On braided simple extensions and braided non-semisimple near-group categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915836483076096 |
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| author | Sebbag, Daniel |
| author_facet | Sebbag, Daniel |
| contents | 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 $\mathcal{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On braided simple extensions and braided non-semisimple near-group categories Sebbag, Daniel Category Theory 18M05, 18M15 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 $\mathcal{C}$. |
| title | On braided simple extensions and braided non-semisimple near-group categories |
| topic | Category Theory 18M05, 18M15 |
| url | https://arxiv.org/abs/2512.12556 |