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Type $G_2$ Quantum Subgroups from Graph Planar Algebra Embeddings

Caleb Kennedy Hill

Abstract

We give graphical presentations for the two quantum subgroups of type $G_2$. To do this we use a method of extending a tensor category by embedding the planar algebra of a $\otimes$-generating object into the graph planar algebra of this object's fundamental graph. This allows the use of computational methods to uncover relations we would have little hope of arriving at otherwise.

Type $G_2$ Quantum Subgroups from Graph Planar Algebra Embeddings

Abstract

We give graphical presentations for the two quantum subgroups of type . To do this we use a method of extending a tensor category by embedding the planar algebra of a -generating object into the graph planar algebra of this object's fundamental graph. This allows the use of computational methods to uncover relations we would have little hope of arriving at otherwise.
Paper Structure (13 sections, 22 theorems, 88 equations, 1 figure, 4 tables)

This paper contains 13 sections, 22 theorems, 88 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebra and Module Objects
  4. Unoriented Planar Algebras
  5. Unoriented Graph Planar Algebras
  6. Cyclic extensions
  7. GPA Embeddings
  8. Level 4 Trivalent Embedding
  9. Extension of level 4
  10. New Relations at Level 4
  11. Level 3 Trivalent Embedding
  12. Extension of level 3
  13. Equivalences

Key Result

Theorem 1

There is an equivalence where $A_4$ is the algebra object corresponding to the level-4 conformal embedding given in Equation eq:conf-embs.

Figures (1)

  • Figure 1: Fusion graphs at level 4 and 3 for $Y_4$ and $Y_3$ (black) and $g_4$ and $g_3$ (orange), respectively. See Figures 21b and 18b, respectively, of g2_graphs.

Theorems & Definitions (44)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Lemma 1
  • Lemma 2
  • Lemma 3: Half-braid
  • Remark 1
  • Definition 5
  • ...and 34 more