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

SIGMA 19 (2023), 056, 22 pages      arXiv:2303.12880      https://doi.org/10.3842/SIGMA.2023.056

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

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.

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

pdf (400 kb)   tex (26 kb)  

References

1. Andruszkiewicz R.R., Brzeziński T., Rybołowicz B., Ideal ring extensions and trusses, J. Algebra 600 (2022), 237-278, arXiv:2101.09484.
2. Baer R., Zur Einführung des Scharbegriffs, J. Reine Angew. Math. 160 (1929), 199-207.
3. Benenti S., Fibrés affines canoniques et mécanique newtonienne, in Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986), Travaux en Cours, Vol. 27, Hermann, Paris, 1988, 13-37.
4. Breaz S., Brzeziński T., Rybołowicz B., Saracco P., Heaps of modules and affine spaces, Ann. Mat. Pura Appl., to appear, arXiv:2203.07268.
5. Brzeziński T., Trusses: between braces and rings, Trans. Amer. Math. Soc. 372 (2019), 4149-4176, arXiv:1710.02870.
6. Brzeziński T., Trusses: paragons, ideals and modules, J. Pure Appl. Algebra 224 (2020), 106258, 39 pages, arXiv:1901.07033.
7. Brzeziński T., Lie trusses and heaps of Lie affebras, Proc. Sci. 406 (2022), PoS(CORFU2021)307, 12 pages, arXiv:2203.12975.
8. Cariñena J.F., Grabowski J., Marmo G., Quantum bi-Hamiltonian systems, Internat. J. Modern Phys. A 15 (2000), 4797-4810, arXiv:math-ph/0610011.
9. Certaine J., The ternary operation $(abc)=ab^{-1}c$ of a group, Bull. Amer. Math. Soc. 49 (1943), 869-877.
10. Gerstenhaber M., On the deformation of rings and algebras, Ann. of Math. 79 (1964), 59-103.
11. Grabowska K., Grabowski J., Urbański P., Lie brackets on affine bundles, Ann. Global Anal. Geom. 24 (2003), 101-130, arXiv:math.DG/0203112.
12. Hochschild G., On the cohomology groups of an associative algebra, Ann. of Math. 46 (1945), 58-67.
13. Kosmann-Schwarzbach Y., Nijenhuis structures on Courant algebroids, Bull. Braz. Math. Soc. (N.S.) 42 (2011), 625-649, arXiv:1102.1410.
14. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
15. Massa E., Vignolo S., Bruno D., Non-holonomic Lagrangian and Hamiltonian mechanics: an intrinsic approach, J. Phys. A 35 (2002), 6713-6742.
16. Prüfer H., Theorie der Abelschen Gruppen, Math. Z. 20 (1924), 165-187.
17. Tulczyjew W.M., Frame independence of analytical mechanics, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 119 (1985), 273-279.
18. Urbański P., Affine Poisson structures in analytical mechanics, in Quantization and Infinite-Dimensional Systems (Bialowieza, 1993), Plenum, New York, 1994, 123-129.
19. Weinstein A., A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys. 2 (1978), 417-420.