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Symmetry, Integrability and Geometry: Methods and Applications, 2023, Volume 19, 056, 22 pp.
DOI: https://doi.org/10.3842/SIGMA.2023.056 (Mi sigma1951) 		 
Affine Nijenhuis Operators and Hochschild Cohomology of Trusses

Tomasz Brzezińskiab, James Papwortha

a Department of Mathematics, Swansea University, Fabian Way, Swansea SA1 8EN, UK
b Faculty of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland
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DOI: https://doi.org/10.3842/SIGMA.2023.056
Abstract: The classical Hochschild cohomology theory of rings is extended to abelian heaps with distributing multiplication or trusses. This cohomology is then employed to give necessary and sufficient conditions for a Nijenhuis product on a truss (defined by the extension of the Nijenhuis product on an associative ring introduced by Cariñena, Grabowski and Marmo in [Internat. J. Modern Phys. A 15 (2000), 4797–4810, arXiv:math-ph/0610011]) to be associative. 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.
Keywords: Nijenhuis operator, Hochschild cohomology, truss, heap, affine space.
Funding agency Grant number
National Science Centre, Poland 2019/35/B/ST1/01115
The research of Tomasz Brzeziński is partially supported by the National Science Centre, Poland, grant no. 2019/35/B/ST1/01115.
Received: April 4, 2023; in final form July 27, 2023; Published online August 4, 2023
Document Type: Article
MSC: 20N10, 16E40, 81R12
Language: English
Citation: Tomasz Brzeziński, James Papworth, “Affine Nijenhuis Operators and Hochschild Cohomology of Trusses”, SIGMA, 19 (2023), 056, 22 pp.
Citation in format AMSBIB
\Bibitem{BrzPap23}
\by Tomasz~Brzezi{\'n}ski, James~Papworth
\paper Affine Nijenhuis Operators and Hochschild Cohomology of Trusses
\jour SIGMA
\yr 2023
\vol 19
\papernumber 056
\totalpages 22
\mathnet{http://mi.mathnet.ru/sigma1951}
\crossref{https://doi.org/10.3842/SIGMA.2023.056}
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