Bundles over Quantum Real Weighted Projective Spaces

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 {\em negative} or {\em odd} class that generalises quantum real projective planes and the {\em positive} or {\em 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 two recent papers [5] and [4] circle actions on three-dimensional (and, briefly, higher dimensional) quantum spaces were revisited. Rather than assigning a uniform grade to each generator, separate 23 generators were given degree by pairwise coprime integers. The zero part of such a grading of 24 the coordinate algebra of the quantum odd-dimensional sphere corresponds to the quantum weighted 25 projective space, while the zero part of such a grading of the algebra of the prolonged even dimensional 26 quantum sphere leads to quantum real weighted projective spaces. 27 In this paper we focus on two classes of algebras O(RP 2 q (l; −)) (l a positive integer) and O(RP 2 q (l; +)) 28 (l an odd positive integer) identified in [4] as fixed points of weighted circle actions on the coordinate 29 algebra O(Σ 3 q ) of a non-orientable quantum Seifert manifold described in [8]. Our aim is to construct 30 quantum U (1)-principal bundles over the corresponding quantum spaces RP 2 q (l; ±) and describe 31 associated line bundles. Recently, the importance of such bundles in non-commutative geometry was 32 once again brought to the fore in [3], where the non-commutative Thom construction was outlined.

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As a further consequence of the principality of U (1)-coactions we also deduce that RP 2 q (l; ±) can be 34 understood as quotients of Σ 3 q by almost free S 1 -actions. 35 We begin in Section 2 by reviewing elements of algebraic approach to classical and quantum bundles. 36 We then proceed to describe algebras O(RP 2 q (l; ±)) in Section 3. Section 4 contains main results 37 including construction of principal comodule algebras over O(RP 2 q (l; ±)). We observe that constructions Since G acts on X we can define the quotient space X/G, The sets xG are called the orbits of the points x. They are defined as the set of elements in X to which x can be moved by the action of elements of G. The set of orbits of X under the action of G forms a partition of X, hence we can define the equivalence relation on X as, x ∼ y ⇐⇒ ∃g ∈ G such that xg = y.
The equivalence relation is the same as saying x and y are in the same orbit, i.e., xG = yG. Given any quotient space, then there is a canonical surjective map which maps elements in X to the their class of orbits. We define the pull-back along this map π to be the set X × Y X := {(x, y) ∈ X × X : π(x) = π(y)}.
As described above, the image of the principal map F G contains elements of X in the first leg and the 86 action of g ∈ G on x in the second leg. To put it another way, the image records elements of x ∈ X in 87 the first leg and all the elements in the same orbit as this x in the second leg. Hence we can identify the 88 image of the canonical map as the pull back along π, namely X × Y X. This is formally proved as a part 89 of the following proposition. 90 Proposition 2.6 G acts freely on X if and only if the map is bijective.

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Proof. First note that the map F G X is well-defined since the elements x and xg are in the same orbit hence 92 map to the same equivalence class under π. Using Proposition 2.5 we can deduce that the injectivity of 93 F G X is equivalent to the freeness of the action. Hence if we can show that F G X is surjective the proof is Definition 2.10 A vector bundle is a bundle (E, π, M ) where each fibre π −1 (m) is endowed with a 123 vector space structure such that addition and scalar multiplication are continous maps.

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Any vector bundle can be understood as a bundle associated to a principal bundle in the following way. Consider a G-principal bundle (X, π, Y, G) and let V be a representation space of G, i.e. a (topological) vector space with a (continuous) left G-action : G × V → V , (g, v) → g v. Then G acts from the right on X × V by (x, v) g := (xg, g −1 v), for all x ∈ X, v ∈ V and g ∈ G.
We can define E = (X × V )/G, and a surjective (continuous map) 125 and thus have a fibre bundle (E, π E , Y, V ). In the case where V is a vector space, we assume that G acts π E (s(y)) = y, i.e. a section is simply a section of the morphism π E . The set of sections of E is denoted by Γ(E).

