Super-Transitivity for Étale Algebra Objects

• The paper introduces super-transitivity for étale algebra objects to quantify the emergence of new simple modules via free-module functors in modular tensor categories.
• Using fusion graph analysis and skein-theoretic presentations, it classifies three infinite families at levels k = N-2, N, and N+2 linked to conformal embeddings.
• It also identifies sixteen exceptional sporadic cases with higher super-transitivity, revealing deep connections to ADE classifications in type A theories.

Super-transitivity for étale algebra objects is a categorical analogue of super-transitivity for subfactors, developed to analyze the emergence and structure of new simple objects (“new stuff”) within the module categories of commutative separable algebras in modular tensor categories, particularly those associated with affine Lie algebras of type A. The notion precisely quantifies the depth at which new irreducibles arise under free-module constructions, reflecting the structure of module categories and their fusion graphs. Recent developments provide a full classification of 1-super-transitive non-pointed étale algebra objects in unitary modular tensor categories $\mathcal{C}(\mathfrak{sl}_N, k)$, as well as a sharp enumeration of all higher super-transitivity exceptions, encapsulating both infinite families and all known sporadic cases (Edie-Michell et al., 16 Jan 2026).

1. Categorical Framework and Étale Algebra Objects

For each $N,k \in \mathbb{N}$, $\mathcal{C}(\mathfrak{sl}_N, k)$ denotes the unitary modular tensor category (MTC) of level-$k$ integrable highest-weight modules over the affine Lie algebra $\widehat{\mathfrak{sl}_N}$. It may also be viewed as the semisimplified, Cauchy-completed category of type-$A$ quantum group representations at $q = e^{2\pi i/(2(N+k))}$, with simple objects indexed by Young diagrams $\lambda$ with at most $N-1$ rows and $\lambda_1 \leq k$. Fusion rules mirror those of classical $\mathfrak{sl}_N$ except where “$k$–box” truncation applies; braiding and twist structures are inherited from the $R$-matrix and ribbon element.

An étale algebra object in a braided tensor category $\mathcal{C}$ is a commutative separable algebra $A \in \mathcal{C}$ satisfying:

• Commutativity with respect to the braiding,
• The splitting of the separability idempotent.

If $\theta_A = \mathrm{id}_A$, the category $\mathcal{C}_A$ of left $A$-modules inherits a rigid, pivotal (unitary) tensor structure. The free and forgetful functors

$$\mathcal{F}_A: \mathcal{C} \longleftrightarrow \mathcal{C}_A: \mathrm{For}$$

form an adjoint pair, providing a mechanism to analyze fusion graphs before and after the extension by $A$.

2. Super-Transitivity: Definition and Operationalization

Given an object $X \in \mathcal{C}$—typically the fundamental vector representation—super-transitivity quantifies when “new” simple $A$-modules appear inside $\mathcal{C}_A$. For an étale algebra $A$ and the free-module functor $\mathcal{F}_A$, denote by

$$\dim \operatorname{End}_{\mathcal{C}}(X^{\otimes n}), \quad \dim \operatorname{End}_{\mathcal{C}_A}(\mathcal{F}_A(X^{\otimes n}))$$

the endomorphism algebra dimensions on $n$-fold tensor powers in the original and extended categories, respectively.

Definition: $A$ is \textbf{$m$-super-transitive} if

$$\dim \operatorname{End}_{\mathcal{C}}(X^{\otimes m}) = \dim \operatorname{End}_{\mathcal{C}_A}(\mathcal{F}_A(X^{\otimes m}))$$

and

$$\dim \operatorname{End}_{\mathcal{C}}(X^{\otimes (m+1)}) < \dim \operatorname{End}_{\mathcal{C}_A}(\mathcal{F}_A(X^{\otimes (m+1))})$$

Equivalently, all simple $A$-modules up to fusion-graph depth $m$ coincide with those of $X$, but at least one strictly new simple arises at depth $m+1$. The $1$-super-transitive case is characterized by agreement to depth 1, with splitting at depth 2.

3. Complete Classification: 1-Super-Transitive Étale Algebra Objects

A sharp classification theorem governs all non-pointed $1$-super-transitive étale algebras in $\mathcal{C}(\mathfrak{sl}_N, k)$ for $N, k \geq 2$ (except $(N, k) = (8, 10), (10, 8)$), identifying three infinite families corresponding to Kac–Wakimoto conformal-embedding levels. Each algebra falls uniquely into one of the following cases:

Case	Level	Embedding and Algebra
(i)	$k = N-2$	$$A_{\mathfrak{sl}_{N(N-1)/2}} \subset \mathcal{C}(\mathfrak{sl}_N, N-2)$$
(ii)	$k = N$	$$A_{\mathfrak{so}_{N^2-1}} \subset \mathcal{C}(\mathfrak{sl}_N, N)$$
(iii)	$k = N+2$	$$A_{\mathfrak{sl}_{N(N+1)/2}} \subset \mathcal{C}(\mathfrak{sl}_N, N+2)$$

In all cases, the simple summands of $A$ are given by the decomposition of the appropriate level-one algebra under conformal inclusion, with further extensions arising only via invertible (pointed) objects.

