Jets of modules in noncommutative geometry

TL;DR

This paper explores the limitations of jet modules in noncommutative geometry, highlighting their restricted role in representing differential operators and the resulting challenges in extending classical Lagrangian formalism.

Contribution

It identifies the constraints of jet modules in noncommutative settings and discusses the implications for formulating Lagrangian mechanics on noncommutative spaces.

Findings

Jets represent only a subset of differential operators in noncommutative geometry

Standard Lagrangian formalism faces difficulties in noncommutative spaces due to jet module limitations

The paper clarifies the scope of jets in noncommutative modules and their impact on geometric formulations

Abstract

Jets of modules over a commutative ring are well known to make up the representative objects of linear differential operators on these modules. In noncommutative geometry, jets of modules provide the representative objects only of a certain class of first order differential operators. As a consequence, a generalization of the standard Lagrangian formalism on smooth manifolds to noncommutative spaces is problematic.