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Proposition 2.12 Sections in a fibre bundle (E, π E , Y, V ) associated to a principal G-bundle X are in bijective correspondence with (continuous) maps f : X → V such that All such G-equivariant maps are denoted by Hom G (X, V ).
A (a)(x, g) = a(xg). This coaction is an algebra map due to the commutativity of the algebras of 143 functions involved. 144 We have viewed the spaces of functions on X and G, next we view the space of functions on Y, where π is the canonical surjection defined above. The map π * is injective, . Therefore, we can identify B with π * (B). Furthermore, a ∈ π * (B) if and only if a(xg) = a(x), for all x ∈ X, g ∈ G. This is the same as for all x ∈ X, g ∈ G, where 1 : G → C is the unit function 1(g) = 1 (the identity element of H). Thus we can identify B with the coinvariants of the coaction A : θ(a ⊗ B a )(x, y) = a(x)a (y), (with π(x) = π(y)).
Note that θ is well defined because π(x) = π(y). Proposition 2.6 immediately yields Proposition 2.13 The action of G on X is free if and only if F G * In view of the definition of the coaction of H on A, we can identify F G X * with the canonical map Thus the action of G on X is free if and only if this purely algebraic map is bijective. In the classical 148 geometry case we take A = O(X), H = O(G) and B = O(X/G), but in general there is no need to 149 restrict oneself to commutative algebras (of functions on topological spaces). In full generality this leads 150 to the following definition.

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Definition 2.14 (Hopf-Galois Extensions) Let H be a Hopf algebra and A a right H-comodule algebra with coaction given by A : is an isomorphism.
Here µ : A ⊗ A → A denotes the multiplication map, η : C → A is the unit map, ∆ : H → H ⊗ H is  Proof.
If a strong connection form ω exists, then the inverse of the canonical map can (see Definition 2.14 ) is the composite while the splitting of the multiplication map (see Definition 2.15 (b)) is given by Conversely, if B ⊆ A is a principal comodule algebra, then ω is the composite where s is the left B-linear right H-colinear splitting of the multiplication B ⊗ A → A.

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Example 2.17 Let A be a right H-comodule algebra. The space of C-linear maps Hom(H, A) is an algebra with the convolution product and unit η • ε. A is said to be cleft if there exists a right H-colinear map j : H → A that has an inverse in the convolution algebra Hom(H, A) and is normalised so that j(1) = 1. Writing j −1 for the convolution inverse of j, one easily observes that is a strong connection form. Hence a cleft comodule algebra is a an example of a principal comodule 174 algebra. The map j is called a cleaving map or a normalised total integral.
where ∆(h) = h (1) ⊗ h (2) is the Sweedler notation for the comultiplication. Next, assume that the lifted canonical map: can is surjective, and write for the C-linear map such that can( (h)) = 1 ⊗ h, for all h ∈ H. Then, by the Schneider theorem [15], A is a principal H-comodule algebra. Explicitly, a strong connection form is where the coaction is denoted by the Sweedler notation A (a) = a (0) ⊗ a (1) ; see [2].

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Having described non-commutative principal bundles, we can look at the associated vector bundles.

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First we look at the classical case and try to understand it purely algebraically. Start with a vector bundle If A is an H-comodule algebra, and B = A coH , then A H V is a left B-module with the action 187 b(a v) = ba v. In particular, in the case of a principal G-bundle X over Y = X/G, for any left One easily checks that the constructed map are mutual inverses. Γ is a projective left B-module, and if V is a finite dimensional vector space, then Γ is a finitely generated projective left B-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 K 0 -group of B. If A is a cleft principal comodule algebra, then every associated module is free, since A ∼ = B ⊗ H as a left B-module and right H-comodule, so that 3. Weighted circle actions on prolonged spheres.

