Integrated differential geometry. Commutative and noncommutative

TL;DR

This paper introduces a unified framework called integrated differential geometry that extends classical differential geometry to all group actions on associative algebras, including noncommutative cases, and defines new cohomological invariants.

Contribution

It generalizes differential geometry to noncommutative settings and introduces integrated de Rham cohomology as a new invariant for group actions on algebras.

Findings

Computed integrated de Rham cohomologies for specific examples

Extended classical geometry concepts to noncommutative algebras

Provided new invariants for analyzing group actions

Abstract

For a manifold M we define a structure on the group action of Diff(M) on the smooth functions on M which reduces to the usual differential geometry upon differentiation at zero along the one-parameter groups of Diff(M). This ``integrated differential geometry'' generalises to all group actions on associative algebras, including noncommutative ones, and defines an ``integrated de Rham cohomology,'' which provides a new set of invariants for group actions. We calculate the first few integrated de Rham cohomologies for two examples;- a discrete group action on a commutative algebra, and a continuous Lie group action on a noncommutative matrix algebra.