  After reviewing some of the fundamental aspects of Drinfel'd doubles and
Poisson-Lie T-duality, we describe the three-dimensional isotropic rigid
rotator on $SL(2,\mathbb{C})$ starting from a non-Abelian deformation of the
natural 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 analyzed
from the Poisson-Lie duality point of view, and a doubled generalized action is
built with $TSL(2,\mathbb{C})$ as carrier space. The aim is to explore within a
simple case the relations between Poisson-Lie symmetry, Doubled Geometry and
Generalized Geometry. In fact, all the mentioned structures are discussed, such
as 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. The
two dual models exhibit many features of Poisson-Lie duals and from the
generalized action both of them can be respectively recovered by gauging one of
its symmetries.
