Non-Degenerate Braided Near-Group Categories

• The category is defined as a braided simple extension of a non-semisimple pointed finite tensor subcategory with a unique simple projective object.
• Key classification results force the self-extension parameter to vanish, ensuring weak integrality and unique modular data via de-equivariantization.
• Explicit constructions employ odd supervector spaces and finite 2-groups, linking modular representation theory with super-categorical frameworks.

A non-degenerate braided non-semisimple near-group category is a highly structured finite tensor category characterized by a near-group fusion rule in the non-semisimple setting and equipped with a non-degenerate braiding. These categories generalize classical (semisimple) near-group categories by allowing the underlying pointed subcategory to be non-semisimple. They constitute a central object of study in the modular representation theory of tensor categories, exhibiting both rich algebraic invariants and intricate relationships to super-categories and doubling procedures.

1. Structural Definitions and the Generalized Near-Group Concept

A finite tensor category $\mathcal{C}$ is a $k$-linear, abelian, rigid monoidal category: it has finitely many isomorphism classes of simple objects, every simple has a projective cover, and the unit object is simple. A non-semisimple near-group category is defined as a braided simple extension $\mathcal{C}$ of a non-semisimple pointed finite tensor subcategory $\mathcal{D}$. Explicitly, $\mathcal{C}$ possesses a decomposition

$\mathcal{C} \cong \mathcal{D} \oplus \mathcal{M}$

with $\mathcal{M}$ a $\mathcal{D}$-bimodule category equivalent to Vec, containing exactly one simple (and automatically projective) object $Q$. The group of invertible objects in $\mathcal{D}$, denoted $\operatorname{Pic}(\mathcal{D}) = G$, parametrizes the pointed part. The generalized fusion rules are of the form

$$Q \otimes Q \cong \bigoplus_{g \in G} P_g \oplus r Q$$

where $P_g$ are the projective covers of the simples $L_g \in \mathcal{D}$, and $r \in \mathbb{N}_{\geq 0}$ is the self-extension parameter. Non-degeneracy is defined by the condition that the Müger center $\mathcal{C}'$ is trivial, i.e., $\mathcal{C}' = \text{Vec}$, up to Lagrangian cases, thus $\mathcal{C}$ is modular in the context of finite non-semisimple tensor categories (Sebbag, 14 Dec 2025).

2. Main Classification Theorems

The central classification results for these categories are as follows:

• Theorem A (Vanishing Self-Extension): If $\mathcal{C}$ is a braided non-semisimple near-group category with generalized fusion rule $(G, r)$, then the extension parameter $r = 0$. Consequently, these categories are weakly integral—the Grothendieck semiring has only roots of unity as dimensions.
• Theorem B (Modularization via Exact Sequences): Every braided non-semisimple near-group category $\mathcal{C}$ possesses a unique symmetric fusion subcategory $\mathcal{E} \cong \mathrm{Rep}(G)$ (with $G$ a finite 2-group) sitting in the Müger center. The de-equivariantization by $G$ yields a modularization exact sequence:

$$\mathrm{Rep}(G) \xrightarrow{i} \mathcal{C} \xrightarrow{F} \mathcal{C}_G =: \mathcal{D}$$

with $\mathcal{D}$ itself a non-degenerate braided non-semisimple near-group category.

• Theorem C (Non-degenerate Classification): Every non-degenerate braided non-semisimple near-group category arises as a braided simple extension of the slightly degenerate pointed category

$\mathcal{D} \simeq s\mathrm{Rep}(W \oplus W^*)$

for $W$ a purely odd supervector space, with symmetric (Lagrangian) subcategory $s\mathcal{V}\text{ec} \simeq s\mathrm{Rep}(W)$. Thus, any such category is determined up to equivalence by the data of a finite 2-group $G$ with central involution and an oddness condition on the dimension invariant $n_p + \log_2 |G|$ (Sebbag, 14 Dec 2025).

