A characterization of coboundary Poisson Lie groups and Hopf algebras

Abstract

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