Noncommutative de Rham cohomology of finite groups
Abstract
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' duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold in general. A short review of the bicovariant (noncommutative) differential calculus on finite G is given for selfconsistency. Exterior derivative, exterior product, metric, Hodge dual, connections, torsion, curvature, and biinvariant integration can be defined algebraically. A projector decomposition of the braiding operator is found, and used in constructing the projector on the space of 2-forms. By means of the braiding operator and the metric a knot invariant is defined for any finite group.
Keywords
Cite
@article{arxiv.math-ph/0211008,
title = {Noncommutative de Rham cohomology of finite groups},
author = {L. Castellani and R. Catenacci and M. Debernardi and C. Pagani},
journal= {arXiv preprint arXiv:math-ph/0211008},
year = {2009}
}
Comments
LaTeX, 25 pages, 4 figures. Added higher order exterior basis and volume forms of quaternion and dihedral groups, corrected sign in eq. (2.37) and (2.40), corrected misprints