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We construct quantum group-valued canonical connections on quantum homogeneous spaces, including a q-deformed Dirac monopole on the quantum sphere of Podles quantum differential coming from the 3-D calculus of Woronowicz on SUq(2) . The construction is presented within the setting of a general theory of quantum principal bundles with quantum group (Hopfâ€¦ (More)

Characteristic properties of corings with a grouplike element are analysed. Associated differential graded rings are studied. A correspondence between categories of comodules and flat connections is established. A generalisation of the Cuntz-Quillen theorem relating existence of connections in a module to projectivity of this module is proven.

For a given entwining structure (A,C)Ïˆ involving an algebra A, a coalgebra C, and an entwining map Ïˆ : C âŠ— A â†’ A âŠ— C, a category MA(Ïˆ) of right (A,C)Ïˆmodules is defined and its structure analysed. In particular, the notion of a measuring of (A,C)Ïˆ to (Ãƒ, CÌƒ)ÏˆÌƒ is introduced, and certain functors between M C A(Ïˆ) and M Ãƒ (ÏˆÌƒ) induced by such a measuring areâ€¦ (More)

Given a ring A and an A-coring C we study when the forgetful functor from the category of right C-comodules to the category of right A-modules and its right adjoint âˆ’ âŠ—A C are separable. We then proceed to study when the induction functor âˆ’ âŠ—A C is also the left adjoint of the forgetful functor. This question is closely related to the problem when A â†’â€¦ (More)

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M2(C) = CZ2 Â·CZ2. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As anâ€¦ (More)

- Tomasz Brzezinski, Piotr M. Hajac
- 2003

Following the idea of Galois-type extensions and entwining structures, we define the notion of a principal extension of noncommutative algebras. We show that modules associated to such extensions via finite-dimensional corepresentations are finitely generated projective, and determine an explicit formula for the Chern character applied to the thus obtainedâ€¦ (More)

We show that certain embeddable homogeneous spaces of a quantum group that do not correspond to a quantum subgroup still have the structure of quantum quotient spaces. We propose a construction of quantum fibre bundles on such spaces. The quantum plane and the general quantum two-spheres are discussed in detail. 0. Introduction A homogeneous space X of aâ€¦ (More)

We develop a generalised gauge theory in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory, embeddable quantum homogeneous spaces and braided group gauge theory, the latter being introduced now by these means. Examplesâ€¦ (More)

To any bimodule which is finitely generated and projective on one side one can associate a coring, known as a comatrix coring. A new description of comatrix corings in terms of data reminiscent of a Morita context is given. It is also studied how properties of bimodules are reflected in the associated comatrix corings. In particular it is shown thatâ€¦ (More)

A relationship between coseparable corings and separable non-unital rings is established. In particular it is shown that an A-coring C has an associative A-balanced product. A Morita context is constructed for a coseparable coring with a grouplike element. Biseparable corings are defined, and a conjecture relating them to Frobenius corings is proposed.