Differential geometry of group lattices
Abstract
In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that firstorder 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 S. There is a distinguished subclass of "bicovariant" Cayley graphs with the property ad(S)S⊂S. We explore the properties of differential calculi which arise from Cayley graphs via the above correspondence. The firstorder calculi extend to higher orders and then allow us to introduce further differential geometric structures. Furthermore, we explore the properties of "discrete" vector fields which describe deterministic flows on group lattices. A Lie derivative with respect to a discrete vector field and an inner product with forms is defined. The LieCartan identity then holds on all forms for a certain subclass of discrete vector fields. We develop elements of gauge theory and construct an analog of the lattice gauge theory (YangMills) action on an arbitrary group lattice. Also linear connections are considered and a simple geometric interpretation of the torsion is established. By taking a quotient with respect to some subgroup of the discrete group, generalized differential calculi associated with socalled Schreier diagrams are obtained.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 April 2003
 DOI:
 10.1063/1.1540713
 arXiv:
 arXiv:mathph/0207014
 Bibcode:
 2003JMP....44.1781D
 Keywords:

 11.15.Ha;
 02.40.Hw;
 02.20.Bb;
 Lattice gauge theory;
 Classical differential geometry;
 General structures of groups;
 Mathematical Physics;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory
 EPrint:
 51 pages, 11 figures