Quantum Principal Bundles

Abstract

This is an exposition (without proofs) of a quantum version ofprincipal bundle theory, where the base is a quantum space andthe group is a compact matrix quantum group in the sense ofWoronowicz.\par After reviewing some facts about compact matrixquantum groups and covariant differential calculi on them, theauthor introduces the definition of a quantum principal bundle,and discusses differential calculi on these objects. Next, heintroduces connections (which always exist) and two specialsubclasses called multiplicative and regular connections whichhave nice features concerning curvature and covariant derivativesbut need not exist in general. The covariant derivative and thehorizontal projection are introduced and an analogue of theBianchi identity is stated.\par For a quantum principal bundlewhich admits a multiplicative regular connection, an analogue ofthe Weil homomorphism is introduced. Finally, some examples arediscussed.\par {For the entire collection see MR\Cite{Keller94:Differential:Universidad}[96b:00020].}