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In this section we recall the definitions of algebras we study in the sequel.  The coordinate algebra of the circle or the group U (1), O(S 1 ) = O(U (1)) can be identified with the * -algebra C[u, u * ] of Laurent polynomials in a unitary variable u (unitary means u −1 = u * ). As a Hopf * -algebra C[u, u * ], is generated by the grouplike element u, i.e.
and thus it can be understood as the group algebra CZ. As a consequence of this interpretation of A 0 is the coinvariant subalgebra of A. Since C[u, u * ] is spanned by grouplike elements, any convolution invertible map j : C[u, u * ] → A must assign a unit (invertible element) of A to u n . Furthermore, colinear maps are simply the Z-degree preserving maps, where deg(u) = 1. Put together, convolution invertible colinear maps j : C[u, u * ] → A are in one-to-one correspondence with sequences (a n : n ∈ Z, a n is a unit in A, deg(a n ) = n).

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Let q be a real number, 0 < q < 1. The coordinate algebra O(S 2n q ) of the even-dimensional quantum sphere is the unital complex * -algebra with generators z 0 , z 1 , . . . , z n , subject to the following relations: is a right CZ 2 -comodule algebra and C[u, u * ] is a left CZ 2 -comodule algebra, hence one can consider the cotensor product algebra O(Σ 2n+1 ) has generators ζ 0 , ..., ζ n and a central unitary ξ which are related in the following way: For any choice of n + 1 pairwise coprime numbers l 0 , ..., l n one can define the coaction of the Hopf for i = 0, 1, ..., n. This coaction is then extended to the whole of O(Σ 2n+1 The algebra of coordinate functions on the quantum real weighted projective space is now defined as the subalgebra of O(Σ 2n+1 q ) containing all coinvariant elements, i.e., In this paper we consider two-dimensional quantum real weighted projective spaces, i.e. the algebras obtained from the coordinate algebra O(Σ 3 q ) which is generated by ζ 0 , ζ 1 and central unitary ξ such that For a pair k, l of coprime integers, the coaction k,l is given on generators by and extended to the whole of O(Σ 3 q ) so that the coaction is a * -algebra map. We denote the comodule . 210 It turns out that the two dimensional quantum real projective spaces split into two cases depending 211 not wholly on the parameter k, but instead whether k is either even or odd, and hence only cases k = 1 212 and k = 2 need be considered [4]. We describe these cases presently.
3.3.1 The odd or negative case.

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For k = 1, O(RP 2 q (l; −)) is a polynomial * -algebra generated by a, b, c − which satisfy the relations: Up to equivalence O(RP 2 q (l; −)) has the following irreducible * -representations. There is a family of one-dimensional representations of labelled by θ ∈ [0, 1) and given by All other representations are infinite dimensional, labelled by r = 1, . . . , l, and given by π r (a)e r n = q 2(ln+r) e r n , π r (b)e r n = q ln+r l m=1 1 − q 2(ln+r−m) 1/2 e r n−1 , π r (b)e r 0 = 0, (12a) where e r n , n ∈ N, is an orthonormal basis for the representation space H r ∼ = l 2 (N).

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The C * -algebra of continuous functions on RP 2 q (l; −), obtained as the completion of these bounded   For k = 2 and hence l odd, O(RP 2 q (l; +)) is a polynomial * -algebra generated by a, c + which satisfy the relations: The embedding of generators of O(RP 2 q (l; +)) into O(Σ 3 q ) or the isomorphism of O(RP 2 q (l; +)) with the coinvariants of O(Σ 3 q (2, l)) is provided by Similarly to the odd k case, there is a family of one-dimensional representations of O(RP 2 q (l; +)) labelled by θ ∈ [0, 1) and given by All other representations are infinite dimensional, labelled by r = 1, . . . , l, and given by π r (a)e r n = q 2(ln+r) e r n , π r (c + )e r n = l m=1 1 − q 2(ln+r−m) 1/2 e r n−1 , π r (c + )e r 0 = 0, where e r n , n ∈ N is an orthonormal basis for the representation space H r ∼ = l 2 (N).

