Noncommutative de Rham cohomology of finite groups

Details Noncommutative de Rham cohomology of finite groups Noncommutative de Rham cohomology of finite groups

2002-11-05

arxiv.org/abs/math-ph/0211008v2

Int.J.Mod.Phys. A19 (2004) 1961-1986

Physics

Mathematical Physics

LaTeX, 25 pages, 4 figures. Added higher order exterior basis and volume forms of quaternion and dihedral groups, corrected si

Scientific paper

10.1142/S0217751X04018403

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.

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