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Lie Affgebras Vis-à-Vis Lie Algebras

Ryszard R. Andruszkiewicz, Tomasz Brzezinski Orcid Logo, Krzysztof Radziszewski

Results in Mathematics, Volume: 80, Issue: 2, Start page: 61

Swansea University Author: Tomasz Brzezinski Orcid Logo

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DOI (Published version): 10.1007/s00025-025-02377-7

Abstract

It is shown that any Lie affgebra, that is an algebraic system consisting of an affine space together with a bi-affine multiplication satisfying affine versions of the antisymmetry and Jacobi identity, is isomorphic to a Lie algebra together with an element and a specific generalized derivation (in...

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Published in: Results in Mathematics
ISSN: 1422-6383 1420-9012
Published: Springer Nature 2025
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URI: https://cronfa.swan.ac.uk/Record/cronfa68951
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spelling 2025-02-28T13:47:53.5780468 v2 68951 2025-02-24 Lie Affgebras Vis-à-Vis Lie Algebras 30466d840b59627325596fbbb2c82754 0000-0001-6270-3439 Tomasz Brzezinski Tomasz Brzezinski true false 2025-02-24 MACS It is shown that any Lie affgebra, that is an algebraic system consisting of an affine space together with a bi-affine multiplication satisfying affine versions of the antisymmetry and Jacobi identity, is isomorphic to a Lie algebra together with an element and a specific generalized derivation (in the sense of Leger and Luks in J Algebra 228:165–203, 2000). These Lie algebraic data can be taken for the construction of a Lie affgebra or, conversely, they can be uniquely derived for any Lie algebra fibre of the Lie affgebra. The close relationship between Lie affgebras and (enriched by the additional data) Lie algebras can be employed to attempt a classification of the former by the latter. In particular, up to isomorphism, a complex Lie affgebra with a simple Lie algebra fibre is fully determined by a scalar and an element of fixed up to an automorphism of , and it can be universally embedded in a trivial extension of by a derivation. The study is illustrated by a number of examples that include all Lie affgebras with one-dimensional, non-abelian two-dimensional, and Lie algebra fibres. Extensions of Lie affgebras by cocycles and their relation to cocycle extensions of tangent Lie algebras is briefly discussed and illustrated by Lie affgebras with the Witt and Virasoro algebra fibres. Journal Article Results in Mathematics 80 2 61 Springer Nature 1422-6383 1420-9012 Lie algebra; Lie affgebra; generalized derivation; quasicentroid 1 3 2025 2025-03-01 10.1007/s00025-025-02377-7 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University SU Library paid the OA fee (TA Institutional Deal) The research of Tomasz Brzeziński and Krzysztof Radziszewski is partially supported by the National Science Centre, Poland, through the WEAVE-UNISONO grant no. 2023/05/Y/ST1/00046. 2025-02-28T13:47:53.5780468 2025-02-24T09:25:01.8592295 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Ryszard R. Andruszkiewicz 1 Tomasz Brzezinski 0000-0001-6270-3439 2 Krzysztof Radziszewski 3 68951__33714__5ff327a9ea7a43da9edbe1df4309f423.pdf 68951.VOR.pdf 2025-02-28T13:41:03.1439198 Output 507255 application/pdf Version of Record true © 2025 The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). true eng http://creativecommons.org/licenses/by/4.0/
title Lie Affgebras Vis-à-Vis Lie Algebras
spellingShingle Lie Affgebras Vis-à-Vis Lie Algebras
Tomasz Brzezinski
title_short Lie Affgebras Vis-à-Vis Lie Algebras
title_full Lie Affgebras Vis-à-Vis Lie Algebras
title_fullStr Lie Affgebras Vis-à-Vis Lie Algebras
title_full_unstemmed Lie Affgebras Vis-à-Vis Lie Algebras
title_sort Lie Affgebras Vis-à-Vis Lie Algebras
author_id_str_mv 30466d840b59627325596fbbb2c82754
author_id_fullname_str_mv 30466d840b59627325596fbbb2c82754_***_Tomasz Brzezinski
author Tomasz Brzezinski
author2 Ryszard R. Andruszkiewicz
Tomasz Brzezinski
Krzysztof Radziszewski
format Journal article
container_title Results in Mathematics
container_volume 80
container_issue 2
container_start_page 61
publishDate 2025
institution Swansea University
issn 1422-6383
1420-9012
doi_str_mv 10.1007/s00025-025-02377-7
publisher Springer Nature
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description It is shown that any Lie affgebra, that is an algebraic system consisting of an affine space together with a bi-affine multiplication satisfying affine versions of the antisymmetry and Jacobi identity, is isomorphic to a Lie algebra together with an element and a specific generalized derivation (in the sense of Leger and Luks in J Algebra 228:165–203, 2000). These Lie algebraic data can be taken for the construction of a Lie affgebra or, conversely, they can be uniquely derived for any Lie algebra fibre of the Lie affgebra. The close relationship between Lie affgebras and (enriched by the additional data) Lie algebras can be employed to attempt a classification of the former by the latter. In particular, up to isomorphism, a complex Lie affgebra with a simple Lie algebra fibre is fully determined by a scalar and an element of fixed up to an automorphism of , and it can be universally embedded in a trivial extension of by a derivation. The study is illustrated by a number of examples that include all Lie affgebras with one-dimensional, non-abelian two-dimensional, and Lie algebra fibres. Extensions of Lie affgebras by cocycles and their relation to cocycle extensions of tangent Lie algebras is briefly discussed and illustrated by Lie affgebras with the Witt and Virasoro algebra fibres.
published_date 2025-03-01T08:22:40Z
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