodule algebras .
4.1. The (Non-)Principality of
Theorem 4.1 is a principal comodule algebra if and only if .
Proof. As explained in  is a prolongation of the -comodule algebra . The latter is a principal comodule algebra (over the quantum real projective plane ) 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:
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,
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,
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:
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
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 , 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.
4.2. The Negative Case
4.2.1. The Principal -Comodule Algebra over
Take the group Hopf -algebra which is generated by unitary grouplike element and satisfies the relation . The algebra is a right -comodule -algebra with coaction
Note that the -degree of the generator is determined by the degree of : the relation and that the coaction must be compatible with all relations imply that . Since has degree zero, must also have degree zero.
The next stage of the process is to find the coinvariant elements of given the coaction defined above.
Proposition 4.2 The fixed point subalgebra of the above coaction is isomorphic to the algebra , generated by , and subject to the following relations
and is central unitary. The embedding of into is given by , and
Proof. Clearly , , and are coinvariant elements of . Apply the coaction to the basis (7) to obtain
These elements are coinvariant, provided . Hence every coinvariant element is a polynomial in , , and . Equations (20) are now easily derived from Equations (6) and (18).
The algebra is a right -comodule coalgebra with coaction defined as
Note in passing that the second and third relations in Equations (20) tell us that the grade of must be double the grade of since and have degree zero, and so
Proposition 4.3 The algebra of invariant elements under the coaction is isomorphic to the .
Proof. We aim to show that the -subalgebra of of elements which are invariant under the coaction is generated by , and . The isomorphism of with is then obtained by using the embedding of in described in Proposition 4.2, i.e., , and .
The algebra is spanned by elements of the type , , where and . Applying the coaction to these basis elements gives Hence is -invariant if and only if . If is even, then is even and
If is odd, then so is and
The case of is dealt with similarly, thus proving that all coinvariants of are polynomials in , , and their -conjugates.
The main result of this section is contained in the following theorem.
Theorem 4.4 is a non-cleft principal -comodule algebra over via the coaction .
Proof. To prove that is a principal -comodule algebra over we employ Proposition 2.16 and construct a strong connection form as follows.
Define recursively as follows.
where and, for all , the deformed or q-binomial coefficients are defined by the following polynomial equality in indeterminate
The map has been designed such that normalisation property, Equation (1a), is automatically satisfied. To check Equation (1b) for given 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 , in Equation (23) to arrive at
which, using Equations (20), simplifies to
Now to start the induction process we consider the case . By Equation (24) providing the basis. Next, we assume that the relation holds for , that is , and consider the case ,
applying the multiplication map to both sides and using the induction hypothesis,
showing Equation (1b) holds for all , where . To show this property holds for each we 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 to gives
This shows that Equation (1c) holds for given by Equation (22b) when . We now assume the property holds for , hence , and consider the case .
hence Equation (1c) is satisfied for all where . The case for is proved in a similar manner, as is Equation (1d). Again, the details are omitted as the process is identical. This completes the proof that is a strong connection form, hence is 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 . Since
and the only invertible elements in the algebraic tensor are scalar multiples of for , we can conclude that the only invertible elements in are the elements of the form . These elements correspond to the elements in , which in turn correspond to in .
Suppose is the cleaving map; to ensure the map is convolution invertible we are forced to put . Since has degree in and has degree in , the map fails to preserve the degrees, hence it is not colinear. Therefore, is a non-cleft principal comodule algebra.
4.2.2. Almost Freeness of the Coaction
At the classical limit, , the algebras represent 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 , or, more precisely, in the surjectivity of the lifted canonical map (Equation (2)). The surjectivity of means the triviality of the cokernel of , thus the size of the cokernel of can be treated as a measure of the size of the stabiliser groups. This leads to the following notion proposed in .
Definition 4.5 Let be a Hopf algebra and let be a right -comodule algebra with coaction . We say that the coaction is almost free if the cokernel of the (lifted) canonical map
is finitely generated as a left -module.
Although the coaction defined in the preceding section is free, at the classical limit represents a singular manifold or an orbifold. On the other hand, at the same limit, corresponds to a genuine manifold, one of the Seifert three-dimensional non-orientable manifolds; see . It is therefore natural to ask, whether the coaction of on which has as fixed points is almost free in the sense of Definition 4.5.
Proposition 4.6 The coaction is almost free.
Proof. Denote by , the -algebra embedding described in Proposition 4.2. One easily checks that the following diagram
where , is commutative. The principality or freeness of proven in Theorem 4.4 implies that , , where is the (lifted) canonical map corresponding to coaction . This means that . Therefore, there is a short exact sequence of left -modules
The left -module is finitely generated, hence so is .
4.2.3. Associated Modules or Sections of Line Bundles
One can construct modules associated to the principal comodule algebra following the procedure outlined at the end of Section 2.2; see Definition 2.21.
