Projective real calculi over matrix algebras

TL;DR

This paper explores real calculi over projective modules and matrix algebras, providing classification insights, concrete examples, and conditions for Levi-Civita connections, advancing the understanding of noncommutative geometric structures.

Contribution

It introduces the concept of quasi-equivalence for matrix representations of real calculi and links the existence of Levi-Civita connections to eigenvectors of anti-hermitian matrices.

Findings

Real calculi over projective modules can be realized as projections of free calculi.

Classification of real calculi over matrix algebras involves quasi-equivalence of representations.

Existence of Levi-Civita connection depends on eigenvectors of specific anti-hermitian matrices.

Abstract

In analogy with the geometric situation, we study real calculi over projective modules and show that they can be realized as projections of free real calculi. Moreover, we consider real calculi over matrix algebras and discuss several aspects of the classification problem for real calculi in this case, leading to the concept of quasi-equivalence of matrix representations. We also use matrix algebras to give concrete examples of real calculi where the module is projective, and show that the existence of a Levi-Civita connection depends on the eigenvectors of specific anti-hermitian matrices in this case.