Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky lattices

Abstract

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.