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Quantum symmetry in multigraphs (part I)

Debashish Goswami, Sk Asfaq Hossain

Abstract

We introduce various notions of quantum symmetry in a directed or undirected multigraph with no isolated vertex and explore relations among them. If the multigraph is single edged (that is, a simple graph where loops are allowed), all our notions of quantum symmetry reduce to already existing notions of quantum symmetry provided by Bichon and Banica. Our constructions also show that any multigraph with at least two pairs of vertices with multiple edges among them possesses genuine quantum symmetry.

Quantum symmetry in multigraphs (part I)

Abstract

We introduce various notions of quantum symmetry in a directed or undirected multigraph with no isolated vertex and explore relations among them. If the multigraph is single edged (that is, a simple graph where loops are allowed), all our notions of quantum symmetry reduce to already existing notions of quantum symmetry provided by Bichon and Banica. Our constructions also show that any multigraph with at least two pairs of vertices with multiple edges among them possesses genuine quantum symmetry.
Paper Structure (22 sections, 26 theorems, 127 equations, 4 figures)

This paper contains 22 sections, 26 theorems, 127 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite quivers or multigraphs:
  4. Compact quantum group:
  5. Co-actions and co-representations:
  6. Quantum automorphisms of single edged (weighted or non-weighted) graphs:
  7. Free wreath product by quantum permutation groups:
  8. Setup and Notations:
  9. Quantum symmetry in multigraphs
  10. Left and right equivariant co-representations on $L^2(E)$:
  11. Induced permutations on $V^s$ and $V^t$:
  12. Induced permutations on $V^s\cap V^t$:
  13. Co-actions on a multigraph
  14. Some useful identities:
  15. The categories $\mathcal{C}^{Ban}_{(V,E)}$, and $\mathcal{C}^{Bic}_{(V,E)}$:
  16. ...and 7 more sections

Key Result

Proposition 2.11

The quotient of a compact quantum group $(\mathcal{A},\Delta)$ by a Woronowicz C* ideal $\mathcal{I}$ has a unique compact quantum group structure such that the quotient map $\pi$ is a homomorphism of compact quantum groups. More precisely, the co-product $\tilde{\Delta}$ on $\mathcal{A}/\mathcal{I} where $a\in \mathcal{A}$.

Figures (4)

  • Figure 1: Two graphs with isomorphic $C(V^s)-L^2(E)-C(V^t)$ bimodule structure.
  • Figure 2: A multigraph with $n$ loops on a single vertex
  • Figure 3: A multigraph with two vertices and $2n$ number of edges.
  • Figure 4: A multigraph version of a triangle with $n$ edges between two vertices.

Theorems & Definitions (76)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 66 more