The proof crucially relies on a depth-three analysis of the fusion graph for $\mathcal{F}_A(X)$. Only for the levels $k \in \{N-2, N, N+2\}$ do the fusion graphs permit $1$-super-transitivity, corresponding to the Kac–Wakimoto conformal embeddings. Planar-algebra presentations using generators (such as the Jones–Wenzl idempotents and an additional splitting idempotent) along with computed two- and three-strand relations identify these module categories with the conformal embedding subcategories. Outside these levels, a twist obstruction prevents further examples, as would-be summands of $A$ fail commutativity or rigidity due to nontrivial twist.

4. Infinite Families and Simple-Current Extensions

Each of the three allowed levels supports an infinite chain of non-pointed $1$-super-transitive étale algebras via Abelian simple-current extensions of the minimal conformal embedding algebra:

• Level $k = N-2$ (D-series): Embedding $$\mathfrak{sl}_N{}_{N-2} \subset \mathfrak{sl}_{N(N-1)/2}{}_1$$ with $A_{\mathfrak{sl}_{N(N-1)/2}}$, and further extensions by cyclic groups of invertibles (order dividing $\dim(X)$), subject to divisibility conditions on $N(N-1)/2$.
• Level $k = N$ (B-series): Embedding $$\mathfrak{sl}_N{}_N \subset \mathfrak{so}_{N^2-1}{}_1$$ with $A_{\mathfrak{so}_{N^2-1}}$, and further extension via pointed modules from the center of $\mathrm{SO}$.
• Level $k = N+2$ (Conjugate D-series): Embedding $$\mathfrak{sl}_N{}_{N+2} \subset \mathfrak{sl}_{N(N+1)/2}{}_1$$ and its simple-current extensions.

These families, well-known in conformal inclusions (Schellekens–Warner, Xu, Kirillov–Ostrik, Edie-Michell–Snyder), are fully captured in this classification. All further 1-super-transitive non-pointed examples arise as such extension algebras.

5. Sporadic Cases with Higher Super-Transitivity

Exactly sixteen exceptional, sporadic non-pointed étale algebra objects in type A are known, all at low rank and level and with super-transitivity strictly greater than 1. These correspond to the ADE classification of conformal embeddings and decompose as follows:

Super-transitivity	Examples
2	$$A_{\mathfrak{so}_5}\subset \mathcal{C}(\mathfrak{sl}_2,10)$$; $A_{E_6}\subset \mathcal{C}(\mathfrak{sl}_3,9)$; $$A_{\mathfrak{so}_{20}}\subset \mathcal{C}(\mathfrak{sl}_4,8)$$; $$A_{\mathfrak{sp}_{20}}\subset \mathcal{C}(\mathfrak{sl}_6,6)$$
3	$A_{E_7}\subset \mathcal{C}(\mathfrak{sl}_3,21)$; $$A_{\text{Schellekens}}\subset \mathcal{C}(\mathfrak{sl}_7,7)$$
4	$A_{G_2}\subset \mathcal{C}(\mathfrak{sl}_2,28)$

These module categories possess skein presentations with quadratic-tangle relations distinct from those found in the infinite families. No sporadic examples beyond these sixteen are currently known or expected, and none are 1-super-transitive—all exhibit the first appearance of new simples at depth 2, 3, or 4, as determined by direct depth and fusion graph analysis.

6. Fusion Graphs, Diagrams, and Skein Presentations

The fusion graphs for $\mathcal{F}_A(X)$ in the infinite 1-super-transitive families coincide up to depth 3 with the known $C_N$, $B_N$, or $D_N$ principal graphs characteristic of the corresponding conformal embedding. Classical Dynkin–ADE or extended Coxeter–Dynkin diagrams provide faithful combinatorial descriptions:

• Infinite families: standard Dynkin diagrams per conformal inclusion.
• Sporadic exceptions: branching at prescribed depth in diagrams such as $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$.

Skein-theoretic presentations are central for explicit computations. The relations required are presentable in the Kazhdan–Wenzl or Liu planar-algebra formalism. For example:

• The Hecke quadratic relation:

$$\includegraphics[height=1em]{Hecke} = (q - q^{-1}) \includegraphics[height=1em]{straight}$$

• The splitting idempotent projector on two strands:

$\includegraphics[height=1em]{split}$

which obeys 4-term and partial-trace relations, fully constraining the module category's structure for each family.

References for explicit skein-theoretic proofs and matrix computations can be found in the works of Edie-Michell and Katumba (Edie-Michell et al., 16 Jan 2026).

7. Implications and Literature Context

The classification of super-transitivity for étale algebra objects in $\mathcal{C}(\mathfrak{sl}_N,k)$ unifies the combinatorial and categorical perspectives on quantum subgroups of type A and provides a comprehensive list of all currently known non-pointed examples. The established families exhaust all possible 1-super-transitive cases, with any further examples constrained by fusion depth, conformal embedding levels, and twist considerations. This work connects the theory of module categories, conformal field theory, skein theory, and the representation theory of quantum groups, consolidating the ADE and principal graph phenomena into a categorical and planar-algebraic context (Edie-Michell et al., 16 Jan 2026).