Riemannian geometry of bicovariant group lattices

Abstract

Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative differential geometry. Despite of the non-commutativity between functions and (generalized) differential forms, for the subclass of "bicovariant" group lattices considered in this work it is possible to understand central geometric objects like metric, torsion and curvature as "tensors" with (left) covariance properties. This ensures that tensor components (with respect to a basis of the space of 1-forms) transform in the familiar homogeneous way under a change of basis. There is a natural compatibility condition for a metric and a linear connection. The resulting (pseudo-) Riemannian geometry is explored in this work. It is demonstrated that the components of the metric are indeed able to properly describe properties of discrete geometries like lengths and angles. A simple geometric understanding in particular of torsion and curvature is achieved. The formalism has much in common with lattice gauge theory. For example, the Riemannian curvature is determined by parallel transport of vectors around a plaquette (which corresponds to a biangle, a triangle or a quadrangle).

Differential Calculi on Quantum Spaces Determined by Automorphisms

“…For differential calculi associated with (a subclass of) Cayley graphs ('group lattices') of finite groups, the corresponding generalized (pseudo-) Riemannian geometry has been developed in [8,9]. This includes (pseudo-) Riemannian geometry of hyper-cubic lattices as a special case.…”

“…Simple examples with nontrivial automorphisms are given by dx x = (x + ℓ) dx [3,4,5] and dx x = q x dx [3,6,7] on the commutative algebra freely generated by x, setting θ 1 = dx. A large class of examples arises from differential calculi on Cayley graphs ('group lattices') of a finite group G, where S ⊂ G \ {e} and φ s = R * s , the pull-back with the right action of s ∈ S on G [8,9]. There are more examples [10], some will be discussed in this work.…”

Difference discrete connection and curvature on cubic lattice

“…Two of us with their collaborator had considered this issue in [18] very briefly in a way different from other relevant proposals (see, e.g. [19], [20], [21], [22], [23]). …”

“…[20], [21], [22], [23], [15], [16]). In this section we define the (difference) discrete connection in simple way similar to that in the continuous case based on the noncommutative differential calculus introduced in the last section.…”

The Newton-Wigner localization concept and noncommutative space

“…For the details we refer the reader to [9][10][11]. Note that that the integration in the second formula in (21) is carried over with the Euclidean volume element dr.…”