  Poisson-Lie T-duality of the Wess-Zumino-Witten (WZW) model having the group
manifold of $SU(2)$ as target space is investigated. The whole construction
relies on the deformation of the affine current algebra of the model, the
semi-direct sum $\mathfrak{su}(2)(\mathbb{R}) \, \dot{\oplus} \, \mathfrak{a}$,
to the fully semisimple Kac-Moody algebra
$\mathfrak{sl}(2,\mathbb{C})(\mathbb{R})$. A two-parameter family of models
with $SL(2,\mathbb{C})$ as target phase space is obtained so that Poisson-Lie
T-duality is realised as an $O(3,3)$ rotation in the phase space. The dual
family shares the same phase space but its configuration space is
$SB(2,\mathbb{C})$, the Poisson-Lie dual of the group $SU(2)$. A parent action
with doubled degrees of freedom on $SL(2,\mathbb{C})$ is defined, together with
its Hamiltonian description.
