{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T15:19:50Z","timestamp":1786979990532,"version":"3.56.0"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T00:00:00Z","timestamp":1676332800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T00:00:00Z","timestamp":1676332800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["PGC2018-094898-B-I00"],"award-info":[{"award-number":["PGC2018-094898-B-I00"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012306","name":"University of Trieste","doi-asserted-by":"crossref","award":["FRA2020-2021"],"award-info":[{"award-number":["FRA2020-2021"]}],"id":[{"id":"10.13039\/501100012306","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2023,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We prove a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level\n                    <jats:italic>m<\/jats:italic>\n                    vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras.\n                  <\/jats:p>","DOI":"10.1007\/s00332-022-09877-8","type":"journal-article","created":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:48:08Z","timestamp":1676594888000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Generalized Nijenhuis Torsions and Block-Diagonalization of Operator Fields"],"prefix":"10.1007","volume":"33","author":[{"given":"Daniel","family":"Reyes","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Piergiulio","family":"Tempesta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giorgio","family":"Tondo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,2,14]]},"reference":[{"key":"9877_CR1","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1007\/BF02517890","volume":"182","author":"OI Bogoyavlenskij","year":"1996","unstructured":"Bogoyavlenskij, O.I.: Necessary conditions for existence of non-degenerate Hamiltonian structures. Commun. Math. Phys. 182, 253\u2013290 (1996)","journal-title":"Commun. Math. Phys."},{"key":"9877_CR2","doi-asserted-by":"publisher","first-page":"1129","DOI":"10.1070\/IM2004v068n06ABEH000511","volume":"68","author":"OI Bogoyavlenskij","year":"2004","unstructured":"Bogoyavlenskij, O.I.: General algebraic identities for the Nijenhuis and Haantjes torsions. Izvestya Math. 68, 1129\u20131141 (2004)","journal-title":"Izvestya Math."},{"issue":"063502","key":"9877_CR3","first-page":"9","volume":"47","author":"OI Bogoyavlenskij","year":"2006","unstructured":"Bogoyavlenskij, O.I.: Block-diagonalizability problem for hydrodynamic type systems. J. Math. Phys. 47(063502), 9 (2006)","journal-title":"J. Math. Phys."},{"key":"9877_CR4","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1007\/s00220-006-0119-9","volume":"269","author":"OI Bogoyavlenskij","year":"2007","unstructured":"Bogoyavlenskij, O.I.: Decoupling problem for systems of quasi-linear PDE\u2019s. Commun. Math. Phys. 269, 545\u2013556 (2007)","journal-title":"Commun. Math. Phys."},{"key":"9877_CR5","doi-asserted-by":"crossref","unstructured":"Bolsinov, A.V., Konyaev, A. Yu., Matveev, V.S.: Nijenhuis geometry. Adv. Math. 394(108001), 52 (2022)","DOI":"10.1016\/j.aim.2021.108001"},{"issue":"013","key":"9877_CR6","first-page":"22","volume":"15","author":"CM Chanu","year":"2019","unstructured":"Chanu, C.M., Rastelli, G.: Block-separation of variables a form of partial separation for natural Hamiltonians. SIGMA 15(013), 22 (2019)","journal-title":"SIGMA"},{"key":"9877_CR7","volume-title":"Methods of Mathematical Physics II","author":"R Courant","year":"1962","unstructured":"Courant, R., Hilbert, D.: Methods of Mathematical Physics II. Interscience Publishers, New York (1962)"},{"key":"9877_CR8","doi-asserted-by":"publisher","first-page":"1197","DOI":"10.1098\/rspa.2005.1627","volume":"462","author":"E Ferapontov","year":"2006","unstructured":"Ferapontov, E., Khusnutdinova, K.: The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability. Proc. Royal Soc. A 462, 1197\u20131219 (2006)","journal-title":"Proc. Royal Soc. A"},{"key":"9877_CR9","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s00208-007-0106-2","volume":"339","author":"EV Ferapontov","year":"2007","unstructured":"Ferapontov, E.V., Marshall, D.G.: Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor. Math. Ann. 339, 61\u201399 (2007)","journal-title":"Math. Ann."},{"issue":"26","key":"9877_CR10","first-page":"22","volume":"32","author":"EV Ferapontov","year":"2022","unstructured":"Ferapontov, E.V., Pavlov, M.V.: Kinetic equation for soliton gas: integrable reductions. J. Nonl. Sci. 32(26), 22 (2022)","journal-title":"J. Nonl. Sci."},{"key":"9877_CR11","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/S1385-7258(56)50046-7","volume":"18","author":"A Fr\u00f6licher","year":"1956","unstructured":"Fr\u00f6licher, A., Nijenhuis, A.: Theory of vector valued differential forms. Part I. Indag. Math. 18, 338\u2013359 (1956)","journal-title":"Part I. Indag. Math."},{"key":"9877_CR12","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1016\/S1385-7258(55)50021-7","volume":"17","author":"J Haantjes","year":"1955","unstructured":"Haantjes, J.: On $$X_{n-1}$$-forming sets of eigenvectors. Indag. Math. 17, 158\u2013162 (1955)","journal-title":"Indag. Math."