Modular $\mathbb{Z}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups

[2411.12251] Modular $\mathbb{Z}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Quantum Algebra arXiv:2411.12251 (math) [Submitted on 19 Nov 2024 ( v1 ), last revised 26 Feb 2026 (this version, v2)] Title: Modular $\mathbb{Z}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups Authors: César Galindo , Simon Lentner , Sven Möller View a PDF of the paper titled Modular $\mathbb{Z}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups, by C\'esar Galindo and 2 other authors View PDF HTML (experimental) Abstract: We explicitly construct nondegenerate braided $\mathbb{Z}_2$-crossed tensor categories of the form $\operatorname{Vect}_\Gamma\oplus\operatorname{Vect}_{\Gamma/2\Gamma}$. They are $\mathbb{Z}_2$-crossed extensions, in the sense of arXiv:0909.3140 , of the braided tensor category $\operatorname{Vect}_\Gamma$ with $\mathbb{Z}_2$-action given by $-\mathrm{id}$ on the finite, abelian group $\Gamma$. Thus, we obtain generalisations of the Tambara-Yamagami categories, where now the abelian group $\Gamma$ may have even order and the nontrivial sector $\operatorname{Vect}_{\Gamma/2\Gamma}$ more than one simple object. The idea for this construction comes from a physically motivated approach in arXiv:2409.16357 to construct $\mathbb{Z}_2$-crossed extensions of $\operatorname{Vect}_\Gamma$ for any $\Gamma$ from an infinite Tambara-Yamagami category $\operatorname{Vect}_{\mathbb{R}^d}\oplus\operatorname{Vect}$, which itself is not fully rigorously defined, and then using condensation from $\operatorname{Vect}_{\mathbb{R}^d}$ to $\operatorname{Vect}_\Gamma$, which we prove commutes with crossed extensions. The $\mathbb{Z}_2$-equivariantisation of $\operatorname{Vect}_\Gamma\oplus\operatorname{Vect}_{\Gamma/2\Gamma}$ yields new modular tensor categories, which correspond to the orbifold of an arbitrary lattice vertex operator algebra under a lift of $-\mathrm{id}$, as discussed in arXiv:2409.16357 . Comments: 48 pages, LaTeX; minor improvements; to appear in Adv. Math Subjects: Quantum Algebra (math.QA) ; Category Theory (math.CT); Representation Theory (math.RT) MSC classes: 18M15, 18M20 Cite as: arXiv:2411.12251 [math.QA] (or arXiv:2411.12251v2 [math.QA] for this version) https://doi.org/10.48550/arXiv.2411.12251 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Adv. Math., 493:110902, 2026 Related DOI : https://doi.org/10.1016/j.aim.2026.110902 Focus to learn more DOI(s) linking to related resources Submission history From: Sven Möller [ view email ] [v1] Tue, 19 Nov 2024 05:53:14 UTC (51 KB) [v2] Thu, 26 Feb 2026 09:47:57 UTC (53 KB) Full-text links: Access Paper: View a PDF of the paper titled Modular $\mathbb{Z}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups, by C\'esar Galindo and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.QA < prev | next > new | recent | 2024-11 Change to browse by: math math.CT math.RT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

Related documents

• SL2 tilting modules in the mixed case#974123
• Square-Free and Dual-Square-Free Abelian Groups#256602
• The Tambara-Yamagami calculus and finite semigroups#1076613
• Gluing abelian categories and stability conditions#662527
• Gluing abelian categories and stability conditions#661477