Quantum Fibre Bundles. An Introduction
An approach to construction of a quantum group gauge theory based on the quantum group generalisation of fibre bundles is reviewed.
A note on coalgebra gauge theory
A generalisation of quantum principal bundles in which a quantum structure group is replaced by a coalgebra is proposed.
Differential smoothness of affine Hopf algebras of Gelfand-Kirillov dimension two
Two-dimensional integrable differential calculi for classes of Ore extensions of the polynomial ring and the Laurent polynomial ring in one variable are constructed. Thus it is concluded that all affine pointed Hopf domains of Gelfand-Kirillov dimension two which are not polynomial identity rings are differentially smooth.
A note on flat noncommutative connections
It is proven that every flat connection or covariant derivative ∇ on a left A-module M (with respect to the universal differential calculus) induces a right A-module structure on M so that ∇ is a bimodule connection on M or M is a flat differentiable bimodule. Similarly a flat hom-connection on a right A-module M induces a compatible left A-action.
The structure of corings with a grouplike element
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.
Integral calculus on (2).
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