Nontrivial Deformation of a Trivial Bundle

The ${\rm SU}(2)$-prolongation of the Hopf fibration $S^3\to S^2$ is a trivializable principal ${\rm SU}(2)$-bundle. We present a noncommutative deformation of this bundle to a quantum principal ${\rm SU}_q(2)$-bundle that is not trivializable. On the other hand, we show that the ${\rm SU}_q(2)$-bundle is piecewise trivializable with respect to the closed covering of $S^2$ by two hemispheres intersecting at the equator.


Introduction and preliminaries
The goal of this paper is to show how a noncommutative deformation can turn a trivializable principal bundle into a nontrivializable quantum principal bundle. This is a peculiar phenomenon because noncommutative deformations usually preserve basic topological features of deformed objects, e.g. K-groups.
On the other hand, this paper exemplifies the general theory of piecewise trivial principal comodule algebras developed in [7,9]. Therefore we follow the notation, conventions and general setup employed therein. To make our exposition self-contained and easy to read, we often recall basic concepts and definitions.
Let π : X → M be a principal G-bundle over M , and G be a subgroup of G. A G -reduction of X → M is a subbundle X ⊆ X over M that is a principal G -bundle over M via the restriction of the G-action on X. Many important structures on manifolds can be formulated as reductions of their frame bundles. For instance, an orientation, a volume form and a metric on a manifold M correspond to reductions of the frame bundle F M to a GL + (n, R), SL(n, R) and O(n, R)-bundle, respectively. See [10] for more details.
An operation inverse to a reduction of a principal bundle is a prolongation of a principal bundle. Let π : X → M be a principal G -bundle over M , and let G be a subgroup of G. Define X × G G := (X × G)/∼, where (x, g) ∼ (xh, h −1 g), for all x ∈ X, g ∈ G and h ∈ G .
is a G-bundle called the G-prolongation of X, with the G-action given by [x, g] An interesting special case is when X = G and M = G/G , that is the homogenous bundle case. It is easy to see that G × G G → G/G is always a trivializable bundle. Indeed, the following G-equivariant bundle maps provide an explicit isomorphism and its inverse: A quantum-group version of the trivializability of G × G G → G/G can be easily checked mimicking the classical argument. In particular, the SU q (2)-prolongation of the standard quantum Hopf fibration is trivializable [5, p. 1104]. However, as the main result of this paper, we show that the SU q (2)-prolongation of the classical Hopf fibration is not trivializable.

Notation
We work over the field C of complex numbers. The unadorned tensor product stands for the tensor product over this field. The comultiplication, counit and the antipode of a Hopf algebra H are denoted by ∆, ε and S, respectively. Our standing assumption is that S is invertible. A right H-comodule algebra P is a unital associative algebra equipped with an H-coaction ∆ P : P → P ⊗ H that is an algebra homomorphism. For a comodule algebra P , we call P co H := {p ∈ P | ∆ P (p) = p ⊗ 1} the subalgebra of coaction-invariant elements in P . A left coaction on V is denoted by V ∆. For comultiplications and coactions, we often employ the Heynemann-Sweedler notation with the summation symbol suppressed:

Reductions and prolongations of principal comodule algebras
Definition 1 ( [4]). Let H be a Hopf algebra, P be a right H-comodule algebra and let B := P co H be the coaction-invariant subalgebra. The comodule algebra P is called principal iff: 2) there exists a left B-linear right H-colinear splitting of the multiplication map B ⊗P → P , 3) the antipode of H is bijective.
Here (1) is the Hopf-Galois (freeness) condition, (2) means equivariant projectivity of P , and (3) ensures a left-right symmetry of the definition (everything can be re-written for left comodule algebras).
A particular class of principal comodule algebras is distinguished by the existence of a cleaving map. A cleaving map is defined as a unital right H-colinear convolution-invertible map j : H → P . Comodule algebras admitting a cleaving map are called cleft. One can show that a cleaving map is automatically injective. However, in general, they are not algebra homomorphisms.
If j : H → P is a right H-colinear algebra homomorphism, then it is automatically convolution-invertible and unital. A cleft comodule algebra admitting a cleaving map that is an algebra homomorphism is called a smash product. All commutative smash products reduce to the tensor algebra P co H ⊗ H, so that smash products play the role of trivial bundles. Here a cleaving map is simply given by j(h) := 1 ⊗ h. A cleaving map defines a left action of H on P co H making it a left H-module algebra: Loosely speaking, J plays the role of the ideal of functions vanishing on a subgroup and I the ideal of functions vanishing on a subbundle. Thus H/J works as the algebra of the reducing subgroup, and P/I as the algebra of the reduced bundle. The coaction-invariant subalgebra P co H remains intact -the base space of a subbundle coincides with the base space of the bundle.
If M is a right comodule over a coalgebra C and N is a left C-comodule, then we define their cotensor product as In particular, for a principal H -comodule algebra P and a Hopf algebra epimorphism H π → H making H a left H -comodule in the obvious way, one proves that the cotensor product P 2 H H is a principal H-comodule algebra with the H-coaction defined by id ⊗ ∆. We call the principal comodule algebra P 2 H H the H-prolongation of P .
We recall now (cf. [7, Definition 3.8]) a quantum version of the notion of piecewise triviality of principal bundles (like local triviality, but with respect to closed subsets).

