Enviar artigo por via de e-mail A canonical basis of a pair of compatible Poisson brackets on a matrix algebra
- Autores: Garazha A.A.1
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Afiliações:
- Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
- Edição: Volume 211, Nº 6 (2020)
- Páginas: 95-106
- Seção: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/142358
- DOI: https://doi.org/10.4213/sm9282
- ID: 142358
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Resumo
Given an arbitrary complex matrix $A$ and a generic matrix $X$ we find a canonical basis for the Kronecker part of the bi-Lagrangian subspace with respect to the corresponding Poisson brackets on the Lie algebra $\mathfrak{gl}_n(\mathbb C)$, and also find a system of functions in bi-involution corresponding to this basis. In particular, for nilpotent matrices $A$ we prove that all nonzero functions obtained by applying the Mishchenko-Fomenko argument shift method to the coefficients of the characteristic polynomial form the Kronecker part of the complete system of functions in bi-involution. Bibliography: 9 titles.
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Sobre autores
Aleksandra Garazha
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Email: garazha.alex.andr@gmail.com
without scientific degree, no status
Bibliografia
- Ф. Р. Гантмахер, Теория матриц, 3-е изд., Наука, М., 1967, 567 с.
- И. К. Козлов, “Элементарное доказательство теоремы Жордана–Кронекера”, Матем. заметки, 94:6 (2013), 857–870
- A. V. Bolsinov, Zhang P., “Jordan–Kronecker invariants of finite-dimensional Lie algebras”, Transform. Groups, 21:1 (2016), 51–86
- Алгебра и геометрия, Семинар кафедры высшей алгебры МГУ им. Ломоносова (электронный ресурс)
- А. С. Мищенко, А. Т. Фоменко, “Уравнения Эйлера на конечномерных группах Ли”, Изв. АН СССР. Сер. матем., 42:2 (1978), 396–415
- V. Futorny, A. Molev, “Quantization of the shift of argument subalgebras in type A”, Adv. Math., 285 (2015), 1358–1375
- A. Molev, O. Yakimova, “Quantisation and nilpotent limits of Mishchenko–Fomenko subalgebras”, Represent. Theory, 23 (2019), 350–378
- B. Kostant, “Lie group representations on polynomial rings”, Amer. J. Math., 85:3 (1963), 327–404
- H. Kraft, “Parametrisierung von Konjugationklassen in $mathfrak{sl}_n$”, Math. Ann., 234:3 (1978), 209–220
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