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Poisson-Lie T-duality of the Wess-Zumino-Witten (WZW) model having the groupmanifold of $SU(2)$ as target space is investigated. The whole constructionrelies on the deformation of the affine current algebra of the model, thesemi-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 modelswith $SL(2,\mathbb{C})$ as target phase space is obtained so that Poisson-LieT-duality is realised as an $O(3,3)$ rotation in the phase space. The dualfamily 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 actionwith doubled degrees of freedom on $SL(2,\mathbb{C})$ is defined, together withits Hamiltonian description.

Poisson-Lie T-Duality of WZW Model via Current Algebra Deformation