3. Fusion Rules, Braiding Structure, and Modular Invariants

With $r = 0$ forced by Theorem A, the fusion ring is determined by

$Q \otimes Q \cong \bigoplus_{g\in G} P_g$

where $Q$ is the unique simple projective over the extended category and each $P_g$ runs over projective covers of simples in $G$. The squared braiding on $Q \otimes L_g$ is a scalar $\lambda_g = \pm 1$, and the braid group relations require that $g \mapsto \lambda_g$ is a character $G \to k^\times$, referred to as the "Q-braiding group" (Editor's term). Modular invariants are fully determined by the quadratic form on $G$ (from the pointed data) and the character $\psi: G \to \{\pm 1\}$ encoding the braiding of $Q$ with the pointed simples. No new continuous invariants arise: associativity and braiding constraints are uniquely pinned once the pointed structure and braiding data are fixed (Sebbag, 14 Dec 2025).

4. Müger Center, Symmetric Subcategories, and Modularization

The Müger center $\mathcal{C}'$ is canonically identified with the symmetric Serre subcategory within the pointed part for which $\lambda_g = 1$. The Picard group $\operatorname{Pic}(\mathcal{C}')$ is always a 2-group, and modularization via de-equivariantization gives a sequence

$$\mathrm{Rep}(G) \rightarrow \mathcal{C} \rightarrow \mathcal{C}_G$$

with $\mathcal{C}_G$ a non-degenerate category with unique new simple projective, and $\operatorname{Pic}(\mathcal{C}_G) = \mathbb{Z}/2$. This modularization ensures that the non-degenerate case fully governs the structure of all braided non-semisimple near-group categories (Sebbag, 14 Dec 2025).

5. Explicit Construction and Parametrization

The non-degenerate case is realized by choosing a purely odd supervector space $W$ and constructing the slightly degenerate pointed super category $s\mathrm{Rep}(W \oplus W^*)$, which admits a unique braided simple extension with a new projective object $Q$, denoted $C(W)$. Central (character) actions of a finite 2-group $G$ then yield all further non-semisimple braided near-group categories by equivariantization: $C(W)^G$ Thus, the general such category is classified up to braided equivalence by a pair $(G, n_p)$, with $G$ a finite 2-group possessing a central involution and $n_p = \log_2(\dim P_1)$, subject to the oddness relation $n_p + \log_2|G|$ odd (Sebbag, 14 Dec 2025).

6. Historical Context and Relation to Semisimple and Fusion Settings

Classical near-group categories arose as fusion categories with group-like simples and a single non-invertible, described by Siehler and further classified in the braided, semisimple context by Siehler and Thornton. The present non-semisimple theory generalizes this by allowing for pointed subcategories that are not semisimple and for extensions of their projective covers. Comparison with the semisimple case reveals that many of the phenomena distinguishing non-semisimple categories—such as the unique simple projective extension and the forced vanishing of the self-extension parameter $r$—do not occur in the semisimple (fusion) world. Moreover, the obstructions to minimal modular extension encountered in Tambara–Yamagami and extraspecial $p$-group fusion categories, as detailed in (Schopieray, 2021), do not impact the non-semisimple near-group categories constructed via super-categories and 2-group equivariantizations. The explicit modular data and $S,T$-matrices are dictated by the underlying supervector space structure and quadratic data (Sebbag, 14 Dec 2025, Schopieray, 2021).

7. Illustrative Examples

• Super-group construction: Let $W$ be a purely odd supervector space of dimension $m$. The category $s\mathrm{Rep}(W \oplus W^*)$ is finite, braided, slightly degenerate, and pointed with Picard group $\mathbb{Z}/2$. Its unique braided simple extension $C(W)$ (with new projective $Q$) provides the non-degenerate prototype.
• Equivariantization: For any finite 2-group $G$ acting centrally on $C(W)$ via a character $G \to \{\pm1\}$, the equivariantization $C(W)^G$ yields all braided non-semisimple near-group categories with Picard group $G'$ containing $\mathbb{Z}/2$.
• Sweedler-type example: The non-semisimple Hopf algebra $\text{Rep}(H_4)$ does not admit a quasitriangular structure, hence it does not yield a braided near-group extension, illustrating the necessity of suitable braiding in the construction.

These examples highlight the rigidity and uniqueness of the structure of non-degenerate braided non-semisimple near-group categories, confirming that the classification exhausts all possibilities up to braided equivalence (Sebbag, 14 Dec 2025).