Quantum Calculi: differential forms and vector fields in noncommutative geometry

TL;DR

This paper advances noncommutative geometry by refining the concept of vector fields, establishing clearer algebraic frameworks, and explicitly constructing bimodules of universal vector fields.

Contribution

It revises and extends previous definitions of noncommutative vector fields, providing new algebraic tools and explicit constructions for bimodules.

Findings

Clarified the correspondence between Cartan pairs and first-order differentials

Constructed bimodules of universal vector fields explicitly

Enhanced the algebraic framework for noncommutative differential calculus

Abstract

In this paper, we revise the concept of noncommutative vector fields introduced previously in Ref. [1,2], extending the framework, adding new results and clarifying the old ones. Using appropriate algebraic tools certain shortcomings in the previous considerations are filled and made more precise. We focus on the correspondence between so-called Cartan pairs and first-order differentials. The case of free bimodules admitting more friendly "coordinate description" and their braiding is considered in more detail. Bimodules of right/left universal vector fields are explicitly constructed.