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The C * -algebra C(RP 2 q (l; +)) of continuous functions on RP 2 q (l; +), obtained as the completion of   In the converse direction, we aim to show that the canonical map is not an isomorphism by showing that the image does not contain 1 ⊗ u, i.e. it cannot be surjective since we know 1 ⊗ u is in the codomain. We begin by identifying a basis for the algebra O(Σ 3 q ) ⊗ O(Σ 3 q ); observing the relations (6a) and (6b) it is clear that a basis for O(Σ 3 q (k, l)) is given by linear combinations of elements of the form, noting that all powers are non-negative. Hence a basis for O(Σ 3 q )⊗O(Σ 3 q ) is given by linear combinations of elements of the form b i ⊗ b j , where i, j ∈ {1, 2, 3, 4}. Applying the canonial map gives where means k,l for simplicity of notation. The next stage is to construct all possible elements in 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 ζ 0 and ζ * 0 to be equal. We now 237 construct all possible elements of the domain which map to 1 ⊗ u after applying the canonical map.  Case 2: use the second relation to obtain ζ * n 0 ζ n 0 (n > 0); this can be done in four ways and nk = 1 =⇒ n = 1 and k = 1.
Note that k = 1 is not a problem provided l is not equal to 1. This is reviewed at the next stage of the 241 proof. The same conclusion is reached in all four cases.
Hence 1 ⊗ u cannot be obtained as an element in the image in this case. Similar calculations for the 243 remaining possibilities show that either 1 ⊗ u is not in the image of the canonical map, or that if 1 ⊗ u is 244 in the image then k = l = 1.

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Case 3: finally, it seems possible that 1 ⊗ u, 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: The first possibility comes out as which implies there are no terms. The same conclusion can be reached for the remaining relations.    Take the group Hopf * -algebra H = CZ l which is generated by unitary grouplike element w and satisfies the relation w l = 1. The algebra O(Σ 3 q ) is a right CZ l -comodule * -algebra with coaction Note that the Z l -degree of the generator ξ is determined by the degree of ζ 1 : the relation ζ * 1 = ζ 1 ξ and 263 that the coaction must to compatible with all relations imply that deg(ζ * 1 ) = deg(ζ 1 ) + deg(ξ). Since ζ 1 264 has degree zero, ξ must also have degree zero.
The next stage of the process is to find the coinvariant elements of O(Σ 3 q ) given the coaction defined 266 above.

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Proposition 4.2 The fixed point subalgebra of the above coaction is isomorphic to the algebra O(Σ 3 q (l; −)), generated by x, y and z subject to the following relations and z is central unitary. The embedding of O(Σ 3 q (l; −)) into O(Σ 3 q ) is given by x → ζ l 0 , y → ζ 1 and 268 z → ξ.

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The algebra O(Σ 3 q (l; −)) is a right O(U (1))-comodule coalgebra with coaction defined as Note in passing that the second and third relations in (20) tell us that the grade of z must be double the grade of y * since xx * and x * x have degree zero, and so The algebra O(Σ 3 q (l; −)) is spanned by elements of the type x r y s z t , x * r y s z t , where r, s ∈ N and t ∈ Z. Applying the coaction ϕ to these basis elements gives x r y s z t → x r y s z t ⊗ u r+s−2t . Hence x r y s z t is ϕ-invariant if and only if 2t = r + s. If r is even, then s is even and x r y s z t = x r y s z (r+s)/2 = (x 2 z) r/2 (y 2 z) s/2 .
The case of x * r y s z t is dealt with similarly, thus proving that all coinvariants of ϕ are polynomials in x 2 z, 278 xyz, y 2 z and their * -conjugates. 279 The main result of this section is contained in the following theorem. Proof. To prove that O(Σ 3 q (l; −)) is a principal O(U (1))-comodule algebra over O(RP q (l; +)) we 283 employ Proposition 2.16 and construct a strong connection form as follows.
where n ∈ N and, for all s ∈ R, the deformed or q-binomial coefficients l m s are defined by the following polynomial equality in indeterminate t The map ω has been designed such that normalisation property (1a) is automatically satisfied. To check property (1b) from equations (22b) and (22c) take a bit more work. We use proof by induction, but first have to derive an identity to assist with the calculation. Set s = q −2 , t = −q −2 y * y in (23) to arrive at (1 + q −2(m−1) (−q −2 y * y)) − 1, which, using (20), simplifies to Now to start the induction process we consider the case n = 1. By (24) (µ • ω)(u) = 1 providing the basis. Next, we assume that the relation holds for n = N , that is (µ • ω)(u N ) = 1, and consider the case n = N + 1, applying the multiplication map to both sides and using the induction hypothesis, showing property (1b) holds for all u n ∈ O(U (1)), where n ∈ N. To show this property holds for each u * n = u −n we adopt the same strategy; this is omitted from the proof as it does not hold further insight, 286 instead repetition of similar arguments.
Property (1c): this is again proven by induction. Applying (id ⊗ ϕ) to ω(u) gives This shows that property (1c) holds for equation (22b) when n = 1. We now assume the property holds hence property (1c) is satisfied for all u n ∈ O(U (1)) where n ∈ N. The case for u * n is proved in a similar 292 manner, as is property (1d). Again, the details are omitted as the process is identical. This completes the 293 proof that ω is a strong connection form, hence O(Σ 3 q (l, −)) is a principal comodule algebra.

