Riemannian geometry of bicovariant group lattices

摘要: Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such calculus geometric structures can be introduced following general recipes noncommutative geometry. Despite noncommutativity between functions and (generalized) forms, for subclass “bicovariant” considered this work it is possible to understand central objects like metric, torsion curvature as “tensors” (left) covariance properties. This ensures that tensor components (with respect basis space 1-forms) transform familiar homogeneous way under change basis. There compatibility condition metric linear connection. The resulting (pseudo-) Riemannian geometry explored work. It demonstrated indeed able properly describe properties geometries lengths angles. A simple g...