Scaling of Poisson spheres and compact Lie groups

Abstract

For $n \ge 2$, we show that on the standard Poisson homogeneous space $\mathbb{S}^{2n−1}$ (including $SU (2) \approx \mathbb{S}3$), there exists a Poisson scaling $\phi_\lambda$ at any scale $\lambda \gt 0$ that is smooth on each symplectic leaf and continuous globally. A generalization to the case of the standard Bruhat-Poisson compact simple Lie groups endowed with a stronger topology is also valid.