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Nijenhuis Operators on Trusses and Lie Affgebras / JAMES PAPWORTH

Swansea University Author: JAMES PAPWORTH

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DOI (Published version): 10.23889/SUThesis.72060

Abstract

Hochschild cohomology theory of rings is extended from the ring case to abelian heaps with distributing multiplication or trusses.This new definition of cohomology is then utilised to give the required conditions for a Nijenhuis product on a truss to be associative, which was previously defined by the...

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Published: Swansea 2026
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Brzeziński, T.
URI: https://cronfa.swan.ac.uk/Record/cronfa72060
first_indexed 2026-06-11T10:56:07Z
last_indexed 2026-06-12T13:21:37Z
id cronfa72060
recordtype RisThesis
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spelling 2026-06-11T12:03:51.5360691 v2 72060 2026-06-11 Nijenhuis Operators on Trusses and Lie Affgebras 410bdbbb6d81a9cfea4312a2fd41cf6f JAMES PAPWORTH JAMES PAPWORTH true false 2026-06-11 Hochschild cohomology theory of rings is extended from the ring case to abelian heaps with distributing multiplication or trusses.This new definition of cohomology is then utilised to give the required conditions for a Nijenhuis product on a truss to be associative, which was previously defined by the extension of the Nijenhuis product on an associative ring introduced by Cari˜nena, Grabowski and Marmo in [Quantum Bi-Hamiltonian Systems, Int. J.Mod. Phys. A 15, 4797–4810, 2000, arXiv:math-ph/0610011].The definition of Nijenhuis product and operators on trusses is then linearised to the case of affine spaces with compatible associative multiplications or associative affgebras. It is shown that this construction leads to compatible Lie brackets on an affine space.This is then taken further, with Lie algebras extended to the affine case using the heap operation, giving them a definition that is not dependent on the unique element 0, but so that they still adhere to antisymmetry and Jacobi properties.It is then looked at how Nijenhuis brackets function on these Lie affgebras and demonstrated how they fulfil the compatibility condition in the affine case. E-Thesis Swansea Mathematics, Algebra 29 4 2026 2026-04-29 10.23889/SUThesis.72060 COLLEGE NANME COLLEGE CODE Swansea University Brzeziński, T. Doctoral Ph.D 2026-06-11T12:03:51.5360691 2026-06-11T11:48:15.2542441 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics JAMES PAPWORTH 1 72060__36930__4880dd3394834783ba9debe94503e1db.pdf 2026_Papworth_J.final.72060.pdf 2026-06-11T11:54:43.7660525 Output 709487 application/pdf E-Thesis – open access true Copyright: the author, James Papworth, 2026. Distributed under the terms of a Creative Commons Attribution 4.0 License (CC BY 4.0) true eng https://creativecommons.org/licenses/by/4.0/
title Nijenhuis Operators on Trusses and Lie Affgebras
spellingShingle Nijenhuis Operators on Trusses and Lie Affgebras
JAMES PAPWORTH
title_short Nijenhuis Operators on Trusses and Lie Affgebras
title_full Nijenhuis Operators on Trusses and Lie Affgebras
title_fullStr Nijenhuis Operators on Trusses and Lie Affgebras
title_full_unstemmed Nijenhuis Operators on Trusses and Lie Affgebras
title_sort Nijenhuis Operators on Trusses and Lie Affgebras
author_id_str_mv 410bdbbb6d81a9cfea4312a2fd41cf6f
author_id_fullname_str_mv 410bdbbb6d81a9cfea4312a2fd41cf6f_***_JAMES PAPWORTH
author JAMES PAPWORTH
author2 JAMES PAPWORTH
format E-Thesis
publishDate 2026
institution Swansea University
doi_str_mv 10.23889/SUThesis.72060
college_str Faculty of Science and Engineering
hierarchytype
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 Hochschild cohomology theory of rings is extended from the ring case to abelian heaps with distributing multiplication or trusses.This new definition of cohomology is then utilised to give the required conditions for a Nijenhuis product on a truss to be associative, which was previously defined by the extension of the Nijenhuis product on an associative ring introduced by Cari˜nena, Grabowski and Marmo in [Quantum Bi-Hamiltonian Systems, Int. J.Mod. Phys. A 15, 4797–4810, 2000, arXiv:math-ph/0610011].The definition of Nijenhuis product and operators on trusses is then linearised to the case of affine spaces with compatible associative multiplications or associative affgebras. It is shown that this construction leads to compatible Lie brackets on an affine space.This is then taken further, with Lie algebras extended to the affine case using the heap operation, giving them a definition that is not dependent on the unique element 0, but so that they still adhere to antisymmetry and Jacobi properties.It is then looked at how Nijenhuis brackets function on these Lie affgebras and demonstrated how they fulfil the compatibility condition in the affine case.
published_date 2026-04-29T14:21:37Z
_version_ 1867797493045002240
score 11.10865

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