Abstract
In 1983 Bogoyavlenski conjectured that, if the Euler equations on a Lie algebra \(\mathfrak{g}_{0}\) are integrable, then their certain extensions to semisimple lie algebras \(\mathfrak{g}\) related to the filtrations of Lie algebras \(\mathfrak{g}_{0}\subset\mathfrak{g}_{1}\subset\mathfrak{g}_{2}\dots\subset\mathfrak{g}_{n-1}\subset\mathfrak{g}_{n}=\mathfrak{g}\) are integrable as well. In particular, by taking \(\mathfrak{g}_{0}=\{0\}\) and natural filtrations of \({\mathfrak{so}}(n)\) and \(\mathfrak{u}(n)\), we have Gel’fand – Cetlin integrable systems. We prove the conjecture for filtrations of compact Lie algebras \(\mathfrak{g}\): the system is integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given.
Notes
The gradient is determined by an invariant metric: \(df(\xi)=\langle\nabla f(x),\xi\rangle\). Also, to simplify notation, the Lie brackets, the Lie – Poisson brackets and the gradients of the functions on \(\mathfrak{g}_{i}\) will be denoted by the same symbols as on \(\mathfrak{g}\), \(i=1,\dots,n\).
In [6] the symplectic case is considered, but the proof can be easily modified to the Poisson case.
\(\mathfrak{g}_{l}(x_{k})\) denotes the isotropy algebra of \(x_{k}\in\mathfrak{g}_{k}\) within \(\mathfrak{g}_{l}\):
$$\displaystyle\mathfrak{g}_{l}(x_{k})=\{\xi\in\mathfrak{g}_{l}|[\xi,x_{k}]=0\},\qquad l\leqslant k.$$Generic means that the dimensions of the isotropy algebras \(\mathfrak{g}_{i}(x_{i})\) and \(\mathfrak{g}_{i-1}(x_{i})\) are minimal.
By \(\langle\cdot,\cdot\rangle\) we also denote an invariant quadratic form on \(\mathfrak{g}^{\mathbb{C}}\), the extension of the invariant scalar product from \(\mathfrak{g}\) to \(\mathfrak{g}^{\mathbb{C}}\).
Again, we use the same symbol for different objects. The restriction of \(\langle\cdot,\cdot\rangle\) to \(\mathfrak{g}\) is a positive definite invariant scalar product we are dealing with. Also, as above, we identify \(\mathfrak{h}\) and \(\mathfrak{h}^{*}\) by \(\langle\cdot,\cdot\rangle\).
References
Bazaîkin, Ya. V., Double Quotients of Lie Groups with an Integrable Geodesic Flow, Siberian Math. J., 2000, vol. 41, no. 3, pp. 419–432; see also: Sibirsk. Mat. Zh., 2000, vol. 41, no. 3, pp. 513-530.
Bogoyavlenskii, O. I., Integrable Euler Equations Associated with Filtrations of Lie Algebras, Math. USSR-Sb., 1984, vol. 49, no. 1, pp. 229–238; see also: Mat. Sb. (N.S.), 1983, vol. 121(163), no. 2(6), pp. 233-242.
Bolsinov, A. V., Compatible Poisson Brackets on Lie Algebras and the Completeness of Families of Functions in Involution, Math. USSR-Izv., 1992, vol. 38, no. 1, pp. 69–90; see also: Izv. Akad. Nauk SSSR Ser. Mat., 1991, vol. 55, no. 1, pp. 68-92.
Bolsinov, A. V., Complete Commutative Subalgebras in Polynomial Poisson Algebras: A Proof of the Mischenko – Fomenko Conjecture, Theor. Appl. Mech., 2016, vol. 43, no. 2, pp. 145–168.
Bolsinov, A. V. and Jovanović, B., Integrable Geodesic Flows on Homogeneous Spaces, Sb. Math., 2001, vol. 192, no. 7, pp. 951–968; see also: Mat. Sb., 2001, vol. 192, no. 7, pp. 21-40.