Definition 4.
An H-comodule algebra P is called piecewise trivial iff there exists a family {π i : P → P i } i∈{1,...,N } , N ∈ N \ {0}, of surjective H-colinear maps such that: 1) the restrictions π i | P co H : P co H → P co H i form a covering, 2) the P i 's are smash products (P i ∼ = P co H i H as H comodule algebras).
Assume also that the antipode of H is bijective. Then, as smash products are principal, it follows from [7, Theorem 3.3] that piecewise trivial comodule algebras are automatically principal. To emphasize this fact and stay in touch with the classical terminology, we frequently use the phrase "piecewise trivial principal comodule algebra". Note also that the consequence of principality of P is that {π i : P → P i } i∈{1,...,N } is a covering of P (see [9]).

Definition 5 ([9]
). Let {π i : P → P i } i∈{1,...,N } , N ∈ N \ {0}, be a covering by right H-colinear maps of a principal right H-comodule algebra P such that the restrictions π i | P co H : P co H → P co H i also form a covering. A piecewise trivialization of P with respect to the covering {π i : P → P i } i∈{1,...,N } is a family {j i : H → P i } i∈{1,...,N } of right H-colinear algebra homomorphisms (cleaving maps).
It is clear that a principal comodule algebra is piecewise trivial if and only if it admits a piecewise trivialization (see the preceding section).

The Peter-Weyl comodule algebra
The Peter-Weyl comodule algebra (see [1] and references therein) extends the notion of regular functions in the C * -algebra of a compact quantum group (linear combinations of matrix coefficients of the finite-dimensional corepresentations) to unital C * -algebras equipped with a compact quantum group action.
Definition 6 (cf. [11]). For a unital C * -algebra A and a compact quantum group (H, ∆), we say that an injective unital * -homomorphism δ : A → A ⊗ min H is a coaction if and only if Here ⊗ min denotes the spatial tensor product of C * -algebras and {·} cls stands for the closed linear span of a subset of a Banach space. We say that a compact quantum group acts on a unital C * -algebra if there is a coaction in the aforementioned sense. One shows that it is an O(H)-comodule algebra which is a dense * -subalgebra of A [11,13].
The Peter-Weyl comodule algebra of functions on a compact Hausdorff space with an action of a compact group is principal if and only if the action is free [1,2]. In other words, the Galois condition of Hopf-Galois theory holds if and only if we have a compact principal bundle.
2 The SU q (2)-prolongation of the classical Hopf f ibration To fix the notation, let us recall definitions of the Hopf algebras O(U(1)) and O(SU q (2)), and the Peter-Weyl comodule algebra P C(U(1)) (C(S 3 )) of functions on the classical sphere S 3 . For details on the latter algebra we refer the reader to [3].
Recall that the * -algebra O(U(1)) of polynomial functions on U(1) is generated by the unitary element u : U(1) x → x ∈ C, and can be equivalently defined as the algebra of Laurent polynomials in u subject to the relation u −1 = u * . The Hopf algebra structure is given by ∆(u) := u ⊗ u, ε(u) := 1 and S(u) := u −1 .
The algebra of polynomial functions on SU q (2) [14] is generated as a * -algebra by α and γ satisfying relations where 0 < q ≤ 1. The Hopf algebra structure comes from the matrix The Hopf * -algebra epimorphism makes O(SU q (2)) into a left and right O(U(1))-comodule algebra via the left and right coactions (π ⊗ id) • ∆ and (id ⊗ π) • ∆ respectively. For q = 1 the Hopf algebra O(SU q (2)) is commutative, and we denote its generators by a and c rather then α and γ. The Peter-Weyl comodule algebra P C(U(1)) (C(S 3 )) is the subalgebra of C(SU (2)) that is the algebraic direct sum of the modules of continuous sections of the complex line bundles L n , n ∈ Z, associated to the Hopf fibration: We have the following proper inclusions of function algebras: O(SU(2)) P C(U(1)) (C(S 3 )) C(S 3 ).
A trivialization associated with the above covering is given by the following cleaving maps, which are clearly algebra homomorphisms: One can argue (cf. [3]) that To see this, first note that π a (C(S 2 )) ∼ = C(D) ∼ = π c (C(S 2 )). Then, for any n ∈ Z, whence f i (P C(U(1)) (C(S 3 ))) = C(D)⊗O(U(1)). Indeed, f i (π i (Γ(L n ))) ⊆ π i (C(S 2 ))⊗u n . On the other hand, consider an arbitrary element y ∈ C(D). Then there exist elements y a , y c ∈ C(S 2 ) such that y = π a (y a ) = π c (y c ). Hence y ⊗ u n = f z (π z (y z z n ω n )), z = a, c.