Continual Lie algebras determined by chain complexes

[2012.07431] Continual Lie algebras determined by chain complexes Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Functional Analysis arXiv:2012.07431 (math) [Submitted on 14 Dec 2020 ( v1 ), last revised 19 May 2026 (this version, v7)] Title: Continual Lie algebras determined by chain complexes Authors: A. Zuevsky View a PDF of the paper titled Continual Lie algebras determined by chain complexes, by A. Zuevsky View PDF HTML (experimental) Abstract: Continual Lie algebras are infinite-dimensional generalizations of Lie algebras with discrete root system by considering continual root systems. In this paper we establish the general relation between chain complexes and continual Lie algebras. The natural orthogonality condition with respect to a product among elements of a chain complex $\mathcal C$ spaces brings about to $\mathcal C$ the structure of a graded algebra with differential relations. We prove the main result of this paper: a chain complex endowed with an appropriate Leibniz-property product of elements of its spaces and the Jacobi identity brings about the structure of a continual Lie algebra with the root space determined by parameters for the complex. That provides a new source of examples of continual Lie algebras. Finally, as an example, we consider the case of Čech-de Rham complex associated to a foliation of a smooth manifold. In a particular case of this chain complex, we derive explicitly the commutation relations for the corresponding continual Lie algebra. Subjects: Functional Analysis (math.FA) Cite as: arXiv:2012.07431 [math.FA] (or arXiv:2012.07431v7 [math.FA] for this version) https://doi.org/10.48550/arXiv.2012.07431 Focus to learn more arXiv-issued DOI via DataCite Submission history From: A Zuevsky [ view email ] [v1] Mon, 14 Dec 2020 11:35:13 UTC (18 KB) [v2] Thu, 28 Sep 2023 14:32:07 UTC (17 KB) [v3] Sun, 31 Mar 2024 10:43:03 UTC (17 KB) [v4] Sat, 7 Sep 2024 17:09:12 UTC (17 KB) [v5] Tue, 22 Jul 2025 09:11:33 UTC (18 KB) [v6] Sat, 11 Apr 2026 21:32:00 UTC (21 KB) [v7] Tue, 19 May 2026 07:37:38 UTC (23 KB) Full-text links: Access Paper: View a PDF of the paper titled Continual Lie algebras determined by chain complexes, by A. Zuevsky View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2020-12 Change to browse by: math 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

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