Geodesic
Affine Nijenhuis Operators and Hochschild Cohomology of Trusses
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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
Mots-clés : affine space. 
@article{SIGMA_2023_19_a55,
     author = {Tomasz Brzezi\'nski and James Papworth},
     title = {Affine {Nijenhuis} {Operators} and {Hochschild} {Cohomology} of {Trusses}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a55/}
}
TY  - JOUR
AU  - Tomasz Brzeziński
AU  - James Papworth
TI  - Affine Nijenhuis Operators and Hochschild Cohomology of Trusses
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2023
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a55/
LA  - en
ID  - SIGMA_2023_19_a55
ER  - 
%0 Journal Article
%A Tomasz Brzeziński
%A James Papworth
%T Affine Nijenhuis Operators and Hochschild Cohomology of Trusses
%J Symmetry, integrability and geometry: methods and applications
%D 2023
%V 19
%U http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a55/
%G en
%F SIGMA_2023_19_a55
Tomasz Brzeziński; James Papworth. Affine Nijenhuis Operators and Hochschild Cohomology of Trusses. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a55/

[1] Andruszkiewicz R.R., Brzeziński T., Rybołowicz B., “Ideal ring extensions and trusses”, J. Algebra, 600 (2022), 237–278, arXiv: 2101.09484 | DOI | MR | Zbl

[2] Baer R., “Zur Einführung des Scharbegriffs”, J. Reine Angew. Math., 160 (1929), 199–207 | DOI | MR

[3] Benenti S., “Fibrés affines canoniques et mécanique newtonienne, in Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986)”, Travaux en Cours, 27, Hermann, Paris, 1988, 13–37 | MR

[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 | DOI | MR | Zbl

[6] Brzeziński T., “Trusses: paragons, ideals and modules”, J. Pure Appl. Algebra, 224 (2020), 106258, 39 pp., arXiv: 1901.07033 | DOI | MR | Zbl

[7] Brzeziński T., “Lie trusses and heaps of Lie affebras”, Proc. Sci., 406 (2022), PoS(CORFU2021)307, 12 pp., arXiv: 2203.12975 | DOI | MR

[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 | DOI | MR | Zbl

[9] Certaine J., “The ternary operation $(abc)=ab^{-1}c$ of a group”, Bull. Amer. Math. Soc., 49 (1943), 869–877 | DOI | MR | Zbl

[10] Gerstenhaber M., “On the deformation of rings and algebras”, Ann. of Math., 79 (1964), 59–103 | DOI | MR | Zbl

[11] Grabowska K., Grabowski J., Urbański P., “Lie brackets on affine bundles”, Ann. Global Anal. Geom., 24 (2003), 101–130, arXiv: math.DG/0203112 | DOI | MR | Zbl

[12] Hochschild G., “On the cohomology groups of an associative algebra”, Ann. of Math., 46 (1945), 58–67 | DOI | MR | Zbl

[13] Kosmann-Schwarzbach Y., “Nijenhuis structures on Courant algebroids”, Bull. Braz. Math. Soc. (N.S.), 42 (2011), 625–649, arXiv: 1102.1410 | DOI | MR | Zbl

[14] Magri F., “A simple model of the integrable Hamiltonian equation”, J. Math. Phys., 19 (1978), 1156–1162 | DOI | MR | Zbl

[15] Massa E., Vignolo S., Bruno D., “Non-holonomic Lagrangian and Hamiltonian mechanics: an intrinsic approach”, J. Phys. A, 35 (2002), 6713–6742 | DOI | MR | Zbl

[16] Prüfer H., “Theorie der Abelschen Gruppen”, Math. Z., 20 (1924), 165–187 | DOI | MR

[17] Tulczyjew W.M., “Frame independence of analytical mechanics”, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 119 (1985), 273–279 | MR | Zbl

[18] Urbański P., “Affine Poisson structures in analytical mechanics”, Quantization and Infinite-Dimensional Systems (Bialowieza, 1993), Plenum, New York, 1994, 123–129 | DOI | MR | Zbl

[19] Weinstein A., “A universal phase space for particles in Yang–Mills fields”, Lett. Math. Phys., 2 (1978), 417–420 | DOI | MR | Zbl