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Following the discussion of Section 3.1, to determine whether the constructed comodule algebra is cleft we need to identify invertible elements in O(Σ 3 q (l, −)). Since

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It is known that every orbifold can be obtained as a quotient of a manifold by an almost free action.

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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 307 can, or, more precisely, in the surjectivity of the lifted canonical map can (2). The surjectivity of can 308 means the triviality of the cokernel of can, thus the size of the cokernel of can can be treated as a 309 measure of the size of the stabiliser groups. This leads to the following notion proposed in [5].

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Definition 4.5 Let H be a Hopf algebra and let A be a right H-comodule algebra with coaction A : A → A ⊗ H. We say that the coaction is almost free if the cokernel of the (lifted) canonical map is finitely generated as a left A-module.

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Although the coaction ϕ defined in the preceding section is free, at the classical limit q → 1
, the * -algebra embedding described in Proposition 4.2. One easily checks that the following diagram where (−) l : u → u l , is commutative. The principality or freeness of ϕ proven in Theorem 4.4 implies that 1 ⊗ u ml ∈ Im(can), m ∈ Z, where can is the (lifted) canonical map corresponding to coaction 1,l . This means that O(Σ 3 q ) ⊗ C[u l , u −l ] ⊆ Im(can). Therefore, there is a short exact sequence of left O(Σ 3 q )-modules is finitely generated, hence so is 318 coker(can).  One can construct modules associated to the principal comodule algebra O(Σ 3 q (l, −)) following the 321 procedure outlined at the end of Section 2.2; see Definition 2.21.

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Every one-dimensional comodule of O(U (1)) = C[u, u * ] is determined by the grading of a basis element of C, say 1. More precisely, for any integer n, C is a left O(U (1))-comodule with the coaction Identifying O(Σ 3 q (l, −)) ⊗ C with O(Σ 3 q (l, −)) we thus obtain, for each coaction n In i ω(u n ) [1] i ⊗ ω(u n ) [2] i . Then For example, for l = 2 and n = 1, using (22b) and (22a) as well as redistributing numerical coefficients we obtain Although the matrix E [1] is not hermitian, the left-upper 2×2 block is hermitian. On the other hand, once Proof. We will prove the formula (27a) as (27b) is proven by similar arguments. Recall that c n = Tr E[n]. By normalisation (22a) of the strong connection ω, obviously c 0 = 1. In view of equation (22b) we obtain the following recursive formula In principle, c n could be a polynomial in a, b and c − . However, the third of equations (20) together with (24) and identification of a as y 2 z yield that is a polynomial in a only. As commuting x and y through a polynomial in a in formula (28) will produce a polynomial in a again, we conclude that each of the c n is a polynomial in a. The second of (20), the centrality of z and the identification of a as y 2 z imply that

