Noncommutative de Rham cohomology of finite groups

TL;DR

This paper explores noncommutative de Rham cohomology on finite groups up to order 8, establishing Poincare' duality and constructing algebraic geometric structures, including a knot invariant, within this framework.

Contribution

It provides a systematic study of noncommutative differential calculus on finite groups, demonstrating Poincare' duality and developing algebraic tools like metrics and invariants.

Findings

Poincare' duality holds for all studied finite groups.

A projector decomposition of the braiding operator is achieved.

A knot invariant is constructed using the braiding operator and metric.

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.