[2303.18065] Basic quasi-reductive root data and supergroups Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Representation Theory arXiv:2303.18065 (math) [Submitted on 31 Mar 2023 ( v1 ), last revised 29 Apr 2026 (this version, v4)] Title: Basic quasi-reductive root data and supergroups Authors: Rita Fioresi , Bin Shu View a PDF of the paper titled Basic quasi-reductive root data and supergroups, by Rita Fioresi and Bin Shu View PDF HTML (experimental) Abstract: We investigate pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd $G$-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup $\mathbb{G}$, that is, $\mathbb{G}_{\text{ev}}$ is isomorphic to $G$, and the quotient $\mathbb{G}/\mathbb{G}_{\text{ev}}$ is $G$-equivariantly isomorphic to $Y$. We prove that, if $Y$ satisfies certain conditions (basic quasi-reductive root data), then the question has a positive answer given by an existence and uniqueness theorem. The corresponding supergroups are said to be basic quasi-reductive, which can be classified, up to isogeny. We then decide the structure of connected quasi-reductive algebraic supergroups provided that: (i) the root system does not contain $0$; (ii) $\mathfrak{g}:=\text{Lie}(\mathbb{G})$ admits a non-degenerate even symmetric bilinear form. (iii) all odd reflections are invertible. Remarkably, those supergroups are exactly basic quasi-reductive supergroups of monodromy type. Comments: Indagationes Mathematicae (2026), in press Subjects: Representation Theory (math.RT) ; Algebraic Geometry (math.AG); Group Theory (math.GR) MSC classes: 17B05, 14M30, 58A50, 14A22 Cite as: arXiv:2303.18065 [math.RT] (or arXiv:2303.18065v4 [math.RT] for this version) https://doi.org/10.48550/arXiv.2303.18065 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Bin Shu [ view email ] [v1] Fri, 31 Mar 2023 13:53:51 UTC (24 KB) [v2] Tue, 5 Sep 2023 03:01:12 UTC (24 KB) [v3] Fri, 18 Jul 2025 02:43:59 UTC (25 KB) [v4] Wed, 29 Apr 2026 23:34:43 UTC (26 KB) Full-text links: Access Paper: View a PDF of the paper titled Basic quasi-reductive root data and supergroups, by Rita Fioresi and Bin Shu View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2023-03 Change to browse by: math math.AG math.GR 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