Geodesic
A canonical basis of a pair of compatible Poisson brackets on a matrix algebra
Sbornik. Mathematics, Tome 211 (2020) no. 6, pp. 838-849 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: bi-Hamiltonian systems, Jordan-Kronecker invariants, argument shift method. 
@article{SM_2020_211_6_a2,
     author = {A. A. Garazha},
     title = {A~canonical basis of a~pair of compatible {Poisson} brackets on a~matrix algebra},
     journal = {Sbornik. Mathematics},
     pages = {838--849},
     year = {2020},
     volume = {211},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_6_a2/}
}
TY  - JOUR
AU  - A. A. Garazha
TI  - A canonical basis of a pair of compatible Poisson brackets on a matrix algebra
JO  - Sbornik. Mathematics
PY  - 2020
SP  - 838
EP  - 849
VL  - 211
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/SM_2020_211_6_a2/
LA  - en
ID  - SM_2020_211_6_a2
ER  - 
%0 Journal Article
%A A. A. Garazha
%T A canonical basis of a pair of compatible Poisson brackets on a matrix algebra
%J Sbornik. Mathematics
%D 2020
%P 838-849
%V 211
%N 6
%U http://geodesic.mathdoc.fr/item/SM_2020_211_6_a2/
%G en
%F SM_2020_211_6_a2
A. A. Garazha. A canonical basis of a pair of compatible Poisson brackets on a matrix algebra. Sbornik. Mathematics, Tome 211 (2020) no. 6, pp. 838-849. http://geodesic.mathdoc.fr/item/SM_2020_211_6_a2/

[1] F. R. Gantmacher, The theory of matrices, v. 1, 2, Chelsea Publishing Co., New York, 1959, x+374 pp., ix+276 pp. | MR | MR | Zbl | Zbl

[2] I. K. Kozlov, “An elementary proof of the Jordan–Kronecker theorem”, Math. Notes, 94:6 (2013), 885–896 | DOI | DOI | MR | Zbl

[3] A. V. Bolsinov, Zhang P., “Jordan–Kronecker invariants of finite-dimensional Lie algebras”, Transform. Groups, 21:1 (2016), 51–86 | DOI | MR | Zbl

[4] Algebra i geometriya, Seminar kafedry vysshei algebry MGU im. Lomonosova (elektronnyi resurs) http://halgebra.math.msu.su/alg-geom/

[5] A. S. Miščenko, A. T. Fomenko, “Euler equations on finite-dimensional Lie groups”, Math. USSR-Izv., 12:2 (1978), 371–389 | DOI | MR | Zbl

[6] V. Futorny, A. Molev, “Quantization of the shift of argument subalgebras in type A”, Adv. Math., 285 (2015), 1358–1375 | DOI | MR | Zbl

[7] A. Molev, O. Yakimova, “Quantisation and nilpotent limits of Mishchenko–Fomenko subalgebras”, Represent. Theory, 23 (2019), 350–378 | DOI | MR | Zbl

[8] B. Kostant, “Lie group representations on polynomial rings”, Amer. J. Math., 85:3 (1963), 327–404 | DOI | MR | Zbl

[9] H. Kraft, “Parametrisierung von Konjugationklassen in $\mathfrak{sl}_n$”, Math. Ann., 234:3 (1978), 209–220 | DOI | MR | Zbl