Maximal Nijenhuis pencils containing the subpencil of symmetric 2 × 2-matrices
https://doi.org/10.22405/2226-8383-2025-26-5-307-312
Abstract
A linear space of operator fields that consists of Nijenhuis operators is called a Nijenhuis
pencil. Maximal (by inclusion) Nijenhuis pencils serve as interesting examples of such pencils.
The case when maximal Nijehuis pencil contains a subpencil of symmetric constant (𝑛 × 𝑛)-
matrices (in some fixed coordinate system) was recently investigated in paper [4], in which the
complete description of such maximal pencils was obtained for 𝑛 ⩾ 3. As it turned out, the case 𝑛 = 2 requires special research. This problem is solved in the present paper.
About the Author
Mikhail Mikhailovich Chernin
Lomonosov Moscow State University; Moscow Center of Fundamental and Applied Mathematics
Russian Federation
Russian Federation
References
1. Bolsinov, A. V., Konyaev, A.Yu. & Matveev, V. S. 2022, “Nijenhuis geometry”, Adv. Math., vol. 394, Article 108001.
2. Konyaev, A.Yu. 2021, “Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators”, Differential Geom. Appl., vol. 74, Article 101706.
3. Bolsinov, A. V., Konyaev A.Yu. & Matveev V. S. 2021, “Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type”, Nonlinearity, vol. 34, no. 8. pp. 5136–5162.
4. Konyaev, A.Yu. 2023, “Symmetric matrices and maximal Nijenhuis pencils”, Sbornik: Math., vol. 214, no. 8, pp. 1101–1110.
Review
For citations:
Chernin M.M. Maximal Nijenhuis pencils containing the subpencil of symmetric 2 × 2-matrices. Chebyshevskii Sbornik. 2025;26(5):307-312. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-5-307-312
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