Abstract:
We introduce a family of compatible Poisson brackets on the space of $2 \times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the $XXX$ Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.
\Bibitem{Tsi08}
\by A.~V.~Tsiganov
\paper The Poisson Bracket Compatible with the Classical Reflection Equation Algebra
\jour Regul. Chaotic Dyn.
\yr 2008
\vol 13
\issue 3
\pages 191--203
\mathnet{http://mi.mathnet.ru/rcd570}
\crossref{https://doi.org/10.1134/S1560354708030052}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2415373}
\zmath{https://zbmath.org/?q=an:1229.70061}
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https://www.mathnet.ru/eng/rcd570
https://www.mathnet.ru/eng/rcd/v13/i3/p191
This publication is cited in the following 12 articles:
A. V. Tsiganov, “On bi-Integrable Natural Hamiltonian Systems on Riemannian Manifolds”, JNMP, 18:2 (2021), 245
A. V. Vershilov, Yu. A. Grigorev, A. V. Tsyganov, “Ob odnoi integriruemoi deformatsii volchka Kovalevskoi”, Nelineinaya dinam., 10:2 (2014), 223–236
A V Tsiganov, “On natural Poisson bivectors on the sphere”, J. Phys. A: Math. Theor., 44:10 (2011), 105203
A. V. Tsiganov, “On the generalized Chaplygin system”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 21, Zap. nauchn. sem. POMI, 374, POMI, SPb., 2010, 250–267
A. V. Tsyganov, “O novom razdelenii peremennykh dlya chastnogo sluchaya volchka Kovalevskoi”, Nelineinaya dinam., 6:3 (2010), 639–652
A. V. Tsiganov, “New variables of separation for particular case of the Kowalevski top”, Regul. Chaotic Dyn., 15:6 (2010), 659–669
A. V. Vershilov, “On the bi-Hamiltonian structure of Bogoyavlensky system on $so(4)$”, Regul. Chaotic Dyn., 15:6 (2010), 670–676
A. V. Tsiganov, “On the generalized Chaplygin system”, J Math Sci, 168:6 (2010), 901
A V Vershilov, A V Tsiganov, “On bi-Hamiltonian geometry of some integrable systems on the sphere with cubic integral of motion”, J. Phys. A: Math. Theor., 42:10 (2009), 105203
A. V. Tsiganov, “Change of the time for the periodic Toda lattices and natural systems on the plane with higher order integrals of motion”, Regul. Chaotic Dyn., 14:4 (2009), 541–549
A.V. Tsiganov, “On bi-hamiltonian structure of some integrable systems”, JNMP, 15:2 (2008), 171
A. V. Tsiganov, “On Maximally Superintegrable Systems”, Regul. Chaotic Dyn., 13:3 (2008), 178–190
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