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The noncommutative geometry of frame bundles

Stefan Wagner

TL;DR

The paper addresses the noncommutative geometry of frame bundles by showing that a free $SO(n)$-action on a unital C$^*$-algebra is determined, up to isomorphism, by its associated vector bundle of type $π$. It develops a construction that, from a correspondence $M$ over a unital C$^*$-algebra $\mathcal{B}$ of type $π$, yields a free C$^*$-dynamical system $(\mathcal{A}_M,SO(n),α_M)$ with fixed point algebra $\mathcal{B}$ and $\Gamma_{\mathcal{A}_M}(π) \cong M$, captured by a unitary tensor functor from a small subcategory of $SO(n)$ representations to $\mathrm{Corr}(\mathcal{B})$. A bijective classification ties equivalence classes of such free actions to equivalence classes of correspondences tensorial of type $π$, and several nontrivial examples—including a quantum projective 7-space and the even part of $\mathcal{O}_2$—demonstrate the framework. The work lays groundwork for extending noncommutative frame bundles to richer geometric data and paves the way for future characteristic-class constructions within noncommutative spin geometry.

Abstract

We apply ourselves to the noncommutative geometry of frame bundles by showing that each C$^*$-algebraic noncommutative principal $\mathrm{SO}(n)$-bundle is, up to isomorphism, uniquely determined by its associated noncommutative vector bundle with respect to the standard representation of $\mathrm{SO}(n)$. For this, we provide a construction procedure, via unitary tensor functors, that for a certain type of correspondence, let's say $M$, attaches a free C$^*$-dynamical system $(\mathcal{A}_M,\mathrm{SO}(n),α_M)$ with the property that its associated noncommutative vector bundle with respect to the standard representation of $\mathrm{SO}(n)$ is isomorphic to $M$.

The noncommutative geometry of frame bundles

TL;DR

The paper addresses the noncommutative geometry of frame bundles by showing that a free -action on a unital C-algebra is determined, up to isomorphism, by its associated vector bundle of type . It develops a construction that, from a correspondence over a unital C-algebra of type , yields a free C-dynamical system with fixed point algebra and , captured by a unitary tensor functor from a small subcategory of representations to . A bijective classification ties equivalence classes of such free actions to equivalence classes of correspondences tensorial of type , and several nontrivial examples—including a quantum projective 7-space and the even part of —demonstrate the framework. The work lays groundwork for extending noncommutative frame bundles to richer geometric data and paves the way for future characteristic-class constructions within noncommutative spin geometry.

Abstract

We apply ourselves to the noncommutative geometry of frame bundles by showing that each C-algebraic noncommutative principal -bundle is, up to isomorphism, uniquely determined by its associated noncommutative vector bundle with respect to the standard representation of . For this, we provide a construction procedure, via unitary tensor functors, that for a certain type of correspondence, let's say , attaches a free C-dynamical system with the property that its associated noncommutative vector bundle with respect to the standard representation of is isomorphic to .
Paper Structure (10 sections, 13 theorems, 35 equations)

This paper contains 10 sections, 13 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Noncommutative frame bundles
  4. Examples
  5. Example: the free module of rank $n$
  6. Example: classical vector bundles
  7. Example: the quantum projective 7-space
  8. Example: the even part of the Cuntz algebra $\mathcal{O}_2$
  9. The construction of the free C*-dynamical systems from the unitary tensor functor
  10. The special case of $\mathop{\mathrm{SO}}\nolimits(2)$

Key Result

Corollary 2.1

Let $\pi$ be the standard representation of $\mathop{\mathrm{SO}}\nolimits(n)$, $n \geq 3$. Each irreducible representation of $\mathop{\mathrm{SO}}\nolimits(n)$ occurs as a subrepresentation of some tensor product representation $\pi^{\otimes k}$, $k \geq 0$ (with $\pi^{\otimes 0} = 1$).

Theorems & Definitions (29)

  • Corollary 2.1: See, e. g., GoWa09
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Definition 3.5
  • ...and 19 more