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.

arXiv commentLaTeX, 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

References in corpus (7)

• An introduction to non-commutative differential geometry on quantum groups
• Noncommutative Geometry of Finite Groups
• Noncommutative geometry and physics: a review of selected recent results
• Gravity on Finite Groups
• Differential Geometry of Group Lattices
• Discretized Yang-Mills and Born-Infeld actions on finite group geometries
• Finite group discretization of Yang-Mills and Einstein actions

Cited by in corpus (2)

• Exact Bianchi Identity in Regge Gravity
• Riemannian Geometry of Bicovariant Group Lattices