Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

Brzeziński, Tomasz; Majid, Shahn

doi:10.1007/pl00005530

Cited by 83 publications

(87 citation statements)

References 30 publications

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“…These representations give a desired trace, and bring us to our first main result: [1,12,3]. Next, the representations [13] …”

Section: Abridged Versionmentioning

confidence: 78%

“…Moreover, one can prove that O(S 2 q,s ) ⊆ O(SU q (2)) is a principal O(SU q (2))/J s -extension [3]. With the help of [2], the latter follows from an explicit construction of a strong connection:…”

Section: Corollary 24 the Image Of The Positive Cone Ofmentioning

confidence: 99%

“…Here J s is the coideal right ideal generated by K, L − s, L * − s, and K, L generate the polynomial algebras O(S 2 q,s ). One can show that O(SU q (2))/J s coincides with O(U (1)) viewed as a coalgebra, and that O(S 2 q,s ) ⊆ O(SU q (2)) is a principal O(SU q (2))/J s -extension [1,12,3]. Next, the representations [13]…”

Section: Abridged Versionmentioning

confidence: 99%

“…Then the classification of SU q (2)-quantum homogeneous spaces yields a two-parameter family of noncommutative two-spheres [13]. The latter form the base of the Hopf fibrations of SU q (2) [3]. Function algebras of total spaces of principal U (1)-bundles always decompose into direct sums of sections of all associated line bundles.…”

Section: Introductionmentioning

confidence: 99%

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Chern numbers for two families of noncommutative Hopf fibrations

Hajac

Matthes

Szymański

2003

Comptes Rendus. Mathématique

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Abstract.We consider noncommutative line bundles associated with the Hopf fibrations of SUq(2) over all Podleś spheres and with a locally trivial Hopf fibration of S 3 pq . These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U (1) with Galois-type extensions encoding the principal fibrations of SUq(2) and S 3 pq . We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them. Abridged versionIn this paper, we combine the algebraic tool of Galois-type extensions with the analytic tool of the noncommutative index formula to study two kinds of examples of quantum fibrations. Our main result is that the line bundles associated to these principal fibrations are mutually non-isomorphic. This gives an estimate of the positive cones of the algebraic K 0 -groups of the base-space quantum spheres.Let B ⊆ P be an inclusion of algebras such that B is the coinvariant subalgebra for some coalgebra C coaction ∆ R : P → P ⊗ C. Using the framework of Galois-type extensions, one can say when such an extension of algebras is principal. (The definition is tuned in such a way that for commutative algebras it coincides with the concept of affine group scheme torsors -the principal bundles of algebraic geometry.) Every principal C-extension B ⊆ P allows one to assign to any finite-dimensional corepresentation of C a finitely generated projective left B-module of colinear homomorphisms Hom C (V ϕ , P ) [2]. Taking its class in K 0 (B) and composing it with the Chern character defines the Chern-Galois character from the space of all finite-dimensional corepresentations of C to the even cyclic homology of B. On the other hand, the K-homology Chern character assigns to finitely summable Fredholm modules cyclic cocycles. In the 1-summable case it takes a particularly simple form, notably it turns a pair of bounded * -representations (ρ 1 , ρ 2 ) into a trace (cyclic 0-cocycle) on B via the formula tr ρ = Tr • (ρ 1 − ρ 2 ). The evaluation of this trace on the Chern-Galois character applied to a corepresentation gives a numerical invariant of the K 0 -class of the module defined by this corepresentation. Moreover, for our examples, the integrality of these invariants (guaranteed by the noncommutative index formula) makes them computable. Here {e k } k≥0 is an orthonormal basis of a separable Hilbert space and f 0 , f 1 are generators of the * -algebra O(S 2 pq ). These representations give a desired trace, and bring us to our first main result: Theorem 0.1 For all µ ∈ Z, the pairing between the cyclic 0-cocycle tr ρ and the K 0 -class of O(S 3 pq ) µ (Chern number) coincides with the winding number µ, i.e., tr ρ , [O(S 3 pq ) µ ] = µ. Our second example is a family of noncommutative Hopf fibrations of SU q (2) over all Podleś quantum spheres S 2 q,s , s ∈ [0, 1]. As handling the generic Podleś spheres requires going beyond the Hopf-Galois theory, they were among main motivating examples driving the development of the theory of pr...

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“…These representations give a desired trace, and bring us to our first main result: [1,12,3]. Next, the representations [13] …”

Section: Abridged Versionmentioning

confidence: 78%

Section: Corollary 24 the Image Of The Positive Cone Ofmentioning

confidence: 99%

Section: Abridged Versionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Chern numbers for two families of noncommutative Hopf fibrations

Hajac

Matthes

Szymański

2003

Comptes Rendus. Mathématique

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“…The need for such a generalisation arises from the theory of quantum and coalgebra principal bundles [1]. As explained in [2] the structure of a classical principal bundle is encoded in the factorisation built on the algebra of functions on the total space of a bundle and the group algebra of the structure group. The action of the structure group determines the twisting Ψ.…”

Section: Introductionmentioning

confidence: 99%

Deformation of Algebra Factorisations

Brzeziński

2001

Communications in Algebra

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On Modules Associated to Coalgebra Galois Extensions

Brzeziński

1999

Journal of Algebra

184

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For a given entwining structure (A, C) ψ involving an algebra A, a coalgebra C, and an entwining map ψ : C ⊗ A → A ⊗ C, a category M C A (ψ) of right (A, C) ψmodules is defined and its structure analysed. In particular, the notion of a measuring of (A, C) ψ to (Ã,C)ψ is introduced, and certain functors between M C A (ψ) and MC A (ψ) induced by such a measuring are defined. It is shown that these functors are inverse equivalences iff they are exact (or one of them faithfully exact) and the measuring satisfies a certain Galois-type condition. Next, left modules E and right modulesĒ associated to a C-Galois extension A of B are defined. These can be thought of as objects dual to fibre bundles with coalgebra C in the place of a structure group, and a fibre V . Cross-sections of such associated modules are defined as module maps E → B orĒ → B. It is shown that they can be identified with suitably equivariant maps from the fibre to A. Also, it is shown that a C-Galois extension is cleft if and only if A = B ⊗ C as left B-modules and right C-comodules. The relationship between the modules E andĒ is studied in the case when V is finite-dimensional and in the case when the canonical entwining map is bijective.

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Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

Cited by 83 publications

References 30 publications

Chern numbers for two families of noncommutative Hopf fibrations

Chern numbers for two families of noncommutative Hopf fibrations

Deformation of Algebra Factorisations

On Modules Associated to Coalgebra Galois Extensions

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