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The construction of a $E_7$-like quantum subgroup of $SU(3)$

Cain Edie-Michell, Lance Marinelli

Abstract

In this short note we construct an embedding of the planar algebra for $\overline{\operatorname{Rep}(U_q(sl_3))}$ at $q = e^{2πi \frac{1}{24}}$ into the graph planar algebra of di Francesco and Zuber's candidate graph $\mathcal{E}_4^{12}$. Via the graph planar algebra embedding theorem we thus construct a rank 11 module category over $\overline{\operatorname{Rep}(U_q(sl_3))}$ whose graph for action by the vector representation is $\mathcal{E}_4^{12}$. This fills a small gap in the literature on the construction of $\overline{\operatorname{Rep}(U_q(sl_3))}$ module categories. As a consequence of our construction, we obtain the principal graphs of subfactors constructed abstractly by Evans and Pugh.

The construction of a $E_7$-like quantum subgroup of $SU(3)$

Abstract

In this short note we construct an embedding of the planar algebra for at into the graph planar algebra of di Francesco and Zuber's candidate graph . Via the graph planar algebra embedding theorem we thus construct a rank 11 module category over whose graph for action by the vector representation is . This fills a small gap in the literature on the construction of module categories. As a consequence of our construction, we obtain the principal graphs of subfactors constructed abstractly by Evans and Pugh.
Paper Structure (6 sections, 2 theorems, 21 equations)

This paper contains 6 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Planar Algebras
  4. The graph planar algebra
  5. Finding the solution
  6. Verifying the solution

Key Result

Theorem 1.2

The map defines a tensor functor

Theorems & Definitions (8)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • proof : Proof of Theorem \ref{['thm:main']}