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After reviewing some of the fundamental aspects of Drinfel'd doubles andPoisson-Lie T-duality, we describe the three-dimensional isotropic rigidrotator on $SL(2,\mathbb{C})$ starting from a non-Abelian deformation of thenatural carrier space of its Hamiltonian description on $T^*SU(2) \simeq SU(2)\ltimes \mathbb{R}^3$. A new model is then introduced on the dual group$SB(2,\mathbb{C})$, within the Drinfel'd double description of$SL(2,\mathbb{C})=SU(2) \bowtie SB(2,\mathbb{C})$. The two models are analyzedfrom the Poisson-Lie duality point of view, and a doubled generalized action isbuilt with $TSL(2,\mathbb{C})$ as carrier space. The aim is to explore within asimple case the relations between Poisson-Lie symmetry, Doubled Geometry andGeneralized Geometry. In fact, all the mentioned structures are discussed, suchas a Poisson realization of the $C$-brackets for the generalized bundle $T\oplus T^*$ over $SU(2)$ from the Poisson algebra of the generalized model. Thetwo dual models exhibit many features of Poisson-Lie duals and from thegeneralized action both of them can be respectively recovered by gauging one ofits symmetries.

T-Duality and Doubling of the Isotropic Rigid Rotator