Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 056, 22 pages      arXiv:2303.12880

Affine Nijenhuis Operators and Hochschild Cohomology of Trusses

Tomasz Brzeziński ab and James Papworth a
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

Received April 04, 2023, in final form July 27, 2023; Published online August 04, 2023

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.

Key words: Nijenhuis operator; Hochschild cohomology; truss; heap; affine space.

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