Dual pairs and regularization of Kummer shapes in resonances (Q2303559)

Recently, a Poisson-geometric structure behind the Kummer shapes was discovered by finding a dual pair of Poisson maps in \(n:m\) resonance [\textit{D. D. Holm} and \textit{C. Vizman}, J. Geom. Mech. 4, No. 3, 297--311 (2012; Zbl 1264.53072)]. This paper offers an alternative view of the above dual pair as well as of the Kummer shapes in \(n:m\), \(n:-m\) and multidimensional resonances. More precisely, the \(n:m\) resonance with arbitrary \((n, m)\in \mathbb{N}^2\) is related with the \(1:1\) resonance. The dual pair for \(1:1\) resonance naturally yields the dual pair for the \(n:m\) resonance. An advantage of the present method is that the reduced dynamics in \(\operatorname{su}(2)^{\ast }\) becomes a standard Lie-Poisson dynamics. A byproduct of this construction is that the Kummer shapes are all regularized to become a sphere.