Dual Pairs and Regularization of Kummer Shapes in Resonances

We present an account of dual pairs and the Kummer shapes for n : m resonances that provides an alternative to Holm and Vizman’s work. The advantages of our point of view are that the associated Poisson structure on su(2)* is the standard (+)-Lie–Poisson bracket independent of the values of (n, m) as well as that the Kummer shape is regularized to become a sphere without any pinches regardless of the values of (n, m). A similar result holds for n : −m resonance with a paraboloid and su(1, 1)* . The result also has a straightforward generalization to multidimensional resonances as well. ©2019 American Institute of Mathematical Sciences