# Differential geometry of group lattices

@article{Dimakis2002DifferentialGO, title={Differential geometry of group lattices}, author={Aristophanes Dimakis and Folkert Muller-Hoissen}, journal={Journal of Mathematical Physics}, year={2002}, volume={44}, pages={1781-1821} }

In a series of publications we developed “differential geometry” on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first-order differential calculi (over the algebra of functions) on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set. A particular class of digraphs are Cayley graphs, also known as group lattices. They are determined by a discrete group G and a finite subset…

## 15 Citations

### Riemannian geometry of bicovariant group lattices

- Mathematics
- 2003

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…

### On a correspondence principle between discrete differential forms, graph structure and multi-vector calculus on symmetric lattices ∗

- Mathematics, Computer Science
- 2007

A bijective correspondence between first order differential calculi and the graph structure of the symmetric lattice is introduced that allows one to encode completely the interconnectionructure of the graph in the exterior derivative, which naturally leads to a discrete version of Clifford Analysis.

### Discrete differential geometry on causal graphs

- Mathematics
- 2004

Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete…

### Noncommutative de Rham cohomology of finite groups

- Mathematics
- 2002

We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S_3, the dihedral group D_4 and the quaternion group Q. Poincare'…

### Automorphisms of associative algebras and noncommutative geometry

- Mathematics
- 2004

A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on…

### Noncommutative Geometries and Gravity

- Mathematics
- 2007

AbstractWe brieﬂy review ideas about “noncommutativityof space-time” and approaches toward a corre-sponding theory of gravity. PACS: 02.40.Gh, 04.50.+h, 04.60.-mKeywords: Noncommutative geometry,…

### Difference discrete connection and curvature on cubic lattice

- Mathematics, Computer Science
- 2006

The difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space and the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice are defined.

### Difference discrete connection and curvature on cubic lattice

- Mathematics, Computer Science
- 2006

The difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space and the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice are defined.

### Generic spectrum of the weighted Laplacian operator on Cayley graphs

- Mathematics
- 2020

In this paper we address the problem of determining whether the eigenspaces of a class of weighted Laplacians on Cayley graphs are generically irreducible or not. This work is divided into two parts.…

### On the topology of a cyclic universe with colour fields

- Mathematics
- 2007

The topology of the universe is discussed in relation to the singularity problem. We explore the possibility that the initial state of the universe might have had a structure with 3-Klein bottle…

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