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Jacobi actions of $$SO(2)\times\mathbb{R}^ 2$$ and $$SU(2,\mathbb{C})$$ on two Jacobi manifolds. (English) Zbl 0842.57034
The actions of two subgroups of $$\text{SO} (2) \times \mathbb{R}$$ on a sphere of the dual space $$G^* = (\text{so} (2) \times \mathbb{R})^*$$ with the Jacobi manifold structure obtained as quotient by the homothety group of the Lie-Poisson structure in $$G^* \smallsetminus \{0\}$$ are studied. It is shown that the natural action of $$\text{SU} (2, \mathbb{C})$$ on the unitary 3-sphere of $$\mathbb{C}^2$$ with the Jacobi structure determined by its canonical contact structure is a Jacobi action that admits a unique $$\text{Ad}^*$$-equivariant momentum mapping (in the sense of J.-M. Sourian and B. Konstant).
Reviewer: M.Rahula (Tartu)
MSC:
 57S25 Groups acting on specific manifolds
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