Jordan--Kronecker invariants of Borel subalgebras of semisimple Lie algebras

K. S. Vorushilov

Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 32-56.

Résumé

In the theory of bi-Hamiltonian systems, the generalized Mischenko–Fomenko conjecture is known. The conjecture states that there exists a complete set of polynomial functions in involution with respect to a pair of naturally defined Poisson structures on a dual space of a Lie algebra. This conjecture is closely related to the argument shift method proposed by A. S. Mishchenko and A. T. Fomenko in [10]. In research works devoted to this conjecture, a connection was found between the existence of a complete set in bi-involution and the algebraic type of the pencil of compatible Poisson brackets, defined by a linear and constant bracket. The numbers that describe the algebraic type of the generic pencil of brackets on the dual space to a Lie algebra are called the Jordan–Kronecker invariants of a Lie algebra. The notion of Jordan–Kronecker invariants was introduced by A. V. Bolsinov and P. Zhang in [2]. For some classes of Lie algebras (for example, semisimple Lie algebras and Lie algebras of low dimension), the Jordan–Kronecker invariants have been computed, but in the general case the problem of computation of the Jordan–Kronecker invariants for an arbitrary Lie algebra remains open. The problem of computation of the Jordan–Kronecker invariants is frequently mentioned among the most interesting unsolved problems in the theory of integrable systems [4, 5, 6, 11]. In this paper, we compute the Jordan–Kronecker invariants for the series $Bsp(2n)$ and construct complete sets of polynomials in bi-involution for each algebra of the series. Also, we calculate the Jordan–Kronecker invariants for the Borel subalgebras $Bso(n)$ for any $n.$ Thus, together with the results obtained in [2] for $Bsl(n)$, this paper presents a solution to the problem of computation of Jordan–Kronecker invariants for Borel subalgebras of classical Lie algebras.

Détail

Keywords: Lie algebras, integrable Hamiltonian systems, argument shift method, Jordan–Kronecker invariants.

Bibliographie

[1] Arkhangelskii A. A., “Vpolne integriruemye gamiltonovy sistemy na gruppe treugolnykh matrits”, Matem. sb., 108(150):1 (1979), 134–142 | MR

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

[3] Bolsinov A., Izosimov A., Kozlov I., “Jordan-Kronecker invariants of Lie algebra representations and degrees of invariant polynomials”, Transform. Groups, 2019 (to appear); arXiv: 1407.1878

[4] Bolsinov A. V., Izosimov A. M., Konyaev A. Yu., Oshemkov A. A., “Algebra i topologiya integriruemykh sistem. Zadachi dlya issledovaniya”, Trudy seminara po vektornomu i tenzornomu analizu, 28, 1012, 119–191

[5] Bolsinov A., Izosimov A., Tsonev D., “Finite-dimensional integrable systems: A collection of research problems”, Journal of Geometry and Physics, 2016 | DOI | MR

[6] Bolsinov A. V., Matveev V. S., Miranda E., Tabachnikov S., “Open Problems, Questions and Challenges in Finite-Dimensional Integrable Systems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018) | MR

[7] Bolsinov A. V., “Complete commutative subalgebras in polynomial Poisson algebras: a proof of the Mischenko-Fomenko conjecture”, Theor. Appl. Mech., 43:2 (2016), 145–168 | DOI | Zbl

[8] Groznova A. Yu., Vychislenie invariantov Zhordana–Kronekera dlya algebr Li malykh razmernostei, vypusknaya kvalifikatsionnaya rabota, MGU, 2018

[9] Korotkevich A. A., “Polnye nabory polinomov na borelevskikh podalgebrakh”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2006, no. 5, 20–25 | MR | Zbl

[10] Mischenko A. S., Fomenko A. T., “Uravneniya Eilera na konechnomernykh gruppakh Li”, Izv. AN SSSR, 42:2 (1978), 396–415 | MR

[11] Rosemann S., Schöbel K., “Open problems in the theory of finite-dimensional integrable systems and related fields”, Journ. Geom. and Phys., 87 (2015), 396–414 | DOI | MR | Zbl

[12] Thompson R., “Pencils of complex and real symmetric and skew matrices”, Linear Algebra Appl., 147 (1991), 323–371 | DOI | MR | Zbl

[13] Trofimov V. V., “Uravneniya Eilera na borelevskikh podalgebrakh poluprostykh algebr Li”, Izv. AN SSSR. Ser. matem., 43:3 (1979), 714–732 | MR | Zbl

[14] Vorontsov A., “Kronekerovy indeksy algebry Li i otsenka stepenei invariantov”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 66:1 (2011), 26–30 | MR | Zbl

[15] Vorushilov K., “Jordan–Kronecker invariants for semidirect sums defined by standard representation of orthogonal or symplectic Lie algebras”, Lobachevskii Journal of Mathematics, 38:6 (2017), 1121–1130 | DOI | MR | Zbl

[16] Vorushilov K. S., “Invarianty Zhordana—Kronekera dlya polupryamykh summ vida $sl(n)+(\mathbb R^n)^k$ i $gl(n)+(\mathbb R^{n})^k$”, Fundament. i prikl. matem., 22:6 (2019), 3–18 | MR