Every one-dimensional comodule of is determined by the grading of a basis element of , say . More precisely, for any integer , is a left -comodule with the coaction
Identifying with we thus obtain, for each coaction
In other words, consists of all elements of of -degree . In particular . Each of the is a finitely generated projective left -module, i.e., it represents the module of sections of the non-commutative line bundle over . The idempotent matrix defining can be computed explicitly from a strong connection form (see Equations (22) in the proof of Theorem 4.4) following the procedure described in . Write . Then
For example, for and , using Equations (22b) and (22a) as well as redistributing numerical coefficients we obtain
Although the matrix is not hermitian, the left-upper block is hermitian. On the other hand, once is completed to the -algebra of continuous functions on (and then identified with the suitable pullback of two algebras of continuous functions over the quantum real projective space; see ), then a hermitian projector can be produced out of by using the Kaplansky formula; see [18, page 88].
The traces of tensor powers of each of the make up a cycle in the cyclic complex of , whose corresponding class in the cyclic homology is known as the Chern character of . Again, as an illustration of the usage of an explicit form of a strong connection form, we compute the traces of for general .
Lemma 4.7 The zero-component of the Chern character of is the class of the polynomial in generator of , given by the following recursive formula. First, , and then, for all positive ,
Proof. We will prove the formula (27a) as (27b) is proven by similar arguments. Recall that . By normalisation (22a) of the strong connection , obviously . In view of Equation (22b) we obtain the following recursive formula
In principle, could be a polynomial in and . However, the third of Equations (20) together with Equation (24) and identification of as yield
that is a polynomial in only. As commuting and through a polynomial in in Equation (28) will produce a polynomial in again, we conclude that each of the is a polynomial in . The second of Equations (20), the centrality of and the identification of as imply that
and in view of Equations (28) and (29) yield Equation (27a).
4.3. The Positive Case
4.3.1. The Principal -Comodule Algebra over
In the same light as the negative case we aim to construct quantum principal bundles with base spaces , and proceed by viewing as a right -comodule algebra, where is a Hopf-algebra of a finite cyclic group. The aim is to construct the total space of the bundle over as the coinvariant subalgebra of . must contain generators and of . Suppose and is a coaction. We require to be compatible with the algebraic relations and to give zero -degree to and are zero. These requirements yield
Bearing in mind that is odd, the simplest solution to these requirements is provided by , , , . This yields the coaction
where ( ) is the unitary generator of . is extended to the whole of so that is an algebra map, making a right -comodule algebra.
Proposition 4.8 The fixed point subalgebra of the coaction is isomorphic to the -algebra generated by and central unitary subject to the following relations:
The isomorphism between and the coinvariant subalgebra of is given by , and .
Proof. Clearly , , and are coinvariant elements of . Apply the coaction to the basis (7) to obtain
These elements are coinvariant, provided in the first case or in the second. Since is odd, must be even and then , hence the invariant elements must be of the form
as required. Equations (30) are now easily derived from Equations (6) and (18).
The algebra is a right -comodule with coaction defined as,
The first relation in Equations (30a) bears no information on the possible gradings of the generators of , however the second relation in Equations (30a) tells us that the grade of must be the same as that of since,
This is consistent with Equations (30b) since the left hand sides, and , have degree zero, as do the right had sides,
The coaction is defined setting the grades of and as 1, and putting the grade of as to ensure the coaction is compatible with the relations of the algebra .
Proposition 4.9 The right -comodule algebra has as its subalgebra of coinvariant elements under the coaction .
Proof. The fixed points of the algebra under the coaction are found using the same method as in the odd case. A basis for the algebra is given by , , where and .
Applying the coaction to the first of these basis elements gives,
Hence the invariance of is equivalent to . Simple substitution and re-arranging gives,
i.e., is a polynomial in and . Repeating the process for the second type of basis element gives the -conjugates of and . Using Proposition 4.8 we can see that and .
In contrast to the odd case, although is a principal comodule algebra it yields trivial principal bundle over .
Proposition 4.10 The right -comodule algebra is trivial.
Proof. The cleaving map is given by,
which is an algebra map since is central unitary in , hence must be convolution invertible. Also, is a right -comodule map since,
completing the proof.
Since is a trivial principal comodule algebra, all associated -modules are free.
4.3.2. Almost Freeness of the Coaction
As was the case for , the principality of can be used to determine that the -coaction on that defines is almost free.
Proposition 4.11 The coaction is almost free.
Proof. Denote by the -algebra embedding described in Proposition 4.8. One easily checks that the following diagram
where is 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 -modules
where is the lifted canonical map corresponding to coaction . The left -module is finitely generated, hence so is .
In this paper we discussed the principality of the -coactions on the coordinate algebra of the quantum Seifert manifold weighted by coprime integers and . We concluded that the coaction is principal if and only if , which corresponds to the case of a -bundle over the quantum real projective plane. In all other cases the coactions are almost free. We identified subalgebras of which admit principal -coactions, whose invariants are isomorphic to coordinate algebras of quantum real weighted projective spaces. The structure of these subalgebras depends on the parity of . For the odd case, the constructed principal comodule algebra is non-trivial, while for the even case, the corresponding principal comodule algebra turns out to be trivial. The triviality of is a disappointment. Whether a different nontrivial principal -comodule algebra over can 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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