},{"key":"9877_CR13","doi-asserted-by":"crossref","unstructured":"Kosmann-Schwarzbach, Y.: Beyond recursion operators, in Proceedings of the XXXVI Workshop on Geometric Methods in Physics, Bia\u0142owie\u017ca, Poland, July 2017, Birkhauser, (2019)","DOI":"10.1007\/978-3-030-01156-7_18"},{"key":"9877_CR14","volume-title":"Computational Linear and Commutative Algebra","author":"M Kreuzer","year":"1993","unstructured":"Kreuzer, M., Robbiano, L.: Computational Linear and Commutative Algebra. Springer, New York (1993)"},{"key":"9877_CR15","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1134\/S0040577918080081","volume":"196","author":"F Magri","year":"2018","unstructured":"Magri, F.: Haantjes manifolds with symmetries. Theor. Math. Phys. 196, 1217\u20131229 (2018)","journal-title":"Theor. Math. Phys."},{"key":"9877_CR16","doi-asserted-by":"publisher","first-page":"391","DOI":"10.2307\/1970051","volume":"65","author":"A Newlander","year":"1957","unstructured":"Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. of Math. 65, 391\u2013404 (1957)","journal-title":"Ann. of Math."},{"key":"9877_CR17","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1016\/S1385-7258(51)50028-8","volume":"54","author":"A Nijenhuis","year":"1951","unstructured":"Nijenhuis, A.: $$X_{n-1}$$-forming sets of eigenvectors. Indag. Math. 54, 200\u2013212 (1951)","journal-title":"Indag. Math."},{"key":"9877_CR18","doi-asserted-by":"publisher","first-page":"390","DOI":"10.1016\/S1385-7258(55)50054-0","volume":"17","author":"A Nijenhuis","year":"1955","unstructured":"Nijenhuis, A.: Jacobi-type identities for bilinear differential concomitants of certain tensor fields I. Indag. Math. 17, 390\u2013397 (1955)","journal-title":"Indag. Math."},{"key":"9877_CR19","doi-asserted-by":"publisher","first-page":"398","DOI":"10.1016\/S1385-7258(55)50055-2","volume":"17","author":"A Nijenhuis","year":"1955","unstructured":"Nijenhuis, A.: Jacobi-type identities for bilinear differential concomitants of certain tensor fields II. Indag. Math. 17, 398\u2013403 (1955)","journal-title":"Indag. Math."},{"issue":"106021","key":"9877_CR20","first-page":"25","volume":"104","author":"D Reyes Nozaleda","year":"2022","unstructured":"Reyes Nozaleda, D., Tempesta, P., Tondo, G.: Classical multiseparable Hamiltonian systems, superintegrability and Haantjes geometry. Commun. Nonl. Sci. Num. Sim. 104(106021), 25 (2022)","journal-title":"Commun. Nonl. Sci. Num. Sim."},{"key":"9877_CR21","doi-asserted-by":"crossref","unstructured":"Tempesta, P., Tondo, G.: Haantjes Algebras and diagonalization. J. Geom. Phys. (preprint, arXiv:1710.04522) 160, 103968, 21 (2021)","DOI":"10.1016\/j.geomphys.2020.103968"},{"key":"9877_CR22","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s10231-021-01107-4","volume":"201","author":"P Tempesta","year":"2022","unstructured":"Tempesta, P., Tondo, G.: Haantjes manifolds and classical integrable systems. Ann. Mat. Pura Appl. 201, 57\u201390 (2022a)","journal-title":"Ann. Mat. Pura Appl."},{"key":"9877_CR23","doi-asserted-by":"publisher","first-page":"1647","DOI":"10.1007\/s00220-021-04233-5","volume":"389","author":"P Tempesta","year":"2022","unstructured":"Tempesta, P., Tondo, G.: Higher Haantjes brackets and integrability. Commun. Math. Phys. 389, 1647\u20131671 (2022b)","journal-title":"Commun. Math. Phys."},{"key":"9877_CR24","doi-asserted-by":"crossref","unstructured":"Tempesta, P., Tondo, G., Polarization of generalized Nijenhuis torsions, arXiv:2209.12716v2,: Contemp. Math, AMS (to appear) (2023)","DOI":"10.1090\/conm\/789\/15848"},{"issue":"023","key":"9877_CR25","first-page":"18","volume":"12","author":"G Tondo","year":"2016","unstructured":"Tondo, G., Tempesta, P.: Haantjes structures for the Jacobi-Calogero model and the Benenti systems. SIGMA 12(023), 18 (2016)","journal-title":"SIGMA"},{"key":"9877_CR26","doi-asserted-by":"publisher","first-page":"1366","DOI":"10.1134\/S004057791809009X","volume":"196","author":"G Tondo","year":"2018","unstructured":"Tondo, G.: Haantjes algebras of the Lagrange top. Theor. Math. Phys. 196, 1366\u20131379 (2018)","journal-title":"Theor. Math. Phys."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-022-09877-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-022-09877-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-022-09877-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,6]],"date-time":"2023-12-06T20:32:29Z","timestamp":1701894749000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-022-09877-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,14]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,4]]}},"alternative-id":["9877"],"URL":"https:\/\/doi.org\/10.1007\/s00332-022-09877-8","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,14]]},"assertion":[{"value":"8 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 December 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 February 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"35"}}