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Quantum symmetry in multigraphs (part II)

Debashish Goswami, Sk Asfaq Hossain

Abstract

This article is a continuation of "Quantum symmetry in multigraphs (part I)". In this article, we give an explicit construction of a non-Bichon type co-action on a multigraph that is, it preserves quantum symmetry of (V,E) in our sense but not always in Bichon's sense. This construction itself is motivated from automorphisms of quantum graphs.

Quantum symmetry in multigraphs (part II)

Abstract

This article is a continuation of "Quantum symmetry in multigraphs (part I)". In this article, we give an explicit construction of a non-Bichon type co-action on a multigraph that is, it preserves quantum symmetry of (V,E) in our sense but not always in Bichon's sense. This construction itself is motivated from automorphisms of quantum graphs.
Paper Structure (5 sections, 8 theorems, 30 equations, 1 figure)

This paper contains 5 sections, 8 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. More Notations
  3. The category $\mathcal{C}_{(V,E)}$ and its universal object $Q_{(V,E)}$
  4. Action of $Q_{(V,E)}$ on $(V,E)$
  5. The non-uniform case

Key Result

Proposition 3.2

For $(\mathcal{A},\Delta, \alpha)\in \mathcal{C}_{(V,E)}$, $\alpha$ induces a quantum permutation$\alpha_V$ of the vertex set $V$ which is given by, where $Q^k_i=\sum^N_{r=1}q^{ks}_{ir}$ for all $i,k\in V$ and $s\in\{1,..,N\}$.

Figures (1)

  • Figure 1: A non-uniform multigraph

Theorems & Definitions (17)

  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 7 more