Scaling of Poisson spheres and compact Lie groups (Q1939228)

In [Differ. Geom. Appl. 9, No. 1--2, 213--238 (1998; Zbl 0930.37032)], \textit{A. Weinstein} proved that there is no nontrivial smooth scaling \((\phi,\lambda)\) of the standard Bruhat-Poisson structure \(\pi\) on \(\mathrm{SU}(2)\), i.e. a diffeomorphism \(\phi\) of \(\mathrm{SU}(2)\) and a scalar \(\lambda\neq0\) such that \(\phi^*\pi=\lambda^{-1}\pi\), other than \((\phi,\lambda)=(\imath,-1)\) where \(\imath(u):=u^{-1}\) is the inversion map on \(\mathrm{SU}(2)\). This result is generalized to all compact groups with Bruhat-Poisson structure . In the present paper, the author proves that if a scaling \(\phi_{\lambda}\equiv(\phi,\lambda)\) is only required to be continuous on the whole manifold but smooth on each symplectic leaf of a Poisson manifold, then it exists for all \(\lambda>0\) on the Poisson homogeneous space \(S^{2n-1}\) of the Bruhat-Poisson \(\mathrm{SU}(n)\). Also, a Liouville vector field generating \(\phi_\lambda\) is explicitly computed for \(\mathrm{SU}(2)\). Furthermore, if a standard Bruhat-Poisson compact simple Lie group \(K\) is endowed with some stronger topology that is still compatible with the original differential structure on each symplectic leaf of \(K\), then a leafwise smooth and globally continuous scaling \(\phi_\lambda\) exists on \(K\) for all \(\lambda>0\).