Hopf $2$-cocycles for certain affine algebraic reductive groups

[2607.18599] Hopf $2$-cocycles for certain affine algebraic reductive groups Skip to main content Search Submit Donate Log in Mathematics > Representation Theory arXiv:2607.18599 (math) [Submitted on 21 Jul 2026 ( v1 ), last revised 13 Oct 2026 (this version, v3)] Title: Hopf $2$-cocycles for certain affine algebraic reductive groups Authors: Shlomo Gelaki View PDF HTML (experimental) Abstract: Let $G$ be a reductive affine algebraic group over $\mathbb C$ with toral identity component $T$ and finite component group $K$. We describe finite indecomposable semisimple module categories over $\Rep(G)$ using equivariant data that incorporate the extension class. We prove that every fiber functor is classical and determine the image of its Hopf $2$-cocycle support in $K$ and its intersection with $T$. This yields a minimality criterion involving a finite stabilizer cocycle and an alternating bicharacter on a sublattice of $\widehat T$ of index at most $|K|$. When $\dim T=1$, every Hopf $2$-cocycle for $G$ has finite support of order at most $|K|^2$. We compute all gauge classes for the split and nonsplit extensions with component group $\mathbb{Z}/2\mathbb{Z}$ and inversion action, and compare the support formulas with the Künneth coordinates for commutative direct products. Finally, for an arbitrary reductive group $G$, a minimal Hopf $2$-cocycle in a regular deformation of the trivial Hopf $2$-cocycle forces $G$ to be a torus. This uses locality and rigidity of algebraic matrix coefficients on the semisimple quotient of $G$. Comments: 28 pages. Exposition improved, some results sharpened, and Section 5 is new Subjects: Representation Theory (math.RT) ; Quantum Algebra (math.QA) MSC classes: 18M20, 16T05, 17B37 Cite as: arXiv:2607.18599 [math.RT] (or arXiv:2607.18599v3 [math.RT] for this version) https://doi.org/10.48550/arXiv.2607.18599 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Shlomo Gelaki [ view email ] [v1] Tue, 21 Jul 2026 00:25:58 UTC (20 KB) [v2] Fri, 28 Aug 2026 12:07:22 UTC (20 KB) [v3] Tue, 13 Oct 2026 17:16:17 UTC (30 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2026-07 Change to browse by: math math.QA References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

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