Noncommutative Riemannian geometry of Kronecker algebras

TL;DR

This paper explores noncommutative Riemannian geometry on Kronecker algebras, extending concepts like metrics and Levi-Civita connections from the noncommutative torus to more general algebraic structures.

Contribution

It introduces a framework for studying derivation-based differential calculi and Levi-Civita connections on Kronecker algebras, expanding noncommutative geometric methods.

Findings

Construction of non-trivial Levi-Civita connections for various derivation Lie algebras.

Analysis of torsion-free bimodule connections compatible with hermitian forms.

Application of the framework to generalized Kronecker algebras and comparison with noncommutative torus.

Abstract

We study aspects of noncommutative Riemannian geometry of the path algebra arising from the Kronecker quiver with N arrows. To start with, the framework of derivation based differential calculi is recalled together with a discussion on metrics and bimodule connections compatible with the *-structure of the algebra. As an illustration, these concepts are applied to the noncommutative torus where examples of torsion free and metric (Levi-Civita) connections are given. In the main part of the paper, noncommutative geometric aspects of (generalized) Kronecker algebras are considered. The structure of derivations and differential calculi is explored, and torsion free bimodule connections are studied together with their compatibility with hermitian forms, playing the role of metrics on the module of differential forms. Moreover, for several different choices of Lie algebras of derivations, non-trivial Levi-Civita connections are constructed.