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.
Categories: hep-th math-ph math.MP