Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky lattices

Details Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky lattices Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky lattices

1996-09-30

arxiv.org/abs/solv-int/9610001v1

J. Math. Phys., 1997, V. 38, p. 4179-4201.

Nonlinear Sciences

Exactly Solvable and Integrable Systems

35 pages, LaTeX

Scientific paper

10.1063/1.532090

A new Lax representation for the Bogoyavlensky lattice is found, its $r$--matrix interpretation is elaborated. The $r$--matrix structure turns out to be related to a highly nonlocal quadratic Poisson structure on a direct sum of associative algebras. The theory of such nonlocal structures is developed, the Poisson property of the monodromy map is worked out in the most general situation. Some problems concerning the duality of Lax representations are raised.

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