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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
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 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.
Keywords: Mathematics, Algebra
College: Faculty of Science and Engineering

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