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This introductory text is the first book about quantum principal bundles and their quantum connections which are natural generalizations to non-commutative geometry of principal bundles and their connections in differential geometry. To make for a more self-contained book there is also much background material on Hopf algebras, (covariant) differential calculi, braid groups and compatible conjugation operations. The approach is slow paced and intuitive in order to provide researchers and students in both mathematics and physics ready access to the material.

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Introduction.- First Order Differential Calculus.- Fodc´s of a Hopf Algebra.- Adjoint Co-action.- Covariant Bimodules.- Covariant Fodc´s.- The Braid Groups.- An Interlude: Some Abstract Nonsense.- The Braided Exterior Algebra.- Higher Order Differential Calculus.- Structures.- Quantum Principal Bundles.- Finite Classical Groups.- Dunkl Operators as Covariant Derivatives in a QPB.- What Next?.

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Stephen Bruce Sontz is a Professor at Centro de Investigacion en Matematicas (CIMAT). He received his Ph.D. from the University of Virginia, USA, in 1994.

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