Quantum Gauge Groups of Quantum Principal Bundles

We review different approaches to gauge groups in noncommutative geometry and study their emergence in the context of noncommutative principal bundles (Hopf–Galois extensions). In interesting examples with quantum base space, infinitesimal gauge transformations form a braided Lie algebra, with triangular R-matrix. The case of the quantum orthogonal frame bundle on the noncommutative 2n-dimensional sphere is presented. Connections are then defined as splittings of the noncommutative Atiyah sequence and their infinitesimal gauge transformations and geometry are studied.