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Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices. (English) Zbl 1053.37061
The paper provides a comprehensive review of multi-Hamiltonian structures of generalized Toda lattices corresponding to the classical simple Lie groups. Generalized Hamiltonians are \(H=\frac12(p,p)+\sum e^{(v_i,p)}\) \((p,q\in\mathbb{R}^N\), moreover \(v_1,\dots,v_N\in\mathbb{R}^N\) is a system of roots of a simple Lie algebra \(\mathcal G\)).
The author recalls the Shouten brackets, symplectic and Poisson manifolds, bi-Hamiltonian systems and master symmetries. Then, the particular case of Toda lattices (where \({\mathcal G} = A_N\)) is discussed with many details including several approaches to the multi-Hamiltonian structures, Lie group symmetries, negative hierarchies, master integrals and symmetries, and Noether symmetries. Analogous results for \(B_N\), \(C_N\) and \(D_N\) lattices follow: they are all bi-Hamiltonian. The lower dimensional cases \(B_2\), \(C_3\), and \(D_4\) are explicitly presented.
The case of exceptional simple Lie groups is still open.

MSC:
37K60 Lattice dynamics; integrable lattice equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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