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Abstract

We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known $π _{+}$). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the $π _{+}$ structure on SU(N) is described in terms of generators and relations as an example.