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The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal

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

In two recent papers [

In this paper we focus on two classes of algebras

We begin in

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.

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.

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 [

For each

The local triviality condition is satisfied if for each

The map

Let

With an eye on algebraic formulation of freeness, the

“

Since

The sets

The equivalence relation is the same as saying

which maps elements in

As described above, the image of the principal map

Take

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

(a)

(b) the action

(c)

(d) the induced map

The first two properties tell us that principal bundles are bundles admitting a principal action of a group

We describe a principal bundle

Any vector bundle can be understood as a bundle associated to a principal bundle in the following way. Consider a

We can define

Conversely, given

To make the transition from algebraic formulation of principal and associated bundles to non-commutative setup more transparent, we assume that

Using the fact that

We have viewed the spaces of polynomial functions on

where

for all

for all

Since

Note that

In view of the definition of the coaction of

Thus the action of

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.

(a)

(b) the multiplication map

As indicated already in [

The following characterisation of principal comodule algebras [

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

Conversely, if

where

and unit

is a strong connection form. Hence a cleft comodule algebra is an example of a principal comodule algebra. The map

In particular, if

where

is surjective, and write

for the

where the coaction is denoted by the Sweedler notation

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

If

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

In the converse direction, define a left

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.

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

The coordinate algebra of the circle or the group

and thus it can be understood as the group algebra

Let

For any choice of

for

The algebra of coordinate functions on the quantum real weighted projective space is now defined as the subalgebra of

In this paper we consider two-dimensional quantum real weighted projective spaces,

The linear basis of

For a pair

and extended to the whole of

It turns out that the two dimensional quantum real projective spaces split into two cases depending on not wholly the parameter

For

The embedding of generators of

Up to equivalence

All other representations are infinite dimensional, labelled by

where

The

For

The embedding of generators of

Similarly to the odd

All other representations are infinite dimensional, labelled by

where

The

The general aim of this paper is to construct quantum principal bundles with base spaces given by

In the converse direction, we aim to show that the canonical map is not an isomorphism by showing that the image does not contain

noting that all powers are non-negative. Hence a basis for

where

or

We see that to obtain identity in the first leg we require the powers of

and

hence no possible terms. A similar calculation for the three other cases shows that

and

Note that

In all possibilities

and

Hence

Also

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

This concludes that

Theorem 4.1 tells us that if we use

Take the group Hopf

Note that the

The next stage of the process is to find the coinvariant elements of

These elements are coinvariant, provided

The algebra

Note in passing that the second and third relations in Equations (20) tell us that the grade of

The algebra

If

The case of

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

Define

where

The map

which, using Equations (20), simplifies to

Now to start the induction process we consider the case

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

showing Equation (1b) holds for all

Equation (1c): this is again proven by induction. Applying

This shows that Equation (1c) holds for

hence Equation (1c) is satisfied for all

Following the discussion of

and the only invertible elements in the algebraic tensor

Suppose

At the classical limit,

Although the coaction

where

The left

One can construct modules associated to the principal comodule algebra

Every one-dimensional comodule of

Identifying

In other words,

For example, for

Although the matrix

The traces of tensor powers of each of the

In principle,

that is a polynomial in

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

In the same light as the negative case we aim to construct quantum principal bundles with base spaces

Bearing in mind that

where

These elements are coinvariant, provided

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

The algebra

The first relation in Equations (30a) bears no information on the possible gradings of the generators of

hence,

This is consistent with Equations (30b) since the left hand sides,

The coaction

Applying the coaction

Hence the invariance of

In contrast to the odd

which is an algebra map since

completing the proof.

Since

As was the case for

where

where

In this paper we discussed the principality of the