Hannover 2003 – wissenschaftliches Programm
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GR: Gravitation und Relativitätstheorie
GR 5: Grundlegende Probleme und alternative Ans
ätze
GR 5.2: Fachvortrag
Dienstag, 25. März 2003, 16:50–17:10, A310
Pseudo-Riemannian geometry of group lattices — •Folkert Müller-Hoissen1 and Aristophanes Dimakis2 — 1Max-Planck-Institut für Strömungsforschung, Bunsenstrasse 10, D-37073 Göttingen — 2Department of Financial and Management Engineering, University of the Aegean, GR-82100 Chios
Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with (non-commutative) differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of algebraic noncommutative geometry. Despite of the non-commutativity between functions and (generalized) differential forms, for the subclass of ‘bicovariant” group lattices 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. Essential features of the resulting (pseudo-) Riemannian geometry and corresponding examples are presented in this talk. A particular example is given by a hypercubic lattice digraph which leads to a discretization of gravity theories.