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Classification of 1-super-transitive quantum subgroups in type A

Cain Edie-Michell, Jacques Katumba

TL;DR

The paper defines 1-super-transitivity for étale algebras in the type A tensor categories C(𝔰𝔩_N, k) and proves a rank-agnostic classification: any 1-super-transitive étale algebra is a simple current extension of one of three étale algebras arising from conformal embeddings with k in {N−2, N, N+2}. The authors develop a framework using the module category C_A, Conformal Embedding presentations SE_N and SD_N^±, and a detailed analysis of End_A spaces and their convolution product to constrain fusion graphs and object structure. They show that in the k = N case the resulting category matches SE_N and that A is a simple current extension of A_{so_{N^2−1}}, while in k = N±2 they identify SD_N^±-type presentations and obtain the corresponding simple current extensions. Together these results establish a concrete, finite, and explicit description of all 1-super-transitive étale algebras in these categories, with two exceptional level-rank pairs excluded from the classification and the door left open for computational completion of remaining cases. The work advances the understanding of étale algebras in braided fusion categories beyond rank constraints and provides rank-agnostic insight into the conformal-embedding origin of non-pointed examples.

Abstract

We define a notion of super-transitivity for ètale algebra objects $A \in \mathcal{C}(\mathfrak{sl}_N, k)$. This definition is a direct analogue of the notion of super-transitivity for subfactors, and measures at what depth the first ``new stuff'' appears in the category of $A$-modules internal to $\mathcal{C}(\mathfrak{sl}_N, k)$. Our main theorem gives a classification of all 1-super-transitive ètale algebra objects in $\mathcal{C}(\mathfrak{sl}_N, k)$ running over all $N,k \in \mathbb{N}$. Our classification captures all known infinite families of non-pointed ètale algebras in $\mathcal{C}(\mathfrak{sl}_N, k)$, and includes all but 16 of the known non-pointed ètale algebra objects in these categories. These remaining 16 known examples have super-transitivities between 2 and 4.

Classification of 1-super-transitive quantum subgroups in type A

TL;DR

The paper defines 1-super-transitivity for étale algebras in the type A tensor categories C(𝔰𝔩_N, k) and proves a rank-agnostic classification: any 1-super-transitive étale algebra is a simple current extension of one of three étale algebras arising from conformal embeddings with k in {N−2, N, N+2}. The authors develop a framework using the module category C_A, Conformal Embedding presentations SE_N and SD_N^±, and a detailed analysis of End_A spaces and their convolution product to constrain fusion graphs and object structure. They show that in the k = N case the resulting category matches SE_N and that A is a simple current extension of A_{so_{N^2−1}}, while in k = N±2 they identify SD_N^±-type presentations and obtain the corresponding simple current extensions. Together these results establish a concrete, finite, and explicit description of all 1-super-transitive étale algebras in these categories, with two exceptional level-rank pairs excluded from the classification and the door left open for computational completion of remaining cases. The work advances the understanding of étale algebras in braided fusion categories beyond rank constraints and provides rank-agnostic insight into the conformal-embedding origin of non-pointed examples.

Abstract

We define a notion of super-transitivity for ètale algebra objects . This definition is a direct analogue of the notion of super-transitivity for subfactors, and measures at what depth the first ``new stuff'' appears in the category of -modules internal to . Our main theorem gives a classification of all 1-super-transitive ètale algebra objects in running over all . Our classification captures all known infinite families of non-pointed ètale algebras in , and includes all but 16 of the known non-pointed ètale algebra objects in these categories. These remaining 16 known examples have super-transitivities between 2 and 4.
Paper Structure (15 sections, 28 theorems, 152 equations)

This paper contains 15 sections, 28 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Categories C(sl(N),k)
  4. Tensor Categories from Ètale Algebras
  5. Ètale Algebras from Conformal Embeddings
  6. Initial Algebra Structure
  7. The case of k=N
  8. Initial Relations
  9. The Convolution Product
  10. A basis for the three box space
  11. Classification in the case of k = N +-2
  12. Initial Relations
  13. Case i)
  14. Case ii)
  15. Matrix Representations

Key Result

Theorem 1.5

Let $N,k \geq 2$ with $(N,k) \notin \{(8,10), (10,8)\}$ and let $A\in \mathcal{C}(\mathfrak{sl}_N, k)$ be a $1$-super-transitive ètale algebra object. Then either:

Theorems & Definitions (64)

  • Definition 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Example 2.1
  • Definition 2.2: sovietHans
  • Proposition 2.3: sovietHans
  • ...and 54 more