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On the existence of noncommutative Levi-Civita connections in derivation based calculi

Joakim Arnlind, Victor Hildebrandsson

Abstract

We study the existence of Levi-Civita connections, i.e torsion free connections compatible with a hermitian form, in the setting of derivation based noncommutative differential calculi over $\ast$-algebras. We prove a necessary and sufficient condition for the existence of Levi-Civita connections in terms of the image of an operator derived from the hermitian form. Moreover, we identify a necessary symmetry condition on the hermitian form that extends the classical notion of metric symmetry in Riemannian geometry. The theory is illustrated with explicit computations for free modules of rank three, including noncommutative 3-tori. We note that our approach is algebraic and does not rely on analytic tools such as $C^\ast$-algebra norms.

On the existence of noncommutative Levi-Civita connections in derivation based calculi

Abstract

We study the existence of Levi-Civita connections, i.e torsion free connections compatible with a hermitian form, in the setting of derivation based noncommutative differential calculi over -algebras. We prove a necessary and sufficient condition for the existence of Levi-Civita connections in terms of the image of an operator derived from the hermitian form. Moreover, we identify a necessary symmetry condition on the hermitian form that extends the classical notion of metric symmetry in Riemannian geometry. The theory is illustrated with explicit computations for free modules of rank three, including noncommutative 3-tori. We note that our approach is algebraic and does not rely on analytic tools such as -algebra norms.

Paper Structure

This paper contains 12 sections, 15 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Noncommutative differential forms
  3. Derivation based calculi
  4. Connections and hermitian forms
  5. Derivation based calculi
  6. Generalized metric symmetry
  7. Index calculus on projective modules
  8. Connections compatible with hermitian forms
  9. Torsion free connections on $\Omega^1_{\mathfrak{g}}$
  10. Levi-Civita connections
  11. A sufficient condition for projective modules
  12. Levi-Civita connections on free modules of rank 3

Key Result

Proposition 3.8

Let $h$ be a left hermitian form on a left $\mathcal{A}$-module $M$. Then for $m\in M$ and $a\in\mathcal{A}$. Moreover, if $h$ is invertible then for $\phi\in M^\ast$ and $a\in\mathcal{A}$.

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • Remark 3.7
  • Proposition 3.8
  • proof
  • ...and 42 more