Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isometry Theorem for Continuous Quiver of Type $\tilde{A}$

Xiaowen Gao, Minghui Zhao

TL;DR

This work proves an Isometry Theorem for continuous quivers of type $\tilde{A}$ by leveraging an equivalence with a type $A$ quiver endowed with a $\mathbb{Z}$-action. Building on the Hanson–Rock decomposition for $\tilde{A}_{\mathbb{R}}$ and the existing Isometry Theorem for ${A}_{\mathbb{R}}$, the authors translate interleaving distances to a bottleneck metric on persistence diagrams in the quotient plane $\mathbb{R}^2/\sim$. The main result asserts $d_{b,\mathbb{R}^2/\sim}(dgm(\mathbf{V}),dgm(\mathbf{W})) = d_{i,\tilde{A}_{\mathbb{R}}}(\mathbf{V},\mathbf{W})$ for pointwise finite-dimensional nilpotent representations $\mathbf{V},\mathbf{W}$, obtained via an equivalence with a quiver with automorphism $\mathbf{Q}$. This extends stability guarantees of persistence diagrams to periodic/quasi-periodic continuous quivers, enabling robust analysis in settings with discrete translational symmetry.

Abstract

The Isometry Theorem for continuous quiver of type $A$ plays an important role in persistent homology. In this paper, we shall generalize Isometry Theorem to continuous quiver of type $\tilde{A}$.

Isometry Theorem for Continuous Quiver of Type $\tilde{A}$

TL;DR

This work proves an Isometry Theorem for continuous quivers of type by leveraging an equivalence with a type quiver endowed with a -action. Building on the Hanson–Rock decomposition for and the existing Isometry Theorem for , the authors translate interleaving distances to a bottleneck metric on persistence diagrams in the quotient plane . The main result asserts for pointwise finite-dimensional nilpotent representations , obtained via an equivalence with a quiver with automorphism . This extends stability guarantees of persistence diagrams to periodic/quasi-periodic continuous quivers, enabling robust analysis in settings with discrete translational symmetry.

Abstract

The Isometry Theorem for continuous quiver of type plays an important role in persistent homology. In this paper, we shall generalize Isometry Theorem to continuous quiver of type .

Paper Structure

This paper contains 10 sections, 9 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Continuous quivers of type ${A}$
  3. Continuous quivers of type ${A}$
  4. Isometry Theorem
  5. Continuous quivers of type $\tilde{A}$
  6. Continuous quivers of type $\tilde{A}$
  7. Isometry Theorem
  8. The proof of the main result
  9. Continuous quivers of type $A$ with automorphism
  10. The proof of the main result

Key Result

Theorem 2.1

Let $\mathbb{V}$ and $\mathbb{W}$ be two pointwise finite-dimensional representations of continuous quiver ${{A}}_{\mathbb{R}}$ of type ${A}$. Then,

Theorems & Definitions (15)

  • Theorem 2.1: 2015Persistence
  • Theorem 3.1: 2020Decomposition
  • Theorem 3.2
  • Remark 4.1
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • proof
  • Lemma 4.4: 2015Persistence
  • Lemma 4.5
  • ...and 5 more