On quantum $G$-structures

We review the classical theory of principal bundles, with particular emphasis on frame bundles and $G$-structures. We then develop the noncommutative framework by introducing the necessary notions of differential calculi, Hopf algebras, quantum principal bundles, and associated quantum vector bundles. Within this setting, we review Majid’s notion of a quantum frame resolution. Building on the theory of reductions of quantum principal bundles, we introduce a notion of quantum $G$-structure as a reduction of a quantum frame resolution and prove that every such reduction naturally inherits the structure of a quantum frame resolution.