On braided simple extensions and braided non-semisimple near-group categories

We study simple extensions of pointed finite tensor categories, that is, tensor categories $\mathcal{C}$ admitting an abelian decomposition $\mathcal{C} \cong \mathcal{D} \oplus \mathcal{M}$ where $\mathcal{D}$ is a pointed tensor subcategory and $\mathcal{M}$ has a unique simple projective object. Such categories provide a natural generalization of near-group categories. Our results concern the braided case. We prove that every non-degenerate braided non-semisimple near-group category is a braided simple extension of $\mathrm{sRep}(W\oplus W^*)$ with non-trivial braiding for which $\mathrm{sRep}(W)$ is Lagrangian. Moreover, any braided non-semisimple near-group category $\mathcal{C}$ arises canonically as an extension of such a category by $\mathrm{Rep}(G)$, where $G$ is the Picard group of a symmetric subcategory determined by the unique simple projective object of $\mathcal{C}$.

💡 Research Summary

The paper investigates a class of finite tensor categories that are built as “simple extensions” of pointed categories. A simple extension consists of a pointed finite tensor subcategory D and an abelian complement M that contains a unique simple projective object Q; the whole category is C = D ⊕ M. When D admits a fiber functor, M is equivalent to Vec, and C inherits a finite tensor structure compatible with the D‑action. The authors focus on the braided case and on categories that generalize the well‑known near‑group fusion categories to the non‑semisimple setting.

In the classical (semisimple) theory a near‑group category C(G,r) has a group G of invertible simple objects and a single non‑invertible simple object Q satisfying Q⊗Q ≅ ⊕_{g∈G} g ⊕ r·Q. The integer r is a key parameter. The authors extend this notion to non‑semisimple categories by allowing Q⊗Q to decompose into projective covers of simple objects rather than direct sums of simples; the parameter r now counts how many copies of the projective cover of Q appear. Such a category is called a non‑semisimple near‑group category with generalized fusion rule (G,r).

The main results are three theorems that together give a complete structural description of braided non‑semisimple near‑group categories.

Theorem 1 shows that if a non‑semisimple near‑group category is braided, then the parameter r must be zero. Consequently the tensor product Q⊗Q contains only invertible objects; there is no contribution from the projective cover of Q. This mirrors the situation in the semisimple case where only the (G,0) rule admits a braiding.

Theorem 2 establishes that any braided non‑semisimple near‑group category C fits into a modularization exact sequence
 Rep(G) → C → D,
where D is a non‑degenerate braided non‑semisimple near‑group category and G is a finite group determined by C. The sequence is exact in the sense of tensor categories (i.e. Rep(G) embeds fully, the quotient functor is surjective and normal, and Rep(G) lands in the Müger center of C). Thus C can be viewed as a central extension of a non‑degenerate core D by a symmetric fusion category Rep(G).

Theorem 3 identifies the core D uniquely: every non‑degenerate braided non‑semisimple near‑group category is a braided simple extension of a category equivalent to sRep(W ⊕ W*), where W is a purely odd super‑vector space. Moreover sRep(W) sits inside D as a Lagrangian (maximally symmetric) subcategory. In other words, D is obtained from the super‑representation category of the exterior algebra on W⊕W* by endowing it with a non‑trivial braiding that makes sRep(W) Lagrangian.

Combining Theorems 2 and 3 yields a full classification: any braided non‑semisimple near‑group category is obtained by first taking a non‑degenerate braided extension of sRep(W⊕W*), then extending further by a symmetric Rep(G) determined by the unique projective object. As corollaries the authors deduce that such categories are never integral (their Frobenius–Perron dimensions are not integers) and they give a concrete description of the possible parameters. Theorem 5 states that the data of a braided non‑semisimple near‑group category can be encoded by a pair (G, nₚ) where G is a 2‑group with a central element of order 2, nₚ∈ℕ determines the dimension of the projective cover of the unit (dim P₁ = 2^{nₚ}), and nₚ + n_g is odd (n_g = log₂|G|). This mirrors known results for braided fusion near‑group categories and provides a categorical proof of earlier observations such as the impossibility of a braided simple extension of Rep(H₄).

The technical backbone of the paper relies on Müger’s centralizer theory, the notion of Lagrangian subcategories, and the process of de‑equivariantization. By embedding Rep(G) as a symmetric subcategory of the Müger center, the authors construct the exact sequence and then apply de‑equivariantization to isolate the non‑degenerate core. The analysis of the projective cover Q and its tensor square is crucial for proving r = 0; the authors show that any non‑trivial contribution from the projective cover would violate the braiding axioms.

Overall, the work extends the classification program for near‑group categories beyond the semisimple realm, showing that braiding imposes very rigid constraints even in the non‑semisimple setting. It identifies sRep(W⊕W*) as the universal building block for non‑degenerate braided extensions and clarifies how symmetric group extensions fit into the picture. The results open the way for further investigations into non‑semisimple braided tensor categories, especially those arising from Hopf algebras with non‑trivial projective objects.