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In the same light as the negative case we aim to construct quantum principal bundles with base spaces O(RP q (l; +)), and procced by viewing O(Σ 3 q ) as a right H -comodule algebra, where H is a Hopfalgebra of a finite cyclic group. The aim is to construct the total space O(Σ 3 q (l, +)) of the bundle over O(RP q (l; +)) as the coinvariant subalgebra of O(Σ 3 q ). O(Σ 3 q (l, +)) must contain generators ζ 2 1 ξ and ζ l 0 ξ of O(RP q (l; +)). Suppose H = CZ m and Φ : O(Σ 3 q ) → O(Σ 3 q ) ⊗ H is a coaction. We require Φ to be compatible with the algebraic relations and to give zero Z m -degree to ζ 2 1 ξ and ζ l 0 ξ are zero. These requirements yield 2 deg(ζ 1 ) + deg(ξ) = 0 mod m, l deg(ζ 0 ) + deg(ξ) = 0 mod m.
Bearing in mind that l is odd, the simplest solution to these requirements is provided by m = 2l, deg(ξ) = 0, deg(ζ 0 ) = 2, deg(ζ 1 ) = l. This yields the coaction where v (v 2l = 1) is the unitary generator of CZ 2l . Φ is extended to the whole of O(Σ 3 q ) so that Φ is an 335 algebra map, making O(Σ 3 q ) a right CZ 2l -comodule algebra.

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The algebra O(Σ 3 q (l, +)) is a right O(U (1))-comodule with coaction defined as, The first three relations (30a) bear no information on the possible gradings of the generators of O(Σ 3 q (l, +)), however the final relation of (30a) tells us that the grade of y * must have the same grade of z since, deg(y * ) = − deg(y ) = deg(y ) + 2 deg(z ), This is consistent with relations (30b) since the left hand sides, x x * and x * x , have degree zero, as do the right had sides, deg(y z ) = deg(y ) + deg(y * ) = deg(y ) + (− deg(y )) = 0 The coaction Ω is defined setting the grades of x and y as 1, and putting the grade of z as −1 to ensure 340 the coaction is compatible with the relations of the algebra O(Σ 3 q (l, +)).

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Proof. The fixed points of the algebra O(Σ 3 q (l, +)) under the coaction Ω are found using the same 344 method as in the odd k case. A basis for the algebra O(Σ 3 q (l, +)) is given by x r y s z t , x * r y s z t , where 345 r, s ∈ N and t ∈ Z.

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Applying the coaction Ω to the first of these basis elements gives, x r y s z t → x r y s z t ⊗ u r+s−t .
Hence the invariance of x r y s z t is equivalent to t = r + s. Simple substitution and re-arranging gives, x r y s z t = x r y s z r+s = (x z ) r (y z ) s , i.e. x r y s z t is a polynomial in x z and y z . Repeating the process for the second type of basis element 347 gives the * -conjugates of x z and y z . Using Proposition 4.8 we can see that a = ζ 2 1 ξ = y z and 348 c + = ζ l 0 ξ = x z .

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In contrast to the odd k case, although O(Σ 3 q (l, +)) is a principal comodule algebra it yields trivial 350 principal bundle over O(RP q (l; +)).

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Proposition 4.11 The coaction 2,l is almost free.  algebras O(RP 2 q (l; ±)) of quantum real weighted projective spaces. The structure of these subalgebras depends on the parity of k. For the odd k case, the constructed principal comodule algebra O(Σ 3 q (l, −)) is 369 non-trivial, while for the even case, the corresponding principal comodule algebra O(Σ 3 q (l, +)) turns out 370 to be trivial. The triviality of O(Σ 3 q (l, +)) is a disappointment. Whether a different nontrivial principal 371 O(U (1))-comodule algebra over O(RP 2 q (l; +)) can be constructed or whether such a possibility is ruled 372 out by deeper geometric, topological or algebraic reasons remains to be seen.