Bolsinov, A. V. and Jovanović, B., Non-Commutative Integrability, Moment Map and Geodesic Flows, Ann. Glob. Anal. Geom., 2003, vol. 23, no. 4, pp. 305–322.
Bolsinov, A. V. and Jovanović, B., Complete Involutive Algebras of Functions on Cotangent Bundles of Homogeneous Spaces, Math. Z., 2004, vol. 246, no. 1–2, pp. 213–236.
Brailov, A. V., Construction of Completely Integrable Geodesic Flows on Compact Symmetric Spaces, Math. USSR-Izv., 1987, vol. 29, no. 1, pp. 19–31; see also: Izv. Akad. Nauk SSSR Ser. Mat., 1986, vol. 50, no. 4, pp. 661-674, 877.
Dragović, V., Gajić, B., and Jovanović, B., Singular Manakov Flows and Geodesic Flows of Homogeneous Spaces of \({\rm SO}(N)\), Transfom. Groups, 2009, vol. 14, no. 3, pp. 513–530.
Dragović, V., Gajić, B., and Jovanović, B., On the Completeness of the Manakov Integrals, J. Math. Sci. (N.Y.), 2017, vol. 223, no. 6, pp. 675–685; see also: Fundam. Prikl. Mat., 2015, vol. 20, no. 2, pp. 35-49.
Kozlov, V. V. and Fedorov, Yu. N., Various Aspects of \(n\)-Dimensional Rigid Body Dynamics, in Dynamical Systems in Classical Mechanics, Amer. Math. Soc. Transl. Ser. 2, vol. 168, Providence, R.I.: AMS, 1995, pp. 141–171.
Gel’fand, I. and Tsetlin, M., Finite-Dimensional Representation of the Group of Unimodular Matrices, Dokl. Akad. Nauk SSSR, 1950, vol. 71, pp. 825–828 (Russian).
Gel’fand, I. and Tsetlin, M., Finite-Dimensional Representation of the Group of Orthogonal Matrices, Dokl. Akad. Nauk SSSR, 1950, vol. 71, pp. 1017–1020 (Russian).
Guillemin, V. and Sternberg, Sh., On Collective Complete Integrability According to the Method of Thimm, Ergodic Theory Dynam. Systems, 1983, vol. 3, no. 2, pp. 219–230.
Guillemin, V. and Sternberg, Sh., The Gel’fand – Cetlin System and Quantization of the Complex Flag Manifolds, J. Funct. Anal., 1983, vol. 52, no. 1, pp. 106–128.
Guillemin, V. and Sternberg, Sh., Multiplicity-Free Spaces, J. Differential Geom., 1984, vol. 19, no. 1, pp. 31–56.
Harada, M., The Symplectic Geometry of the Gel’fand – Cetlin – Molev Basis for Representations of \({\rm Sp}(2n,{\mathbb{C}})\), J. Symplectic Geom., 2006, vol. 4, no. 1, pp. 1–41.
Heckman, G. J., Projections of Orbits and Asymptotic Behavior of Multiplicities for Compact Connected Lie Groups, Invent. Math., 1982, vol. 67, no. 2, pp. 333–356.
Jovanović, B., Geometry and Integrability of Euler – Poincaré – Suslov Equations, Nonlinearity, 2001, vol. 14, no. 6, pp. 1555–1567.
Jovanović, B., Integrability of Invariant Geodesic Flows on \(n\)-Symmetric Spaces, Ann. Global Anal. Geom., 2010, vol. 38, no. 3, pp. 305–316.
Jovanović, B., Geodesic Flows on Riemannian g.o. Spaces, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 504–513.
Krämer, M., Multiplicity Free Subgroups of Compact Connected Lie Groups, Arch. Math. (Basel), 1976, vol. 27, no. 1, pp. 28–36.
Lompert, K. and Panasyuk, A., Invariant Nijenhuis Tensors and Integrable Geodesic Flows, SIGMA Symmetry Integrability Geom. Methods Appl., 2019, vol. 15, Paper No. 056, 30 pp.
