We review quantum gravity model building using the new formalism of 'quantum Riemannian geometry' to construct this on finite discrete spaces and on fuzzy ones such as matrix algebras. The formalism starts with a 'differential ...

We provide a systematic treatment of boundaries based on subgroups K ⊆ G for the Kitaev quantum double D(G) model in the bulk. The boundary sites are representations of a -subalgebra Ξ ⊆ D(G) and we explicate its structure ...

We show that the Ehresmann-Schauenburg bialgebroid of a quantum principal bundle P or Hopf Galois extension with structure quantum group H is in fact a left Hopf algebroid L(P,H). We show further that if H is coquasitriangular ...

We study geodesics flows on curved quantum Riemannian geometries using a recent formulation in terms of bimodule connections and completely positive maps. We complete this formalism with a canonical operation on ...