Divergences on projective modules and non-commutative integrals

TL;DR

This paper introduces a method to construct differential calculi and divergences on finitely generated projective modules over algebras, exploring properties of associated integrals with applications to noncommutative geometry.

Contribution

It presents a new approach to define differential structures and integrals on projective modules, including explicit examples like noncommutative algebras and supergeometry.

Findings

Derived a formula for integration by parts in noncommutative settings

Constructed examples of inner calculi and Berezin integrals

Analyzed properties of divergences and associated integrals

Abstract

A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a noncommutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.