Manakov, S. V., Note on the Integration of Euler’s Equations of the Dynamics of an \(n\)-Dimensional Rigid Body, Funct. Anal. Appl., 1976, vol. 10, no. 4, pp. 328–329; see also: Funktsional. Anal. i Prilozhen., 1976, vol. 10, no. 4, pp. 93-94.
Mikityuk, I. V., Integrability of the Euler Equations Associated with Filtrations of Semisimple Lie Algebras, Math. USSR-Sb., 1986, vol. 53, no. 2, pp. 541–549; see also: Mat. Sb., 1984, vol. 125(167), no. 4, pp. 539-546.
Mikityuk, I. V., Integrability of Geodesic Flows for Metrics on Suborbits of the Adjoint Orbits of Compact Groups, Transform. Groups, 2016, vol. 21, no. 2, pp. 531–553.
Mykytyuk, I. V. and Panasyuk, A., Bi-Poisson Structures and Integrability of Geodesic Flows on Homogeneous Spaces, Transform. Groups, 2004, vol. 9, no. 3, pp. 289–308.
Mishchenko, A. S. and Fomenko, A. T., Euler Equations on Finite-Dimensional Lie Groups, Math. USSR-Izv., 1978, vol. 12, no. 2, pp. 371–389; see also: Izv. Akad. Nauk SSSR. Ser. Mat., 1978, vol. 42, no. 2, pp. 396-415, 471.
Mishchenko, A. S. and Fomenko, A. T., Generalized Liouville Method of Integration of Hamiltonian Systems, Func. Anal. Appl., 1978, vol. 12, no. 2, pp. 113–121; see also: Funktsional. Anal. i Prilozhen., 1978, vol. 12, no. 2, pp. 46-56.
Nekhoroshev, N. N., Action–Angle Variables and Their Generalization, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 180–198; see also: Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 181-198.
Panyushev, D. I. and Yakimova, O. S., Poisson-Commutative Subalgebras of \(\mathcal{S}(\mathfrak{g})\) Associated with Involutions, Int. Math. Res. Not., 2021, vol. 2021, no. 23, pp. 18367–18406.
Panyushev, D. I. and Yakimova, O. S., Reductive Subalgebras of Semisimple Lie Algebras and Poisson Commutativity, arXiv:2012.04014 (2020).
Reiman, A. G., Integrable Hamiltonian Systems Connected with Graded Lie Algebras, J. Math. Sci., 1982, vol. 19, pp. 1507–1545; see also: Zap. Nauchn. Semin. LOMI AN SSSR, 1980, vol. 95, pp. 3-54.
Sadetov, S. T., A Proof of the Mishchenko – Fomenko Conjecture (1981), Dokl. Akad. Nauk, 2004, vol. 397, no. 6, pp. 751–754 (Russian).
Thimm, A., Integrable Geodesic Flows on Homogeneous Spaces, Ergodic Theory Dynam. Systems, 1981, vol. 1, no. 4, pp. 495–517.
Trofimov, V. V., Euler Equations on Borel Subalgebras of Semisimple Lie Groups, Math. USSR-Izv., 1980, vol. 14, no. 3, pp. 653–670; see also: Izv. Acad. Nauk SSSR. Ser. Mat., 1979, vol. 43, no. 3, pp. 714-732.
Trofimov, V. V. and Fomenko, A. T., Algebra and Geometry of Integrable Hamiltonian Differential Equations, Moscow: Faktorial, 1995 (Russian).
ACKNOWLEDGMENTS
We are grateful to the referees for a thorough review and constructive comments that have greatly improved quality of the paper.
Funding
This research is supported by Project 7744592 MEGIC, Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics, of the Science Fund of Serbia.
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MSC2010
37J35, 17B63, 17B80, 53D20
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Jovanović, B., Šukilović, T. & Vukmirović, S. Integrable Systems Associated to the Filtrations of Lie Algebras. Regul. Chaot. Dyn. 28, 44–61 (2023). https://doi.org/10.1134/S1560354723010045
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DOI: https://doi.org/10.1134/S1560354723010045