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Almost Differentially Nondegenerate Nijenhuis Operators

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Abstract

The paper is devoted to the study of Nijenhuis operators of arbitrary dimension \(n\) in a neighborhood of a point at which the first \(n-1\) coefficients of the characteristic polynomial are functionally independent and the last coefficient (the determinant of the operator) is an arbitrary function. We prove a theorem on the general form of such Nijenhuis operators and also obtain their complete description for the case in which the determinant has a nondegenerate singularity.

DOI 10.1134/S106192082204001X

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References

  1. A. Bolsinov, V. Matveev, and A. Konyaev, “Nijenhuis Geometry”, Advances in Mathematics, 394 (2022).

    Article  MATH  Google Scholar 

  2. A. Bolsinov, V. Matveev, and A. Konyaev, “Applications of Nijenhuis Geometry II: Maximal Pencils of Multi-Hamiltonian Structures of Hydrodynamic Type”, Nonlinearity, 34 (2021), 5136–5162.

    Article  ADS  MATH  Google Scholar 

  3. A. Bolsinov, V. Matveev, and A. Konyaev “Nijenhuis Geometry III: gl-Regular Nijenhuis Operators”, arxiv: 2007.09506v1 [math.DG], 18 Jul 2020.

  4. A. Konyaev, “Nijenhuis Geometry II: Left-Symmetric Algebras and Linearization Problem for Nijenhuis Operators”, Differential Geom. Appl., 74 (2021).

    Article  MATH  Google Scholar 

  5. A. Konyaev, “Completeness of Commutative Sokolov–Odesskii Subalgebras and Nijenhuis Operators on gl(n)”, Sb. Math., 211:4 (2020), 583–593.

    Article  MATH  Google Scholar 

  6. A. Bolsinov, V. Matveev, S. Tabachnikov, and E. Miranda, “Open Problems, Questions and Challenges in Finite-Dimensional Integrable Systems”, Philos. Trans. R. Soc. A, 376 (2018).

    Article  MATH  Google Scholar 

  7. D. Akpan, “Singularities of Two-Dimensional Nijenhuis Operators”, European J. Math., 8 (2022), 1328–1340.

    Article  MATH  Google Scholar 

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Acknowledgments

The author thanks A.A. Oshemkov, E.A. Kudryavtseva, A.Yu. Konyaev, V.A. Kibkalo, V.N. Zavyalov, and all participants of the seminar “Algebra and Geometry of Integrable Systems” for valuable advice.

Funding

The author was supported by the scholarship of Theoretical Physics and Mathematics Advancement Foundation “BASIS.”

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    D. Zh. Akpan

  2. Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

    D. Zh. Akpan

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  1. D. Zh. Akpan
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Correspondence to D. Zh. Akpan.

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Akpan, D.Z. Almost Differentially Nondegenerate Nijenhuis Operators. Russ. J. Math. Phys. 29, 413–416 (2022). https://doi.org/10.1134/S106192082204001X

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  • DOI: https://doi.org/10.1134/S106192082204001X

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